FIR Filters

# FIR Filters Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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FIR Filters

Fourier series Method

1. Frequency response of a discrete-time filter is a periodi 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.

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

T = sampling period

The F.S co-efficient 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 no. of co-efficient, 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 bi = h(i-M) 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 bM

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(ejw) 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.

Ex:

Design a LPF (FIR) filter with frequency response

Ex:

Design LPF that approximate following freq response.

H(F)    = 1     0≤ F ≤ 1000Hz

= 0 else where 1000 ≤ F≤ Fs/2

When the sampling frequency is 8000 SPS. The impulse response duration is to be
limited to 2.5ms

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

Ex:

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

= 0 else where

Fs = 800SPS

Ti = 20 ms

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)

The document FIR Filters Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Signals and Systems.
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