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FIR and IIR Transfe

the input output relation of

an LTI system is:

[ ] [ ] [k

y n h k x n

=

=

the output in thez domain is:

Where

so we can write the transfer function in the familiar for

( ) [ ] n

n

H z h n z

=

=

( ) ( ). ( )Y z H z X z =

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so we can write the transfer function in the familiar for

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FIR and IIR Transfe

recall the LTI IIR difference equation:0

[ ]N

k k

a y n k =

applying thez transform to both sides of the differenc

and making use of the linearity and time shifting propez transform gives:

M

0 0

( )N M

k

k k

k k

a z Y z b z

= =

=

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n E

n Ex

Consider the M-point moving-average FIR filter with an impulse res

=][nh

otherwise,010,/1 n

( ) ( )

1

10

1 1 1( )

[ 1 ]1

M MMn

Mn

z zH z z

M M z zM z

=

= = =

Observe the following

The transfer function hasMzeros on

the unit circle at* ,

There are M-1 poles at z= 0 and a single

pole at z= 1

The pole at z= 1 exactly cancels thezero at z = 1

kjez /2= 10 k

-0.5

0

0.5

1

ImaginaryPart

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Ideal

deal

An ideal filter is a digital filter designed to pass signal co

certain frequencies without distortion, which therefore h

frequency response equal to 1 at these frequencies, and hfrequency response equal to 0 at all other frequencies

The range of frequencies where the frequency response t

value of one is called thepassband

The range of frequencies where the frequency response t

value of zero is called thestopband

The transition frequency from a passband to stopband reg

thecutoff frequency Note that an ideal filter cannot be realized Why?

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Ideal

deal

We note the following about the impulse response of an

hLP[n] is not absolutely summable

The corresponding transfer function is therefore not BIBO stable

hLP[n] is not causal, and is of doubly infinite length

The remaining three ideal filters are also characterized by doubly in

impulse responses and also are not absolutely summable

Thus, the ideal filters with the idealbrick wallfrequency

cannot be realized with finite dimensional LTI filter

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Zero

ero

Phase F

hase F

Now, for non-real-timeprocessing of real-valued input signals of fin

phase filtering can be implemented by relaxing the causality require

A zero-phase filtering scheme can be obtained by the following proc

Process the input data (finite length) with a causal real-coefficient filter H

Time reverse the output of this filter and process by the same filter.

Time reverse once again the output of the second filter

x[n] v[n] u[n] w[H(z) H(z)

][][],[][ nwnynvnu ==

),()()( XHV =)()()(

UHW =

( )*)( VU = ( ) *( ) *(Y W H = =

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In M

n M

The function filtfilt() implements the zero-phase filtering sch

filtfilt()

Zero-phase digital filtering

y = filtfilt(b,a,x) performs zero-phase digital filtering by processi

data in both the forward and reverse directions. After filtering in

direction, it reverses the filtered sequence and runs it back throu

The resulting sequence has precisely zero-phase distortion and

filter order. filtfilt minimizes start-up and ending transients by m

conditions, and works for both real and complex inputs.

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Linear

inear

Note that a zero-phase filter cannot be implemented for rapplications. Why?

For a causal transfer function with a nonzero phase respophase distortion can be avoided by ensuring that the tranhas (preferably) a unity magnitude and alinear-phase chthe frequency band of interest

Note that this phase characteristic is linear for all in [0 2].

Recall that the phase delay at any given frequency 0was If we have linear phase, that is, ()=-, then the total delay at

Real time means causal operation. No time to revers

jeH =)( 1)( =H == )()(H

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What is the Big D

hat is the Big D

The deal is huge!

If the phase spectrum is linear, then the phase delay is in

the frequency, and it is the same constant for all freque In other words, all frequencies are delayed by seconds,

equivalently, the entire signal is delayed by seconds.

Since the entire signal is delayed by a constant amount, there is no

If the filter does not have linear phase, then different freq

components are delayed by different amounts, causing si

distortion.

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Take Home Me

ake Home Me

If it is desired to pass input signal components in a certai

range undistorted in both magnitude and phase, then the

function should exhibit a unity magnitude response and aresponse in the band of interest|HLP()|

HLP()

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Generalized Linear

eneralized Linear

Now consider the following system, where G() is real (i.e., n

From our previous discussion, the term e-j simply introdelay, that is, normally independent of frequency. Now,

If G() is positive, the phase term is ()=-, hence the system h

If G()

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Approximately Linear

pproximately Linear

Consider the following transfer functions

Note that above a certain frequency, say c, the magnitude is very cis most of the signal above this frequency is suppressed So if the ph

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Linear Phase F

inear Phase F

It is typically impossible to design a linear phase IIR filte

designing FIR filters with precise linear phase is very ea

Consider a causal FIR filter of length M+1 (order M)

This transfer function has linear phase, if its impulse response h[n]symmetric

or anti-symmetric

Nn

n zhzhhznhzH =

++++== ]2[]1[]0[][)(

210

MnnMhnh = 0],[][

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LINEARPHASE

A linear phase solution for the causal FIR filter function

( )j H e =

requires: where c and are con

is the amplitude response (zero-phase response), a real fu

( )( ) ( )j j cH e e H +=

for a real FIR the filter function has to be even:

which implies: or:( ) ( ( ) )( ) ( )j c j ce H e H + +=

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LINEARPHASE

This linear phase filter description can be generalised in

for four type of FIR filters:

Type 1: symmetric sequence of odd length

Type 2: symmetric sequence of even length

Type 3: anti-symmetric sequence of odd length

Type 4: anti-symmetric sequence of even length

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LINEARPHASE

It is possible to generate other type of FIR transfer func

type transfer function. Take a given type I transfer func

From this function we can generate at least three types

/2

/2

/2 /2

( ) ( )

( ) ( 1) ( )

( ) ( 1) ( )

N

N

N N

E z z H z

F z H z

G z z H z

=

=

=

( )

( )

( )

E

F

G

with amplituderesponses:

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Linear Phase F

inear Phase F

There are four possible scenarios: filter length even or od

impulse response is either symmetric or antisymmetric

FIR II: even length, symmetric FIR I: odd le

FIR

antFIR IV: even length,

antisymmetricN

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FIR I and FIR I

IR I and FIR I

For symmetric coefficients, we can show that the frequency respons

following form:

FIR II (M is odd, the sequence is symmetric and of even length)

Note that this of the form H()=e-jG(), where =M/2, and G() is th

summation term) Output is delayed by M/2 samples!

FIR I (M is even, sequence is symmetric and of odd length)

( )

=

=

21

0

2

2cos][2)(

M

i

Mj

iM

iheH

G()

+=

2

1

2

2cos][2]

2[)(

MM

j iMihMheH

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FIR III and FIR IV

IR III and FIR IV

For antisymmetric sequences, we have h[n]=-h[M-n], which gives u

the summation expression:

FIR IV (M is odd, the sequence is antisymmetric and of even length

FIR III (M is even, the sequence is antisymmetric and of odd length

( )

=

=

+ 21

0

22

2sin][2)(

M

i

Mj

iM

iheH

=

=

+

12

1

22

2sin][2)(

M

i

Mj

iM

iheH

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Zeros Poles of an FIR

eros Poles of an FIR

E

Zero at =/2

Zero at =

Zero at =7/10

not on unit circle!

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Zero Locat

ero Locat

Linear Phase inear Phase

For linear phase filters, the impulse response satisfies

We can make the following observations as facts

=

==M

n

nznMhzHnMhnh

0

)()(][][ )(= HzzH M

1. If z0 is a zero of H(z), so too is 1/z0=z0-1

2. Real zeros that are not on the unit circle, always occur in pairs

such as (1-z-1) and (1--1z-1), which is a direct consequence ofFact 1.

3. If the zero is complex, z0=ej, then its conjugate e-j is also

zero. But by the above statements, their symmetric

counterparts -1ej and -1e-j must also be zero. So, complex

zeros occur in quadruplets4. If M is odd (filter length is even), and symmetric (that is,

h[ ] h[M ]) th H( ) t h t 1

http://engineering.rowan.edu/

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Digital Signal Processing, 2010 Robi Polikar, Rowan University

Linear Phase ilter

Zero Locations

1 1

Type 2Type 1

1 1

1 1

Type 4Type 3

1 1

Type 1 FIR filter: Either an even

number or no zeros at z = 1 and

z=-1

Type 2 FIR filter: Either an even

number or no zeros at z = 1, and

an odd number of zeros at z=-1Type 3 FIR filter: An odd number of

zeros at z = 1 and and z=-1

Type 4 FIR filter: An odd number of

zeros at z = 1, and either an even

number or no zeros at z=-1

The presence of zeros at z=1

leads to some limitations on the use of

these linear-phase transfer functions

for designing frequency-selective filters

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A Type 2 FIR filter cannot be used

to design a highpass filter since it

always has a zero at z=-1

A Type 3 FIR filter has zeros at

both z = 1 and z=-1, and hence

cannot be used to design either alowpass or a highpass or a

bandstop filter

A Type 4 FIR filter is not

appropriate to design a lowpass

filter due to the presence of a zero

at z = 1

Type 1 FIR filter has no such

restrictions and can be used to

design almost any type of filter

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