Subjects signal processing

Power Spectral Density Feeb39

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Power Spectral Density Feeb39


1. The problem asks to find the power spectral density (PSD) of a stochastic process, which describes how the power of a signal or time series is distributed over frequency. 2. The PSD is typically found by taking the Fourier transform of the autocorrelation function $R_X(\tau)$ of the process: $$ S_X(f) = \int_{-\infty}^{\infty} R_X(\tau) e^{-j 2 \pi f \tau} d\tau $$ 3. Important rules: - The autocorrelation function must be known or given. - The PSD is a real, even function of frequency. 4. Since the previous question is not provided here, assume the autocorrelation function $R_X(\tau)$ is known. 5. Substitute $R_X(\tau)$ into the integral and compute the Fourier transform. 6. Simplify the integral using properties of $R_X(\tau)$ (e.g., evenness, exponential form). 7. The result is the power spectral density $S_X(f)$, which quantifies power distribution over frequency. Without the explicit $R_X(\tau)$, the exact PSD cannot be computed here, but the method above applies to any given autocorrelation function.