Question

Does the quantization of the coefficients in the direct form structure destroy the linear phase property of FIR linear phase un-quantized filters? Why ?

Answer 1

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Soln:  

Because for FIR systems we must only be concerned with the locations of zeros of the system function H(z).

since for causal FIR systems, all the poles are at z=0.

and because Direct-Form structure is commonly used in implementation of FIR systesm.

lets system function H(z) for Direct-form FIR system expressed in the form of

H(z) =

Now lets assume that the coefficient for impulse responce (h[n]) are quantized

which results in a new coefficent as

Here = quantized error in coefficients of

now taking fourier transform of above quation we get system function for quantized system

=

because z-transform is linear opration, system function for quantized system is than

where

we know that at   system function become frequency responce  

hence it proved that the system function and therefore also the frequency responce of the quantized system is linearly related to the quantization error in the impulse responce coefficient.

NOTE:

Also we know that in parralal configuration the total implulse responce as well as total system function (Total frequency responce) equal to the the sum of individual impulse responces as well as sum of system function (frequency responce) of individual system.

So here impulse responce of quantized system looks like total impulse responce of parallal system of h[n] &h[n]

h[n] =un-quantized impulse respoce

h[n] =quantized impulse respoce

similarly for system function of quantized system function also looks like total system function of parallal system H(z) & H(z)

thats why we can say that

quantization of the coefficients in the direct form structure destroy the linear phase property of FIR linear phase un-quantized filters.