FIR Filters | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

In digital signal processing, there are two main types of filters: FIR and IIR. 

FIR filters have a finite impulse response, which means they settle to zero in a fixed amount of time. This is different from IIR filters, which can keep responding indefinitely. 

An FIR filter of order N takes N+1 samples to settle to zero.

Let us study Fourier Series Method to design an FIR Filter.

Fourier series Method

1. Frequency response of a discrete-time filter is a periodic function with period Ωs (sampling freq).

2. From the F.S analysis we know that any periodic function can be expressed as a linear combination of complex exponentials.

Desired Frequency Response

The desired freqency response of a discrete time filter can be represented by F.S as

where, T = sampling period

Fourier Series Coefficient

The F.S coefficient or impulse response samples of filter can be obtained using

clearly if we wish to realize this filter with impulse response h(n), then it must have finite number of coefficients, which is equivalent to truncating the infinite expansion of H(e ^jΩT) ,which leads to approximation of (e ^jΩT ) which is denoted by 

We choose m=n-1/2,   in order to keep ‘n’ no of samples in h(n).

However, this filter can’t be physically realizable due to the presence of +ve powers of Z, means that the filter must produce an output that is advanced in time with respect to the i/p. this difficulty can be overcome by introducing a delay  M=N-1/2, samples.

Therefore H(z) = Z^-M H₁(z) = Z^-M 

H(z) = h(-M)Z⁰ + h(-M+1) Z^-1 +…. +h(M) Z^-2M

Let b_i = h(i-M) for  i=0 to 2M

H(z) =  be the transfer function of discrete filter that is physically realizable.

Properties

1. N=2M+1, impulse response co-eff, bi = 0 to 2M.

2. h(n) is symmetric about b_M

Ex:   M=4

3. The duration of impulse response is Ti = 2MT

4. Its magnitude and time delay function can be found in the following way

This implies that magnitude response of the filter we have desired approximates the desire magnitude response. The time delay of H(e^jw) is a constant M. thus sinusoids of different frequencies are delayed by the same amount as they are processed by the filter, we have designed. Consequently, this is a linear phase filter, which means that it does not introduce phase distortion.

Example 1:

Design a LPF (FIR) filter with frequency response

Solution:

Example 2:

Design LPF that approximate following frequency response.

H(F)    = 1     0≤ F ≤ 1000Hz

                                     = 0 else where 1000 ≤ F≤ Fs/2

When the sampling frequency is 8000 SPS (Samples per second). The impulse response duration is to be limited to 2.5ms.

Solution:

 N=21

OR

h(0) = 0.25      h(6) = -0.05305

h(1) = 0.22508   h(7) = -0.03215

h(2) = 0.15915     h(8) = 0

h(3) = 0.07503    h(9) = 0.02501

h(4) = 0    h(10) = 0.03183

h(5) = -0.04502

bi = h(i-10)

FIR HPF

FIR BPF

Example:

Desing a BPF for H(f) = 1 160≤ F ≤ 200Hz

                                                 = 0 else where

Fs = 800 SPS and Ti = 20 ms

Solution:

 N = 17

h(0) = 0.1      h(4) = 0.07568

h(1) = 0.01558    h(5) = 0.06366

h(2) = -0.09355  h(6) = -0.05046

h(3) = -0.04374   h(7) = -0.07220   h(8) = 0.02338

bi = h(i-8)    h(-n) = h(n)