Design of FIR Digital Filters; - · PDF fileTitle: Microsoft PowerPoint - DSP-LECT-13-16.ppt...

2006 Copyright 2006 ©Andreas Spanias 13-16-1

Design of FIR Digital Filters;LINEAR PHASELecture 13-15

Andreas [email protected]


FIR Digital FiltersAdvantages:

Linear Phase DesignQuite Efficient for designing notch filtersAlways Stable

Disadvantages:Requires High Order for Narrowband Design

Applications:Speech Processing, TelecommunicationsData Processing, Noise Suppression, RadarAdaptive Signal Processing, Noise Cancellation, Echo

Cancellation, Multipath channels


FIR Digital Filters

L

ii inxbny

0)()(

nynx

T T

0b

Lb...

.....1b

.

LL zzXbzzXbzXbzY )(...)()()( 1

10

LL zbzbb

zXzYzH ...

)()()( 1

10


FIR Filter Frequency Response

jLL

jj ebebbeH ...)( 10

sff2

o

o

foldover


FIR Filter Design

1. LINEAR PHASE DESIGN

2. FOURIER SERIES DESIGN

3. ZERO PLACEMENT

4. FREQUENCY SAMPLING

5. LEAST SQUARES

6. IMPLEMENTATIONS


LINEAR PHASE DESIGN

Linear Phase (constant time delay) FIR filter design is important in pulse transmission applications where pulse dispersion must be avoided. The frequency response function of the FIR filter iswritten as:

jLL

jjj ebebebbeH ...)( 2210

where

))(arg()(,)()( jj eHeHM

)()()( jj eMeH


GROUP DELAY

The time delay or group delay of a filter is defined as

dd )(

therefore if is a linear function of then is aconstant.

is given in terms of samples


LINEAR PHASE AND IMPULSE RESPONSE SYMMETRIES

It can be shown that linear phase is achieved if

)()( nLhnh

where h(n) is the impulse response of the filter. For L = odd

21

0

)( ))(()(

L

n

nLn zznhzH

21

0

2

2cos)(2)(

L

n

Ljj nLnheeH


LINEAR PHASE DESIGN

if we define the pseudomagnitude

21

01 2

cos)(2)(

L

n

j nLnheH

Then

0)(,2

0)(,2)(

1

1

j

j

eHL

eHL

hence the phase response is piecewise linear.


SYMMETRIC AND ANTI-SYMMETRIC LINEAR PHASE FILTERS

Two Anti-symmetries for L=even or L=odd for

)()( nLhnh

)()( nLhnh

Two Symmetries for L=even or L=odd for


EXAMPLES OF SYMMETRIES

nh 4L

0

nh

0

nh

0

nh

0

4L

3L

3L


EXAMPLES OF PHASE AND SYMMETRY IN h(n)

Normalized frequency (Nyquist == 1)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-200

-150

-100

-50

0

50

Phas

e (d

egre

es)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-100

0

100

Normalized frequency (Nyquist == 1)

Pha

se(d

egre

es)

nh

0

nh

0

4L

4L


Note that for )()( nLhnh

then )()( 1zHzzH L

Hence for L=odd and z = -1 then

)1()1()1( HH L

Thus the filter must have a zero at and is therefore notadequate for high-pass filtering.

HPF USING CERTAIN LINEAR PHASE FILTERS


Note that for )()( nLhnh

then )()( 1zHzzH L

Hence any L and z = 1 then

)1()1( HH

Thus the filter must have a zero at and is therefore notadequate for low-pass filtering.

0

LPF USING CERTAIN LINEAR PHASE FILTERS


Design of FIR Digital FiltersLecture 14 - FIR DESIGN USING THE FOURIER SERIES



Design Using the Fourier Series

In filter design, there is an ideal transfer function Hd(z) that isapproximated by H(z). For example for a low-pass filter, an ideal frequency response is given below:

We like to minimize the mean square error, i.e.,

deHeH jjd

2)()(

21

zHd

c c


Design Using the Fourier Series (Cont.)

Recalling that in general2

1

)()(n

ni

jij eiheH

then

deiheHn

ni

jijd

22

1

)()(2

1

Minimization of the integral above leads to an h(n) sequence that isprecisely equal to the sequence of Fourier series coefficients characterizing Hd(z). The resultant impulse response sequence is a sampled truncated sinc function.


Fourier Series Design Example

For the ideal low pass filter the impulse response sequence is an infinite length sampled sinc function. Lets say the sampling frequency is 8KHz and we wish to have a cutoff frequency at 2KHz. This results in

28000200022

s

cc f

f

That is

zHd

2n

2n


Fourier Series Design Example (Cont.)The ideal impulse response hd(n) is given by

,...2,1,0,2

sinc21)( nnnh d

For an FIR filter of 11 coefficients

,...0,0,51,0,

31,0,1,

21,1,0,

31,0,

51,0,0...)(nh

This impulse response is not causal, however a shift operator of 5delays (z-5) will convert it into a causal one.


REALIZATION

. .5/1 3/1 /1 21/

...

.1z 1z 1z 1z 1z

1z 1z 1z 1z 1z

ny

nx


Fourier Series Design Example (Cont.)

51,

31,1,

21,1,

31,

51

10865420 bbbbbbb

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-600

-400

-200

0


Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-60

-40

-20

0

20


Mag

nitu

de R

espo

nse

(dB

)


Fourier Series Design Example L=32

16,15,....0,....,15,16,2

)16(sinc21 nnbn

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2000

-1500

-1000

-500

0


Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-80

-60

-40

-20

0

20


Mag

nitu

de R

espo

nse

(dB

)


Fourier Series Design Example L=64

32,31,....0,....,31,32,2

)32(sinc21 nnbn

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3000

-2000

-1000

0


Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50


Mag

nitu

de R

espo

nse

(dB

)


Truncating with Hamming Window L=64

Hamming(L)*2

)32(sinc21 nbn

0 0.1 0.2 0 .3 0.4 0 .5 0.6 0.7 0 .8 0.9 1-4000

-3000

-2000

-1000

0

Norm aliz ed frequenc y (Ny qu is t = = 1)

Pha

se (

degr

ees)

0 0 .1 0.2 0 .3 0.4 0 .5 0.6 0.7 0 .8 0.9 1-150

-100

-50

0

50

Norm aliz ed frequenc y (Ny qu is t = = 1)

Mag

nitu

de R

espo

nse

(dB

)


F.S. Design Rectangular vs Hamming Window - L=64

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-150

-100

-50

0

50


Mag

nitu

de R

espo

nse

(dB

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-100

-50

0

50


Mag

nitu

de R

espo

nse

(dB

)

Hamming

Rectangular

~45 dB

~13 dB


Truncating in time- frequency convolution and F.S. design

•The main-lobe width determines transition characteristics

•The sidelobe level determines rejection characteristics

Ideal LPFNarrow mainlobe=Narrower transition

Wide mainlobe =Wide transition


Notes On Fourier Series Design

•The design performed in the previous example involved truncationof an ideal symmetric impulse response.A symmetric impulse response produces a linear phase response.

•Truncation involves the use of a window function which is multiplied with the impulse response. Multiplication in the time domain maps into frequency-domain convolution and the spectralcharacteristics of the window function affect the design.

•The main-lobe width determines transition characteristics

•The sidelobe level determines rejection characteristics


Design of FIR Digital FiltersLecture 15 - FIR DESIGN USING THE KAISER WINDOW



2fH

dp11dp1

dsLPFIdeal

ffp fcPassband bandTransition Stopband

RegionsTolerance

DEFINING DESIGN SPECIFICATIONS

fs


DESIGN USING THE KAISER WINDOW

The Kaiser window is parametric and its bandwidth as well as its sidelobeenergy can be designed. Mainlobe bandwidth controls the transition characteristics and sidelobe energy affects the ripple characteristics.

10,)(

1

)(0

2/12

0

LnI

nI

nw

= L/2 ; associated with the order of the filter

is a design parameter that controls the shape of the window

I0(.) is a zeroth Bessel function of the first kind


DESIGN USING THE KAISER WINDOW (Cont.)

I xk

x k

k0

2

11

12

( )!

25 terms from the Bessel function are sufficient


EXAMPLES OF KAISER WINDOW

0 20 40 60 80 100 120 1400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 0

1


KAISER WINDOW DESIGN EQUATIONSGiven fp, fs, T and dp, ds determine the FIR filter coefficients.

TffA

dpds

ps )(2log20

),min(

10

The filter order is )2(285.2

8AL

0 1 1 0 2 8 7 5 00 5 8 4 2 2 1 0 0 7 8 8 6 2 1 2 1 5 00 2 1

0 4

. ( . ) ,

. ( ) . ( ) ,,

.

A AA A A

A

and the kaiser parameter is given by


DESIGN PROCEDURE

1. Determine the cutoff frequency for the ideal Fourier Seriesmethod.

ff f

cs p

22. Design the ideal LPF using the Fourier Series.3. Design the Kaiser window4. Shift and truncate the ideal impulse response

LnLnhnwnh dLPF 0,2

)()(

Note that this procedure can be generalized for the design of BPF, HPF, and BSF.


Design by Zero-Placement

As zeros are placed towards the unit circle the frequency response magnitude decreases at and in the vicinity of the frequency of the zeros. .

o

o

foldover


Design by Zero-Placement

Example: Design a linear phase FIR filter for 60Hz interferencecancellation, that will pass a 10Hz signal of interest withoutattenuation. The sampling frequency is 500Hz.

256

500602

25500102

60

10

A second order filter is sufficient, since only a zero pair on the unit circle is required for 60Hz cancellation. The 10Hz responseis adjusted with a gain factor.


Design by Zero-Placement (Cont.)

A linear phase (steady state) design means symmetric impulseresponse:

0)(

1)(606060

101010

2010

2010

jjj

jjj

ebebbeH

ebebbeH

and

77.2,9.1 10 bb


Design by Zero-Placement (Cont.)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-50

0

50

100

150


Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-60

-40

-20

0

20


Mag

nitu

de R

espo

nse

(dB

)

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

O

O

jjj eeeH 29.177.29.1)(


Frequency Sampling Methods for FIR Filter design

The Frequency Sampling Method (FSM) involves: a) uniform sampling of a desired continuous frequency response function at Npoints, b) applying the N-point inverse Discrete Fourier Transformto obtain an N-point impulse response.

The FSM guarantees that the FIR frequency response matches that of the desired filter at the sampled points, however the response between the sampled points is different. If the desired filter function is due to an infinite impulse response then the FSM suffers also from time-domain aliasing.


Min-Max and Parks-McClellan Optimum FIR Design

The Parks-McClellan design is based on Min-Max

Equiripple and linear phase design is possible

This class of methods involve minimizing the maximum errorbetween the designed FIR filter frequency response and a prototype

)(maxmin,...,1,0),(

j

LiiheE

where

E e W e H e H ej jd

j j( ) ( )( ( ) ( ))


FIR Filter Design Using MATLAB+

IN THE MATLAB SP TOOLBOX

cremez - Complex and nonlinear phase equiripple FIR filter design.fir1 - Window based FIR filter design - low, high, band, stop, multi.fir2 - Window based FIR filter design - arbitrary response.fircls - Constrained Least Squares filter design - arbitrary response.fircls1 - Constrained Least Squares FIR filter design - low and highpassfirls - FIR filter design - arbitrary response with transition bands.firrcos - Raised cosine FIR filter design.intfilt - Interpolation FIR filter design.kaiserord - Window based filter order selection using Kaiser window.remez - Parks-McClellan optimal FIR filter design.remezord - Parks-McClellan filter order selection.

+ MATLAB is registered trade mark of the MathWorks


FIR Filter Realizations

Direct Realizations L

i

ii zbzH

0

)(

...

...

nx

ny

1z 1z 1z

0b1b 2b 1Lb Lb

•Require multiply accumulate instructions


FIR Filter Cascade Realizations

Cascade Realizationsq

iiii zbzbbzH

1

22

110 )()(

...1z

1z

1z

1z

10b

11b

12b

20b

21b

22b

nx

•Reduced Effects from Coefficient Quantization and round-off•In Fixed-Point implementation signal scaling must be done carefully at each stage


Transform-Domain FIR Filter Realizations

A Transform domain realization is possible using the overlapand save and the FFT. This yields computational savings for highorder implementations. Input data is organized in 2N-point blocks and blocks are shifted N points at a time. The data blocks andN zero-padded coefficients are transformed and multiplied and theresults is inverse transformed. The last N-points are selected as theresult. The blocks are updated and the process is repeated.

F F T IF F Tx y

-

X k (0 )

X k (1 )

X k (L )

B k (0 )

B k (1 )

B k (L )

Select Last N points

x(n)

y(n)


Transform-Domain FIR Filter Realizations (2)

Implements an N-th order filter with two 2-N point FFTs

For processing N points complexity is reduced fromO(N2) to O(Nlog2N)

F F T IF F Tx y

-

X k (0 )

X k (1 )

X k (L )

B k (0 )

B k (1 )

B k (L )

Select Last N points

x(n)

y(n)


Simple and Smart FIR Filter Realizations

L

i

izL

zH0

1)(

A simple L-tap filter can be built such that no multiplies are not used

This is a LPF with linear phase – if L is radix 2 the division can be implemented with shifts - below an example with L=16

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1- 2 0 0

- 1 0 0

0

1 0 0

N o r m a l i z e d fr e q u e n c y ( N y q u i s t = = 1 )

Pha

se (

degr

ees)

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1- 6 0

- 4 0

- 2 0

0

N o r m a l i z e d fr e q u e n c y ( N y q u i s t = = 1 )

Mag

nitu

de R

espo

nse

(dB

)


Reducing Complexity by realizing FIR as IIR

)1(11)( 1

)1(

0 zLzz

LzH

LL

i

i

An FIR filter with all its coefficients equal can be realized bylooking at geometric series convergence and pole-zero cancellation

The IIR filter can be implemented using a simple difference equationwith long delay. Precision may become a problem as the pole zerocancellation is on the unit circle.

1( ) ( ( ) ( 1)) ( 1)y n x n x n L y nL


Pole-zero cancellation in FIR to IIR transformation

)1(11)( 1

)1(

0 zLzz

LzH

LL

i

i

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

Real part

Imag

inar

y pa

rt

Pole zero cancellation at z=1


Modifying Filter Response by Subtractive Operations

)()(1 jHPF

jLPF eHeH

__ =HPFLPFALL-PASS

c cc c

Using simple operations like this we can transform prototypeLPF to HPF or BPF, etc.


Implementing Efficiently Digital Cross-Over Using Subtractive Operations

LPF to tweeter

to wooferLinear Phase LPFWith delay L/2 samples

-+

could also implement delay compensation z –L/2


Frequency Sampling

/20 /4

Ideal frequency response

N=16 samples

/40

Sampled ideal frequency response

/40

Interpolated frequency response

(a)

(b)

(c)

12 /

0( ) ( ) ( )k

Nj j kn N

d d dn

H k H e h n e

( ) ( ) 0 1dk

h n h n kN n N ; aliasing

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Design of FIR Digital Filters; - · PDF fileTitle: Microsoft PowerPoint - DSP-LECT-13-16.ppt...

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