Homework 7

Homework 7 - EE 4541 Digital Signal Processing Fall 2007...

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Unformatted text preview: EE 4541 Digital Signal Processing Fall 2007 Problem Set 7 Due: October 25, 2007 1. In the system shown below, H ( z ) = 1 - e j / 4 z -1 1 - e - j / 4 z -1 Determine the output signal y a (t ) . x a (t ) ( )( ) C/D x[n] h[n] y[n] D/C 225 150 0 150 225 F (Hz) T=1 /200 s T=1 /200 s y a (t ) 3e - j 2 / 3 2e - j / 4 X a ( jF ) 3e j 2 / 3 2e j / 4 EE 4541 Digital Signal Processing Fall 2007 Problem Set 7 Due: October 25, 2007 2. Consider the single-pole lowpass filter and its impulse response: L 1 H a ( s ) = ha (t ) = e -t u (t ) s + a. What is the dc gain? What is the 3dB corner frequency for this filter? At what time does the analog impulse response decay to 1/e of its initial value? At what frequency does the frequency response go to zero? b. Define h1 [n] = Tha (nT ) , find the corresponding transfer function H 1 ( z ) . What is the dc gain? What is the 3dB frequency? How many samples are needed before the impulse response decays to 1/e of its initial value? At what frequency does the frequency response go to zero? 2 z -1 , find the corresponding transfer c. Consider the transformation, s = T z +1 function H 2 ( z ) = H a ( s ) s = 2 z -1 . What is the dc gain? What is the 3dB frequency? T z +1 How many samples are needed before the impulse response decays to 1/e of its initial value? At what frequency does the frequency response go to zero? d. Bonus (20 points): Can you explain and differences between the results obtained the two methods? 3. An analog signal of the form x a (t ) = a(t ) cos(2000 t ) is bandlimited to the range 900 F 1100 Hz. It is used as an input to the system used in the figure below. a) Determine and sketch the spectra for the sequences x[n] and w[n] . Assume X ( j ) to have a general shape within the frequency band where it has nonzero values. b) We wish to design a lowpass filter so that v[n] a[n] using a window design method with a Hamming window. Suggest appropriate values for p and s such that p equals the highest frequency in A e j and s equals the lowest frequency of the nearest image resulting from the demodulation. c) Bonus (20 points): Determine the sampling rate of the A/D converter that would allow us to eliminate the frequency conversion. ( ) EE 4541 Digital Signal Processing Fall 2007 Problem Set 7 Due: October 25, 2007 ^ x a (t ) x[n] w[n] v[n] a (t ) A/D X H e j D/A ( ) 1 Fs = = 2500 Ts cos(0.8n) Fs = 1 = 2500 Ts ...
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