5.4.1. Suppose X(t) is WSS with µX = 0 and RX(τ ) = 2e
−2|τ|
, and suppose X(t) is the input of
a LTI system with impulse response h(t) = δ(t) − e
−3tu(t). The output is denoted Y (t).
Also define Z = Y (t) and W = Y (t − 1).
(a) Determine SX(f) and PX.
(b) Determine SX,Y (f) and SY (f).
(c) Determine RY (τ ) and PY .
(d) Determine E[Z], E[W], E[Z
2
], E[W2
], and E[ZW].
(e) Determine Var(Z), Var(W), and Cov(Z, W).
![5.4.1. Suppose X(t) is WSS with µX = 0 and RX(τ ) = 2e −2|τ| , and suppose X(t) is the input of a LTI system with impulse response h(t) = δ(t) − e −3tu(t). The output is denoted Y (t). Also define Z = Y (t) and W = Y (t − 1). (a) Determine SX(f) and PX. (b) Determine SX,Y (f) and SY (f). (c) Determine RY (τ ) and PY . (d) Determine E[Z], E[W], E[Z 2 ], E[W2 ], and E[ZW]. (e) Determine Var(Z), Var(W), and Cov(Z, W).](https://i0.wp.com/electricalstudent.com/wp-content/uploads/2022/08/541.png?w=1170&ssl=1 "5.4.1. Suppose X(t) is WSS with µX = 0 and RX(τ ) = 2e −2|τ| , and suppose X(t) is the input of a LTI system with impulse response h(t) = δ(t) − e −3tu(t). The output is denoted Y (t). Also define Z = Y (t) and W = Y (t − 1). (a) Determine SX(f) and PX. (b) Determine SX,Y (f) and SY (f). (c) Determine RY (τ ) and PY . (d) Determine E[Z], E[W], E[Z 2 ], E[W2 ], and E[ZW]. (e) Determine Var(Z), Var(W), and Cov(Z, W).")
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