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

Document Description: FIR Filters for Electronics and Communication Engineering (ECE) 2022 is part of Signals and Systems preparation. The notes and questions for FIR Filters have been prepared according to the Electronics and Communication Engineering (ECE) exam syllabus. Information about FIR Filters covers topics like and FIR Filters Example, for Electronics and Communication Engineering (ECE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for FIR Filters.

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FIR Filters

Fourier series Method

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

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

FIR Filters Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) T = sampling period

The F.S co-efficient or impulse response samples of filter can be obtained using

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

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 

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

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

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

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

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

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

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

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

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

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 

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

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

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

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

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

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

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

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

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

FIR Filters Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) N=21

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

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

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

OR

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

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

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

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

FIR HPF

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

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

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

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

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

FIR BPF

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

Ex:

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

= 0 else where

Fs = 800SPS

Ti = 20 ms

FIR Filters Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) N = 17

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

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

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

bi = h(i-8)    h(-n) = h(n)

 

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