A REPORT ON

DIGITAL FILTER DESIGN USING MATLAB

A partial fulfillment of

COMPUTER ORIENTED PROJECT (BITS GC 331)

Under the guidance of Prof.S.P.Dubey

By

CH.Yeswanth

2004P3PS093

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI

GOA CAMPUS

April,2007

TABLE OF CONTENTS

1. INTRODUCTION

a. Definition of a filter

b. Operation of a basic Digital Filter

c. Recursive & Non-recursive filters

d. Introduction to MATLAB

2. DIGITAL FILTER DESIGN

3. FINITE IMPULSE RESPONSE (FIR) FILTER DESIGN

a. Properties of Linear-phase FIR Filter

b. Windows Design Techniques

c. Frequency Sampling Design Techniques

d. Optimal Equiripple Design Technique

e. MATLAB implementation

4. INFINITE IMPULSE RESPONSE (IIR) FILTER DESIGN

a. Basic design of IIR filter

b. IIR filter design steps

c. Characteristics of Prototype Analog Filters

d. Analog-to-digital Filter Transformations

e. Frequency-band Transformations

f. MATLAB implementation

5. COMPARISON OF FIR vs. IIR FILTERS

a. MATLAB applications

6. CONCLUSION

1. INTRODUCTION

a. Definition of a filter:

The function of a filter is to remove unwanted parts of the signal, such as random noise,

or to extract useful parts of the signal, such as the components lying within a certain

frequency range.

The following block diagram illustrates the basic idea.

There are two main kinds of filter, analog and digital. They are quite different in their

physical makeup and in how they work.

An analog filter uses analog electronic circuits made up from components such as

resistors, capacitors and op-amps to produce the required filtering effect

A digital filter uses a digital processor to perform numerical calculations on sampled

values of the signal. The processor may be a general-purpose computer such as a PC, or a

specialized DSP (Digital Signal Processor) chip.

b. Operation of a basic Digital Filter:

The basic theory of the operation of digital filters is essential to an understanding of how

digital filters are designed and used.

Suppose the "raw" signal which is to be digitally filtered is in the form of a voltage

waveform described by the function

FILTER Raw (unaltered)

signal

Filtered

signal

V= X(t)

where t is time.

This signal is sampled at time intervals h (the sampling interval). The sampled value at

time t = ih is

Xi= X(ih)

Thus the digital values transferred from the ADC to the processor can be represented by

the sequence

X0 , X1 ,X2 , X3 , ...

corresponding to the values of the signal waveform at

t = 0, h, 2h, 3h, ...

and t = 0 is the instant at which sampling begins.

At time t = nh (where n is some positive integer), the values available to the processor,

stored in memory, are

X0 , X1 , X2, X3 , ... X n

The digital output from the processor to the DAC consists of the sequence of values

Y0 , Y1 , Y2 , Y3 , ... Y n

In general, the value of Yn is calculated from the values X0, X1, X2, X3, .... Xn. The way

in which the y's are calculated from the x's determines the filtering action of the digital

filter.

c. Recursive & Non-recursive filters:

From above, it is clear that the values of output variables

A recursive filter is one which in addition to input values also uses previous output

values. These, like the previous input values, are stored in the processor's memory.

The word recursive literally means "running back", and refers to the fact that previously-

calculated output values go back into the calculation of the latest output. The expression

for a recursive filter therefore contains not only terms involving the input values

(xn, xn-1, xn-2, ...) but also terms in yn-1, yn-2, ...

Some people prefer an alternative terminology in which a non-recursive filter is known as

an FIR (or Finite Impulse Response) filter, and a recursive filter as an IIR (or Infinite

Impulse Response) filter.

d. Introduction to MATLAB:

MATLAB is an interactive, matrix-based system for scientific and engineering numeric

computation and visualization. Its strength lies in the fact that complex numerical

problems can be solved easily and in a fraction of the time. It is also powerful in the

sense that by using its relatively simple programming capability, MATLAB can be easily

extended to create new commands and functions.

The DSP applications are primarly algorithms that are implemented either on a DSP

processor or in software, a fair amount of programming is required. Using interactive

software, such as MATLAB, it is now possible to place more emphasis on learning new

and difficult concepts than on programming algorithms.

2. DIGITAL FILTER DESIGN

The design of a digital filter is carried out in three steps:

i. Specifications: they are determined by the applications

ii. Approximations: once the specification are defined, we use various concepts

and mathematics that we studied so far to come up with a filter description

that approximates the given set of specifications.

iii. Implementation: The product of the above step is a filter description in the

form of either a difference equation, or a system function H(z), or an impulse

response h(n). From this description we implement the filter in hardware or

through software on a computer.

Specifications are required in the frequency-domain in terms of the desired magnitude

and phase response of the filter. Generally a linear phase response in the passband is

desirable.

• In the case of FIR filters, It is possible to have exact linear phase

• In the case of IIR filters, a linear phase in the passband is not achievable.

• Hence we will consider magnitude-only specifications.

Absolute specifications: Provide a set of requirements on the magnitude response

function |H(ejw)|. Generally used for FIR filters.

Relative specifications: Provide requirements in decibels (dB), given by

This is used for both FIR and IIR filters.

0|)(|

|)(|log20

max

10 ≥−=jw

jw

eH

eHscaledB

3. FINITE IMPULSE RESPONSE (FIR) FILTER DESIGN

a. Properties of Linear-phase FIR Filter:

Let h(n), n=0,1,…,M-1 be the impulse response of length (or duration) M. Then the

system function is

Which has (M-1) poles at the origin (trivial poles) and M-1 zeros located anywhere in

the z-plane. The frequency response function is

For impulse response, h(n), if we impose a linear-phase constraint

Where alpha is a constant phase delay, then the h(n) must be symmetric about

alpha,which is index of symmetry, that is

There are two possibilities of symmetry, that is when M is odd and M is even.

If the phase response satisfy the condition

Which is a straight line but not through the origin. In this case alpha is not a constant

phase delay, but

Is constant, which is the group delay. Therefore alpha is a constant group delay. In this

case, as a group, frequencies are delayed at a constant rate. For this type of linear phase one can show that

∑∑−

=

−−−−−

=

− ==1

0

1)1(1

0

)()()(M

n

nMMM

n

n znhzznhzH

ππ ≤<−=∑−

=

− wenheHM

n

jwnjw ,)()(1

0

ππα ≤<−−=∠ wweH jw ,)(

2

1,10),1()(

−=−≤≤−−=

MwithMnnMhnh α

ππαβ ≤<−−=∠ wweH jw ,)(

α−=∠

dw

eHd jw )(

2,

2

1,10),1()(

πβα ±=

−=−≤≤−−−=

MwithMnnMhnh

This means that the impulse response h(n) is antisymmetric. The index of symmetry is

still alpha=(M-1)/2. Once again we have two possible types, one for M odd and one for

M even.

b. Windows Design Techniques:

There are several design methods, to design a FIR filter in MATLAB. One of the most

useful technique is Window design technique.

The basic idea is to choose a proper ideal frequency-selective filter (which always has a

noncausal, infinite-duration impulse response) and then truncate (or window) its impulse

response to obtain a linear-phase and causal FIR filter.

An ideal LPF of bandwidth wc<pi is given by

From this the ideal impulse response is

To obtain a causal and linear-phase FIR filter h(n) of length M, we must have

This operation is called “Windowing”.

Depending on how we define w(n) above, we obtain different window design.

≤<

≤⋅=

−

π

α

||0

||,1)(

ww

wweeH

c

c

wj

jw

d

[ ])(

)](sin[)(

2

1)()( 1

απα

π

π

π −−

=== ∫+

−

−

n

nwcdweeHeHFnh jwnjw

d

jw

dd

2

1,

0

10)()(

−=

−≤≤

=M

andelsewhere

Mnnhnh

d α

−≤≤

=

=

otherwise

Mnovertorespectwithfunctionsymmetricsomenw

nwnhnh d

,0

10)(

)()()(

α

There are different window design methods:

• Rectangular window:

This is the simplest window function but provides the worst performance from the

viewpoint of stopband attenuation.

• Bartlett window:

Bartlett suggested a more gradual transition in the form of a triangular window

• Hanning window:

This is a raised cosine window function.

• Hamming window:

This is similar to Hanning window except that it has a small amount of

discontinuity and is given by

• Blackman window:

This is also similar to the previous two but contains a second harmonic term and

is given by

( )( )

( )( )

1,)(2

1)(

2

1)(

sin

sin)(

sin

sin)(

,0

10,1)(

2

2

2

2 21

>>=≈

=⇒

=

−≤≤

=

∫∫+

−

+

−

− −

MdWdWwH

wWeeW

otherwise

Mnnw

cc

M

ww

r

ww

rr

w

wM

r

jw

w

wMjw

λλπ

λλπ ππ

−≤≤−

−−

−≤≤

−

=

otherwise

MnM

M

n

Mn

M

n

nw

,0

12

1,

1

22

2

10,

1

2

)(

−≤≤

−

−=

otherwise

MnM

n

nw

,0

10,1

2cos15.0

)(

π

−≤≤

−

−=

otherwise

MnM

n

nw

,0

10,1

2cos46.054.0

)(

π

• Kaiser window:

This is an adjustable window function that is widely used in practice. The window

function is given by

Where I0[] is the modified zero-order Bessel function. The parameter beta

controls tha minimum stopband attenuation As and can be chosen to yield

different transition widths for near-optimum As.

Kaiser has developed empirical design equations, given below

Window name Transistion

Approximation

Width

Exact values

Minimum

stopband

attenuation

Rectangular 4π/M 1.8π/M 21dB

Bartlett 8π/M 6.1π/M 25dB

Hann 8π/M 6.2π/M 44dB

Hamming 8π/M 6.6π/M 51dB

Blackman 12π/M 11π/M 74dB

−≤≤

−

+

−

−=

otherwise

MnM

n

M

n

nw

,0

10,1

4cos08.0

1

2cos5.042.0

)(

ππ

10,][

1

211

)(0

2

0

−≤≤

−−−

= MnI

M

nI

nwβ

β

<<−+−

≥−=

+∆

−≈

−=∆=

5021)21(07886.0)21(5842.0

50),7.8(1102.0

136.14

95.7

2.

,,,

4.0

sss

ss

s

ps

spsp

AAA

AAparameter

f

AMorderfilter

wwfwidthtransitionNorm

AandRwwGiven

β

π

c. Frequency Sampling Design Techniques:

In this design approach we use the fact that the system function H(z) can be obtained from

the samples H(k) of the frequency response H(ejw).

The basic idea is, for a given ideal lowpass filter Hd(ejw), choose the filter length M and

then sample Hd(ejw) at M equispaced frequencies between 0 and 2pi. The actual response

is the interpolation of the samples is given by

The approximation error that is difference between the ideal and the actual response—is

zero at the sampled frequencies. The approximation error at all other frequencies

depends on the shape of the ideal response; that is, the sharper the ideal response, the

larger the approximation error. The error is larger near the band edge and smaller within

the band.

Two Design Approaches:

• Naïve design method:

Use the basic idea literally and provide no constraints on the approximation error,

that is, we accept whatever error we get from the design.

• Optimum design method:

To try minimize error in the stop band by varying values of the transition band

samples.

∑ ∑−

=

−

=−

−−

−

−==

1

0

1

0/211

)(1)()(

M

n

M

kMkj

Mn

ez

kH

M

zznhzH π

)]([)( kHIDFTnh =

d. Optimal Equiripple Design Technique:

The disadvantages of the window design and the frequency sampling designis that, we

cannot specify the band frequencies wp and ws precisely in the design. We cannot specify

both delta1 and delta2 ripple factors simultaneously. The approximation error—that is,

the difference between the ideal response and the actual response—is not uniformly

distributed over the band intervals.

The techniques to eliminate the above three problems are:

• For linear-phase FIR filter it is possible to derive a set of conditions for which it can

be proved that the design solution is optimal in the sense of minimizing the maximum

approximation error (sometimes called the minimax or the Chebyshev error).

• Filters that have this property are called equiripple filter because the approximation

error is uniformly distributed in both the passband and the stopband.

• This results in low-order filter.

Parks-McClellan Algorithm:

It was solved by Remez.

1. Estimate the filter length order M by (7.48)

2. Guess extrema frequencies wi (i= 1:L+2)

3. Find an Lth polynomial that fits these points

4. Determine new wi’s by interpolation of the polynomial

5. Iteration from beginning

6. Determing a(i) and Emax by min(max(E(w)))

π

δδ

δδδ

δ

δδδ

2,

6.14

log20ˆ

10*)1(1

log20

110

110

1

1log20

2110

20/

12

1

210

20/

20/

1

1

110

ps

A

s

R

R

p

wwf

fM

A

R

s

p

p

−=∆

∆

−=

+=⇒+

−=

+

−=⇒

+−

−=

−

e. MATLAB implementation:

MATLAB provides several functions to implement window functions, as given below

• W = boxcar(M) returns the M-point Rectangular window function in array W.

• W = Bartlett(M) returns the M-point Bartlett window function in array W.

• W = hann(M) returns the M-point Hann window function in array W.

• W = hamming(M) returns the M-point Hamming window function in array W.

• W = blackman(M) returns the M-point Blackman window function in array W.

• W = Kaiser(M) returns the beta-valued M-point Kaiser window function in array

W.

Using the above window functions we write a MATLAB program to design a FIR digital

filter from given specifications.

Example: Design a digital lowpass filter with Wp = 0.2*pi, Ws = 0.3*pi, As = 50dB.

Use all window design methods such that user can select any window at a time for the

above specifications.

MATLAB code:

%This program uses the window method to design low pass FIR %filter. %The following windows are supported: Rectangular, Hamming, %Hanning, Bartlett, Blackman and Kaiser. % wtype: 1 for Rectangular window % 2 for Barlett window % 3 for Hanning window % 4 for Hamming window % 5 for Blackman window % 6 for Kaiser window clear all; wtype = 5; wp = 0.2*pi; ws = 0.3*pi; As = 50; tr_width = ws-wp; if wtype==1 M = ceil(1.8*pi/tr_width)+1

n = [0:1:M-1]; wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_rect = (boxcar(M))'; h = hd .* w_rect; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000; % plots subplot(2,2,2); stem(n,w_rect); title('Rectangular window') axis([0 M-1 0 1.1]); xlabel('n'); ylabel('w(n)')

elseif wtype==2 M = ceil(6.1*pi/tr_width)+1 n = [0:1:M-1];

wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_bart = (bartlett(M))'; h = hd .* w_bart; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000;

% plots subplot(2,2,2); stem(n,w_bart); title('Bartlett window') axis([0 M-1 0 1.1]); xlabel('n'); ylabel('w(n)')

elseif wtype==3 M = ceil(6.2*pi/tr_width)+1 n = [0:1:M-1];

wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_hann = (hanning(M))'; h = hd .* w_hann; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000;

% plots subplot(2,2,2); stem(n,w_hann); title('Hanning window')

axis([0 M-1 0 1.1]); xlabel('n'); ylabel('w(n)') elseif wtype==4 M = ceil(6.6*pi/tr_width)+1 n = [0:1:M-1];

wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_ham = (hamming(M))'; h = hd .* w_ham; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000;

subplot(2,2,2); stem(n,w_ham); title('Hamming window') axis([0 M-1 0 1.1]); xlabel('n'); ylabel('w(n)')

elseif wtype==5 M = ceil(11*pi/tr_width)+1 n = [0:1:M-1];

wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_black = (blackman(M))'; h = hd .* w_black; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000;

% plots subplot(2,2,2); stem(n,w_black); title('Blackman window') axis([0 M-1 0 1.1]); xlabel('n'); ylabel('w(n)')

elseif wtype==6 M = ceil((As-7.95)/(2.285*tr_width)+1)+1 n = [0:1:M-1]; beta = 0.1102*(As-8.7)

wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_kai = (kaiser(M,beta))'; h = hd .* w_kai; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))';

mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000; % plots subplot(2,2,2); stem(n,w_kai); title('Kaiser window') axis([0 M-1 0 1.1]); xlabel('n'); ylabel('w(n)')

end subplot(2,2,1); stem(n,hd); title('Ideal Impulse Response') axis([0 M-1 -0.1 0.3]); xlabel('n'); ylabel('hd(n)') subplot(2,2,3); stem(n,h); title('Actual impulse response') axis([0 M-1 -0.1 0.3]); xlabel('n'); ylabel('h(n)') subplot(2,2,4); plot(w/pi,db); title('magnitude response in db');grid axis([0 1 -100 10]); xlabel('frequency in pi units'); ylabel('decibels')

Output:

For wtype=1, that is Rectangular window

M =19

For wtype=2, that is Bartlett window

M = 62

For wtype=3, that is Hanning window

M = 63

For wtype=4, that is Hamming window

M = 67

For wtype=5, that is Blackman window

M = 111

For wtype=6, that is Kaiser window

M = 61

Beta = 4.5513

4. INFINITE IMPULSE RESPONSE (IIR) FILTER DESIGN

a. Basic design of IIR filter:

IIR filter have infinite-duration impulse responses, hence they can be matched to analog

filters, all of which generally have infinitely long impulse responses.

The basic technique of IIR filter design transforms well-known analog filters into digital

filters using complex-valued mappings. The advantage of this technique lies in the fact

that both analog filter design (AFD) tables and the mappings are available extensively in

the literature.

The basic technique is called the A/D filter transformation. However, the AFD tables are available only for lowpass filters. We also want design other frequency-selective filters

(highpass, bandpass, bandstop, etc.). To do this, we need to apply frequency-band

transformations to lowpass filters. These transformations are also complex-valued

mappings.

There are two approaches to this basic technique of IIR filter design:

Approach 1:

This approach used in MATLAB.

Approach 2:

This is a study-oriented approach.

Design analog

lowpass filter

Apply Freq. band

transformation

s-->s

Apply filter

transformation

s-->z

Desired

IIR filter

Design analog

lowpass filter

Apply filter

transformation

s-->z

Apply Freq. band

transformation

z-->z

Desired

IIR filter

b. IIR Filter Design Steps:

• Design analog lowpass filter

• Study and apply filter transformations to obtain digital lowpass filter

• Study and apply frequency-band transformations to obtain other digital filters

from digital lowpass filter.

c. Characteristics of Prototype Analog Filters:

IIR filter design techniques rely on existing analog filter filters to obtain digital filters.

We designate these analog filters as prototype filters.

Three prototypes are widely used in practice:

• Butterworth lowpass

• Chebyshev lowpass (Type I and II)

• Elliptic lowpass

I. Butterworth:

This filter is characterized by the property that its magnitude response is flat in both

passband and stopband.

The magnitude-squared response of an N-order lowpass filter, where Omega_c is the

cutoff frequency in rad/sec.

II. Chebyshev-I:

• Chebyshev-I filters have equiripple response in the passband.

• Chebyshev-II filters have equiripple response in stopband.

• Butterworth filters have monotonic response in both bands.

• We note that by choosing a filter that has an equripple rather than a monotonic

behavior, we can obtain a low-order filter.

N

c

a jH2

2

1

1|)(|

ΩΩ

+

=Ω

• Therefore Chebyshev filters provide lower order than Buttworth filters for the

same specifications.

Where N is the order of the filter, Epsilon is the passband ripple factor and Tn(x) is

the Nth-order Chebyshev polynomial.

III. Chebyshev-II:

It has a monotone passband and an equiripple stopband, which implies that this filter

has both poles and zeros in the s-plane.

Therefore the group delay characteristics are better (and the phase response more

linear) in the passband than the Chebyshev-I prototype.

IV. Elliptical:

These filters exhibit equiripple behavior in the passband as well as the stopband. They

are similar in magnitude response characteristics to the FIR equiripple filters.

Therefore elliptic filters are optimum filters in that they achieve the minimum order N

for the given specifications.

These filters, for obvious reasons, are very difficult to analyze and therefore, to

design. It is not possible to design them using simple tools, and often programs or

tables are needed to design them.

∞<<

≤≤=

ΩΩ

+

=Ω

−

−

xx

xxNxT

T

jH

N

c

N

a

1)),(cosh(cosh

10)),(coscos()(

1

1|)(|

1

1

22

2

ε

( )( ) 122

2

/1

1|)(| −

ΩΩ+=Ω

cN

a

TjH

ε

ΩΩ

+

=Ω

c

N

a

U

jH22

2

1

1|)(|

ε

Where N is the order, epsilon is passband ripple, Un() is the Nth order Jacobian elliptic

function. For computation of filter order N, we have

d. Analog-to-digital Filter Transformations:

After discussing different approaches to the design of analog filters, to transform them

into digital filters. These transformations are derived by preserving different aspects of

analog and digital filters.

• Impulse invariance transformation: It preserves the shape of the impulse response

from Analog to Digital filter. Convert a differential equation representation into a corresponding difference equation and also preserves the shape of the step

response. The digital filter impulse response to look similar to that of a

frequency-selective analog filter.

Frequency-domain aliasing formula.

• Bilinear Transformation: This mapping is the best transformation method.

It preserves the system function representation from Analog to Digital domain. It is a stable design and there is no aliasing. There is no restriction on the type of

filter that can be transformed.

e. Frequency-band Transformations:

Using the results of Low-pass filter and Frequency Band Transformation, we can design

other kinds of filters, such as High-pass, Band-pass, Bnad-stop filters.

( )( ) ∫

−=

−=

Ω

Ω=

−

−=

2/

0 2221

2

1

2

1

sin1)(,

1,,

1)(

1)( π

θ

θε

x

dxK

Akk

kKkK

kKkKN

s

p

∑∞

−∞=

−=k

a kT

jsHT

zHπ21

)(

2/1

2/1

1

121

1

sT

sTz

z

z

Ts

−+

=⇒+−

= −

−

Let HLP(Z) be the given prototype lowpass digital filter, and let H(z) be the desired

frequency-selective digital filter. Define a mapping of the form

Given that HLP(Z) is a stable and causal filter, we also want H(z) to be stable and causal.

This imposes the following requirements:

• G( ) must be a rational function in 1/z so that H(z) is implementable.

• The unit circle of the Z-plane must map onto the unit circle of the z-plane.

• For stable filters, the inside of the unit circle of the Z-plane must also map onto

the inside of the unit circle of the z-plane.

The general form of the function G() that satisfies the above requirements is a rational

function of the all-pass type given by

Now by choosing an appropriate order n and the coefficients alpah_k, we can obtain a

variety of mappings.

f. MATLAB implementation:

Now, we design an IIR digital filter in MATLAB software using above steps and

methods.

Example: Design an IIR digital filter with Butterworth, Chebyshev, Inverse Chebyshev

or Elliptic characteristic using bilinear transformation

MATLAB code:

%output arguments: %b,a - coefficients of digital filter %ftype - 'low' for lowpass filter % 'high' for highpass filter % 'bandpass' for bandpass filter % 'stop' for bandstop filter %ctype - 1 for Butterworth filter % 2 for Chebyshev filter

1||,1

)(1

1

1

11 <−

−Π±==

−

−

=−−

k

k

kn

kz

zzGZ α

αα

)(

1111|)()(thatsuch)( −− =

−− ==zGZLP ZHzHzGZ

% 3 for Inverse Chebyshev filter % 4 for Elliptic filter %steps of designing a digital filter implied in the called functions: % step 1: estimate the minimum order of the filter from specifications % step 2: get analog, pre-warped frequencies % step 3: convert to low-pass prototype estimate % step 4: Get n-th order analog lowpass prototype with desired filter %characters % step 5: Transform to lowpass, bandpass, highpass, or bandstop of desired Wn % step 6: Use Bilinear transformation to find discrete equivalent: clear all; format; Fs = 100000; %sampling frequency(Hz). Wp = [20500 23500]/(Fs/2); %passband edge frequency normalised by Fs/2. Ws = [19000 25000]/(Fs/2); %stopband edge frewquency normalised by Fs/2. Rp = 0.25; %passband attenuation in dB. Rs = 45; %stopband attenuation in dB ctype = 4; %character of filter ftype = 'bandpass'; %type of filter if ctype==1 [n,Wn]=buttord(Wp,Ws,Rp,Rs); [b,a]=butter(n,Wn,ftype); elseif ctype==2 [n,Wn]=cheb1ord(Wp,Ws,Rp,Rs); [b,a]=cheby1(n,Rp,Wn,ftype); elseif ctype==3 [n,Wn]=cheb2ord(Wp,Ws,Rp,Rs); [b,a]=cheby2(n,Rs,Wn,ftype); elseif ctype==4 [n,Wn]=ellipord(Wp,Ws,Rp,Rs); [b,a]=ellip(n,Rp,Rs,Wn,ftype); end %Output the result disp('Numerator coefficients (in descending powers of z):'); disp(b); disp('Denominator coefficients (in descending powers of z):'); disp(a);

freqz(b,a,1024,Fs); %Impulse invariance method of anolog-to-digital filter conversion %a,b -- s-plane coefficients %az,bz -- digital filter coefficients clear all; b = 1; a = [1.84496 1.920675 1]; [bz,az]=impinvar(b,a) %get z-plane coefficients using impulse Inv. freqz(bz,az,1024);

Output:

The is given for elliptical bandpass filter.

Numerator coefficients (in descending powers of z):

Columns 1 through 5

0.0061 -0.0083 0.0236 -0.0221 0.0351

Columns 6 through 9

-0.0221 0.0236 -0.0083 0.0061

Denominator coefficients (in descending powers of z):

Columns 1 through 5

1.0000 -1.4483 4.4832 -4.2207 6.6475

Columns 6 through 9

-3.9458 3.9187 -1.1828 0.7634

bz =

0 0.5677

Example: Designing an elliptical lowpass filter using both bilinear and impulse invariant

transformations.

MATLAB code:

% Comparison of magnitude and phase responses of an analogue % and equivalent BZT and impulse invariant discrete-time filters % and pole-zero diagrams (elliptic low pass filter) % Fs=10000; FN=Fs/2; fp=1000; fs=3000; wp=fp*2*pi; ws=fs*2*pi; Ap=1; As=60; % % Calculate filter coefficients and frequency responses [N, wc]=ellipord(wp, ws, Ap, As,'s'); % analog filter [B, A]=ellip(N, Ap, As, wc, 's'); [bBZT, aBZT]=bilinear(B,A,Fs); % BZT filter [bIIT,aIIT]=impinvar(B,A,Fs); % Impulse invariance filter % % Compute frequency response [Ha, wa]=freqs(B,A); [HBZT, fBZT]=freqz(bBZT, aBZT, 512, Fs); [HIIT, fIIT]=freqz(bIIT, aIIT, 512, Fs); % % Plot magnitude frequency responses % figure(1); % Plot analogue magnitude response plot(wa/(2*pi), 20*log10(abs(Ha))) hold on figure (1); plot(fBZT, 20*log10(abs(HBZT)), 'r:') % Plot BZT magnitude response hold on figure (1); plot(fIIT, 20*log10(abs(HIIT)), 'g:') % Plot Impinv magnitude response legend('Analog', 'BZT', 'Imp Invar'); axis([0 10000 -120 0]) ylabel('Magnitude (dB)') xlabel('Frequency (Hz)') title('Filter magnitude responses') hold off; % % Plot phase responses

% figure(2); plot(wa/(2*pi), angle(Ha)*180/pi) % Plot analogue phase response hold on figure (2); plot(fBZT, angle(HBZT)*180/pi, 'r:') % Plot BZT phase response hold on figure(2); plot(fIIT, angle(HIIT)*180/pi, 'g.') %ImpInvar phase response legend('Analog', 'BZT','Imp Invar'); axis([0 10000 -360 360]) ylabel('Phase (Degrees)') xlabel('Frequency (Hz)') title('Filter Phase Responses') hold off % % Plot pole-zero diagrams % figure (3); zplane(bBZT, aBZT) title('Pole-zero diagram - BZT filter') figure (4); zplane(bIIT, aIIT) xmin=-1; xmax=1; ymin=-1; ymax=1; % Scale the z-plane for the Impulse Inva. axis([xmin xmax ymin ymax]) title('Pole-zero diagram - impulse invariance filter')

Output:

5. COMPARISION OF FIR vs. IIR FILTERS

So far we have seen many techniques for designing both FIR and IIR filters. In practice

one wonders about which filter should be chosen for a given application and which

method should be used to design it. Some meaning comparisons can be attempted if we

focus on the minimax optimal (or equiripple) filters. In the case of FIR filters these

optimal filters are the equiripple filters designed via the Parks-McClellan algorithm,

while in the case of IIR filters these are the elliptical filters.

One basis of comparison is the number of multiplications required to compute one output

sample in the sample realization of these filters. For FIR filters the standard realization is

the linear-phase direct form, while for elliptic filters cascade forms are widely used. Let

M be the length of a linear phase FIR filter (assume M be odd). Then we need

1

2 2

M M+; (for large M)

multiplications per output sample. Let N (assume be even) be the order of an elliptic filter

with the cascade form realization. Then there are N/2 second-order sections, each

requiring 3 multiplications. There are an additional three multiplications in the overall

structure for a total of

3

3 32 2

N N+ ; (for large N)

multiplications per output sample.

Now if we assume that each filter meets exactly the same specifications i.e. ω p, ω s,

δ 1 (or passband ripple), and δ 2 (or stopband attenuation) for a lowpass filter, then these two filters are equivalent if

1 3 3 13 3

2 2

M N M

N N

+ += ⇒ = + ; (for large N)

Then means that if the ratio M/N = 3, then two filters are roughly efficient. However, an

equiripple FIR filter is more efficient if M/N < 3, or an elliptic IIR filter is more efficient

if M/N >3.

a. MATLAB applications:

Consider an example of filter which passes all frequencies upto 100Hz and attenuates

above 175 Hz, with a passband ripple of 0.34dB and stopband attenuation of 40dB.

Design two FIR filters (both Parks-McClellan algorithm and Hamming window method)

(i) Use the Parks-McClellan algorithm to design the minimum-length filter that meets the

specifications.

(ii) Using same length (M) obtained in part (i), design an FIR lowpass filter using the

window method with a Hamming window. Does this filter meet the specifications? If not,

find the smallest length Hamming window design that has the desired characteristics.

Design two IIR filters both Butterworth and Elliptic filters.

Compare the magnitude responses of each filter.

Answer:

The MATLAB codes are given below along with magnitude responses of each filter.

Clearly in IIR filter, Elliptic gives more accurate response for desired specifications than

Butterworth.

In FIR filter, Hamming window having M=26, does not has the required response, but it

satisfies for order M=45. Both Hamming and Parks-McClellan algorithm satisfies the

specifications, but equiripple filter design by Parks-McClellan algorithm is preferred as it

has order of M = 26. This is because of Optimal Equiripple Design.

MATLAB code:

Design 1: IIR Elliptic filter

Fs=1000; FN=Fs/2; fp=100; fs=175; wp=fp*2*pi; ws=fs*2*pi; Ap=0.34; As=40; [N, wc]=ellipord(wp, ws, Ap, As,'s'); % analog filter [B, A]=ellip(N, Ap, As, wc, 's'); [bBZT, aBZT]=bilinear(B,A,Fs); % BZT filter [Ha, wa]=freqs(B,A); [HBZT, fBZT]=freqz(bBZT, aBZT, 512, Fs); % Plot magnitude frequency responses

figure(1); % Plot analogue magnitude response plot(wa/(2*pi), 20*log10(abs(Ha))) hold on figure (1); plot(fBZT, 20*log10(abs(HBZT)), 'r:') % Plot BZT magnitude response legend('Analog', 'BZT', 'Imp Invar'); axis([0 1000 -120 0]) ylabel('Magnitude (dB)') xlabel('Frequency (Hz)') title('Filter magnitude responses') hold off figure(2); plot(wa/(2*pi), angle(Ha)*180/pi) % Plot analogue phase response hold on figure (2); plot(fBZT, angle(HBZT)*180/pi, 'r:') % Plot BZT phase response legend('Analog', 'BZT','Imp Invar'); axis([0 1000 -360 360]) ylabel('Phase (Degrees)') xlabel('Frequency (Hz)') title('Filter Phase Responses')

Output:

Design 2: IIR Butterworth filter

Fs=1000; FN=Fs/2; fp=100; fs=175; wp=fp*2*pi; ws=fs*2*pi; Ap=0.34; As=40; [N, wc]=buttord(wp, ws, Ap, As,'s'); % analog filter [B, A]=butter(N, Ap, As, wc, 's'); [bBZT, aBZT]=bilinear(B,A,Fs); % BZT filter [Ha, wa]=freqs(B,A); [HBZT, fBZT]=freqz(bBZT, aBZT, 512, Fs); % Plot magnitude frequency responses % figure(1); % Plot analogue magnitude response plot(wa/(2*pi), 20*log10(abs(Ha))) hold on figure (1); plot(fBZT, 20*log10(abs(HBZT)), 'r:') % Plot BZT magnitude response legend('Analog', 'BZT', 'Imp Invar'); axis([0 1000 -120 0]) ylabel('Magnitude (dB)') xlabel('Frequency (Hz)') title('Filter magnitude responses') hold off figure(2);

plot(wa/(2*pi), angle(Ha)*180/pi) % Plot analogue phase response hold on figure (2); plot(fBZT, angle(HBZT)*180/pi, 'r:') % Plot BZT phase response legend('Analog', 'BZT','Imp Invar'); axis([0 1000 -360 360]) ylabel('Phase (Degrees)') xlabel('Frequency (Hz)') title('Filter Phase Responses')

Output:

Design 3: FIR Equiripple filter using Parks-McClellan algorithm

wp = 0.2*pi; ws = 0.35*pi; delta1 = 0.02; delta2 = 0.0102; [N,f,m,weights] = firpmord([wp,ws]/pi,[1,0],[delta1,delta2]); h = firpm(N,f,m,weights); [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000;wsi = ws/delta_w+1; wpi = wp/delta_w; Asd = -max(db(wsi:1:501)) N = N+1 h = firpm(N,f,m,weights); [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); Asd = -max(db(wsi:1:501)) N = N+1 h = firpm(N,f,m,weights); [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); Asd = -max(db(wsi:1:501)) M = N+1 n = [0:1:M-1]; subplot(2,1,1); stem(n,h); title('Actual impulse response') axis([0 M-1 -0.1 0.3]); xlabel('n'); ylabel('h(n)') subplot(2,1,2); plot(w/pi,db); title('magnitude response in db'); grid axis([0 1 -100 10]); xlabel('frequency in pi units'); ylabel('decibels')

Output:

Design 4: FIR filter using Hamming window of order M=26

wp = 0.2*pi; ws = 0.35*pi; tr_width = ws-wp; M = 26; n = [0:1:M-1]; wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_ham = (hamming(M))'; h = hd .* w_ham; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000; subplot(2,1,1); stem(n,h); title('Actual impulse response') axis([0 M-1 -0.1 0.3]); xlabel('n'); ylabel('h(n)') subplot(2,1,2); plot(w/pi,db); title('magnitude response in db'); grid axis([0 1 -100 10]); xlabel('frequency in pi units'); ylabel('decibels')

Output:

Design 5: FIR filter using Hamming window of order M=45

wp = 0.2*pi; ws = 0.35*pi; tr_width = ws-wp; M = ceil(6.6*pi/tr_width)+1 n = [0:1:M-1]; wc = (ws+wp)/2; alpha = (M-1)/2; n = [0:1:(M-1)]; m = n-alpha; fc = ws/pi; hd = fc*sinc(fc*m); w_ham = (hamming(M))'; h = hd .* w_ham; [H,w] = freqz(h,[1],1000,'whole'); H = (H(1:1:501))'; w = (w(1:1:501))'; mag = abs(H); db = 20*log10((mag+eps)/max(mag)); pha = angle(H); grd = grpdelay(h,[1],w); delta_w = 2*pi/1000; subplot(2,1,1); stem(n,h); title('Actual impulse response') axis([0 M-1 -0.1 0.3]); xlabel('n'); ylabel('h(n)') subplot(2,1,2); plot(w/pi,db); title('magnitude response in db'); grid axis([0 1 -100 10]); xlabel('frequency in pi units'); ylabel('decibels')

Output:

Clearly from the above, FIR equiripple filter has order M=26 and IIR elliptic filter has

order N=6. As M/N > 3, IIR elliptic filter is desirable for the given specifications.

6. CONCLUSION

• All Window design methods are used but, according to the need of use, required

order of filter, we have choose different windows which are appropriate to our

need.

• To obtain Gibbs phenomenon Bartlett window is used. Kaiser window is widely

used, as the beta parameter, gives the transition width, from the order of filter.

Other windows are also used in various applications.

• The design of IIR filters becomes easy when the required analog design is done.

The advantage of this technique lies in the fact that both analog filter design

(AFD) tables and the mappings

• For FIR filter design, linear phase constraint is necessarily required, to make the

design easy. But for IIR filter design this cannot be done.

• For most applications IIR elliptic filter are desirable from the computational point

of view. If we take into account the phase equalizers, then FIR filter designs look

good because of their exact linear-phase characteristics.

• The most favorable conditions for FIR filters are large values of δ 1 , small values

of δ 2 and large transition width. • MATLAB is a widely used software for Filter designing. This design method can

be implemented in the software itself as a part of certain real systems.