Homework 3

# Homework 3 - EE 4541 Digital Signal Processing Fall 2007...

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Unformatted text preview: EE 4541 Digital Signal Processing Fall 2007 Problem Set 3 Due September 27, 2007 1. Consider the filter with transfer function given by: 1 + z -1 H ( z) = 1 - 0.9 z -1 Determine the impulse response for a causal implementation. Is the filter stable or unstable? ii. Sketch the magnitude response of the filter. Is the filter Lowpass, Highpass, Bandpass, Bandstop, or allpass? n iii. Sketch the magnitude response for the filter g[n] = (- 1) h[ n] . Comment on your result. 2. Let x[n] be a purely real sequence. You are given the following information about x[n] and must determine what it is. Even if you are unable to specify x[n] fully, you may receive partial credit by describing which features of x[n] are determined by each clue. i. x[n] is a causal sequence. ii. iii. iv. v. vi. Let v[ n] = x[n + 2] . Its DTFT, V e j , is purely real. lim X ( z ) = 2 z i. ( ) 1 2 1 2 - X (e ) e d = 0 j j x[ 2] > 0 - X (e ) j 2 d = 9 3. The magnitude responses of 4 LTI systems are given in the figure: (a) 30 40 30 20 10 10 0 -4 8 6 4 2 2 0 -4 1 -2 0 2 Frequency, radians 4 0 -4 -2 0 2 Frequency, radians 4 0 -4 5 4 3 (b) 20 -2 0 2 Frequency, radians (c) 4 -2 0 2 Frequency, radians (d) 4 EE 4541 Digital Signal Processing Fall 2007 Problem Set 3 Due September 27, 2007 i. State whether these filters act as LPF, BPF, or HPF. Also, which filter has the lowest cutoff frequency and which one has the highest. Which of the 4 magnitude responses matches: Filter 1 represented by the difference equation: y[n] = x[n] + x[ n - 1] + x[n - 2] + x[n - 3] + x[ n - 4] Filter 2 represented by the difference equation: y[ n] = 1.6 y[ n - 1] - 0.64 y[n - 1] + x[n] Filter 3 represented by the difference equation: y[n] = x[n] + x[n - 1] + x[n - 2] + x[ n - 3] + x[ n - 4] + x[ n - 5] + x[n - 6] + x[n - 7] Filter 4 represented by the difference equation: y[ n] = 1.4 y[ n - 1] - 0.49 y[ n - 1] + x[ n] + x[n - 1] + x[n - 2] ii. iii. iv. In each case, explain clearly the basis of your choice, i.e., no credit for mere guessing. 4. 5. 6. 7. 8. Problem 3.41 P&M. Problem 4.17 P&M. Problem 4.18 P&M (Grad Students only). Problem 4.22 P&M. Use Assignment3.m to generate the figures below and answer the following questions: i. What is the 90% bandwidth (in Hz) of the aortic pressure waveform? See Section 4.2.9 P&M for definition. You may have to use the zoom and the Datatip. You can also use the MATLAB command find, but this will require a little work. ii. What is the 90% bandwidth (in Hz) of the filtered waveform? iii. What is the 6dB bandwidth of the MA filter used? iv. Describe the effects of the filter on the PSD of the signal. In particular, what frequency components are affected (if any)? v. Can you estimate the SNR for both signals? vi. Repeat (i. v.) the above for the ecg waveform. EE 4541 Digital Signal Processing Fall 2007 Problem Set 3 Due September 27, 2007 ...
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