DSP 6 - FIR Filter Design

© 2010 Trity Technologies Sdn Bhd. All Rights Reserved

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Practical Signal Processing Concepts and Algorithms using MATLAB

FIR Filter Design

FIR Filter Design© 2010 Trity Technologies Sdn Bhd. All Rights

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Section Outline

• FIR design methods• Windowing• Standard band filters• Arbitrary response filters• Multiband filters• Raised cosine filters• Frequency domain filtering

I II III IV


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FIR Filter Design

FIR filters have the following advantages:

• Exactly linear phase is possible

• Always stable, even when quantized

• Design methods are generally linear

• Efficient hardware realizations

• Startup transients have finite duration

FIR filters have the following disadvantages:

• Higher filter order than IIR filters

• Corresponding greater delays


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FIR Methods

Filter Method Description Filter Functions

Windowing Apply window to truncated inverse Fourier transform of desired filter

fir1, fir2, kaiserord

Multiband with Transition Bands

Equiripple or least squares approach over frequency sub-bands

firls, firpm, firpmord

Constrained Least Squares

Minimize squared integral error over entire frequency range subject to maximum error constraints

fircls, fircls1

Arbitrary Response

Arbitrary responses, including nonlinear phase and complex filters

cfirpm

Raised Cosine Lowpass response with smooth, sinusoidal transition

firrcos


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Method Description Functions

WindowingApply window to truncated inverse Fourier transform of desired "brick wall" filter

fir1, fir2, kaiserord

Multiband with Transition Bands

Equiripple or least squares approach over sub-bands of the frequency range

firls, remez, remezord

Constrained Least Squares

Minimize squared integral error over entire frequency range subject to maximum error constraints

fircls, fircls1

Arbitrary Response

Arbitrary responses, including nonlinear phase and complex filters

cremez

Raised Cosine Lowpass response with smooth, sinusoidal transition

firrcos


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Impulse Response Revisited

M

mm mnxbny

0

)()(FIR filter:

>> b = [b0 b1 … bM]; a = 1; no need to specify

stem(b) impz(b,1)

FIR filter coefficients give the impulse response


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Linear Phase Filters

Impulse response is symmetric about its midpoint:

Except for cfirpm, all of the FIR filter design functions in the Signal Processing Toolbox design linear phase filters only.

constant phase delay constant group delay

no phase distortion

–φ(ω)/ω –dφ(ω)/dω


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FIR Filter Types

Filter Type

Filter Order

Symmetry Response H(0) Response H(1)(Nyquist)

Type I Even No restriction No restriction

Type II Odd No restriction H(1) = 0

Type III Even H(0) = 0 H(1) = 0

Type IV Odd H(0) = 0 No restriction

1,...,1),2()( nkknbkb

1,...,1),2()( nkknbkb

fir1,fir2,firls,firpm,fircls,fircls1,firrcos

No highpass or bandstop filters

odd length,even symmetry

even length,even symmetry

odd length,odd symmetry

even length,odd symmetry

I II III IV

length of impulse response - 1

1,...,1),2()( nkknbkb

1,...,1),2()( nkknbkb


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Digital filters with finite-duration impulse response (all-zero, or FIR filters) have both advantages and disadvantages compared to infinite-duration impulse response (IIR) filters.

FIR filters have the following primary advantages:

• They can have exactly linear phase.

• They are always stable.

• The design methods are generally linear.

• They can be realized efficiently in hardware.

• The filter startup transients have finite duration.

The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter.


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• Example 1Example 1

Design a 48th-order FIR bandpass filter with passband 0.35 0.65:

• SolutionSolution>>b = fir1(48,[0.35 0.65]);

>>freqz(b,1,512)

• Example 2Example 2::

Design a lowpass filter with the following specifications using the optimal design method :

>>rp = 0.01; % Passband ripple

>>rs = 0.1; % Stopband ripple

>>fs = 8000; % Sampling frequency

>>f = [1500 2000]; % Cutoff frequencies

>>a = [1 0]; % Desired amplitudes


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• SolutionSolution

>>[n,fo,ao,w] = remezord(f,a,dev,fs);

>> dev=[0.01 0.1]

>>[n,fo,ao,w]=remezord([1500 2000],[1 0],dev,8000);

% approximate order, normalized frequency band edges, frequency band amplitudes, and weights that meet input specifications f, a, and dev.%

>>b=remez(n,fo,ao,w);

%use n, fo, ao and w to design the filter b which approximately meets the specifications given by remezord input parameters f, a, and dev.%

>>freqz(b,1,1024,8000);

>>title('Lowpass Filter Designed to Specifications');


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Window-Based Design

fft

ifft

H(f) h(n)

hw(n)HW(f)

h.*w


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Windowing Functions

bartlett, barthannwin, blackman, blackmanharris, bohmanwin, chebwin, flattopwin, gausswin, hamming, hann, kaiser, parsenwin, rectwin, triang, tukeywin

.* =

W = gausswin(2*length(y),3);w = W(length(y)+1:end);

y = sin(n); wy = w'.*y;


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Windowing and Spectra

Truncated Signal and DFT:

Truncated, Windowed Signal and DFT

wider main lobe

lower side lobes>> windft


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Window Visualization Tool

>> wvtool(kaiser(32,1),kaiser(32,5),kaiser(32,10))


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Window Design and Analysis Tool

>> wintool


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Example: Lowpass Filter

)(sinc2

1)(

2

1)( 000

0

ndedeHnh njnj

>> b = ...0.4*sinc(...0.4*(-25:25));>> fvtool(b,1)

>> bw = ...b.*hamming(51)';>> fvtool(bw,1)

Right click on the y-axis label in FVTool and choose Magnitude squared


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Standard Band FIR Design

>> b = fir1(n,Wn,'stop',kaiser(n+1,beta));


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Arbitrary Response FIR Filters

>> b = fir2(n,f,m,window);


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Multiband Filters

>> b = firls(n,f,a)


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Raised Cosine Filters

>> b = firrcos(n,F0,df,fs);


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Frequency Domain Filtering

y(n) long x(n)

finite buffer

Overlap-add method (fftfilt):

• Partion x(n) into data blocks

• FFT to the frequency domain

• Convolve blocks efficiently with b using multiplication

• IFFT back to the time domain

• Assemble shifted partial convolutions using superposition

b


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Section Summary

• FIR design methods• Windowing• Standard band filters• Arbitrary response filters• Multiband filters• Raised cosine filters• Frequency domain filtering

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DSP 6 - FIR Filter Design

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