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Let X(t) be a wide sense stationary (WSS) random process with power spectral density Sx(f). If Y(t) is the process defined as Y(t) = X(2t - 1), the power spectral density Sy(f) is

1. Sy(f) = 1/2 Sx(f/2) e^-jπf
2. Sy(f) = 1/2 Sx(f/2) e^-jf/2
3. Sy(f) = 1/2 Sx(f/2)
4. Sy(f) = 1/2 Sx(f/2) e^-j2πf

Correct answer: Sy(f) = 1/2 Sx(f/2)

Solution

The correct option is right because the transformation Y(t) = X(2t - 1) involves a time-scaling and shifting operation, which results in the power spectral density being scaled by a factor of 1/2 and the frequency being halved, leading to Sy(f) = 1/2 Sx(f/2).

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