FIR Filter Design

A Finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of finite duration, because it settles to zero in finite time.
FIR has some useful properties which make it preferable to an IIR.
The term digital filter arises because these filters operate on discrete – time signals.
Require no feedback. This means that any rounding errors are not compounded by summed iterations. The same relative error occurs in each calculation. This also makes implementation simpler.

Also, it has some disadvantages, the main disadvantage of FIR filters is that considerably more computation power in a general purpose processor is required compared to an IIR filter with similar sharpness or selectivity, especially when low frequency (relative to the sample rate) cutoffs are needed. However, many digital signal processors provide specialized hardware features to make FIR filters approximately as efficient as IIR for many applications.

The term finite impulse response arises because the filter out-put is computed as a weighted, finite term sum, of past, present, and perhaps future values of the filter input, i.e.,

An FIR filter is based on a feed-forward difference equation as showed by equation (2)
The meaning of feed forward is that there is no feedback of past or future outputs to form the present output, just input related terms.

Filter Design
An FIR filter is designed by finding the coefficients and filter order that meet certain specifications, which can be in the time-domain (e.g. a matched filter) and/or the frequency domain (most common).

Several FIR filter design methods:

1. Window Design Method
2. Frequency Sampling Method
3. Weighted Least Squares Design
4. Parks- McClellan method (also known as the Equiripple, Optimal, or Minimax method)

Window Design Method
I used in my implementation in Matlab this method. The window design method is also advantageous for creating efficient half-band filters, because the corresponding sinc function is zero at every other sample point (except the center one). The product with the window function does not alter the zeros, so almost half of the coefficients of the final impulse response are zero. An appropriate implementation of the FIR calculations can exploit that property to double the filter's efficiency.

Frequency Response
The filter's effect on the sequence x[n] is described in the frequency domain by the convolution theorem:

where operator’s F and respectively denote the discrete-time Fourier transform (DTFT) and its inverse. Therefore, the complex-valued, multiplicative function H(ω) is the filter's frequency response. It is defined by a Fourier series:

where the added subscript denotes 2π-periodicity. Here ω represents frequency in normalized units (radians/sample).

Why We Use FIR as a Filter?

FIRR is a lowpass filter which is an also Digital and Digital filter is more stable than analog filters. In addition, digital filters do not take up space like analog filters because they run on the microprocessor. So, it uses frequently.

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