Wdigfreqresp6

Wdigfreqresp6 - ECE 421 - Sum 2010 Detailed Solutions - Set...

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Unformatted text preview: ECE 421 - Sum 2010 Detailed Solutions - Set 6: Digital Filter Frequency Response 1 6-1. Let the sampling interval in a digital system be T s = 0.1 milliseconds. A pure sinusoid x t = sin 2 f c t is sampled by this system. Compute the radian frequency c (units = radians/samp. interval) if the Hz frequency f c is given below: (a) f c = 200 Hz, (b) f c = 1 KHz, (c) f c = 5 KHz DETAILED SOLUTION: A sinewave with digital frequency c radians per sampling interval has the mathematical form x [ m = sin c m a Using the definition of a digital signal x [ m obtained from an arbitrary analog signal gives x [ m = x mT s = x t t = mT s b With x t as given in the problem statement, equation b then produces x [ m = sin 2 f c t t = mT s = sin 2 f c m 10 4 c Rewriting the result of c in a slightly different from will simplify our work to follow. x [ m = sin 2 f c 10 4 m d Therefore, the problems in parts (a), (b), and (c) can now be solved by substituting the appropriate value of f c into d and simplifying. (a) f c = 200 Hz Substitute f c = 200 into d and simplify. This procedure gives x [ m = sin 2 200 10 4 m = sin : 04 m e Comparing equation a with equation e shows that the radian frequency for part (a) is given by c = : 04 f ECE 421 - Sum 2010 Detailed Solutions - Set 6: Digital Filter Frequency Response 2 6-1. (cont.) (b) f c = 1 KHz Substitute f c = 1000 into d and simplify. This procedure gives x [ m = sin 2 10 3 10 4 m = sin : 2 m g Equation g shows that the radian frequency for part (b) is therefore given by c = : 2 h (c) f c = 5 KHz Substitute f c = 5000 into d and simplify. This procedure gives x [ m = sin 2 5 10 3 10 4 m = sin m i Equation i shows that the radian frequency for part (c) is thus given by c = j ECE 421 - Sum 2010 Detailed Solutions - Set 6: Digital Filter Frequency Response 3 6-2. A digital filter impulse response h [ m has the analytical form h [ m = a m u [ m . Compute the digital frequency response H if a has the values given below. Also accurately plot j H j for each case. (a) A = : 45, (b) A = : 72, (c) A = : 3 e j : 1 , (d) A = : 5 j : 2 DETAILED SOLUTION: To work this problem, find the filter transfer function H z and then compute the frequency response H by evaluating H z on the unit circle. The transfer function H z is the z-transform of the h [ m given in the problem statement: h [ m = A m u [ m H z = 1 1 Az 1 = z z A a Thus, for this problem we have H z = z z A b The digital filter frequency response, H , is found by evaluating H z on the unit circle: H = H z z = e j = z z A z = e j c Performing the evaluation in equation c then gives H = e j e j A d Therefore, to find the frequency response requested in parts (a) - (d) of this problem, we simply need to substitute the appropriate value of A into equation d . We can also find a general form for the magnitude....
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Wdigfreqresp6 - ECE 421 - Sum 2010 Detailed Solutions - Set...

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