APPLYING DIVERSITY TO MITIGATE INTERFERENCE IN UNDERWATER ACOUSTIC COMMUNICATION NETWORKS

Many acoustic channels suffer from interference which is neither narrowband nor impulsive. This relatively long duration partial band interference can be particularly detrimental to system performance. In undersea networks, many dropped messages are lost due to partial band interference which corrupts different portions of the received signal depending on the relative position of the interferers, information source and receivers due to the slow speed of propagation. A survey of recent work in interference mitigation and orthogonal frequency division multiplexing (OFDM) provides motivation to develop a spatial diversity receiver for use in underwater networks. The spatial diversity receiver for underwater communications identifies portions of the signal suffering from interference on different receivers, removes these portions of the signal and then optimally combines the remaining clean portions of the signal. The gain of the spatial diversity combining strategy is investigated as a function of signal to interference ratio (SIR), signal to noise ratio (SNR), interference bandwidth and time duration. Analytic results of performance for receivers on additive white Gaussian noise channels suffering time orthogonal interference demonstrate the effectiveness of the spatial diversity combining strategy as compared to conventional maximum ratio combining. Simulation results on time invariant channels confirm the effectiveness of the algorithm under more complex channel conditions. Comparison of the spatial diversity receiver performance to a recently developed single receiver parameterized interference cancellation algorithm is made using experimental data collected on the Atlantic Undersea Test and Evaluation Center (AUTEC) network which consists of multiple distributed cabled hydrophones. Both techniques are effective strategies for combating interference but approach the problem of interference in fundamentally different ways: the spatial diversity receiver blanks the interference while the cancellation algorithm coherently removes it from the desired signal. Spatial diversity reconstruction is effective and realizes the most gain at low SIR and moderate SNR while parameterized interference cancellation is most effective at moderate SIRs and SNRs. The two approaches are complimentary and an effective multi-channel receiver strategy would be to adaptively utilize both techniques.

tion source and receivers due to the slow speed of propagation. A survey of recent work in interference mitigation and orthogonal frequency division multiplexing (OFDM) provides motivation to develop a spatial diversity receiver for use in underwater networks. The spatial diversity receiver for underwater communications identifies portions of the signal suffering from interference on different receivers, removes these portions of the signal and then optimally combines the remaining clean portions of the signal. The gain of the spatial diversity combining strategy is investigated as a function of signal to interference ratio (SIR), signal to noise ratio (SNR), interference bandwidth and time duration. Analytic results of performance for receivers on additive white Gaussian noise channels suffering time orthogonal interference demonstrate the effectiveness of the spatial diversity combining strategy as compared to conventional maximum ratio combining. Simulation results on time invariant channels confirm the effectiveness of the algorithm under more complex channel conditions. Comparison of the spatial diversity receiver performance to a recently developed single receiver parameterized interference cancellation algorithm is made using experimental data collected on the Atlantic Undersea Test and Evaluation Center (AUTEC) network which consists of multiple distributed cabled hydrophones. Both techniques are effective strategies for combating interference but approach the problem of interference in fundamentally different ways: the spatial diversity receiver blanks the interference while the cancellation algorithm coherently removes it from the desired signal. Spatial diversity reconstruction is effective and realizes the most gain at low SIR and moderate SNR while parameterized interference cancellation is most effective at moderate SIRs and SNRs. The two approaches are complimentary and an effective multi-channel receiver strategy would be to adaptively utilize both techniques.

ACKNOWLEDGMENTS
The contribution that this work makes to the existing ocean of knowledge about underwater communications is only possible because of the support, expertise, guidance, love, and just plain presence of many people in my life. I am grateful for all that I have been blessed to receive. Calling to mind these people, and the journey they have accompanied me on, is at once inspiring and humbling.
My adviser, Professor Swaszek, is bright, witty, thoughtful and caring. Somehow he always managed to coax me back from the acute corner of the 13-sided polygon, where I found myself struggling, to the middle of a sandbox where I could play -and somehow always done with humor and patience. Johnson, for all of the support you have provided. Thank you to Joe, Tod, Phillip iv and Chris for the opportunity to do interesting technical work and the mentoring you provided along the way. Joe, I am especially grateful for our friendship.
Sound propagating in the ocean bends, reflects, scatters; its path winds, is rarely straight. Life is similarly non-linear, full of randomness and the unexpected.
There are people who make life seem simpler and fuller and more beautiful: I am grateful to my sister and brother, Sarah, Ari, Leslie and Gladys. I am blessed with many truly remarkable friendships. There are those friends who I don't see for All of these people nourish me, seeing something in me that I sometimes struggle to see myself.
Finally, I am grateful to Kim  Acoustic Communication Channels and Interference

Background
While humans have recognized for millennia that sound can be heard underwater, the ability to communicate underwater over long distances is a recent achievement. Overviews of the challenges and technical accomplishments in underwater communications may be found in [1][2][3][4][5][6][7][8][9][10][11][12]. One of the first Underwater Acoustic Communication systems, developed by the Submarine Signal Company in the early 1900s, attempted to warn ships of navigation hazards by detecting signals transmitted from underwater bells placed in the vicinity of lightships or lighthouses on shipboard receivers. Unfortunately the receivers suffered from such high ambient noise that they could not reliably detect the transmitted signal. Requested by the Submarine Signal Company to improve its receiver, Reginald A.
Fessenden developed an electro-mechanical device, eventually known as the Fessenden Oscillator, capable of both producing and detecting sound. In 1913, he used the device to transmit Morse code messages over several miles between two tugboats in Boston Harbor [13].
Today, a reliable acoustic data link is of great importance for the development of ocean-observation systems and sensor networks. Applications abound, ranging from the oil industry to aquaculture, including instrument monitoring, pollution control, climate recording, search and survey missions, the study of marine life, to military applications [12]. The current vision is for the development of an undersea sensor network, specifically a collection of bottom-mounted instruments and tetherless moving platforms in which different kinds of oceanographic data will be exchanged among the members in real time [14]. Major motivations for the use of wireless communications in the undersea network include 1) reduction in deployment costs and 2) the operational flexibility provided by the ability to monitor and react to sensor information in real or near-real time [12].
Research into the physical layer of undersea networks is extremely active with focus areas including: the improvement of single carrier modulation/detection with more powerful coding techniques and turbo equalization; the exploration of multi-carrier modulation as an alternative to single carrier; the extension of single-input/single-output (SISO) systems to multi-input/multi-output (MIMO) configurations that provide spatial multiplexing and the ability to send parallel data streams from multiple transmitters [12].
Yet, the deployment of undersea networks is in its infancy. A few short duration experimental networks have been deployed. For example, the Seaweb deployment in 2000 involved 17 nodes spread over 16 km 2 for multiple days [7]. The Massachusetts Institute of Technology and the Commonwealth Scientific and Industrial Research Organisation (CISRO) of Australia deployed a network consisting of both fixed and autonomous nodes in very near-shore areas of Australia [15].
To date the only long term undersea network is operated by the U. S. Navy in the Tongue of the Ocean [16]. It consists of 96 hydrophones placed over a 100 km 2 area and is shown in Fig. 1. As currently configured, approximately 97 percent of transmitted messages are successfully decoded; of the remaining three percent, many are corrupted by acoustic interference arising from active acoustic emissions.
For example, Fig. 2 shows the impact of interference on a received data packet. In panel (a), the data packet was received without interference and successfully decoded in contrast to panel (b) where interfering signals are clearly evident and the message was lost. Years of extensive observations of activities in the vicinity of the network demonstrate that the widely separated hydrophones suffer from partialband interference emanating from multiple spatially separated sources. The nature of this interference is different from the impulsive or narrowband interference typically encountered in other applications. Furthermore, unlike RF communications and acoustic array processing applications where interference is highly correlated in time among the various receivers, in the acoustic network interference affects different portions of the received signals due to the wide separation of the receivers and the low speed of propagation. The degradation in the received signal is highly variable, depending on the relative position of the interferers, information source and receivers as well as the channel conditions. While successful steps to mitigate interference have recently been reported [17], utilizing the spatial diversity implicit in the undersea network to mitigate interference has not yet been attempted. The motivation of this work is to examine the potential benefits that leveraging spatial diversity in underwater acoustic networks might provide.

Challenges of the Underwater Channel
While the spatial diversity implicit in the network offers the opportunity to mitigate interference, the diversity receiver must address the challenges the underwater acoustic channel presents, namely: 1) severely limited operational bandwidth, 2) significant multipath spread, 3) non-stationary statistics, and 4) the need for explicit time synchronization in mobile applications (see e.g. [3,6,19,20]). As is the case at AUTEC, many acoustic channels suffer from interference which is neither narrowband nor impulsive [17]. This relatively long duration partial band interference can be particularly detrimental to communication systems that do not account for it. The challenges the underwater channel presents are manifold, but the diversity inherent in the channel offers opportunities to ameliorate them.
Diversity techniques are well known to improve the reliability of communications over fading multipath channels and rely on the notion that errors occur in reception when the channel attenuation is large; that is, when the channel is in a (a) Andros Island (b) (c) Figure 1: The AUTEC acoustic network. AUTEC is located off the coast of Andros Island in the Bahamas. The squares with circles (in panel (c)) indicate the hydrophone locations. Panel (a) is from [18]. deep fade [21]. Diversity techniques may be classified as either explicit, where the message is intentionally transmitted through different uncorrelated parts of the signal space, or implicit, where the message occupies the entire signal spectrum and the receiver tries to identify it by compensating for the channel distortion [22].
The central idea behind both approaches is to supply the receiver with multiple replicas of the same information transmitted over independently fading portions of the channel thereby reducing the probability that all signal components will fade simultaneously.
Diversity techniques can be employed on selective channels, where selectivity refers to the differential enhancement or attenuation exhibited in the received signal resulting from transmission through the channel with respect to a particular parameter. For example, a frequency selective channel attenuates different frequency bands of the transmitted signal differently, and its frequency response varies over its bandwidth. The underwater acoustic channel exhibits time, frequency and spa-tial selectivity as a consequence of the physics of sound propagation in the ocean.
Frequency selectivity arises from the absorption losses which increase with signal frequency. The coherence bandwidth of the channel, the range of frequencies over which the channel gain is almost constant, is on the order of 10 Hz at a transmit frequency of 10 kHz. Time and spatial selectivity result from time-varying multipath propagation, and the low speed of sound (1500 mps). Multipath formation is governed by sound reflection and refraction. Reflection occurs at the surface and bottom and off any objects in the medium. Refraction results from the spatial variability of sound speed which depends on the temperature, salinity, and pressure. The multipath delay spread, the time delay between the first and last significant arrival of a transmitted signal, can range from 10 milliseconds in shallow water at short ranges to hundreds of milliseconds in deep ocean channels. The multipath structure is highly environmentally dependent and the impulse response may be sparse exhibiting a few large amplitude taps separated by many taps of small magnitude.
Changes in the propagation medium and transmitter/receiver motion induce time variability in the channel. Changes in the propagation medium occur over different time scales ranging from seasonal temperature changes to much faster fluctuations that affect the signal such as surface wave action. Wave motion causes scattering of the signal and rapid fluctuations in signal path length induces Doppler spreading. The slow speed of sound relative to the speed of mobile transmitters results in significant variability to the channel through the Doppler effect which causes frequency shifting as well as frequency spreading. The resulting channel exhibits significant time variability. Coherence time, a measure of the time scale over which the channel changes significantly, can range from seconds in extremely stable situations to 100 milliseconds in rapidly fluctuating channels.

The Physics of Underwater Sound Propagation and Underwater Communication Channel Models
This section introduces the wave equation which governs the propagation of sound in water and describes typical acoustic communication channel models. No acoustic communication systems attempt to solve the wave equation directly. However, all acoustic communication systems must operate in an environment governed by it.

The Wave Equation
The ocean forms an acoustic waveguide bounded above by the sea surface and below by the sea floor. The sound speed of the medium, discontinuities within it and boundary conditions interact in a rich and complex manner to affect the transmission of sound in the waveguide. Despite the richness and complexity of these interactions, mathematical models exist which can predict and explain the observed sound propagation behavior to a certain extent. The starting point for these models is the solution of the wave equation with various simplifying assumptions. While the focus of this work is the development of an acoustic communications receiver which avoids solving the wave equation directly, understanding it provides insight into the physics of how spatial variability arises in the propagation of sound. The following discussion draws on the work in [23] and [24].
These equations relate time and spatial changes in density ρ, particle velocity v and pressure p. The operator [·] S is the thermodynamic quantity calculated at constant entropy. The speed of sound in an ideal fluid is related to the density and pressure or bulk modulus of elasticity through The ambient quantities of the quiescent (time independent) medium are identified by the subscript 0. Small perturbations for the pressure and density are denoted by primes, that is p = p 0 + p and ρ = ρ 0 + ρ . The particle velocity v due to density and pressure fluctuations is much smaller than the speed of sound.
Retaining only the first order terms in the hydrodynamic equations leads to the linear wave equation. To the lowest order Eqs. (1)-(4) become The material properties ρ 0 and c 2 may be assumed independent of time provided the time scale of acoustic propagation is much shorter than the time scale of oceanographic changes. Taking the partial derivative of Eq. (5) with respect to time and the divergence of Eq. (6) yields: and combining Eqs. (8) and (9) produces: Applying the linearized adiabatic equation of state, Eq. (7), removing the primes and letting ρ 0 = ρ results in the linearized wave equation for pressure: Note that the derivation assumes that density is constant with respect to the time scale of interest but may vary spatially. If the density exhibits no spatial variability, then the standard form of the wave applies, While the wave equation describes the physics of sound propagation, solving it is not an easy task. A number of computer algorithms such as Kraken [25][26][27][28][29][30] or Bellhop [31] have been developed to do this. However, the results rely on the estimates of spatially and time varying parameters of the ocean which are difficult to obtain accurately. Attempts to incorporate detailed knowledge of the environment into signal processing algorithms have met with only limited success.
Processing algorithms that avoid solving the wave equation directly but account for the general characteristics of underwater sound propagation have been more fruitful. The channel model discussed in the next section enables the development of such algorithms.

Model of the Underwater Communication Channel
The underwater acoustic channel can be modeled as a linear, time-varying system which is described by the channel impulse response h(t; τ ), where t is the "short-time" variable corresponding to time variations of the impulse response due to moving platforms and physical boundaries and τ is the "age" (or elapsed-time) variable corresponding to the multipath delay for a fixed value of t. In the absence of noise, the received bandpass signalỹ(t) is determined by the convolution of the channel impulse response with the transmitted bandpass signals(t): Alternatively, Eq. (14) may be interpreted as a system with impulse response h(t; τ ) at time t to an impulse that is applied at time t − τ .
A reasonable model for the underwater acoustic channel (and for other physical channels such as the ionosphere at frequencies below 30 MHz and mobile cellular radio) is a special case of Eq. (14) where the impulse response has the form Here, the channel consists of N pa discrete paths with path specific time varying amplitudes A p (t) and delaysτ p (t). Further simplifying assumptions can be made depending on the relative time duration of the signal and the coherence time of the channel. When the signal is short compared to the coherence time, the following assumptions may be adopted: AS1): The path amplitudes are constant within a short time period AS2): The delay variation over the time duration of the signal can be approximated by its first-order Taylor series expansion whereτ p is the initial delay and a p is the first order derivative ofτ p (t) and T is the period of the signal.
The parameter a p is known as the Doppler scaling factor and can be related to the relative speed v p of the transmitter and receiver along the pth path as a p = vp c . Positive values of a p occur for positive relative Doppler shifts resulting in signal contraction while negative values of a p indicate opening relative Doppler causing signal dilation. The Doppler scale factor for a vessel traveling at fifteen knots transmitting acoustic communications to a fixed sensor is approximately 5.1 × 10 −3 (7.7 mps / 1500 mps). The scale factor is much larger in underwater communications compared to mobile radio where the speed of propagation is high and typical receiver speeds (29 mps ≈ 65 mph) produce a Doppler scale factor on the order of 1 × 10 −7 . Based on assumptions AS1 and AS2, the impulse response of the underwater channel with independent Doppler scaling on each path is modeled as The received bandpass signal,ỹ(t), is related to the transmitted bandpass signal,s(t), through the convolution of the transmitted signal with the channel impulse response with the addition of noise,ñ(t) and (possibly) interferenceĩ(t) : The resulting time-varying channel with path-specific Doppler scales is parameterized by N pa triplets {A p , a p ,τ p }. The received signal in the frequency domain is found via Fourier transform. Applying the Fourier transform pairs (20) toỹ (t) yields whereÑ (f ) andĨ(f ) are the Fourier transforms ofñ(t) andĩ(t) respectively.
If the Doppler fluctuations remain relatively constant over a signal period, the channel is parameterized by its common Doppler scale factor, a, and the N pa pairs of amplitudes and delays, {A p ,τ p }. In this case, the received signal is given bỹ The frequency domain received signal is found as above,

Introduction to Orthogonal Frequency Division Multiplexing (OFDM)
Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier communications modulation method commonly used in applications such as digital television, wireless networks, audio broadcasting and 4G mobile applications.
Techniques have been developed for it that successfully deal with challenges of the underwater acoustic channel. This section introduces the basic mathematics of OFDM modulation and describes the approach developed in [32] that has been successful in compensating for the Doppler induced signal contraction and dilation.
OFDM divides the channel into multiple parallel narrowband channels through the use of an orthogonal partition [33]. In OFDM, the partitioning is accomplished via the Fourier transform which results in the data being transmitted on multiple mutually orthogonal sinusoids. An example of an OFDM time domain signal comprised of four subcarriers is shown in Fig 3. The subcarriers are shown in color, and the transmitted waveform resulting from adding all of the subcarriers is shown in black. The subcarrier data rate is lower than the total data rate and the subchannel bandwidth is less than the total system bandwidth.
For a given total system bandwidth, the number of carriers determines the bandwidth of the subchannels. Typically the number of carriers is selected to ensure that the bandwidth of the subchannels is less than the coherence bandwidth of the channel so that the subchannels experience relatively flat fading and inter-symbol  Zero Padded OFDM (ZP-OFDM). The two variants differ in the treatment of the interval between data blocks. In CP-OFDM, the guard interval occurs before the data block and is filled with samples from the end of the transmit data sequence as shown in Fig 6. By appending a cyclic prefix to the channel input sequence, the linear convolution of the transmit signal with the channel impulse response becomes a circular convolution (see chapter 12 of [33]). On the receiver side, the samples corresponding to the cyclic prefix are discarded for CP-OFDM and the samples    corresponding to the OFDM data block are processed via Fourier transform.
In ZP-OFDM, the guard interval occurs after the data block and is filled with zeros. In contrast to the CP-OFDM receiver, the ZP-OFDM receiver retains the samples from the guard interval and processes the sequence with a method known as overlap and add. The overlap and add method adds samples from the guard interval to received samples corresponding to the beginning of the OFDM data block and then takes the Fourier transform of the newly formed sequence. The work in [35] compared the merits of CP-OFDM and ZP-OFDM and found that ZP-OFDM was preferable to CP-OFDM because: ZP-OFDM guarantees symbol recovery because linear channel equalization is possible irrespective of the location of channel nulls.
ZP-OFDM provides more flexibility to pursue a variety of different approaches to channel estimation and equalization.  ZP-OFDM with semiblind pilot-based channel estimation tracks channel variations better than CP-OFDM.
According to [35], the one drawback of ZP-OFDM relates to Peak-to-Average power ratio induced clipping. Because ZP-OFDM introduces slightly more nonlinear distortions, it needs slightly more increased power backoff than CP-OFDM.
The power backoff issue was addressed in [36] where use of a nonbinary LDPC code in conjunction with ZP-OFDM was found to reduce the Peak-to-Average power ratio. In an interference environment, the retention of guard samples required by the overlap and add processing is an additional drawback for ZP-OFDM. Interference occurring on these samples is retained for ZP-OFDM whereas for CP-OFDM interference occurring in the guard band would be discarded along with the cyclic prefix.
The work that follows uses ZP-OFDM. It was chosen because it is by far the prevailing choice for underwater acoustic communications. Furthermore, the interference cancellation technique developed in [17] was applied to ZP-OFDM.
Thus, use of ZP-OFDM enables a more direct comparison of this work with that in the literature.
The basic receiver processing chain for a ZP-OFDM signal received on an underwater channel is shown in Fig. 8. The bandpass filter (BPF) removes noise outside the frequency band of interest prior to synchronization of the received signal through the use of the known packet preamble. Doppler scale coarse estimation and resampling removes the main Doppler effect prior to partitioning the packet into blocks. After downshifting to baseband and low pass filtering (LPF), the time domain waveform is transferred to the frequency domain via overlapping, adding and taking the Fast Fourier Transform (FFT). After compensating for any residual Doppler effect (e.g. carrier frequency offset (CFO) estimation and compensation), the channel is estimated and the signal is equalized. Symbol detection and decoding complete the processing steps. Some receivers employ "turbo equalization" if not all of the parity checks are satisfied on the first attempt. In this technique, information from satisfied parity checks is fed back to assist the equalization process where they serve as additional pilot tones. The next sections discusses the mathematics of these operations.

Transmitted OFDM Signal
Let T denote the OFDM symbol duration and T g the guard interval between blocks. The OFDM block duration is then T = T + T g . The frequency spacing is Δf = 1/T . The kth subcarrier is at the frequency where f c is the center frequency and K is assumed even. The signal bandwidth  and define S A and S N as the nonoverlapping sets of active and null subcarriers.

The active S A and null S N subcarriers satisfy S
The transmitted bandpass signal is given bỹ where g(t) is the pulse shaping filter with Fourier transform G(f ). In this work, is a rectangular window of duration T .

Channel Model and Received Signal
Assuming that the channel exhibits independent Doppler on each path, the channel model of Eq. (21) applies, and the received bandpass signal when no interference is present is given bỹ The Doppler effect inỹ(t) is removed through the two step Doppler compensation technique described in [32]. The main Doppler effect of the received signal is removed through resampling using a factor (1 +â) resulting in the resampled signalz rs (t) =ỹ( t (1+â) ) . The estimateâ of the Doppler scaling factor a is found by comparing the time duration of the received packetT rx with the known time duration of the transmitted packet T tx , where the received packet time duration is estimated by cross correlating the received signal with the known preamble and postamble. The resampled signalz rs (t) may contain a residual Doppler error conventionally called the carrier frequency offset (CFO) in RF communications. After bandpass to baseband downshifting of z rs (t), the carrier frequency offset caused by the residual Doppler effect is compensated for by multiplying the baseband signal z rs (t) by e −j2πˆ t . The estimate of the carrier frequency offsetˆ is found using the method in [32] which takes the estimate as the frequencyˆ which minimizes the energy on the null carriers in z(t) = e −j2πˆ t z rs (t). After the two step Doppler compensation, the baseband receive signal is given by . (27) and the equivalent Doppler compensated bandpass signal is Substituting forỹ(t) from the channel model with path-specific Doppler scales (e.g. Taking the Fourier transform ofz(t) yields the received signal response as a function of frequency: (31) Applying the Fourier transform pairs Frequency translation (32) to s(( Thus, 1+ap τp , (34) and letting b p = (a p −â)/(1 +â) so that 1/(1 + b p ) = (1 +â)/(1 + a p ) produces The OFDM frequency measurement on the mth subcarrier is z into Eq. (35) to yield the frequency domain sample on the mth subcarrier Reordering the summations produces Noting that f m = f c + m/T produces and letting yields Defining the channel mixing matrix, produces the result, which can be written in vector-matrix notation as and succinctly written as The channel mixing matrix, H, is specified by N pa triplets {ξ p , τ p , b p } and may be written as where Λ(τ ) is a K × K diagonal matrix with the mth diagonal entry, and Γ(b, ) is a K × K matrix with the (m, k)th entry, The channel mixing matrix H is typically a banded matrix with the off-diagonal terms equal to zero for diagonal d > D for some D. In situations with little Doppler or where the Doppler compensation technique can eliminate its effect, H is well approximated by its main diagonal and equalization may be adequately performed by assuming a channel mixing matrix with a single main diagonal. This is the assumption of the equalizer in [32]. The off-diagonal terms in H model inter-carrier interference; that is, the terms H [m, k] where m = k describe the contribution to the measurement on carrier m due to the symbol s[k] transmitted on carrier k. When the inter-carrier interference is more substantial, the equalizer must account for it as, for example, in the equalizer of [37] which progressively increases the number of diagonals in H considered in the equalization process.

Survey of Recent Work in OFDM and Interference Mitigation
Acoustic communications have historically used a single carrier frequency.
However, within the last decade research has focused on multi-carrier modulation and OFDM in particular, (see [32,33, and references therein). OFDM di-vides the doubly (time-and frequency-) selective underwater channel into multiple orthogonal subbands. The bandwidth of each subband is chosen to be less than the coherence bandwidth of the channel so that each subband experiences relatively flat fading, thereby simplifying the channel equalization problem. Additionally, Inter-Symbol Interference (ISI) is mitigated since the transmission scheme may be viewed as transmitting many parallel low-rate narrowband signals rather than one fast-rate wideband signal [31]. The effects of ISI may be further reduced by inserting a guard interval between OFDM symbols that is filled with either a cyclic prefix (CP-OFDM) or a zero prefix (ZP-OFDM). Trade-offs between the two prefix approaches are discussed in [35].
Although OFDM mitigates ISI, inter-carrier interference (ICI) resulting from Doppler induced shifting of signals from one subband into another can severely degrade performance. Motion-induced Doppler distortion presents a major obstacle in applying OFDM to underwater channels since both the channel impulse response and the non-uniform effects of relative motion on subbands must be estimated.
Solutions to this problem are found in two broad approaches [16]: 1) adaptive synchronization, which requires few subbands dedicated to channel estimation but relies on coherence between adjacent OFDM blocks, and 2) non-adaptive synchronization, which does not rely on channel coherence but requires null subcarriers to gain robustness to fast channel variations. Selection and implementation of an appropriate technique depends on the coherence time and coherence bandwidth of the channel as well as making appropriate trade-offs among computational complexity, required channel estimation accuracy, and data rate. Distributing pilot tones evenly throughout the band and performing interpolation is more efficient than periodically dedicating an entire OFDM symbol to channel estimation [60]. Iterating the equalization and decoding tasks, that is performing turbo equalization, can yield large performance improvements [61], and reduced-complexity approximate algorithms are available as in [62] for example.
Through appropriate Doppler compensation of the received signal and judicious assignment of codewords to data slots within an OFDM packet, explicit time and frequency diversity can be achieved. Assigning parts of a codeword to subbands separated by more than the coherence bandwidth of the channel achieves frequency diversity while using an interleaving depth greater than the coherence time of the channel provides time diversity. Experimental results with OFDM have achieved data rates on the order of tens of kilobits per second but none of the algorithms have attempted to account for interference while taking advantage of spatial diversity. The algorithms in [17] and [38] address the possibility of interference but do not provide spatial diversity while the systems developed in [22] achieve time, frequency, and spatial diversity for a vertical line array but do not address interference. The asynchronous multiuser OFDM algorithm proposed in [63] views interfering users as a single external interference which the algorithm parameterizes and uses in an iterative channel estimation, data detection and interference mitigation scheme on a single receiver. Observations from past field experiments indicate that significant improvement in the reliability of message reception can be realized by mitigating interference. Discussion of the interference typical in the underwater environment is available in [64] and [65] .
Interference mitigation has a long history in RF communications but the interference is typically impulsive or narrowband [66] and the algorithms are typically developed for a single receiver. Partial band interference is not addressed [17].
Optimum reception is possible given the probability density function of the impulsive noise, but typically the impulsive noise statistics are not available to the receiver [68]. Without a priori knowledge of the impulsive noise statistics, the traditional approach to dealing with impulsive noise has been to mark those symbols hit by impulses as erasures rather than correcting them directly since more erasures than errors can be corrected by the same code [68]. The marking of erasures and error-and-erasure decoding have traditionally been performed separately in cascade [68]. The work in [67] applied the traditional cascade approach but explored using error correction coding (soft error protection) and adaptively adjusting the erasures marking threshold (hard error protection) to combat short time duration, high magnitude interference. Supplying channel state information to the decoder enhanced the performance of the applied forward error correction (FEC) code. The erasures marking threshold was adjusted on each of the multicarrier subchannels to protect those subchannels most susceptible to interference. The authors found that combining both the soft and hard error protection was most effective. A low computational complexity algorithm to cancel the impact of impulse noise in the time domain was proposed in [72]. It employed threshold blanking to cancel the impact of impulse noise on the traditional time domain least squares OFDM channel estimate. In [69], a frequency domain strategy to compensate for impulsive noise after OFDM demodulation and channel equalization was proposed as an alternative to the more conventional approach of suppressing the impulsive noise in the time domain. A support vector machine (SVM) algorithm for coherent robust demodulation of OFDM in the presence of impulse noise interfering with the pilot tones was developed in [71]. Compressed sensing techniques were applied to the impulse noise cancellation problem in [73]. Recently, [76] explored bandpass and baseband clipping approaches for suppressing impulsive noise in Direct Sequence Spread Spectrum Systems in underwater communications.
A narrowband interference canceller was developed in [77] by using a model of the power spectral density of the narrowband interferer as prior information. The canceller estimates the linear minimum mean-square error of the spectral leakage by measuring the narrowband interference on a few active or null OFDM subcarriers. [78] and [79] propose mitigating narrowband interference in OFDM through successive cancellation by applying hard or soft decisions based on the known or estimated second order statistics of the received signal. Using the structure provided by FEC coding, each subcarrier is used to predict the narrowband interference contribution on the considered subcarrier in order to subtract out the contribution in an iterative fashion. The most reliable subcarriers are decoded first.
Joint approaches to equalization and interference mitigation may work iteratively such as in [70] or by expanding the states of the decoding algorithm as in [81] and [83]. In [70] applying a clipping and nulling technique at the front of an iterative algorithm significantly reduced the symbol error rate (SER). The iterative decoder further improved performance through a novel syndrome decoding technique which segregated the decoding based on the noise level of the received components. The joint approach to erasure marking and decoding (joint erasure marking Viterbi algorithm (JEVA)) proposed in [68] was motivated by the observation that separation of erasure marking and channel decoding leads to less accurate detection of the impulse noise corrupted symbols because the erasure marker does not take the code structure into account. For a given number of erasures, the JEVA automatically finds the most likely erasure pattern based on the most likely trellis path under the erasure constraint. The algorithm effectively mitigates impulsive noise at the expense of higher complexity. In [83], a simple pilot aided interference detector removes in-band interference. Residual interference that cannot be detected by the pilot aided interference detector is removed by a joint interference detection and decoding scheme. By exploiting the code structure in interference detection, the joint scheme detects most of the symbols suffering residual interference without requiring knowledge of the interference distribution.
A message-passing approach to jointly estimate the channel and mitigate strong co-channel interference of similar form as the desired signal was proposed in [87]. The algorithm, based on belief propagation (BP), which performs statistical inference on graphical models by propagating locally computed beliefs, effectively exploited the non-Gaussian statistics of the interference enabling its detection and suppression without requiring specific knowledge of the interfering signal. Two blind algorithms to mitigate multiple interferers were proposed in [88]: one for the case of strong signal and one for the case of strong interference. Both algorithms find the coefficients of the linear Minimum Mean Square Error filter based on the autocorrelation matrix function of the channel outputs.
Generally, joint approaches provide better performance at the cost of additional computational complexity. Limitations on system performance may be found in [89] for OFDM systems subject to impulsive noise and for multicarrier and single carrier quadrature amplitude modulation (QAM) systems in [90]. The capability of Low Density Parity Check (LDPC) and turbo coding to mitigate burst errors is discussed in [91].
In mobile radio, optimal receiver combining (ORC) [92] is a multi-receiver technique that has been used for years to combat interference. Its performance bounds were developed in [93], but the channel is much less complicated than in underwater communications: it contains no multipath and Doppler effects may be safely ignored. The ORC algorithm assumed the channel was known and the combining was performed on the time domain received waveform. Its original application was to Phase-Shift Keying (PSK) modulation.
Some more recent work has addressed diversity combining in the underwater environment. A receiver that addressed the challenges of the underwater channel and performed joint diversity combining of multiple channel receptions with channel equalization was proposed in [94]. More recently, [95] proposed a multiple-input multiple output system which used space-time coding and iterative decoding techniques to obtain high data rates over shallow-water, medium range channels in the absence of interference. In the 200-300 Hz frequency range, [96] exploited beam diversity among sensors separated by hundreds of meters to improve communication performance.
Whereas processing time and computational complexity are critical drivers in many RF applications (such as mobile phones, digital audio and video broadcasting), underwater communication applications exist (such as submarine to shore message trafficking) where decoding delays are tolerable and vast amounts of computational power are available. Thus, the additional available decoding time and computing power may be employed to untangle the more complicated channel and Doppler effects.

Motivation for the Research
The motivation for this research is to examine approaches to leverage the spatial diversity of underwater acoustic communications networks suffering from interference. The AUTEC network [21] consisting of 96 fiber-optically connected hydrophones that receive signals over a time-varying multipath channel in the presence of interference provides the impetus to examine a unique communica- of the interference reduces the error rate. Reducing the message error rate to less than one percent will enhance safety on the AUTEC range and enable more users to operate on the range simultaneously. The AUTEC range is currently an overbooked Navy resource. Significant scheduling flexibility and cost reduction can be realized by developing the capability to communicate effectively in this congested acoustic environment.
The application of this work extends beyond the AUTEC network to cabled acoustic networks in general and other situations where interference corrupts receptions on widely separated receivers. As we continue to explore and operate in the oceans, the number of activities grows. Many of these activities, such as oil exploration and mapping of the ocean floor using active sonar, generate signals of interest to the user of the application at hand but create interference from  the perspective of other users. Communications is an essential aspect of many of these operations. For example, autonomous undersea vehicles require a reliable communications link to send and receive data to remote users. As these activities increase, understanding the limits interference places on performance and developing approaches to mitigate its effects are important areas of research.

Summary
After providing some background information about the early development of underwater acoustic communications and the AUTEC acoustic network, this chapter surveyed the challenges of the acoustic channel and reviewed recent work in interference mitigation and OFDM as motivation for the development of a spatial diversity receiver that will be introduced in the next chapter. The wave equation governs the propagation of sound underwater, but solving it directly is difficult.
Instead, practical underwater communication algorithms are built on the simpler channel models discussed in section 1.3.2. As human activities in the ocean increase, its acoustic environment becomes more congested, motivating the need to develop techniques to mitigate interference and understand the limits interference places on performance.

Introduction
After providing an example of the spatial diversity exhibited at AUTEC, section 2.3 introduces the concept of a spatial diversity receiver. Section 2.4 applies the concept of a spatial diversity receiver to the OFDM modulation scheme and describes an algorithm capable of mitigating partial band, partial block interference in OFDM signals. After describing the approach to using spatial diversity to mitigate interference, the parameterized interference cancellation algorithm from [17] is presented. Section 2.7 compares the approaches of the two algorithms.

Spatial Diversity at AUTEC
Acoustic propagation is much more environmentally variable than RF, largely due to the slow ∼ 1500 mps speed of propagation and its strong dependence on ocean temperature, pressure and salinity. AUTEC has a downward-refracting acoustic environment with surface water temperature of typically 26 • C (80 • F) and the thermocline appearing between 120 and 210 meters (400 to 700 feet).
The propagation of sound under these environmental conditions exhibits shadows zones. Fig. 11 illustrates this variability from a networking point of view: thirty-nine messages were transmitted from the location marked by the blue star, numbers of receptions at receiver hydrophones are indicated by the number adjacent to each phone, and the relative signal strength and quality, as assessed by the AUTEC ACOMMs processing algorithm, are represented by color. Green circles represent receptions with high signal-to-noise ratio (SNR), red circles indicate correctly-decodable receptions with degraded SNR and/or enhanced or dynamic multipath and black circles represent low SNR receptions that are unusable. (The small black dots indicate hydrophones that were not part of the experiment). As expected, receivers close to the transmitter receive the strongest signals, although even in this case the number of high-quality receptions drops off precipitously with range: a receiver less than four miles away only received seven high-quality receptions. The range to some receivers that were unable to decode any transmissions is in many cases comparable to hydrophones with excellent reception. This patterning is very typical of the oceanic area and arises from spatial fluctuations of three-dimensional temperature distribution in the ocean. Correlation of reception intensity or quality between successive packet transmissions is typically low.
The acoustic environment at AUTEC is congested. Active sonars, mammal vocalizations and snapping shrimp are some of the sources that interfere with underwater acoustic communications. Currently 97 percent of the transmitted messages are correctly decoded. Reducing the error rate to below one percent will enhance range safety and enable more platforms to operate simultaneously on the range.

A Spatial Diversity Receiver to Mitigate Interference
The concept of combining multiple receptions of the same transmitted signal is not new. In a network, information is available for combining from multiple layers: 1) the received waveform, 2) the detected received symbols and 3) the decoded information. In general, a diversity processor could operate on any of these layers or among the different layers. This work focuses on combining the received waveforms. The concept of combining received waveforms has been a standard approach for decades. Maximum ratio combining (MRC) weights the received waveforms based on the signal-to-noise ratio at each receiver (see chapter 7, section 5 of [33]). Optimal ratio combining (ORC) [92] also takes advantage of the differential fading received signals experience but additionally considers  Green circles represent receptions with high SNR, red circles indicate correctly-decodable receptions with degraded SNR and/or enhanced or dynamic multipath and black circles represent low SNR receptions that are unusable. (The small black dots indicate hydrophones that were not part of the experiment.) the impact of co-channel interferers in calculating the weighting of the received waveforms. ORC was specifically developed to combat co-channel interference on Rayleigh fading channels for mobile radio. It was originally applied to Phase-Shift-Keying (PSK) modulation and assumed the channel was known. Whereas ORC takes advantage of the differential fading of the virtually synchronously received interference among the receivers, the algorithm described here relies on the slow speed of signal propagation underwater which results in the received interference exhibiting some degree of time orthogonality. Underwater receptions also experience differential fading and when the interference is not time orthogonal, the algorithm makes use of this in a manner similar to ORC.
Receivers that seek to combine information at the time-domain waveform level face some common challenges. Obviously, the combining must occur on waveforms containing the same transmitted message. Once the signals containing the same message have been selected the receiver must: Properly time align the received waveforms Estimate and equalize the channel through which the waveform was transmitted Determine and apply the appropriate weights to the received signal Re-equalize the combined waveform Data detection and decoding can then be performed. Proper synchronization and equalization of the received waveforms are critical steps and errors in these processes lead to performance degradation. The waveform reconstruction algorithm described below assumes that the correct waveforms have been selected for combining and that these waveforms have been properly time aligned.
In the context of the literature, the algorithm can be seen as building upon ORC in taking explicit advantage of the time orthogonality of the interference.
Recalling that the ORC technique was originally applied to PSK modulation assuming the channel was known provides another vantage point. From this perspective, the work extends the ORC concept to OFDM modulation and offers an approach to dealing with channel estimation and equalization that is suitable to OFDM modulation.

Spatial Diversity Reconstruction for OFDM
The OFDM spatial diversity receiver for underwater communications identi- nal (e.g. it occupies the same time bins on all receivers), averaging is performed. At the input to the SDR processing the received baseband time domain signal is given by and its frequency domain samples on each of the K subcarriers are contained in  the vector where H is the channel mixing matrix, w is the additive noise and v is the interference. The interference has time duration T I (which contaminates N I baseband samples), center frequency f Ic and bandwidth B I 1 . It occupies the frequency band

Frequency Domain Interference Detection
LetS v denote the subcarriers in the noise only band (S v := {m : f m ∈ B I }), and let S v denote the subcarriers in the band which potentially suffers from  [17] and discussed in Section 2.6.2. The GRLT interference detector is left for future work.

Channel Equalization
Next the channel H is estimated and equalized. A number of channel estimation and equalization algorithms are available such as [37,50,52,61,62,87].
This work used the least squares algorithm from [32] which is applicable provided the number of pilot tones K p = |S P | is greater than the length of the channel, e.g. K p ≥ N pa + 1. The K p pilot tones at the subcarrier indices p 1 , p 2 , ..., p Kp i.e.

{s[p i ]}
Kp i=1 are known to the receiver and the channel taps can be found based on the least squares (LS) formulation Following the design principles in [32], this work spaced unit amplitude pilot tones equally within the K subbands and the equalizer assumed that the channel length was L baseband samples. Because the pilots are equispaced, F H F = K p I L+1 and since they are of unit amplitude, D H The frequency domain estimate of the channel is obtained fromĥ LS using This equalizer ignores any residual Inter-Carrier Interference remaining after compensating for the Doppler effect. More sophisticated equalization algorithms exist which allow for off diagonal terms in the channel matrix H. This simpler algorithm adequately equalized the signals transmitted during the at-sea experiments reported in the next chapter. Since the received signal suffers interference on only some of the received pilot tones, a weighted least squares approach could be reasonably expected to improve equalization performance. However, since the standard LS approach performed adequately on the data, no effort was made to implement a weighted least squares equalizer.

Time Domain Waveform Reconstruction
Provided interference is detected on any reception, the received frequency There are N = K windows whose output is given by Time domain interference is declared in the window where q achieves its maximum.
The time domain reconstruction is a weighted combination of the received waveforms which takes into account where the interference occurs as well as the signal, noise and interference power on the receivers. An example of the weighting scheme for three receivers is illustrated in Fig 15. The calculation of the weights Letᾱ r be the ratio of the signal power to noise variance on the rth receiver outside of the interference band calculated as whereσ 2 sr andσ 2 B Ir are estimates of the signal power and noise variance on the rth receiver obtained by averaging over the active and null subcarriers outside of the interference band as follows: Let α r be similarly defined asᾱ r but occurring inside the interference band. Thus, Let i r denote the indicator function of interference in the samples z r on the rth receiver:

Re-equalization, Data Detection and Decoding
Taking the Fourier transform of the reconstructed time domain signal z SDR produces the reconstructed signal in the frequency domain z SDR . After running the equalizer (from section 2.4.2) on z SDR , the symbols are extracted. Finally, the LDPC decoder is run on the detected symbols producing the decoded message.

Sources of Error
Equalization plays a critical role in the Spatial Diversity Reconstruction algorithm since the reconstruction process operates on equalized waveforms. The received waveform z r is modeled as: where A errr is the amplitude error and φ errr is the phase error on receiver r. In panel respectively. Looking beyond a single carrier frequency, Fig. 18 shows the results of a 20 • phase mismatch applied to the OFDM waveform used in the simulations and experiments discussed in the next chapter.
In order to gain understanding of the effects mismatches in the equalization process induce in overall performance, the SDR algorithm was applied to an OFDM waveform in which errors in amplitude A err and phase φ err were artificially inserted on two receivers as follows: The amplitude error was varied between 0 and 0.9 and the phase mismatch was varied between 0 and 180 degrees. The length of the interference was N I = 342 samples, approximately one third of the received signal, and was separated by 171 samples on the two receivers. The SDR algorithm was applied to all combinations of received signals suffering from the range of amplitude and phase mismatches.      indicates that phase mismatches are more detrimental to performance, but that relatively large phase mismatches are tolerable. Phase mismatches of less than 90 • did not produce errors.

Mitigation of Partial Band Partial Block Interference through Coherent Cancellation
The work in [17] developed an iterative single receiver parameterized interference cancellation (PIC) algorithm to mitigate partial band interference of known time duration, T I , and effective interference bandwidth, B I . Since the interference is time and band limited 1 , it can be represented by a Fourier series. The algorithm employs an iterative approach to estimate the N I = B I T I complex coefficients as well as the time delay (with respect to the start of the OFDM block) of the interference. On each iteration, the estimates of the interference and detected symbols from the previous iteration are used to construct a GLRT to determine if interference is still present. If residual interference is detected, its model parameters are calculated with a maximum-likelihood approach. The modeled interference is then coherently subtracted from the desired signal, the channel is equalized, and symbol estimation is again attempted. This is repeated until the parity checks are     Figure 19: Signal constellation resulting from SDR processing with phase mismatch.
The received signal constellation resulting from a phase offset of +10 • on receiver 1 and −10 • degrees on receiver 2.  satisfied or some maximum preset iteration limit is reached.

Interference Parameterization
The interference parameterization from [17] is repeated below to provide context and completeness. LetĨ(t) denote the bandpass waveform of the interference having center frequency f Ic , bandwidth B I and time duration T I , and let I(t) denote the baseband waveform. Since I(t) is time limited, it adopts a Fourier series representation as where c l is the coefficient on the basis e where N I = B I T I and N I is assumed to be even without loss of generality. Therefore, Eq. (65) can be rewritten as The corresponding bandpass signal is wheref l = f Ic + l T I . The Fourier transform ofĨ(t) in the frequency band can be expressed as Note that the interference overlaps the received OFDM signal with an unknown delay. Define τ I as the delay of the interference relative to the starting point of the OFDM block in which it resides. After the pre-processing of the OFDM receiver, the interference component at the mth subcarrier is Following the derivation to Eq. (35) in chapter 1, we can formulate ν[m] as where Stacking interference components at all subcarriers into a vector ν yields where Λ(τ I ) is a K × K diagonal matrix, Γ I is a K ×N I matrix, and u is an The received signal in the presence of interference is then formulated as

Interference Detection and Estimation
Assume that both channel and symbol estimates (Ĥ andŝ) are available.
Here,w denotes the equivalent additive noise within the frequency band, which consists of the ambient noise and the residual noise due to imperfect channel and information symbol estimates.
Assume that the noise samples are independent and follow a complex Gaussian distribution CN (0, σ 2 B I I M I ), where σ 2 B I denotes the noise variance. The likelihood function of the measurement componentz in the presence of interference is Let H 0 and H 1 denote the absence and presence of interference, respectively. To detect the presence of interference in one particular OFDM block, define the GLRT statistic L(z) = max (77) where Γ th is a predetermined threshold.
Define an objective function to be maximized over {τ I , u} Typically for an optimization problem, the maximization must be carried out jointly over the parameters, in this case u and τ I . However, successful results were achieved in [17] by finding the optimal estimate of Fourier coefficients u and then maximizing over the delay, τ I . This was accomplished by setting J u to zero to yield the optimal estimate of u aŝ substitutingû into Eq. (78) to obtain and solving Eq. (80) by one dimensional grid search over the delay. Based on the estimated parameters {û,τ I }, the test statistic is evaluated as The test statistic L(z) is the ratio of the energy of the estimated interference to the energy of the equivalent noise. The test threshold Γ th is determined based on the predetermined probability of false alarm P F A or the probability of detection P D . In our experience with the algorithm, we noted that there was no loss in performance in applying the cancellation algorithm in the absence of interference. The cost of false alarms is negligible. However, not applying the algorithm when interference is present significantly degrades performances. Consequently, it is best to set the threshold to maximize probability of detection.

Channel Estimation, Equalization and LDPC Decoding
If the presence of interference is declared from the GLRT detector, the desired OFDM component can be obtained by subtracting the estimated interference from the received signalz = z − Λ(τ I )Γ Iû = Hs +w (82) wherew denotes the equivalent noise which consists of the ambient noise and the residual interferencew If no interference is detected (either absence of interference or missed detection), simply setû = 0 in Eq. (82). Based on the observation vectorz and the symbol vectorŝ, the channel estimator in [32] is used to estimate the channel matrix H. This estimator assumes a diagonal channel matrix and is appropriate for the channel observed in the AUTEC-0514 experiment. The work in [17] used the more sophisticated channel estimator of [54] in conjunction with the MMSE estimator of [61]. The LMMSE estimate of the information symbols is fed into the LDPC decoder to obtain hard and soft decisions on the information symbols. If all the parity checks are not satisfied, another iteration of interference cancellation is performed, in which information symbols satisfying the parity check equations are used as additional pilot symbols.

Noise Variance Estimation
Due to the partial-band property of the interference, the noise variance σ 2 w is estimated separately for the noise within and outside of the interference band B I based on the frequency measurements at the null subcarriers. Based on the estimates of the channel matrix and the transmitted symbols, the variance of the equivalent noise outside of the interference band, which consists of the ambient noise and the residual ICI due to the banded assumption of the channel matrix, can be estimated aŝ For the equivalent noise within the interference band, which consists of the ambient noise, the residual ICI and the residual interference, the noise variance can be estimated as Since the implementation of the equalizer assumed a diagonal channel, D is taken as zero in Eqs. (84) and (85). The estimated variance is then used for interference detection and information symbol estimation.

Comparison of Spatial Diversity Reconstruction and Parameterized Interference Cancellation
The SDR algorithm and the PIC algorithm address the problem of interference in fundamentally different ways. Interfering signals with different time durations and bandwidths but a time bandwidth product less than the selected value N I will still be estimated accurately and subtracted. In contrast, SDR requires the time duration and bandwidth to be known separately and its performance is best when applied to signals with the specified time duration and bandwidth.
While SDR requires more detailed prior knowledge than PIC, its performance is less dependent on the SIR as will be seen in the results presented in the next chapter. PIC relies on accurately estimating the channel and transmitted symbols to aid in estimating and canceling the interference. As the interference gets stronger, channel estimation and symbol detection degrade resulting in poorer estimates of the interference. At loud enough interference levels, the algorithm fails. Because SDR blanks the interference rather than attempting to cancel it, the algorithm is less susceptible to interference induced equalization errors. In contrast to PIC, there is no low SIR threshold where SDR abruptly fails. Its performance is more strongly a function of the background noise and the degree of time orthogonality of the interference than the level of the interference.

Spatial Diversity Reconstruction
Parameterized Interference Cancellation Requires multiple receivers.
Single receiver algorithm which might be extended to multiple receivers. However, the approach to extending the algorithm and the performance gained from additional receivers are unclear.
Relies on the time orthogonality of interference for blanking.
Relies on estimating the time arrival and Fourier series coefficients of the interference to coherently remove it from the desired signal.
Performance is not limited by SIR. Performance degrades at low SIR.
A priori knowledge facilitates frequency domain interference detection and sets the size of time domain blanking window.
A priori knowledge sets the number of coefficients used to estimate interference.
Imprecise prior knowledge leads to performance loss.
Prior knowledge does not need to be particularly precise. A loose upper bound on the time bandwidth product N I = B I T I is sufficient.
Single pass receiver: interference detection, excision, channel estimation and data detection are separate processes.
Interference detection, interference reconstruction and cancellation, channel estimation and data detection are coupled to form an iterative receiver.
Analytic expression for performance under certain conditions. 1 No analytic expression for expected performance.
Requires operations in the time and frequency domains to take advantage of the time orthogonality of the interference.
Operates exclusively in the frequency domain but does not leverage the time orthogonality of the interference.

Summary
Many acoustic channels suffer from interference which is neither narrowband nor impulsive. This relatively long duration partial band interference can be par-ticularly detrimental to system performance. However, some parameters of the interference are usually known or can be acquired and exploited. For a single receiver, prior knowledge can facilitate interference cancellation. For multiple receivers, the prior knowledge can be used to exploit the detection and excision of portions of the received signals suffering from interference so that they can be combined more effectively. While the single receiver technique relies on accurately estimating and subtracting the interference, the multi-receiver technique exploits the time orthogonality of the interference due to the slow speed of sound propagation in water and the geographical extent of the network to combine clean portions of the received signals.

Introduction
This chapter reports the performance of the Spatial Diversity Reconstruction algorithm under different channel conditions. In Section 3.

Performance on Additive White Gaussian Noise Channel with Interference
The performance of the SDR and MRC algorithms over an impulse channel with additive white Gaussian noise and suffering from interference which is time orthogonal is derived in this section. The expected variance of the resulting signal is found. This variance can be used as the basis for developing an adaptive combining receiver which applies the SDR or MRC technique depending on which provides the best performance. The performance is derived under the following assumptions:

AS-1
The channels are impulsive.

AS-2
The noise is additive white Gaussian noise.

AS-3
The noise and interference are independent processes.

AS-4
The interference is time and band limited. 1

AS-5
The interference is contiguous and occurs only once within the OFDM block.

AS-6
The interference is additive white Gaussian noise.

AS-7
The interference is time orthogonal among the receivers.
As the assumptions above are relaxed, the analytic expressions of performance provided below become an upper bound of performance. The purpose of channel equalization is to remove intersymbol interference making the channel appear "impulse-like" to the follow-on processing. To the extent that the channel equalizer does not invert the channel response, that is H eq H = I, and intersymbol interference remains there is a degradation in performance which can be categorized as loss due to equalization error. Noise whitening is also inherent is most communication receivers. [57] provided a noise whitening approach for underwater OFDM noting that significant gains can be realized by applying prewhitening before channel estimation. Thus, to the extent that the noise is not white, performance loss should be expected. Communication systems are also typically designed assuming that the noise is independent. As discussed previously, the OFDM carrier spacing is chosen so that carriers experience independent fading and considerable work has been done to mitigate inter-carrier interference (ICI). From one perspective, these mitigation techniques aim to restore the "noise independence" of the carriers. In applications, noise and interference are independent processes by definition. Thus, AS-1 through AS-3 are reasonable assumptions, and further, the extent to which they do not hold, performance typically degrades. Communication systems must deal with different types of interference. For many applications AS-4 and AS-5 hold exactly, while AS-6 and AS-7 are more tenuous but enable an analytic evaluation of performance under optimal conditions. Performance in non-Gaussian interference will be worse for both SDR and MRC because the weighting scheme for both algorithms is optimized for Gaussian noise.
Therefore, considering all of the above, the BER equations derived in this section provide an upper bound for performance.

Additive White Gaussian Noise and Time Orthogonal Interference Channel Model
Let z r denote the rth of R received baseband signal vectors in the frequency domain and let z r be its time domain inverse. That is, z r and z r are K−point interference v is given by where i r • v r models the interference; i r indicates the frequency bins suffering interference and v r is the random variable taking on the value of the interference.
The indicator function i r is defined as . . .

i[n]
. . . . Assume that the interference is time limited to N I samples and time orthogonal among the receivers. Therefore, it can be completely excised (blanked) from the reconstructed waveform. Since the interference is time orthogonal among R receivers and limited to N I samples RN I ≤ K. Further assume the interference is band limited to K I frequency bins.

Spatial Diversity Reconstruction Performance
Assuming the SDR algorithm clairvoyantly blanks the interference on the R receivers, the reconstructed waveform results from the weighted average of K−RN I time domain samples from all R receivers and the weighted average of RN I samples from R − 1 receivers. Fig. 22 illustrates this for the case of three receivers. In the figure time domain samples suffering interference are indicated in red while samples suffering from only noise are shown in green. The reconstructed sample in the nth bin is given by: where is the weighting factor for the nth sample. The noise variance in the nth time sample is given by Taking the Fourier transform of the reconstructed waveform to obtain the OFDM symbol spreads the noise among all the received frequency bins. The noise variance in each frequency bin is given by: Another form of Eq. (90) The variance contribution from samples suffering from no interference on any receiver, (that is group g = 0), is while the variance contribution from group g is and the total noise variance in each frequency bin is given by: Substitution of Eqs. (94) and (95) into Eq. (96) and recalling that σ 2 w n = σ 2 w k K , yields an expression of the SDR frequency domain noise variance in terms of the frequency domain noise variance on each receiver, The uncoded Bit Error Rate of the reconstructed received signal for BPSK is given by: where Q(x) is the right-tail probability (e.g. complementary cumulative distribution function) for an N (0, 1) random variable.

Maximum Ratio Combining Performance
In Maximum Ratio Combining, the interference is not removed from the received signal. The receptions are combined according the received noise levels.
The weights ω r are calculate as The noise variance in the frequency bins not suffering from interference is given by and the noise variance where interference is present is given by The uncoded BER of the MRC received signal for BPSK is given by: An "equivalent" variance for MRC σ 2 MRC k eq under interference conditions is found by inverting Eq. (102) yielding The equivalent MRC variance σ 2 MRC k eq and the SDR variance σ 2 SDR k can be compared to determine when to implement each combining strategy: Thus, Eqs. (96) and (103) provide the framework for developing an adaptive combining receiver which accounts for the time duration and bandwidth of the interference as well as the SIR and SNR. The development of such a receiver is left for future work. The implemented receiver transitioned between the MRC and blanking strategies based on the significance level of the Kolmogorov-Smirnov test.

Example: Two Receivers on Additive White Gaussian Noise Channel with Interference
For the special case of two receivers suffering time orthogonal interference on an additive white Gaussian noise channel, the variance of the noise in the kth bin for the SDR algorithm from Eq. (96) is If the noise level on the two receivers is the same, γ 0,1 = γ 0,2 = 1 2 and the variance is and recalling that σ 2 resulting in a BER of For MRC, the weights are the same (ω 1 = ω 2 = 1 2 ) and the variance of the combined signal is and the BER is given by For the special case, when the variance of the noise σ 2 w k = 1, the BER performance of the two algorithms is Plots of the performance from Eq. (108) in blue and Eq. (110) in red are shown

Example: Three Receivers on Additive White Gaussian Noise Channel with Interference
The analytic expressions for performance, e.g. Eqs. (98) and (102) simulated receptions at each different SNR and SIR level were performed. Simulation A held the SNR on each receiver constant at 0, 3, and 6 dB respectively and varied the SIR level from -10 to 10 dB. Each receiver experienced the same SIR as it was varied. Simulation B held the SIR on each receiver constant at -3, 0, and 3 dB while varying the SNR from -10 to 6 dB. Each receiver experienced the same SNR as it was varied.    interference. These receptions were processed using the MRC algorithm to provide a baseline. Fig. 25 shows the results from simulation A and Fig. 26 shows the re- The performance of blanking is constant with SIR since it removes the portions of the signals suffering from interference (see Fig. 25).
For a given noise level, there is an interference level at which it is better to apply MRC instead of blanking. As noted above, this is the interference level where σ 2 MRC k eq < σ 2 SDR k (see Fig.25).
For a given interference level, there is greater gain in applying blanking   Figure 25: Performance on AWGN channel as a function of SIR. Analytic and Simulation results for different combining strategies on AWGN Channel with and without interference. Clairvoyant blanking of interference applies the SDR algorithm with exact knowledge of the samples suffering from interference. The interference is time orthogonal with a duration of 256 samples. The SNR is 0, 3, and 6 dB respectively for each receiver. The SIR is the same for all channels varying from -10 to 10 dB. The range of SIRs was chosen to demonstrate the behavior of the combining strategies, not because it represents the expected operating regime of the receiver.   Figure 26: Performance on AWGN channel as a function of SNR. Analytic and Simulation results for different combining strategies on AWGN Channel with and without interference as a function of SNR. Clairvoyant blanking of interference applies the SDR algorithm with exact knowledge of the samples suffering from interference. The interference is time orthogonal with a duration of 256 samples. The SIR is -3, 0 and 3 dB respectively for each receiver. The SNR is the same for all channels varying from -6 to 6 dB. The range of SNRs was chosen to demonstrate the behavior of the combining strategies, not because it represents the expected operating regime of the receiver.
instead of MRC as the SNR level increases (see Fig. 26).
The performance of blanking on a channel suffering interference does not achieve the performance of MRC on a channel without interference. That is, blanking cannot completely mitigate all the effects of the interference (see Figs. 25 and 26).
The performance of all the combining strategies tends to converge as SNR gets very low (see Fig. 26).

Simulation Results on a Time Invariant Channel
The spatial diversity reconstruction algorithm was tested on a time invariant channel using simulated data. The geometry for the simulation, shown in Fig. 27, postulated a source equidistant from two receivers and an interfering signal closer to receiver 1 than receiver 2 causing the interference to arrive at receiver 1 before receiver 2. The simulation further postulated that the interference arrives in the first half of the OFDM symbol period on receiver 1 whereas it corrupts the second half of the received signal on receiver 2. The simulated time-invariant channels are shown in Fig. 28. Channel 1 is taken from [21]. Simulation results were obtained for two cases: 1) a priori known channel impulse response in which the receptions were equalized by inverting the channel response; that is, a zero-forcing equalizer was employed; 2) unknown channel impulse response in which a minimum mean square error equalizer estimated the channel response based on pilot tones using the estimator from [32]. Table 3 lists the simulation parameters. in Table 3. The interference is thus orthogonal in the time domain on the two receivers but overlaps in the frequency domain.
The simulated time domain interference is sampled, overlapped and added, and an FFT is taken to produce frequency domain interference which is then scaled to the appropriate SIR and added to the background noise, which is modeled as complex white Gaussian noise with an SNR of 7.9 dB. After adding the simulated noise to an OFDM symbol vector, the waveform reconstruction algorithm was run at an SNR of 7.9 dB for SIRs varying from -10 to 2 dB. The Monte Carlo simulation was stopped when either 500,000 bits had been processed or 250 errors were made.

Example of SDR on Experimental Data
Examples of the received data packets are shown in Independently processing the third block from the channels shown in Fig. 31 without interference suppression resulted in thirteen errors on receiver 68 and eight errors on receiver 76 whereas applying the parameterized interference mitigation algorithm resulted in no errors. Processing the third blocks using maximum ratio combining (MRC) resulted in three errors while applying spatial diversity reconstruction (SDR) resulted in no errors. While the interference mitigation techniques both reduce the BER to zero, applying the error correction code also reduces the error rate to zero. Thus, the example is not the most convincing demonstration of the power of these techniques because the interference level is not high enough to drive the error rate above the correction capability of the code. Unfortunately, the interference level in this example is the highest observed during the experiment.
Consequently, in order to investigate the performance of the algorithms over a wider range of SIR and SNR levels, pseudo-experimental data was generated from actual received data.

Statistical Performance of Algorithms Based on Pseudo-Experimental Data
In order to test the performance of the algorithms at different interference and noise levels, a data set of 1,500 synthesized received signals was created from the interference and noise sampled during the experiment. The pseudo-experimental interference was created by windowing the appropriate portions of the received direct path LFM signals and amplifying them to the appropriate interference level.
Similarly, noise from portions of the experiment when no signal or interference was present was amplified to the appropriate level to vary the signal to noise ratio.
The amplified interference and noise was added to received OFDM blocks in which  the symbols were all detected correctly prior to decoding. For testing of the SDR and MRC algorithms, the interference was time orthogonal on the two receivers.
A detailed explanation of the steps used to generate the pseudo-experimental data is provided in Appendix A.
The results of processing the pseudo-experimental signals prior to decoding are shown in Fig. 32 for single channel processing, MRC and SDR. Clairvoyant blanking (shown in cyan circles) always excised the correct portion of the signal suffering interference whereas the SDR algorithm (shown in blue stars) had to detect the presence of interference as well as select the portion of the received signals to excise. The SDR algorithm applied the blanking strategy when interference was high and the MRC strategy otherwise. At low SIRs (when the interference is large), there is a clear benefit to excising the interference for all values of SNR, but more dramatic gain is realized at higher SNRs as evidenced by comparing the reduction in BER seen as SNR increases from 0.00 to 6.00 dB for the same SIR level. For example, at -6.00 dB SIR and 0.00 dB SNR, the BER is 0.0979 for SDR and 0.1153 for MRC; SDR performance is better by 0.0174. In comparison for the same interference level (-6.00 dB SIR) but a higher SNR of 6.00 dB, the BER is 0.0208 for SDR and 0.0548 for MRC; SDR performance is better by 0.0340.
As the interference power approaches the noise power, the benefit of excising the interference diminishes and the loss from not averaging over the noise becomes apparent. This is demonstrated at higher SIRs in the results for clairvoyant blanking where, even with perfect knowledge of the time window suffering interference, performance is degraded as compared to MRC. These results are consistent with the theoretical results presented earlier. The interference detection threshold can be set to transition between the SDR and MRC combining strategies. The blanking strategy was employed when the significance level of the Kolmogorov-Smirnov hypothesis test exceeded ks th . By only declaring interference when it is sufficiently large, the SDR algorithm exhibits the same performance as MRC at high SIRs and the benefits of the clairvoyant blanking at low SIR. Even though the threshold employed was not adjusted for SNR, the SDR algorithm achieves close to the best possible combining performance across the range of SNRs tested. At 0.00 and 3.00 SNR, the SDR algorithm closely follows the best combining method: blanking at low SIR and MRC at high SIR. At an SNR of 6.00 dB, a small performance gap exists between SDR and MRC as the SIR approaches zero. Note that the lower the SIR, the easier it is to detect the interference and correctly remove the corrupted portion of the signal. The effectiveness of the time and frequency domain interference detectors is borne out in the experimental results. The results show that, when the algorithm is blanking the interference, its performance is very close to the clairvoyant detector as illustrated by the proximity of the SDR and clairvoyant blanking curves in Fig. 32 at low SIRs. The single channel PIC algorithm (orange triangles) is effective at moderate SIR levels. This is the region where Fig. 32 indicated the most effective strategy was to transition from blanking to averaging (MRC). The PIC technique is ineffective at low SIRs; precisely the region where blanking gains are most dramatic.
As discussed in the previous chapter, SDR requires more detailed prior knowledge than PIC. However, as seen in Fig. 33, SDR performance is less dependent on the SIR than PIC since PIC relies on accurately estimating the channel and transmitted symbols to aid in estimating and canceling the interference. As the interference gets stronger, channel estimation and symbol detection degrade resulting in poorer estimates of the interference. At loud enough interference levels, the algorithm fails. Because SDR blanks the interference rather than attempting to cancel it, the algorithm is less susceptible to interference induced errors as shown in both Figs. 32 and 33. In contrast to PIC, at 3.00 dB SNR or above, there is no low SIR threshold where SDR fails. Its performance is less a function of SIR and more dependent on the background noise and the degree of time orthogonality of the interference.
The two approaches are complimentary in this sense: SDR is not limited by low SIR, but it requires multiple receivers; PIC is limited at low SIR but operates on a single receiver. In an undersea network one effective multi-channel receiver strategy would be to employ PIC on receptions suffering from moderate interference and transition to the multi-channel SDR technique when PIC fails. Alternatively, the SDR strategy could be employed as the default and PIC could be attempted in cases when the lack of time orthogonality and severity of the interference make blanking or averaging ineffective operations. In these cases coherent cancellation could be attempted. Chapter 5 discusses more ideas for future work and how these strategies might be combined.

Summary
This chapter reported on the performance of the Spatial Diversity Reconstruction algorithm under different channel conditions. Section 3.

Experiment Objectives
Many communications signals suffer from interference which is neither impulsive nor narrowband. However, some of the interfering waveform parameters such as its bandwidth or time duration are known a priori. Such interfering waveforms may be termed structured acoustic interference (SAI). The goal of the AUTEC-1214 experiment was to assess the performance of the Spatial Diversity Recon- EO-1 requires the SDR and PIC algorithms to successfully decode received OFDM signals suffering from SAI which are not decodable by other methods. In other words, the number of bit errors prior to decoding is too high for the error correction code to correct. Additionally, the SDR and PIC algorithms should not degrade the receiver performance when interference is not present. Following the work in [17], the transmitted message was encoded with a half rate LDPC code. Since [41] observed that typically no decoding errors occurred whenever the uncoded BER was below 0.1, demonstrating the effectiveness of the interference mitigation strategies requires generating received data sets with BERs greater than 0.1.
In order to satisfy EO-2, the data collected during the experiment must contain multiple receptions of the same OFDM waveform corrupted by partial-band, partial-block duration SAI over a range of SIR and SNR levels. Linear Frequency Modulated (LFM) signals of varying time durations and bandwidths were selected for use as the SAI because of their prevalent use in active SONAR applications.
Since the SDR algorithm is inherently a multi-receiver algorithm, EO-2 requires multiple receptions of the same OFDM waveform corrupted by the same partial-band, partial-block duration interference at different times. EO-2 requires a data set of received OFDM signals corrupted by interference parameterized by:

EO-2-P5 Time orthogonality of the interference
The SIR and SNR of the received signal are functions of the transmit power of the signal and interference as well as being highly dependent on the background ocean noise and the propagation of sound from transmitter to receiver. Clearly, the latter two quantities are not within the control of the experiment. Background noise level fluctuates due to changing weather conditions (such thunderstorms) and shipping traffic. Since the received power is strongly dependent on the range from the transmitter to the receiver and the AUTEC network has significant spatial extent, the experiment design opted to transmit at maximum power and rely on the variation in range from the transmitters to receivers to facilitate the collection of data at various SIR and SNR rather than varying the transmit power. The transmit power was not varied during the AUTEC-0514 experiment and receptions occurred at various SNRs as evidenced by the spectrograms and time series of the received data packets shown in Fig 34. The scaling in all the panels is the same and the relative positions of the receivers is shown in Fig. 35. The spectrogram for each of the five receivers, (61,67,68,75,78) Ideally the performance of the algorithms should be assessed directly on the received data without the need to create "pseudo-experimental" data by adding background noise or interfering signals received at other times during the experiment. However, should such "cutting and pasting" be necessary, the setup of the experiment should make it as easy as possible.

Experiment Procedure
Nine stereo wavefiles containing different LFM interfering signals on the left channel and the same OFDM data packets on the right channel and were created The scaling in all the panels is the same. The variation in received SNR is readily apparent by comparing the colors of the received data blocks in the spectrograms and the amplitude of the data blocks in the time series shown below each spectrogram.  Table 5: Distance from receivers to transmitters 69 and 75. Data recorded on receivers indicated with an asterisk (*) aided in constructing the channel model for planning the December experiment.
The experiment consists of transmitting an interfering signal and a desired signal from two different nodes on the AUTEC range and recording the signals received on the other nodes. The signals are transmitted by playing the stereo wavefiles on a compact disc with the left and right output sent to the desired transmit nodes. For this experiment, the left channel contained the interference and the right channel had the desired signal. The data is recorded at 96 kHz on Alesis HD24XR 24-Track recorders and transferred to external hard drives as wavefiles which can then be read into MATLAB ® for data analysis. The AUTEC range limits the play time of a single CD to 30 minutes.
The transmitters and receivers used in the test are shown in Fig. 35. Bidirectional (transmit and receive) node 75 was selected to transmit the desired signal while node 69 was selected to transmit interference. Nodes 75 and 69 were selected as the transmit nodes because they were used in the May 2014 experiment.
The distance between the transmit nodes and the receive nodes is shown in Table 5.

OFDM Communication Signal and LFM Interference
Nine stereo wavefiles containing the same OFDM data packets on the right channel and different LFM interfering signals on the left channel were created using MATLAB ® . The nine files (or songs) make up the CD played during the experiment; Table 6  The OFDM data packet consists of an initial channel probe, a channel probe guard period, four OFDM data blocks and associated guard intervals, a final channel probe, and a three second blanking period. The channel probes are 100 millisecond LFM signals sweeping down from 14 kHz to 8 kHz. Each OFDM block is identical, containing the same data message encoded with a half rate binary LDPC code. The message and parity check bits are interleaved over the data subcarriers.
The interference consists of upswept LFM signals.
The transmitted wavefiles are expected to generate the four types of interference listed below: Surface bounce receptions of the channel probe (Probe SB ).
Direct path receptions of the transmitted LFM interference (LFM DP ).
Surface bounce receptions of the transmitted LFM interference (LFM SB ).  The selection of the OFDM signal parameters was based on the following (see Table 8 for a summary): The packet probe was successfully used in AUTEC-0514 for packet synchronization. There was no compelling reason to modify it.
The center frequency is driven by transmit voltage response of the AUTEC transducers. The center frequency is identical to the AUTEC-0514 experiment and the AUTEC-10 experiment reported in [17].

Number and allocation of subcarriers matched those for the simulation results
reported in [17] and the simulations conducted prior to the AUTEC-0514 experiment. The uniform distribution of pilot subcarriers enables the use of the Doppler compensation algorithm of [32] which was implemented and tested on simulated data prior to the AUTEC-0514 experiment.   The guard interval and the number of blocks per packet were selected jointly.
The guard interval of 10.0 milliseconds is more than twenty times the direct path channel length observed during the AUTEC-0514. The use of a shorter guard interval results in the OFDM data blocks being more tightly packed within the packet and a shorter overall transmit time for the packet. Consequently, more packets can be transmitted during the experiment. Based on the channel model, the fourth OFDM block is likely to suffer from surface bounce interference from the channel probe on receivers less than 5,000 meters from the OFDM transmitter. This is not detrimental to the experiment objectives since the channel probe acts as an additional interfering source.
On more distant receivers, earlier blocks suffer from inter-block interference as can be seen in Table 7. Blocks added to the packet after the fourth block would suffer from inter-block interference on all the receivers. Since mitigation of inter-block interference is not an objective of the experiment, the number of blocks in a packet was limited to four.
The symbol constellation was chosen because it is the simplest to implement.
The mapping from the binary LDPC code to the symbol constellation is straightforward.
The three second blanking period between packets is more than twice the expected time difference of arrival between the direct path and surface bounce receptions. It ensures that the multipath receptions of the previously transmitted packet clears the channel before the next packet is transmitted.
The selection of the LFM interference parameters was based on the following (see Table 8 for a summary): The center frequency was selected to match the center frequency of the OFDM signal so that the calculation of the degree of frequency overlap between the signal and interference is obvious.
Sweep direction. The interference and channel probe sweep in opposite directions to minimize the correlation of these signals which facilitates synchronization of the received OFDM data packets.
Start time of first interference transmission was selected to enable the first two OFDM data packets to be received without suffering interference.
The bandwidth, duration and repetition rate were selected jointly to meet the following criteria: -Produce an uncoded bit error rate of greater than 0.10 on blocks suffering from interference.
-Produce received signals in which the same interfering signal (e.g. LFM signal with the same bandwidth and duration) corrupts different time domain portions of the OFDM block.
The channel model was used as a tool to aid in selecting the signal parameters discussed above. Some of the data generated using the channel model and analyzed   Figure 36: Uncoded bit error rate on simulated channel with no interference mitigation.
The BERs shown here are due to interference only. The effects of the underwater channel and additive noise (sea noise) are not included. Thus, the BER observed during the experiment will likely be higher.
generate received blocks with BERs greater than 0.10 because sea noise and the effects of the underwater acoustic channel will drive the BER higher. portions of the received block will suffer interference. Since each block carries the same message, recorded data from a single receiver should satisfy the requirement to characterize performance as a function of the time orthogonality of the interference (EO2.2). This is an improvement from the AUTEC-0514 experiment where for a given interfering signal on a given receiver, the interference corrupted the same portions of the received signal as can be seen in the received signals shown in Fig. 34.

Data Collected
A quick look at the data collected during AUTEC-1214 shows that the experimental design improved upon the AUTEC-0514 experiment. Fig. 38 shows data packet receptions on receivers 67,68,74,76,77 and 81 from the AUTEC-1214 experiment. The vertical dashed black lines delineate the OFDM blocks. As expected, the LFM interference corrupts different portions of each block. This is an improvement over the AUTEC-0514 experiment where the interference corrupted the same portion of the OFDM block on each receiver because it was transmitted synchronously with the OFDM data packets. Furthermore, the noise level and interference level vary among the receivers confirming that transmitting all the data at full power was a viable approach to generating a data set with varying SNR and SIR due to the spatial extent of the network. Unlike the AUTEC-0514 experiment, the direct path reception was not the strongest multipath reception.
As shown in Fig. 39, the surface bounce path carried the strongest received signal.
The surface bounce channel observed during the AUTEC-1214 experiment is considerable longer than the direct path channel observed during AUTEC-0514 as can be seen in Fig. 40  time synchronization. Ten milliseconds of time (equivalent guard interval used in AUTEC-1214) is shown before and after each block. As discussed above and clearly shown in the figure, the direct path channel observed in AUTEC-0514 was much shorter than ten milliseconds. Unfortunately, the surface bounce channel observed in AUTEC-1214 is longer than the guard interval as clearly evidenced by the signal from the previous block bleeding over into the guard interval and interfering with the selected block. The reception of the selected block also extends to occupy the entire guard interval and part of the next block creating inter-block interference.
The longer channel and the inter-block interference will complicate channel equalization. Processing the data from AUTEC-1214 is left for future work. Planning for the next experiment should anticipate that the direct path signal may not be the strongest and allow for a longer channel impulse response to avoid inter-block interference.

Channel Model
This section describes the channel model developed to aid in the design of the AUTEC-1214 experiment. The development of a channel model facilitated the development of the transmit waveforms and transmit schedule to satisfy the test objectives. Specifically, modeling aided the selection of the OFDM symbol duration, guard interval, number of blocks per packet, the blanking period between packets, the LFM interference duration, bandwidth and repetition rate.
The Sonar Simulation Toolkit (SST) [97] and the Acoustic Channel Simulator [98] were considered as modeling tools for developing the transmit waveforms for the December test. Both models provide sophisticated, state-of-the-art simulations of the underwater acoustic channel. However, there is a considerable learning curve associated with each and since the purpose of employing the models was to design an experiment and work with the real data obtained from the experiment rather than simulated data, the option of developing a very simple channel model leveraging data collected during the AUTEC-0514 was pursued instead.
Analysis of the AUTEC-0514 data showed that the interaction of the direct path and first surface bounce reception had the largest effect on the received signal.
Thus, the key concern from the perspective of experimental design is obtaining a reasonable estimate of the time of the direct path and surface bounce arrivals from the two transmitters to the various receivers. There is some variability among the sound velocity profile from AUTEC-0514 and the historical sound velocity profiles reviewed from November 2014 and December 2013 as shown in Fig. 41. However, the variation is not so great as to invalidate the modeling approach since only a rough estimate of the channel delays is required. Note that the SVP observed during AUTEC-1214 is quite similar to the SVP from the AUTEC-0514 experiment.
Consequently, a simple two tap channel model was developed from each transmitter to the receivers with the amplitude of the direct path and surface bounce taps

Physical Basis for Channel Model
The observation from AUTEC-0514 that the important ray paths are either direct ray (e.g. refracted rays) or surface bounce rays (e.g. refracted surface reflected) is characteristic of deep ocean propagation. The geometry is shown in There are two energy paths connecting the transmitter and receiver: the direct path (DP) and the surface bounce path (SB). Assuming specular reflection at the sea surface, the reflected path appears to originate from the image of the transmitter at (0, −z t ) [23]. The analytic solution may be found by either the image method discussed in section 1.4.2 or by applying Green's theorem to the boundary problem of a point source in a fluid half-space in section 2.3.4 of [23]. For the purposes of designing the experiment, we mention the above to demonstrate that there is a physical basis for the selected model. The model assumed an effective direct path sound speed c DP and a separate surface bounce sound speed c SB . The time of arrival for the direct and surface bounce paths from a transmitter at position (x t , y t , z t ) to a receiver at position (x r , y r , z r ) is given by: respectively, and the time difference of arrival of the direct path and surface bounce arrivals is simply: Based on the direct path and surface bounce TDOA on receivers 67,68,76,78 and 82 from AUTEC-0514, an effective surface bounce sound speed and direct path sound speed of 1510 and 1490 meters per second was determined. Applying these sound speeds to Eqs. (113) and (114) determined the delays for the channel model shown in Fig. 42 where all delays are referenced to the earliest direct path arrival, which occurs on receiver 76.

Summary
This chapter described the planning for the for the AUTEC-1214 experiment. Unlike AUTEC-0514, the message bits and parity check bits are interleaved across the data subcarriers. Therefore, the effects of received interference will be distributed across the structure of the code.
A quick look at the received data from AUTEC-1214 shows that the interference did occur in different time slots within the OFDM packets on each channel. Unfortunately, the strongest received signal occurred via a path with a longer impulse response than the guard interval resulting in all the receptions being contaminated with inter-block interference. This will complicate processing the received data, which is left to future work.

CHAPTER 5
Conclusions and Future Work

Conclusions
Many acoustic channels suffer from interference which is neither narrowband nor impulsive. This relatively long duration partial band interference can be particularly detrimental to system performance. In operational networks, many dropped messages are lost due to partial band interference which corrupts different portions of the received signal depending on the relative position of the interferers, information source and receivers due to the slow speed of propagation. A survey of recent work in interference mitigation and orthogonal frequency division multiplexing (OFDM) as well as observations from the Atlantic Undersea Test and Evaluation Center (AUTEC) provided motivation to develop a spatial diversity receiver for use in underwater networks.
The application of this work extends beyond the AUTEC network to cabled acoustic networks in general and other situations where interference corrupts receptions on widely separated receivers. As we continue to explore and operate in the oceans, the number of activities grows. Many of these activities generate signals of interest to the user of the application at hand, but create interference from the perspective of other users. Communications is an essential aspect of many of these operations. For example, autonomous undersea vehicles require a reliable communications link to send and receive data to remote users. As the undersea environment becomes more acoustically congested, understanding the limits interference places on performance and developing approaches to mitigate its effects are important areas of research. This research examined approaches to leverage the spatial diversity of underwater acoustic communications networks suffering from interference.
The concept of combining multiple receptions of the same transmitted signal is not new. Information is available for combining at multiple different levels: 1) the received waveform, 2) the detected symbols and 3) the decoded information.
In general, a diversity processor could operate on any of these levels or among the different levels. This work focused on combining the received waveforms. The concept of combining received waveforms has been a standard approach for decades.
Maximum ratio combining (see chapter 7, section 5 of [33]) weights the received waveforms based on the signal-to-noise ratio at each receiver. Optimal ratio combining (ORC) [92] also takes advantage of the differential fading received signals experience, but additionally, considers the impact of co-channel interferers in calculating the weighting of the received waveforms. ORC was specifically developed to combat co-channel interference on Rayleigh fading channels for mobile radio. Whereas ORC takes advantage of the differential fading of the virtually synchronously received interference among the receivers, the algorithm developed here relies on the slow speed of signal propagation underwater which results in the received interference exhibiting some degree of time orthogonality. Underwater receptions also experience differential fading and when the interference is not time orthogonal, the algorithm makes use of this in a manner similar to ORC.
The spatial diversity receiver for underwater communications identifies portions of the signal suffering from interference on different receivers, removes these portions of the signal and then optimally combines the remaining clean portions of the signal. Analytic results of performance for receivers on additive white Gaussian noise channels suffering time orthogonal interference demonstrated the effectiveness of the spatial diversity combining strategy as compared to conventional maximum ratio combining. Simulation results on time invariant channels confirmed the effectiveness of the algorithm under more complex channel conditions.
Comparison of the spatial diversity receiver performance to the recently developed single receiver parameterized interference cancellation algorithm was made using results from an experiment conducted at the AUTEC network which consists of multiple distributed cabled hydrophones that receive data transmitted over a timevarying multipath channel in the presence of partial band interference produced by interfering active sonar signals.
The spatial diversity receiver algorithm and the parameterized interference cancellation algorithm address the problem of interference in fundamentally differ-

Future Work
The development of a receiver that effectively uses the SDR algorithm under extremely high interference conditions and the PIC algorithm when the interference is less severe is a logical next step. The approach for combining multiple receivers using the PIC algorithm could also be explored. That is, how is the combining done most effectively? At the waveform level after the interference has been subtracted? By sharing information about satisfied parity checks so that additional tones can be used for equalization among independently operating equalizers? Is the decoding done independently on each receiver and then the information shared at the output or are the symbols derived from a weighted combination of the re- Figure 46: Spatial diversity receiver.
ceived waveforms supplied to a single decoder? Fig. 46 shows one example of how the PIC algorithm might be extended to multiple receivers. The white boxes indicate added functionality: inter-receiver data alignment and multi-receiver LDPC decoding.
The problem of selecting the correct waveforms to combine needs to be addressed. A practical receiver must ensure that the source of the combined waveforms or extracted data is the same. Study of transmit schemes that facilitate information combining in single and multi-user environments could be undertaken as part of this work.
For SDR, the following modifications to the current implementation would likely result in performance gains: Replacing the least squares equalizer with a weighted least squares equalizer. In this work, no attempt was made to mitigate detected interference before equalization. Performance improvement especially at low SIR and low SNR would likely be realizable if the equalization process accounted for the presence of interference.
Development of an adaptive SDR detector based the analytic performance equations.
Extension of the algorithm to handle multiple different interfering signals of different time durations and bandwidths.
From a theoretical perspective, a next logical step for SDR is to derive an analytic expression for an additive white Gaussian noise channel without the requirement that the interference be time orthogonal.      (A.1)

APPENDIX
The signal power in each block was estimated as: .2) and the SNR in the block was then calculated as SNR blk = 10 log 10 (P sig /σ 2 noise )) − 10 log 10 (Kf sym ); (A.3) where f sym is the symbol rate. The resulting signalμ was processed using the OFDM processing string and the power of the interference was estimated aŝ where the noise estimateσ 2 noise was calculated using Eq. A.1. The faired interference plus noise vector v 1,f and the extracted block of sea noise were scaled using the same scale factor φ = P int to produce the scaled interference vectorṽ s = φv 1,f and scaled noise vectorũ s = φũ. The MATLAB ® code in Section A.2 combines the error free received data blocks (1,752), scaled noise blocks (4,868), and interference and noise pairs (819) to test the algorithms at various SNR and SIR levels.