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# Question 3

Design a simple digital filter whose trasnfer function corresponds to a delay of two sample times, i.e.

 y[k] = x[k-2] (22)

Assume a sampling frequency fs = 48 kHz.

3.1 Specify H(z).

3.2 Find the frequency response (amplitude and phase) of this filter.

3.3 Find the impulse response h[k] of this filter by taking the inverse z-transform of H(z).

3.4 Show by convolution that the output of the filter y[k] is x[k-2] as expected.

Solution.

3.1

Z-transform of y[k] is

 = (23) = (24) = z-2X(z) (25)

Thus H(z) is

 H(z) = Y(z)/X(z) = (26)

3.2

The frequency response H(f) is

 H(f) = (27) = (28) = (29) = (30) = (31) = (32)

The magnitude response is

 |H(f)| = (33)

and phase response is

 arg(H(f)) = (34)

where

 d1 = (35) = (36)

3.3

From the z-transform pair

 (37)

The impulse response of the filter is

 (38)

3.4

Given filter impulse response h[k] and filter input sequence x[n], the filter out is

 y[k] = x[k]*h[k] (39) = (40) = (41) = (42) = x[k-2] (43)

Next: About this document ... Up: Solution to Assignment 4 Previous: Question 2
Hyun Ho Jeon
2001-02-12