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Unformatted text preview: 9.7 Problems 325 The power spectral density of X ( t ) is S XX ( w ) = N / 2, and the filter impulse responses are given by h 1 ( t ) = braceleftBig 1 ≤ t < 1 otherwise h 2 ( t ) = braceleftbigg 2 e- t t ≥ otherwise Determine the following: a. The mean E [ Y i ( t ) ] and second moment E [ Y 2 i ( t ) ] of the output signal Y i ( t ) , for i = 1 , 2 b. The crosscorrelation function R Y 1 Y 2 ( t , t + τ ) 9.7 A wide-sense stationary process X ( t ) is the input to a linear system whose impulse response is h ( t ) = 2 e- 7 t , t ≥ 0. If the autocorrelation function of the process is R XX ( τ ) = e- 4 | τ | and the output process is Y ( t ) , find the following: a. The power spectral density of Y ( t ) b. The cross-spectral power density S XY ( w ) c. The crosscorrelation function R XY ( τ ) 9.8 A linear system has a transfer function given by H ( w ) = w w 2 + 15 w + 50 Determine the power spectral density of the output when the input function is a. a stationary random process X ( t ) with an autocorrelation function R XX ( τ ) = 10 e-| τ | . b. white noise that has a mean-square value of 1.2 V 2 /Hz. 9.9 A linear system has the impulse response h ( t ) = e- at , where t ≥ 0 and a > 0, find the power transfer function of the system. 9.10 Consider the system with the impulse response h ( t ) = e- at , where t ≥ 0 and a > 0. Assume that the input is white noise with power spectral density N / 2. What is the power spectral density of the output process? 9.11 The power transfer function of a system is given by | H ( w ) | 2 = 64 [ 16 + w 2 ] 2 Use Table 8.1 to obtain the impulse function h ( t ) of the system. 326 Chapter 9 Linear Systems with Random Inputs 9.12 A wide-sense stationary process X ( t ) has the autocorrelation function given by R XX ( τ ) = cos ( w τ ) The process is input to a system with the power transfer function | H ( w ) | 2 = 64 [ 16 + w 2 ] 2 a. Find the power spectral density of the output process. b. If Y ( t ) is the output process, find the cross-power spectral density S XY ( w ) ....
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This note was uploaded on 01/05/2010 for the course STAT 350 taught by Professor Carlton during the Fall '07 term at Cal Poly.
- Fall '07