Response to the following problem:
Let X(t) denote a (real, zero-mean, WSS) band pass process with autocorrelation function RX(τ ) and power spectral density SX( f ), where SX (0) = 0, and let (t) denote the Hilbert transform of X(t). Then Xˆ (t) can be viewed as the output of a filter, with impulse response 1 πt and transfer function - jsgn( f ), whose input is X(t). Recall that when X(t) passes through a system with transfer function H( f ) and the output is Y (t), we have SY( f ) = SX ( f )|H( f )|2 and SXY ( f ) = SX ( f )H∗( f ).
1. Prove that R(τ ) = RX (τ ).
2. Prove that RX (τ ) = -Rˆ X (τ )
3. If Z(t) = X(t) + j (t), determine SZ ( f ).
4. Define X1(t) = Z(t)e- j2π f0t . Show that Xl(t) is a low pass WSS random process, and determine SX1( f ). From the expression for SX1( f ), derive an expression for RX1(τ ).