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Wpzfilt7

# Wpzfilt7 - ECE 421 Sum 2010 Detailed Solutions Set 7 Filter...

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ECE 421 - Sum 2010 Detailed Solutions - Set 7: Filter Design by Pole-Zero Placement 1 7-1. The poles and zeros of a digital filter are POLES: z 0 8 e j π 6 , ZEROS: z 0 z 0. With G 1, compute the simplest form of the difference equation which implements this filter. DETAILED SOLUTION: The transfer function H z in terms of the poles p 1 and p 2 and the zeros z 1 and z 2 is given by H z G z z 1 z z 2 z p 1 z p 2 a Substitute into a the values of the poles and zeros from the problem statement, as well as G = 1, giving H z z 0 z 0 z 0 8 e j π 6 z 0 8 e j π 6 b Multiply out the factors in c , and use Euler’s formula to obtain H z z 2 z 2 1 6 z cos π 6 0 64 z 2 z 2 1 386 z 0 64 c Substitute H z Y z X z in c and write the right side of c using negative powers of z : Y z X z 1 1 1 386 z 1 0 64 z 2 d Cross-multiplying terms in d then gives Y z 1 386 z 1 Y z 0 64 z 2 Y z X z e Taking the inverse Z -transform term-by-term of e then provides y m 1 386 y m 1 0 64 y m 2 x m f Finally, solving f for y m then gives the desired form for the difference equation: y m 1 386 y m 1 0 64 y m 2 x m g

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ECE 421 - Sum 2010 Detailed Solutions - Set 7: Filter Design by Pole-Zero Placement 2 7-2.
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Wpzfilt7 - ECE 421 Sum 2010 Detailed Solutions Set 7 Filter...

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