Question

g) 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

• In FIR filter, the filter response to an impulse signal becomes practically zero as ‘t’ approaches ‘T’ i.e., t > T. Thus, these are finite duration filters in contrast to the Infinite Impulse Response or IIR filters. The impulse response of IIR filters run indefinitely.

• The direct-form FIR filters have no feedback or closed loops; hence these are non-recursive in nature although the generalized structured FIR filters can be implemented either recursively or non-recursively.

• Non-Recursive FIR filters: Consider an FIR non-recursive filter or the Direct form FIR filter with an output ‘B[k]’ obtained when an input signal ‘A[k]’ is applied to the filter having an impulse response ‘h[k]’, The output can be expressed as

Now, let the filter coefficients are quantized. Then the quantized impulse response h₁[k] of the filter is given by

where q[k] is the quantization error and q[k] = h₁[k] - h[k]

Using the equation of B[k] and h₁[k] above, the output of the filter with the quantization error can be represented as

Thus, input A[k] is now filtered by the error q[k] and is then superimposed to the output to provide the desired output as B₁[k].

In this case, the resultant transfer function H₁(e^iΩ) is given by

From equation (4), we conclude that the quantization of the filter coefficients distorts the linear phase property of FIR linear phase un-quantized filters. These linear distortions can be accommodated to some extent in re-designing the filter although it is not possible to minimize the quantization error magnitude |Q(e^iΩ)| to an arbitrarily small quantity with every possible finite step of quantization. As a result, the attenuation level required to be achieved for the FIR filter is now limited with the quantized coefficients. To tackle such an issue, it is desired to normalize the FIR filter coefficients h[k] prior to the process of quantization. This way, it is possible to maintain a small relative quantization noise power.

• Recursive FIR filters: The effect of quantization is more in recursive filters due to the feedback path as compared to the non-recursive filters. The desired filter characteristics are severely degraded with instability.