HW11 - II. (3pts) Consider the following transfer functions...

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Homework #11 Due: Thursday, May 6, 2010 Note : Use of Matlab (or any other software) is not permitted. I. (see Exercise 4.11) A 1-periodic function on R having the Fourier series: −∞ = = k ikx k e c x f π 2 ) ( (with F[k] replaced by c k ) takes the values: < < < < = 1 10 1 if 0 10 1 0 if 1 ) ( x x x f Sketch the graph of f on [0,1] and corresponding graphs of the functions represented by the following Fourier series: a. (1pt) −∞ = = k ikx k e c x f 2 1 ) ( b. (1pt) () −∞ = = k ikx k k e c x f 2 2 1 ) ( c. (1pt) ∑∑ −∞ = −∞ = = k ikx m k m m e c c x f 2 3 ) ( d. (1pt) −∞ = = k ikx k e c x f 2 2 4 ) ( e. (1pt) −∞ = + = k ikx k k e i c c x f 2 5 5 5 2 ) ( f. (1pt) −∞ = = k ikx k e c x f 2 3 6 ) ( g. (1pt) () −∞ = + = k ikx k k e c c x f 2 7 ) ( Hints. 1) Try to use properties of Fourier series you learned in class or from previous homework (see Problem 4.10) 2) Write (-1) k =e 2 π ik(1/2) when you analyze (b).
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Unformatted text preview: II. (3pts) Consider the following transfer functions of linear-time invariant filters. In each case determine if they can be used as low-pass filters in a 2-scale perfect reconstruction filter bank, and if so, determine the high-pass filter pair g( ). a. 2 ) ( 4 2 i i e e h + = (in time-domain 2 1 2 1 = = h h and remaining coefficients are zero) b. 2 1 ) ( 6 i e h + = (in time-domain 2 1 3 = = h h and remaining coefficients are zero) c. 3 1 ) ( 2 2 i i e e h + + = (in time-domain 3 1 1 1 = = = h h h and remaining coefficients are zero) Total : 10 pts...
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HW11 - II. (3pts) Consider the following transfer functions...

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