Invariance property of the determinant of a matrix whose elements are a sum of sinusoids and its application

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We show that for a signal composed of a sum of m sinewaves with arbitrary frequencies, phases and amplitudes, there exists an instantaneous non-linear function of the signal, which takes on a constant magnitude anywhere on the time axis. This non-linear function is the determinant of a matrix. In the case of continuous-time signals, this matrix is composed of (2m-1) derivatives of the signal. For discrete time signals, the matrix is embedded with (2m-1) signal samples. Using this property, we show that we can determine the frequencies, phases and amplitudes of the m sinewave components. This is accomplished by adding a known auxiliary probe signal to the original signal before computing the determinant function. The nulls in the determinant function, as the frequency of the probe is varied, reveal the frequencies of the m sinewaves. The amplitudes and phases are computed using a ratio of two determinant values.

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AEU. Archiv fur Elektronik und Ubertragungstechnik





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