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FIR Filter Design

Chapter 12.8

Lecture 16

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2 Types of Filters

Finite Impulse Response (FIR) Easy to design Always stable!

Phase distortion is LINEAR Large and slow to implement y[n] = b

0x[n] + b

1x[n-1] + .+ b

mx[n-m]

Infinite Impulse Response (IIR) Harder to design Can be unstable More efficient (faster)!

y[n] = -a1y[n-1] + + a

Ny[n-N] + b

0x[n] + .+ b

mx[n-m]

No poles:

always stable

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Design an FIR LPF

Start with Ideal Filter:

Obtain the time domain impulse response

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Now, lets observe this filter in the time

domain using MATLAB:

Let 1 = /3

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MATLAB CODE

close all clear all %note: we must truncate a sinc function! So, let's begin by

observing the

%function from n = -m:m for different values of m n = -10:10; h = sin(pi/3*n)./(pi*n); h(11) = 1/3;

figure, stem(n, h) a = 1; figure, freqz(h, a)

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Example

Design a LPF-FIR filter if 1= /3, m = 10

0db in pass band Ringing in stop band

fB = 0.3

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Example 2

Design a LPF-FIR filter if 1= /3, m = 20

0db in pass band

More ringing in the stop band

Steeper cut off

Ringing in pass band (wellsee this better in a bit)

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Example 3 Design a LPF-FIR filter if 1 = /3, m = 50

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Design a FIR BPF

How do you think we could adjust the LPF

into a BPF?

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Example Design a BPF-FIR filter if

0= 0.6,

1= /6,

m = 20, 40, 100

close all clear all

%note: we must truckate a sinc function! So, let'sbegin by observing the

%function from n = -m:m for different values of m n = -20:20; h = (1./(pi.*n)).*(sin(pi/6.*n)).*(cos(0.6*pi.*n)); h(21) = 0.1667; figure, stem(n, h)

a = 1; figure, freqz(h, a)

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BPF with

0 = 0.6, 1 = /6, m = 20

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BPF with

0 = 0.6, 1 = /6, m = 40

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BPF with

0 = 0.6, 1 = /6, m = 100

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Design an FIR HPF

n = -10:10;

h = (1./

(pi.*n)).*(sin(2*pi/3.*n)).*(cos(pi.*n));

h(11) = 2/3;

figure, stem(h)

a = 1;

figure, freqz(h, a)

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HPF with

1 = /3, m = 10

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Improving FIR Filter Look again at LPF results

What would we really like to see? Wheredoes this problem come from?

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Improving FIR Filter: Ideal vs Real

Ideal LPF

Real LPF

How do we improve?

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Types of windows

What if a more gradual window is used? Triangle:

Von Hann (aka: the raised cosine window)

Hamming Window (an improved Von Hann Window)

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ExampleDesign a 51-term FIR LPF with 1=0.3

Using a rectangular window:

h[n] =

h[0] =

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Rectangular Window

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Using a Triangular window

h[n] =

h[n] =

ExampleDesign a 51-term FIR LPF with 1=0.3

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Triangular Window

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Using a Von Hann window

h[n] =

h[n] =

ExampleDesign a 51-term FIR LPF with 1=0.3

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Von Hann Window

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Using a Hamming window

h[n] =

h[n] =

ExampleDesign a 51-term FIR LPF with 1=0.3

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Hamming Window

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Closer comparison

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Log Comparison