The Design of Linear Phase Finite Impulse Response Filter

A Finite Impulse Response system is described by the following difference equation:

And its system function is given by,

To design a FIR filter with desired properties, the coefficients of the FIR filter are required. For this purpose, the coefficients of a causal FIR that closely approximates the desired frequency response specification is determined.

Linear Phase Finite Impulse Response Filter

The coefficients are approximated for causal FIR. these are then used as the coefficients of the required FIR.

Why IIR requires less memory and has lower computational complexity than FIR?

Are we using Linear phase FIR filter for approximation?

This line is better written as we considering a linear phase FIR filter

A FIR filter has a linear phase if all the frequency comopnents of a signal are shifted in time by the same amount, resulting in a constant group delay, i.e.

Why does FIR filter need to be linear phase? Or is linear phase just a propery of FIR filter?

Does symmetricity and anti-symmetricity arise due to the linear phase?

Incorporating symmetry (+) and anti-symmetry (-) conditions, for odd M,

Subtituting by in the last term in the above equation,

Similarly, for even ,

Design of a Linear Phase FIR Filter: Windowing

Consider a problem of designing a FIR filter with a desired frequency response and impulse response of an ideal system.

The impulse response is generally infinitely long. So, we need to design a FIR filter of order , which approximates the infinitely long with a finite sequence , where except for .

Let be the approximated frequency response of the designed FIR filter. Then, a possible criterion for this approximation can be defined as:

By using Parseval’s identity,

For a FIR filter of order , the optimal value of is,

This can be represented as a product of with a finite duration rectangular window :

So, what kind of effect does this window have on the desired frequency response ? We can find this out by analyzing the convolution of and , as multiplcation in time domain is equivalent to convolution in frequency domain.

Here,

This equation equals zero whenever

where, is any non-zero integer.

Plot Graph of W

From the above plot, it can be observed that the width of the main-lobe is . As increases, the main-lobe of the rectangular window becomes narrower.

The convolution of with has a smoothing effect on . The smoothing effect reduces as increases.