The Equivalence of Correlation Coefficient of Two Power Spectra and Coherence in Real-valued Signals

In the analysis of signals, understanding the relationship between the correlation coefficient of two power spectra and the coherence of corresponding time signals is article explores the conditions under which the two metrics are equivalent, focusing particularly on real-valued signals.

Definitions and Basic Concepts

Power Spectrum: The power spectrum of a signal provides a distribution of power into frequency components. For a signal x_t, the power spectrum S_x(f) is typically obtained via the Fourier transform.

Coherence: Coherence, denoted by gamma;2(f), measures the degree of linear correlation between two signals x_t and y_t at a specific frequency f. It is defined as:

gamma;2(f) ?X(f)Y(f)* / (?X(f)22

where X(f) and Y(f) are the Fourier transforms of x_t and y_t, respectively.

Relationship Between Power Spectrum Correlation and Coherence

For Real-valued Signals: When the signals are real, the coherence can be interpreted as a normalized measure of the cross-spectral density relative to the individual power spectra. The correlation coefficient of the power spectra essentially captures the same relationship in the frequency domain.

This means that if two signals x_t and y_t are real-valued, the correlation coefficient derived from their power spectra will match the coherence measure computed from the time-domain signals at each frequency, under the assumption of stationary processes and proper normalization of the power spectra.

Mathematical Formulation

Let Z_X(λ) and Z_Y(λ) represent the frequency analysis of the time series X_t and Y_t respectively. The coherency at frequency λ is given by:

Rλ Cov[ZX(λ), ZY(λ)] / sqrt{Var[ZX middot; Var[ZY]}}

Covariances can be interpreted as inner products. In the systems and signal processing literature, in the absence of a noise model for X_t and Y_t, coherence is simply a deterministic inner product as in the provided notation. Cross-spectral density is equivalent to the Fourier transform of the cross-covariance in signal processing terminology.

Probability and Deterministic View in Time Series Analysis

When X_t and Y_t are considered as random processes, the coherence is defined in terms of the covariance, which is a probabilistic quantity. This distinction between a deterministic and probabilistic view is crucial in time series analysis, as highlighted in the works of Brillinger.

References

[1] Brillinger, D. R. (1996). Time Series: Data Analysis and Theory. SIAM.

[2] Giri, N. N. (1977). Mathematical Statistics. Springer.