A9_R1

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Signals and Systems Issued: 15 October 2014 Fall 2014 Due: 22 October 2014 Homework Assignment #9 1. A digital filter has the transfer function H ( z ) = K ( z 2 + 1)( z + 1) ( z 2 + a 2 )( z + a ) , where K is chosen so that H (1) = 1. The sampling rate of the filter is T = 0.000025 s. (a) Plot the frequency response magnitude of the filter for a = 0, a = 0.5, and a = 0.9. (b) Show that the filter will produce zero forced response to an input at 10000 Hz. (c) Suppose the input to the filter is given by x ( t ) = cos(2 π⋅ 3000 t ) + cos(2 π⋅ 10000 t ), t = kT , where the first term (at 3000 Hz) represents the desired signal and the second term (at 10000 Hz) represents some unwanted noise. Plot the input to the filter. Use a computer simulation to calculate and plot the output of the filter for a = 0.5. (d) Suppose the noise component is changed to 9000 Hz, making the input to the filter x ( t ) = cos(2 π⋅ 3000 t ) + cos(2 π⋅ 9000 t ), t = kT . Plot the input to the filter. Use a computer simulation to calculate and plot the output of the filter for a = 0.5. The filter is tuned to reject any noise at 10000 Hz. Does it still work when the noise is at
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