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MUS421/EE367B Lecture 6FIR Digital Filter Design

Julius O. Smith III ([email protected] )Center for Computer Research in Music and Acoustics (CCRMA)

Department of Music , Stanford UniversityStanford, California 94305

March 28, 2005

Outline

Ideal Lowpass Filter

Optimal Least Squares in Time Domain

Window Method

Optimal Methods

Case Study: FIR Hilbert-Transform Design

1

FIR Digital Filter Design

In a previous lecture, we looked at many windows, anexamined their properties. Today, we are going to seehow these windows can be used to design Finite ImpuResponse (FIR) digital lters.

FFT processors implement long FIR lters moreefficiently than any other method (using Overlap-Add

We need exible ways to design all kinds of FIR lteruse in FFT processors.

2


Amplitude Response:

Frequency0 f s/ 20

1

G a

i n

f c

Gain = 1 forf < f c Gain = 0 forf > f c

3

Impulse Response of Ideal LPF

hideal(n)= DTFT 1n H ideal=

12

c

ce jnd

=sin(cn)

n= 2f csinc(2f cn), n Z

Problems:

Innitely long

Non-causal

Cannot be shifted to make it causal

Cannot be implemented in practice

Conclusion:

We must accept some compromise(s) in the designany practical lowpass lter.

Managing such trade-offs is the topic of digital lter design.

4
http://www-ccrma.stanford.edu/~%7B%7Djosmailto:[email protected]://www-ccrma.stanford.edu/http://www.stanford.edu/group/Music/http://www.stanford.edu/http://www.stanford.edu/http://www.stanford.edu/group/Music/http://www-ccrma.stanford.edu/mailto:[email protected]://www-ccrma.stanford.edu/~%7B%7Djos


2/19

Optimal (but Poor) Least-SquaresImpulse Response Design

Let

h(n) = ideal lter impulse response.

h(n) = length L causal FIR lter (to be designed).

Sum of squared errors :

J 2(h) =

n= h(n) h(n)

2=

L 1

n=0h(n) h(n)

2+ c2

wherec2= 0n= |h(n)|

2 + n= L |h(n)|2 does not

depend onh. Note that J 2(h) c2.Result: The error is minimized (in the least-squares sense) by simply matching the rst L terms in the desired impulse response.Optimal least-squares FIR lter:

h(n) =h(n), 0 n L 10, otherwise

Also optimal under anyL p norm with any error weighting:

J p(h) =L 1

n=0

w(n) h(n) h(n) p

+ c p c p

5

Length L = 30 Least-Squares LPF Design

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

Normalized Frequency (cycles/sample)

M a g n

i t u

d e

( L i n e a r )

Length 30 FIR Amplitude Response

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

50

40

30

20

10

0


M a g n

i t u

d e

( d B )

Desired cut-off frequency isf c = 1/ 4

Stopband attenuation only around 10 dB

Passband gain has a resonance near cut-off

Filter isquite poor for audio use

H = H asincL

6

Length L = 71 Least-Squares LPF Design

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5


M a g n

i t u

d e

( L i n e a r )


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

70

60

50

40

30

20

10

0


M a g n

i t u

d e

( d B )

Stopband attenuation still only around 10 dB Corner-frequency resonance ishigher , though narrower Gibbs phenomenon in the frequency domain

What were doing wrong:

Desired lter specication asks for too much(e.g, an innite roll-off rate).

Time-domain-least-squares is a poor choice of errorcriterion for audio work.

7

Length L = 71 Optimal Chebyshev LPF Design

remez(70,[0 0.5 0.6 1],[1 1 0 0]);

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5


M a g n

i t u

d e

( L i n e a r )


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

70

60

50

40

30

20

10

0


M a g n

i t u d e

( d B )

Stopband attenuation better than 60 dB

No corner-frequency resonance

Transition regionfrom passband to stopband iscritical

Error is equiripple in both stopband (visible) andpassband (not visible)

8


3/19

Impulse response is slightly impulsive at theendpoints (not shown).

9

Lowpass Filter Specications

s

p

TW

Ideal lowpassfilter response1

0

1 +p

1 p

0 p sFrequency

Amplitude

Lowpass Filter Specifications

Pass-band Stop-band

Transitionband

s : stopband ripple ( 0.001 = 60 dB typical)

p : passband ripple ( 0.1 dB typical)

s: stopband edge frequency

p: passband edge frequency

TW: transition width= s p SBA: stop-band attenuation= 20log(s)

10

Example Ripple Calculations in Matlab

Lets rst consider the passband ripple spec, 0.1 dB.Converting that to linear ripple amplitude gives, inMatlab,

format long;dp=10^(0.1/20)-1dp =

0.01157945425990

Lets check it:

>> 20*log10(1+dp)ans =

0.10000000000000>> 20*log10(1-dp)ans =

-0.10116471483635

Ok, close enough. Now lets set the stopband ripple to1/10 times the passband ripple and see where we are:

>> ds=dp/10;>> 20*log10(ds)ans =

-58.7262381688205211

So, thats about 60 dB stop-band rejection, which is ntoo bad. Now lets set the stopband ripple to 1/100times the passband ripple:

>> ds=dp/100;>> 20*log10(ds)ans =

-78.72623816882052

So now were close to 80dB SBA, which is getting inthe high delity zone.


The ideal lowpass lter is dened by the followingspecications:

s = p = c= 2f c TW = 0

p = s = 0 SBA=

12


4/19

The Window Method for FIR FilterDesign

In practical FFT processors, we are limited to anite duration impulse response.

An obvious method for FIR lter design is to simplywindow the ideal impulse response, just like wewindow ideal sinusoids in spectrum analysis:

hw(n) = w(n) hideal(n)

wherew is a zero phase window of lengthM (odd).

We saw that the rectangular window is optimal in thetime-domain least-squares sense, but a poor choicefor typical audio lter design. What about otherwindows?

Windowing ismultiplication in the time domainconvolution in the frequency domain.

Thus, in the window method, theideal frequency response is convolved with the window transform:

H w() = (W H ideal)()The convolution smears the response, introducingdeviations in both the pass-band and stop-band.

13

Example

A typical windowed ideal lowpass lter response isdepicted in the following diagram:

-60 -40 -20 0 20 40 60-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Length 101 Lowpass FIR Filter - Hamming Window

time (samples)

A m p

l i t u

d e

14

At this point in the design, we have a non-causal(zero phase) lter.

In order to implement this in a real time situation, weneed to shift the lter by an amount equal to half itslength.

hcausal(n) = h n M 1

2

This shift in the time domain results in alinear phase term in the frequency domain:

H causal() = e j M 12 H ()

15

Examples

length 25 FIR lter

c = / 4

hideal(n) =sin(cn)

n n Z

We will examine several windows:

0 10 20-0.1

0

0.1

0.2

0.3Rectangular

time (samples)

A m p

l i t u

d e

0 10 20-0.1

0

0.1

0.2

0.3Hanning

time (samples)

A m p

l i t u

d e

0 10 20-0.1

0

0.1

0.2

0.3Hamming

time (samples)

A m p

l i t u

d e

0 10 20-0.1

0

0.1

0.2

0.3Kaiser

time (samples)

A m p

l i t u

d e

16


5/19

Amplitude Response of Examples

-2 0 2

-60

-40

-20

0

Rectangular

Frequency (radians/sample)

A m p

l i t u

d e -

d B

-2 0 2

-60

-40

-20

0

Hanning


A m p

l i t u

d e -

d B

-2 0 2

-60

-40

-20

0Hamming


A m p

l i t u

d e -

d B

-2 0 2

-60

-40

-20

0Kaiser


A m p

l i t u

d e -

d B

17

Effect of Changing Filter Length, M = 11, 15, 41

3 2 1 0 1 2 370

60

50

40

30

20

10

0

Hamming Window


A m p

l i t u

d e

d B

18

Other Types of Filters

Ideal highpass, bandpass, and bandstop lters can all beconsidered as modications of an ideal lowpass lter.

A highpass lter can be constructed by shifting a lowpasslter by an amount equal to half the sampling rate.

0

0|H |

|H |

2

2

2

2

This is equivalent to a modulation bye jn in the timedomain. Hence,

hhp(n) = hlp(n)e jn

= hlp(n)( 1)n

19

Summarizing:

To design a high-pass lter, rst design a nitelength, causal lowpass lter

Use a lowpass cutoff frequency equal to (highpass cutoff)

Design the lowpass lter as in the previous sect

Modulate the impulse reponse by( 1)n to exchangeDC andf s/ 2

Alternatively, if the passband of the prototype lowpasslter H () is very at about unity gain, a highpass ltcan be designed by simply subtracting from1:

H hp() = 1 H lp() (n) h lp(n)

A bandpass can be derived by designing the approprialowpass lter (with cutoff frequency equal to half thewidth of the bandpass), and thenmodulating it to thedesired center frequencyc. In the frequency domain,this means shifting two copies of the lowpass prototypone to the left by c, and the other to the right byc.The two shifted copies are added together to give abandpass lter having a real impulse response.

20


6/19

0

0|H |

|H |

c

0 ul

c

0

To design a bandpass lter:

Design a lowpass prototype lter of the desired width(cutoff c = u 0)

Shift to the left and right by0 and sum:hbp(n) = 2 cos(n0)hlp(n)

A bandstop lter can be constructed by rst designing abandpass with similar specications.

In the frequency domain, we desire a lter with a transferfunction equal toH bs = 1 H bp. Hence, in the timedomain we have:

hbs(n) =1 hbp(0) n = 0 hbp(n) n = 1, 2, . . . ,

M 12

21

Summary of the Window Method

Basic Idea:

Start with Ideal Lowpass-Filter Impulse Response

Sinc functionsin(t/T )/ (t/T ) Innite duration Non-causal

Window the innite duration sinc to get a niteduration lter

Windowlength determines transition width Windowtype determines the sidelobe level (SLL Rectangular window yields the optimal lter in

unweighted least squares sense

Shift the noncausal (zero phase) lter to get a caus(linear phase) lter

The linear phase slope equals half the lter leng

Other types of lters (highpass, bandpass, bandstopare designed by rst designing a low pass, andapplying transformations

22

Important Points (Window Method)

Easy to design (seefir1 and fir2 in Octave andMatlab SPTB)

Can design extremely long FIR lters withoutnumerical problems

Not usually optimum in any sense

General Remarks (Window-Method)

Transition width TW determined by window lengthM

As window gets longer, its main lobe narrows Narrower main lobe narrower transition band Tighter specs demand longer lters Length M 1/ TW in general

Pass-band and stop-band ripple are determined bywindow side lobes

We do not have individual control over passbandand stopband ripple(This is a limitation of the window method)

Ripple determined by window used Ripple normally largest around transition band

23

Optimal FIR Digital Filter Design

Optimal Chebyshev FIR Design

L p Norm Minimization

Least squares problems Min-Max problems

Least Squares Optimization

PseudoInverse

FIR Filter Design

Linear Phase

Complex Filter Design Other Applications

Books including digital lter design:

Boyd and Vandenberghe

Parks and Burrus

Rabiner and Gold

Oppenheim and Schafer

Steiglitz24


7/19

Strum and Kirk

See Music 421 References1

1 http://ccrma.stanford.edu/jos/refs421/

25

Optimal Chebyshev Filters

Minimize themaximumof the error

Stopband and passband errors areequiripple

Remez Multiple Exchange (Parks-McClellanalgorithm)

Available in Octave and the Matlab SP Tool Box:remez(), cremez()

Problems with Remez Exchange

Convergenceunlikely for FIR lters longer than a fehundred taps or so

Fundamentallyreal polynomial approximation on thunit circle

Applies only tolinear phase lters Cannot design individual (non-minimum phase)

sub-lters in polyphase lter bank form

Really need optimal weightedcomplex approximatione.g., minimum phase FIR lter design

26

Example: Chebyshev window and its transform:

-100 -50 50 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (samples)

A m p

l i t u

d e

Chebyshev Window: M = 101, Ripple = -40 dB

0 0

27

-3 -2 -1 1 2 3

-70

-60

-50

-40

-30

-20

-10

freq

d B M a g n

i t u

d e

DFT of Chebyshev Window: M = 11, Ripple = -40 dB0

0

28
http://ccrma.stanford.edu/~%7B%7Djos/refs421/http://ccrma.stanford.edu/~%7B%7Djos/refs421/


8/19

Lp norms

The L p norm of anN -dimensional vectorx is dened as

x p=

N 1

i=0

|xi | p1 p

Special cases

L1 norm

x 1=

N 1

i=0

|xi |

Sum of the absolute values of the elements City block distance

L2 norm

x 2= N 1

i=0

|xi |2

Euclidean distance Minimized by Least Squares techniques

29

L -norm

x = lim

p

N 1

i=0

|x i| p1 p

In the limit asp , the L p norm of x isdominated by the maximum element of x.

30

Filter Design using Lp Norms

In terms of L p norm minimization, the FIR lter designproblem becomes:

minh

W (k) [H (k) D(k)] p

Where

k = suitable discrete set of frequencies

H (k) = frequency response of our FIR lter

D(k) = desired (complex) frequency response

W (k) = (optional) weighting functionLets look at some specic cases:

31

Least-Squares Linear-Phase FIR Filter Design

Let the FIR lter length beL + 1 samples, withL even,and suppose well initially design it to be centered abthe time origin (zero-phase). Then the frequencyresponse is given on our frequency gridk by

H (k) =L/ 2

n= L/ 2

hne jkn

Enforcingeven symmetry in the impulse response, i.e.,hn = h n, gives azero phase FIR lter which we canlater right-shiftL/ 2 samples to make a causal,linear phase lter. In this case, the frequency response reducto a sum of cosines :

H (k) = h0 + 2L/ 2

n=1

hn cos(kn), k = 0, 1, 2, . . . , N

or in matrix form:H (0 )H (1 )...

H (N 1 )

=

1 2cos(0 ) . . . 2cos[0 (L 1)]1 2cos(1 ) . . . 2cos[1 (L 1)]...1 2cos(N 1 ) . . . 2cos[N 1 (L 1)]

A

h0h1...

hL/

x(Note that Remez exchange algorithms are also based this formulation internally.)

32


9/19

Matrix Formulation: Optimal L2 Design, Contd

In matrix notation, our lter design problem can be stated

minx

Ax b 22

where, for zero-phase lters,

A =

1 2cos(0) . . . 2cos[0(L 1)]1 2cos(1) . . . 2cos[1(L 1)]

...1 2cos(N 1) . . . 2cos[N 1(L 1)]

x = h

and b = [D(k)] is the desired frequency response at thespecied frequencies.

The functionfirls in the Matlab Signal Processing ToolBox implements frequency-domain weighted-least-squaresFIR lter design (not available in Octave as of 2/2003).

33

Least Squares Optimization

x = argminx

Ax b 2 = arg minx Ax b22

Hence we can minimizeAx b 22 = (Ax b)

T (Ax b)

Expanding this, we have:

(Ax b)T (Ax b) = (bT xT AT )(Ax b)

This is quadratic inx, hence it has aglobal minimumwhich we can nd by taking the derivative, setting it tzero, and solving forx. Doing this yields:

AT Ax AT b = 0

These are the famousnormal equations whose solution given by:

x = (AT A) 1AT b

The matrixA = (AT A) 1AT

is known as thepseudo-inverse of the matrixA.34

Geometrical Interpretation of Least Squares

Typically, the number of frequency constraints is muchgreater than the number of design variables (lter taps).In these cases, we have anoverdetermined system of equations (more equations than unknowns). Therefore,we cannot generally satisfy all the equations, and we areleft with minimizing some error criterion to nd theoptimal compromise solution.

In the case of least-squares approximation, we areminimizing theEuclidean distance , which suggests thefollowing geometrical interpretation:

Ax

column-space of A

Ax = b + eMinimizex e 2 = b Ax 2

35

Ax

column-space of A

This diagram suggests that the error vectorb Ax isorthogonal to the column space of the matrixA, hence imust be orthogonal to each column inA.

AT (b Ax) = 0 AT Ax = AT b

This is how theorthogonality principle can be used toderive the fact that the best least squares solution isgiven by

x = (AT A) 1AT b = Ab

Note that the pseudo-inverseA projects the vector bonto the column space of A.

In Matlab:

x = A \ b

x = pinv(A) * b

36


10/19

Chebyshev Optimal Linear Phase lter Design

Consider theL -norm minimization problem:

minx

Ax b

We said earlier that the norm of a vector is themaximum element of that vector, hence minimizing the norm is minimizing the maximum element of a vector:

minx

maxk

|aT k x bk|

whereaT k denotes the kth row of the matrixA.

We can then write this as:

min ts.t. |aT k x bk| < t

If we introduce a new variablex = xt , then

t = f T x = [ 0 . . . 0 1 ]x, and our optimization problembecomes:

minx

f T x

s.t. [ aT k 0 ] x bk < xf T x

37

Hence we are minimizing alinear objective , subject to aset of linear constraints . This is known as alinear programming problem (linprog in Matlab).

Linear

Linear

Objective

Constraints

38

More General Real FIR Filters

So far we have looked at the design of linear phase FIRlters. In this case,A, x and b are all real.

Sometimes we may want to design lters and specify boththe magnitude and the phase of the frequency response.

Examples:

Minimum phase lters

Inverse lters

Fractional delay lters

Interpolation polyphase sublters

39

Complex FIR Filter Design

In linear-phase lter design, we assumed symmetry oflter coefficients [h(n) = h( n)]

The lter frequency response became asum of cosines (zero phase)

The matrix A was real

The desired magnitude responseb was real

The nal zero-phase lterx could be right-shiftedL/ 2 samples to get a corresponding causallinear-phase FIR lter

Now we would like to specify acomplex frequencyresponse. This means that:

b is complex

A is complex

We still wantx (our lter coefficients) to be real

If we try to use \ orpinv in Matlab, we will generallyget a complex result forx

40


11/19

Summarizing our problem:

minx

Ax b 2

where, A C NxM , bC Nx 1, and x R Mx1

Hence we have,

minx

[R (A) + j I (A)]x [R (b) + j I (b)] 22which can be written as:

minx

R(A)x R (b) + j [ I (A)x + I (b)] 22or

minx

R (A) I (A) x

R (b) I (b)

2

2

which is written in terms of onlyreal variables.

Hence, we can use the standard least squares solvers inMatlab and end up with areal solution.

Related paper

Design of Fractional Delay Filters Using ConvexOptimization (Mohonk-97, Music 421 handout):

http://ccrma.stanford.edu/~jos/eps/mohonk97.ps.gz

41

Optimal FIR Filters Arbitrary Magnitude andPhase Speciciations

cremez (Matlab Signal Processing Toolbox) performscomplex L FIR lter design:

Documentedonlineat The Mathworks(search forcremez )

Convergence theoretically guaranteed forarbitrary magnitude and phase specications versus frequenc

Reduces to Parks-McClellan algorithm (Remez secalgorithm) as a special case.

Written by Karam and McClellan. See Design of Optimal Digital FIR Filters with Arbitrary Magnituand Phase Responses, by Lina J. Karam and JameH. McClellan, ISCAS-96. The paper may bedownloaded at

http://www.eas.asu.edu/~karam/papers/iscas96 km.html

42

Second-Order Cone Problems (SOCP)

A linear function is minimized over the intersection of an affine set and the product of second-order(quadratic) cones

Nonlinear, convex problem including linear and(convex) quadratic programs as special cases

Solved by efficient primal-dual interior-point methods

Number of iterations required to solve a problemgrows at most as thesquare root of the problem size

Typical number of iterations ranges between 5 and50, almost independent of the problem size

See Applications of second-order cone programming, byM. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret,Linear Algebra and its Applications , 284:193-228,November 1998, Special Issue on Linear Algebra inControl, Signals and Image Processing. Available online:

http://www.stanford.edu/~boyd/socp.html

For more on this topic, see also Prof. Boyds course onconvex optimization: EE 364 2 and text book (Boyd andVandenberghe, Cambridge U. Press, 2004).

2 http://www.stanford.edu/class/ee364/

43

Hilbert Transform Filter Design Example:A Case Study in FIR Filter Design

Design Example:

Ideal Single-Sideband Filter (Hilbert transformer)

Window Method using Kaiser Window

Optimal Chebyshev using Remez Exchange

44
http://ccrma.stanford.edu/~%7B%7Djos/gz/mohonk97.ps.gzhttp://www.mathworks.com/access/helpdesk/help/toolbox/signal/filter11.shtmlhttp://www.eas.asu.edu/~%7B%7Dkaram/papers/iscas96_km.htmlhttp://www.stanford.edu/~%7B%7Dboyd/socp.htmlhttp://www.stanford.edu/class/ee364/http://www.stanford.edu/class/ee364/http://www.stanford.edu/~%7B%7Dboyd/socp.htmlhttp://www.eas.asu.edu/~%7B%7Dkaram/papers/iscas96_km.htmlhttp://www.mathworks.com/access/helpdesk/help/toolbox/signal/filter11.shtmlhttp://ccrma.stanford.edu/~%7B%7Djos/gz/mohonk97.ps.gz


12/19

Problem Statement

Design a negative-frequency lter which converts areal signal into its complex analytic signalcounterpart by ltering out negative frequencies.

Equivalently, design a single sideband (SSB) lterwhich outputs a single-sideband signal in response toa complex-conjugate pair of sidebands (a Hermitianspectrum).

We could also call it a positive-frequency-pass lter.

The imaginary part of such a lter is called aHilbert transform lter .

The ideal Hilbert transform is anallpass lter thatdelays positive-frequency components by 90 degrees,and advances negative-frequency components by 90degrees.

45

Hilbert Transform

y(t) = (h i x)(t) (Hilbert transform of x)

h i(t)= 1t (Hilbert transform kern

H i()=

j, > 0 j, < 0

1, = 0

(Hilbert frequency respo

xa(t)= x(t) + jy (t) (Analytic signal fromx)

= 1

0X ()e jtd (Note: Lower limit usua

Proof:

X a() = X () + jY () (By linearity of Fourier

= X + + X (Apply frequency respon+ j [ jX + + jX ]

= 2X + ()

46

Kaiser Window

Kaiser window in causal, linear-phase form:

w = kaiser(M,beta); % M=length=257, beta=8

Zero-pad by a factor of 8 and convert to zero-phase form(N = FFT size = 2048):

wzp = [w((M+1)/2:M),zeros(1,N-M),w(1:(M-1)/2)];

Kaiser window transform:

W = fft(wzp);

47

0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5160

140

120

100

80

60

40

20

Kaiser Window Transform, FFT size = 2048, Window length = 257


G a

i n ( d B )

0

0

48


13/19

Kaiser window transform, close-up on main lobe

0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05120

100

80

60

40

20

Close Up on Kaiser LF Response, FFT size = 2048, Window length = 257


G a

i n ( d B )

0

0

49

Oversimplied Window Method

Sample the ideal Hilbert-transform kernelh(t) = 1/t

to get

h i(n)=

1nT

(Sampled Hilbert transform ke

hw(n)= w(n)h i(n) (Windowed ideal impulse resp

H w() = (W H )(n) (Smoothed ideal frequency re

Design Parameters:

fs = 22050; % sampling rate (Hz)T = 1/fs; % sampling period (sec)M = 257; % FIR filter length = window lengthN = 8*(M-1); % for interpolated spectral displaysbeta = 8; % beta for Kaiser window

Filter Design:

n = [-N/2+0.5:N/2-0.5]; % Time axis (avoid t=0)hi = T ./ (pi*n*T); % Sampled Hilbert kernel

hr = sin(pi*n) ./ (pi*n); % 1/2 sample delay filterh = (hr + j*hi)/2; % Sampled ideal final filterplot(f,fftshift(max(-100,20*log10(abs(fft(h)))))); grid;...

50

FFT of Truncated(2048) Ideal Impulse Response

-0.6 -0.4 -0.2 0.2 0.4 0.6-100

-80

-60

-40

-20

20FFT of Truncated Analytically Specified "Ideal" Impulse Response


G a

i n ( d B )

0

0

h = [h(N/2+1:N),h(1:N/2)]; % zero-phase form (almost)hw = wzp .* h; % Apply window to ideal impulse responseHw = fft(hw); % Frequency response we really get

% Compute total stopband attenuation:ierr = norm(Hw(N/2+2:N))/norm(Hw)

= 0.0378 = 3.8 percent

For spectral displays, rotate buffer so negative frequenciesare on the left and DC is atk = N/ 2:

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Hwp = [Hw(N/2+2:N), Hw(1:N/2+1)]; % Neg. freqs on leftHwpn = abs(Hwp); Hwpn = Hwpn/max(Hwpn);

plot(f,20*log10(Hwpn)); grid;...

FIR Frequency Response by the Window Method

0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5140

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Length 257 AnalyticDerived FIR Frequency Response, FFT size = 2048


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Zoom in on low-frequency transition band

0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05140

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20AnalyticDerived LowFrequency Transition Band, f1/fs=0.023926 (marked by "+")


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Poorly positioned transition region from positive tonegative frequencies.

Need to rstbandpass-lter the ideal impulse

response. High-frequency transition identical, by symmetry

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Ideal Bandlimited Impulse Response

Since we are working with sampled data, the idealsingle-side-band lter impulse response is

h(n) = IDTFT n(u)=

12

u()e jnd

whereu() is the unit step function in the frequencydomain:

u() = 1, 00, < 0

We can evaluate the IDTFT directly as follows:

h(n) =1

2

0e jnd =

12jn

e jn

0=

e jn 12jn

=( 1)n 1

2jn=

0, n even, n = 0 j 1n , n odd

For n = 0, going back to the original integral gives

h(0) =1

2

0

e j 0d =1

2

.

Thus, the real part of h(n) is (n)/ 2, and the imaginarypart is 1/ (n ) for odd n and zero otherwise (as a resultof bandlimiting). There is no aliasing. However, we a

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need a transition band . We can work this out analytically,or we can simply draw it in the frequency domain, takea (large) inverse DFT, and window that. Since the latterapproach is more general, well do that.

Window Method applied to Bandlimited IdealImpulse Response

Further Design Parameters:

f1 = 530; % lower passband limit (Hz)N = 2^(nextpow2(8*fs/f1)) % FFT size from xition widthif N


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Desired Impulse Response

h = ifft(H); % zero-phase form

% measure of round-off error:ierr = norm(imag(h(1:2:N)))/norm(h) % zero?

ierr =

4.1958e-15

% rough estimate of time-aliasing error:aerr = norm(h(N/2-N/32:N/2+N/32))/norm(h)

aerr =

4.8300e-04

% for plots, prefer negative times on the left:hp = [h(N/2+2:N), h(1:N/2+1)];

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Real Part of Desired Impulse Response

150 100 50 50 100 1500.1

0.1

0.2

0.3

0.4

0.5

0.6Ideal SSB Filter Impulse Response, Real Part, FFT size = 2048

Time (samples)

A m p l i t u

d e

0

0

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Imaginary Part of Desired Impulse Response

150 100 50 50 100 1500.4

0.3

0.2

0.1

0.1

0.2

0.3

0.4Ideal SSB Filter Impulse Response, Imaginary Part, FFT size = 2048

Time (samples)

A m p l i t u d e

0

0

Apply window to ideal impulse response:

hw = wzp .* h; % Final FIR coefficientsHw = fft(hw); % Frequency response well see

Total stopband energy:

ierr = norm(Hw(N/2+2:N))/norm(Hw)disp(sprintf([Stop-band energy is %g percent of,...

total spectral energy], 100*ierr));

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Final FIR Filter Frequency Response

0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5180

160

140

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Length 257 FIR filter Frequency Response, FFT size = 2048


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60


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Zoom-in on Transition Region of Final FrequencyResponse

0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05140

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20LowFrequency Transition Band, f1/fs=0.023926 (marked by "+")


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0

0

Transition band better positioned

100 dB stob-band attenuation

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Remez Multiple Exchange Method

Remez exchange algorithm for optimal Chebyshev(equiripple) FIR lter design.

Original reference: J. H. McClellan, T. W. Parks, aL. R. Rabiner, A Computer Program for DesigninOptimum FIR Linear Phase Digital Filters,IEEE Trans. Audio Electro., vol AU-21, Dec. 1973,

pp. 506-526. Text reference: L. R. Rabiner and B. Gold,Theory

and Application of Digital Signal Processing,Prentice-Hall, 1975.

Pertinent recent reference: A. Reilly, G. Frazer, andB. Boashash, Analytic Signal GenerationTips anTraps, IEEE Tr. Signal Processing , vol. 42, no. 11,Nov. 1994.

Design an optimal lowpass lter of the desired width(takes quite a while):

hrm = remez(M-1, [0,(f2-fs/4)/fn,0.5,1], ...[1,1,0,0], [1,10]);

The weighting [1,P] says make the passband ripple Ptimes that of stopband. For steady-state audio spectra,

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passband ripple can be 0.1 dB or more. However,consider an FM signal in which a sinusoid is sweepingback and forth in the passband. In that case, thepassband ripple generates AM sidebands, so a spec morelike that in the stopband may be called for. Here, weallow the passband ripple to be 10 times the stopbandripple, as a compromise.Modulate it up to where its a single-sideband lter. I.e.,right-shift by/ 2 in the frequency domain. I.e., rotatethe frequency response counterclockwise along the unitcircle by 90 degrees:

hr = hrm .* j .^ [0:M-1];

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Optimal Chebyshev FIR(257) FrequencyResponse

0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5200

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100

50

50Length 257 Optimal Chebyshev FIR SingleSidebandFilter Freq. Response


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Zoom-In on Transition Band

0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05200

150

100

50



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0

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Passband Ripple

0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.324

3

2

1

1

2

3

4x 10

4 Passband Ripple


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Initial Impulse Response

5 10 15 20 25 30 35 40 45 508

6

4

2

2

4

6

8x 10

4 First 50 impulseresponse coefficients

Index

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0

0

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Summary of Trade-Offs

Window-method lter:

Stopband droops which wastes lter taps if theworst-case stopband attenuation is the only stopbanspec. However, such roll-off is natural in thefrequency domain and corresponds to a smootherpulse in the time domain.

Passband is degraded by early roll-off. Edge is off

Filter length can be thousands of taps as needed fofull audio band.

Optimal Remez-Exchange lter:

Stopband is ideal, equiripple.

Transition region close tohalf that of the windowmethod.

Pass-band is ideal, equiripple.

Time-domain impulses frompassband ripple are smal

Design time orders of magnitude larger than that fwindow method.

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Fails to converge for lters much longer than 256taps. (Need to increase working precision to getlonger lters.)

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Matlab hilbert() function

The Matlab Hilbert function returns an analytic signalgiven its real part.

Nh = M-2; % This looks bestdelta = [1,zeros(1,Nh)]; % zero-phase impulsehh = hilbert(delta); % zeros negative-freq FFT binsHh = fft(hh); % FIR frequency response

0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5350

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250

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100

50

Frequency Response of Length 256 hilbert() using Matlab


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0

Looks pretty good, but lets zero-pad the impulseresponse to interpolate the frequency response:

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hhzp = [hh(Lh/2+1:Lh), zeros(1,N-Lh), hh(1:Lh/2)];Hhzp = fft(hhzp); % Frequency response

0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5350

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Interpolated Frequency Response of Length 256 hilbert() using Matlab


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0.05 0.04 0.03 0.02 0.01 0.01 0.02 0.03 0.04 0.05350

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50



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Problems:

Transition bandwidth only 2 bins wide.

Massive time aliasing.

Only justiable for periodic signals with period exequal to lter length Nh. In this case only, timealiasing is equivalent to overlap-add from adjacentperiods.

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More generally, any real signal given to hilbert() must beinterpreted as precisely one period of a periodic signal.

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WinFlt 4up

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