This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 421  Sum 2010 Detailed Solutions  Set 6: Digital Filter Frequency Response 1 61. 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 61. (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 62. 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 ztransform 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....
View
Full
Document
 Summer '08
 HALLEN
 Frequency, Signal Processing

Click to edit the document details