# Digital filters: Design of FIR filters

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Digital filters: Design of FIR filters
احسان احمد عرساڻي Lecture 23-24

Introduction to FIR filters
These have linear phase No feedback Output is function of the present and past inputs only These are also called ‘all-zero’ and ‘non-recursive’ filters These do not have any poles

Applications Where: highly linear phase response is required
Need to avoid complicated design

FIR Filter Design Methods
Windows Frequency-sampling

FIR Filter Design: Windows Method
Start from the desired frequency response Hd(ω) Determine the unit (sample) pulse reponse hd(n)=F-1{Hd(ω)} hd(n) is generally infinite in length Truncate hd(n) to a finite length M

Truncating hd(n) Take only M terms N=0 to N=M-1 Remove all others

Truncating hd(n) Take only M terms Remove all others
N=0 to N=M-1 Remove all others Multiplying hd(n) with a rectangular window

Determine H(ω) Take Fourier transform of h(n) Therefore, compute:
Hd(ω) and W(ω) Hd(ω) depends on the required response hd(n)

Computing W(ω) W(ω)=F{w(n)} w(n) is a rectangular pulse

Example A low-pass linear phase FIR filter with the frequency response Hd(ω) is required Hd(n) happens to be non-causal having infinite duration

The impulse response hd(n)

Windowing the hd(n)

The truncated hd(n)

Example A low-pass linear phase FIR filter with the frequency response Hd(ω) is required

Frequency of oscilation increases with M
Magnitude of oscillation doesn’t increase or decrease with M Oscillations occur due to the Gibbs phenonmenon caused by the multipli- cation of the rectangular window with Hd(ω)

Frequency of oscilation increases with M
Magnitude of oscillation doesn’t increase or decrease with M Oscillations occur due to the Gibbs phenonmenon caused by the multipli-cation of the rectangular window with Hd(ω)

Other windows

Other windows

Spectrum of Kaiser window
(Cycles per sample)

Spectrum of Hanning window

Spectrum of Hamming Window
(Cycles per sample)

Spectrum of Blackman Window
(Cycles per sample)

Spectrum of Tukey Window
(Cycles per sample)

Windows’ characteristics

The FIR filter’s response with Rectangular window
M=61

FIR filter’s response with Hamming window

FIR filter’s response with Blackman window

FIR filter’s response with Kaiser window
M=61

Using the FIR filter

Blackman’s filter output