Finite-duration impulse response (FIR) filter (Order= M -1, Length= M ) The impulse response

Finite-duration impulse response (FIR) filter (Order=M-1, Length=M)

• The impulse response

• The system function

• The difference equation (Linear convolution)

• The frequency response (DTFT)

else,0

10,)(

Mnbnh n

1

0

11

110)(

M

n

nn

MM zbzbzbbzH

1

0110 )()1()1()()(

M

kkM knxbMnxbnxbnxbny

1

0

)()(M

n

jnnezebzHH j

Linear-phase FIR filters

Linear-phase FIR filter (Order=M-1, Length=M)

•

• = 0, /2. is a constant

• Four types


,)(H

Type-1 linear-phase FIR filter (Symmetrical, M odd)

• = 0. = (M-1)/2 (integer)

• Symmetric about (the index of symmetry)

•


)]2

1(cos[)(2)2

1()(

)()(

23

0

nMnhMhH

eHHM

nr

jr

10 ),1()( MnnMhnh

Type-2 linear-phase FIR filter (Symmetrical, M even)

• = 0. = (M-1)/2 (non-integer)

•


)]2

1(cos[)(2)(

)()(12/

0

nMnhH

eHHM

nr

jr

10 ),1()( MnnMhnh

Type-3 linear-phase FIR filter (Antisymmetrical, M odd)

• = /2. = (M-1)/2 (integer)

•


0)2/)1((10 ),1()(

MhMnnMhnh

)]2

1(sin[)(2)(

)()(

23

0

nMnhH

eHHM

nr

jr

Type-4 linear-phase FIR filter (Antisymmetrical, M even)

• = /2. = (M-1)/2 (non-integer)

•


10 ),1()( MnnMhnh

)]2

1(sin[)(2)(

)()(12/

0

nMnhH

eHHM

nr

jr

Design Specs (LPF)

FIR LPF filter specifications: (a) Absolute (b) Relative

• Band [0,p]: pass band

• Band [s, ]: stop band

• Band [p, s]: transition band

• 1: Absolute ripple in pass band

• 2: Absolute ripple in stop band

• Rp: Relative ripple in pass band (in dB)

• As: Relative ripple in stop band (in dB)

Design Specs (LPF)

0)(

)(log20 scale dB

max10

HH

1

110 1

1log20

Rp

1

210 1

log20

As

Window Design Techniques

-30 -20 -10 0 10 20 30-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Ideal LPF c=/4

Shifting

-30 -20 -10 0 10 20 30-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Ideal LPF c= /4-shifted

-30 -20 -10 0 10 20 30-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25LPF by Window Design Technique


General Design Procedures:

: Ideal frequency response (given)

• Step 1

• Step 2

• Step 3

)(IDEALH

21

)()(

MeHH j

IDEALd

deHHIDTFTnh njddd

)(21)]([)(

10 ),()()( M-nnwnhnh d

Window function

• symmetric about over

• 0 otherwise

10 M-n


Rectangular Window

otherwise,0

10,1)(

Mnnw

• Exact transition width = s - p = 1.8/M

• Min. stopband attenuation = 21dB

• MATLAB function: w=boxcar (M)


Bartlett Window



• MATLAB function: w=bartlett (M)

otherwise,0

12

1,1

222

10,1

2

)( MnMM

n

MnM

n

nw


Hann Window



• MATLAB function: w=hann (M)

otherwise,0

10)],1

2cos(1[50)( MnM-π n.nw


Hamming Window



• MATLAB function: w=hamming (M)

otherwise,0

10)],1

2cos(46.0540)( MnM-π n.nw

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Hamming Window: M=45


Blackman Window

• Exact transition width = s - p = 11/M


• MATLAB function: w=blackman (M)

otherwise,0

10)],1

4cos(08.0)1

2cos(5.0420)( MnM-π n

M-π n.nw

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Blackman Window: M=45


LPF Design

c

cLPFH

,0,1

)(

c

cj

d

eH

,0,

)(

)()](sin[

21

)(21)]([)(

)(

nnde

deHHIDTFTnh

cnj

njddd

c

c

function hd=ideal_lp(wc,M)%hd: ideal LPF impulse response between 0 and M-1%wc: cut-off frequencies in radians%M: length of the filter alpha=(M-1)/2;n=[0:M-1];m=n-alpha;fc=wc/pi;hd=fc*sinc(fc*m);


LPF Design

%Example 1 in FIR filter design wp=0.2*pi; ws=0.3*pi; tr_width=ws-wp;M=ceil(6.6*pi/tr_width)+1;n=[0:M-1];wc=(ws+wp)/2; %ideal cutoff frequencyhd=ideal_lp(wc,M);w_hamming=(hamming(M))';h=hd.*w_hamming; figure(1);stem(n,h); title('h(n)')figure(2);freqz(h,[1])

Example: p=0.2

s=0.3

As=50 dB


0 10 20 30 40 50 60 70-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3LPF: h(n)


0 0.2 0.4 0.6 0.8 1-2000

-1500

-1000

-500

0

Normalized Frequency ( rad/sample)

Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1-150

-100

-50

0

50


Mag

nitu

de (

dB)

LPF


BPS example:

1s=0.2, 1p=0.35, 2p=0.65, 2s=0.8

As=60 dB

Two transition band widths must be the same!


%Example 2 in FIR filter design% BPS design wp1=0.35*pi; ws1=0.2*pi; wp2=0.65*pi; ws2=0.8*pi; %only one transition bandwidth allowed in window designtr_width=min(wp1-ws1,ws2-wp2); M=ceil(11*pi/tr_width)+1;n=[0:M-1]; wc1=(ws1+wp1)/2; %ideal cutoff frequency 1wc2=(ws2+wp2)/2; %ideal cutoff frequency 2 hd=ideal_lp(wc2,M)-ideal_lp(wc1,M);w_blackman=(blackman(M))';h=hd.*w_blackman; figure(1);stem(n,h); title('h(n)')figure(2);freqz(h,[1])


0 10 20 30 40 50 60 70 80-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5BPS: h(n)


0 0.2 0.4 0.6 0.8 1-6000

-4000

-2000

0

2000


Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1-150

-100

-50

0

50


Mag

nitu

de (

dB)

BPS


Example: Digital differentiator

%Example 2 in FIR filter design% Digital differentiator design M=21;alpha=(M-1)/2;n=0:M-1;hd=(cos(pi*(n-alpha)))./(n-alpha);hd(alpha+1)=0;w_ham=(hamming(M))';h=hd.*w_ham; [H,W]=freqz(h,[1]);plot(W/pi,abs(H)); title('Digital differentiator: |H(\omega)|')


0 5 10 15 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Digital differentiator: h(n)


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3Digital differentiator: |H()|


>> t=linspace(-2,2,1000);>> xt=sin(2*pi*t);yt=conv(xt,h);>> subplot(2,1,1);plot(xt);title('x(t)=sin(2\pit)')>> subplot(2,1,2);plot(yt(22:end));title('y(t)=Dx(t)')

0 200 400 600 800 1000-1

-0.5

0

0.5

1x(t)=sin(2t)

0 200 400 600 800 1000-0.04

-0.02

0

0.02

0.04y(t)=Dx(t)

Finite-duration impulse response (FIR) filter (Order= M -1, Length= M ) The impulse response

Documents

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