Copyright ? 2001,S,K,Mitra 1
Linear-Phase FIR Transfer
Functions
? It is nearly impossible to design a linear-
phase IIR transfer function
? It is always possible to design an FIR
transfer function with an exact linear-phase
response
? Consider a causal FIR transfer function H(z)
of length N+1,i.e.,of order N,
? ? ?? Nn nznhzH 0 ][)(
Copyright ? 2001,S,K,Mitra 2
Linear-Phase FIR Transfer
Functions
? The above transfer function has a linear
phase,if its impulse response h[n] is either
symmetric,i.e.,
or is antisymmetric,i.e.,
NnnNhnh ???? 0],[][
NnnNhnh ????? 0],[][
Copyright ? 2001,S,K,Mitra 3
Linear-Phase FIR Transfer
Functions
? Since the length of the impulse response can
be either even or odd,we can define four
types of linear-phase FIR transfer
functions
? For an antisymmetric FIR filter of odd
length,i.e.,N even
h[N/2] = 0
? We examine next the each of the 4 cases
Copyright ? 2001,S,K,Mitra 4
Linear-Phase FIR Transfer
Functions
Type 1,N = 8 Type 2,N = 7
Type 3,N = 8 Type 4,N = 7
Copyright ? 2001,S,K,Mitra 5
Linear-Phase FIR Transfer
Functions
Type 1,Symmetric Impulse Response with
Odd Length
? In this case,the degree N is even
? Assume N = 8 for simplicity
? The transfer function H(z) is given by
1 2 3( ) [ 0 ] [ 1 ] [ 2 ] [ 3 ]H z h h z h z h z? ? ?? ? ? ?
87654 87654 ????? ????? zhzhzhzhzh ][][][][][
Copyright ? 2001,S,K,Mitra 6
Linear-Phase FIR Transfer
Functions
? Because of symmetry,we have h[0] = h[8],
h[1] = h[7],h[2] = h[6],and h[3] = h[5]
? Thus,we can write 8 1 7( ) [ 0 ] ( 1 ) [ 1 ] ( )H z h z h z z? ? ?? ? ? ?
45362 432 ????? ????? zhzzhzzh ][)]([)]([
)]([)]([{ 33444 10 ??? ???? zzhzzhz
]}[)]([)]([ 432 122 hzzhzzh ????? ??
Copyright ? 2001,S,K,Mitra 7
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
then given by
? The quantity inside the braces is a real
function of w,and can assume positive or
negative values in the range ??w?0
)3c o s (]1[2)4c o s (]0[2{)( 4 w?w? w?w hheeH jj
]}4[)c o s (]3[2)2c o s (]2[2 hhh ?w?w?
Copyright ? 2001,S,K,Mitra 8
Linear-Phase FIR Transfer
Functions
? The phase function here is given by
where b is either 0 or ?,and hence,it is a
linear function of w in the generalized sense
? The group delay is given by
indicating a constant group delay of 4 samples
b?w??w? 4)(
4)( )( ???w? ww?dd
Copyright ? 2001,S,K,Mitra 9
Linear-Phase FIR Transfer
Functions
? In the general case for Type 1 FIR filters,
the frequency response is of the form
where the amplitude response,also
called the zero-phase response,is of the
form
)()( 2/ w? w?w HeeH jNj~
)(wH~
)(wH~ ? w???
?
2/
1 22
)c o s (][2][
N
n
NN nnhh
Copyright ? 2001,S,K,Mitra 10
Linear-Phase FIR Transfer
Functions
? Example - Consider
which is seen to be a slightly modified version
of a length-7 moving-average FIR filter
? The above transfer function has a symmetric
impulse response and therefore a linear phase
response
][)( 6215432121610 ?????? ??????? zzzzzzzH
Copyright ? 2001,S,K,Mitra 11
Linear-Phase FIR Transfer
Functions
? A plot of the magnitude response of
along with that of the 7-point moving-
average filter is shown below
)(zH 0
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / ?
M
a
gni
t
ude
m odi f i e d f i l t e r
m ovi ng- a ve r a ge
Copyright ? 2001,S,K,Mitra 12
Linear-Phase FIR Transfer
Functions
? Note the improved magnitude response
obtained by simply changing the first and the
last impulse response coefficients of a
moving-average (MA) filter
? It can be shown that we an express
which is seen to be a cascade of a 2-point MA
filter with a 6-point MA filter
? Thus,has a double zero at,i.e.,
(w = ?)
)()()( 54321611210 11 ?????? ???????? zzzzzzzH
1??z)(zH 0
Copyright ? 2001,S,K,Mitra 13
Linear-Phase FIR Transfer
Functions
Type 2,Symmetric Impulse Response with
Even Length
? In this case,the degree N is odd
? Assume N = 7 for simplicity
? The transfer function is of the form
321 3210 ??? ???? zhzhzhhzH ][][][][)(
7654 7654 ???? ???? zhzhzhzh ][][][][
Copyright ? 2001,S,K,Mitra 14
Linear-Phase FIR Transfer
Functions
? Making use of the symmetry of the impulse
response coefficients,the transfer function
can be written as
)]([)]([)( 617 110 ??? ???? zzhzhzH
)]([)]([ 4352 32 ???? ???? zzhzzh
)]([)]([{ ///// 2525272727 10 ??? ???? zzhzzhz
)}]([)]([ //// 21212323 32 ?? ???? zzhzzh
Copyright ? 2001,S,K,Mitra 15
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
given by
? As before,the quantity inside the braces is a
real function of w,and can assume positive
or negative values in the range
)c o s(]1[2)c o s(]0[2{)( 25272/7 www?w ?? hheeH jj
)}c o s(]3[2)c o s(]2[2 223 ww ?? hh
??w?0
Copyright ? 2001,S,K,Mitra 16
Linear-Phase FIR Transfer
Functions
? Here the phase function is given by
where again b is either 0 or ?
? As a result,the phase is also a linear function
of w in the generalized sense
? The corresponding group delay is
indicating a group delay of samples
b?w??w? 27)(
27
27)( ?w?
Copyright ? 2001,S,K,Mitra 17
Linear-Phase FIR Transfer
Functions
? The expression for the frequency response
in the general case for Type 2 FIR filters is
of the form
where the amplitude response is given by
)(wH~ ? ?w??
?
?
?2/)1(
1 2
1
2
1 ))(c o s(][2 N
n
N nnh
)()( 2/ w? w?w HeeH jNj~
Copyright ? 2001,S,K,Mitra 18
Linear-Phase FIR Transfer
Functions
Type 3,Anti-symmetric Impulse Response
with Odd Length
? In this case,the degree N is even
? Assume N = 8 for simplicity
? Applying the symmetry condition we get
)]([)]([{)( 33444 10 ??? ???? zzhzzhzzH
)}]([)]([ 122 32 ?? ???? zzhzzh
Copyright ? 2001,S,K,Mitra 19
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
given by
? It also exhibits a generalized phase response
given by
where b is either 0 or ?
)3s i n (]1[2)4s i n (]0[2{)( 2/4 w?w? ??w?w hheeeH jjj
)}s i n (]3[2)2s i n (]2[2 w?w? hh
b??w??w? ?24)(
Copyright ? 2001,S,K,Mitra 20
Linear-Phase FIR Transfer
Functions
? The group delay here is
indicating a constant group delay of 4 samples
? In the general case
where the amplitude response is of the form
4)( ?w?
)()( 2/ w? w?w HjeeH jNj~
)(wH~ ? w??
?
2/
1 2
)s in (][2
N
n
N nnh
Copyright ? 2001,S,K,Mitra 21
Linear-Phase FIR Transfer
Functions
Type 4,Anti-symmetric Impulse Response
with Even Length
? In this case,the degree N is odd
? Assume N = 7 for simplicity
? Applying the symmetry condition we get
)]([)]([{)( ///// 2525272727 10 ??? ???? zzhzzhzzH
)}]([)]([ //// 21212323 32 ?? ???? zzhzzh
Copyright ? 2001,S,K,Mitra 22
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
given by
? It again exhibits a generalized phase
response given by
where b is either 0 or ?
)si n (]1[2)si n (]0[2{)( 25272/2/7 ww??w?w ?? hheeeH jjj
)}si n (]3[2)si n (]2[2 223 ww ?? hh
b??w??w? ?227)(
Copyright ? 2001,S,K,Mitra 23
Linear-Phase FIR Transfer
Functions
? The group delay is constant and is given by
? In the general case we have
where now the amplitude response is of the
form
27)( ?w?
)()( 2/ w? w?w HjeeH jNj~
)(wH~ ? ?w??
?
?
?2/)1(
1 2
1
2
1 ))(si n (][2 N
n
N nnh
Copyright ? 2001,S,K,Mitra 24
Linear-Phase FIR Transfer
Functions
General Form of Frequency Response
? In each of the four types of linear-phase FIR
filters,the frequency response is of the form
? The amplitude response for each of
the four types of linear-phase FIR filters can
become negative over certain frequency
ranges,typically in the stopband
)()( 2/ w? bw?w HeeeH jjNj~ ~
)(wH
Copyright ? 2001,S,K,Mitra 25
Linear-Phase FIR Transfer
Functions
? The magnitude and phase responses of the
linear-phase FIR are given by
? The group delay in each case is
)(|)(| w?w HeH j
?
?
?
?
?
?w? )(
0)(f o r,
0)(f o r,
2
2
?w??b??
?wb??
w
w
H
H
N
N~
~
~
2)( N?w?
Copyright ? 2001,S,K,Mitra 26
Linear-Phase FIR Transfer
Functions
? Note that,even though the group delay is
constant,since in general is not a
constant,the output waveform is not a
replica of the input waveform
? An FIR filter with a frequency response that
is a real function of w is often called a zero-
phase filter
? Such a filter must have a noncausal impulse
response
|)(| wjeH
Copyright ? 2001,S,K,Mitra 27
Zero Locations of Linear-
Phase FIR Transfer Functions
? Consider first an FIR filter with a symmetric
impulse response,
? Its transfer function can be written as
? By making a change of variable,
we can write
??
?
?
?
? ??? N
n
nN
n
n znNhznhzH
00
][][)(
][][ nNhnh ??
nNm ??
???
?
?
?
??
?
? ??? N
m
mNN
m
mNN
n
n zmhzzmhznNh
000
][][][
Copyright ? 2001,S,K,Mitra 28
Zero Locations of Linear-
Phase FIR Transfer Functions
? But,
? Hence for an FIR filter with a symmetric
impulse response of length N+1 we have
? A real-coefficient polynomial H(z) satisfying
the above condition is called a mirror-image
polynomial (MIP)
)()( 1??? zHzzH N
)(][ 10 ?? ?? zHzmhNm m
Copyright ? 2001,S,K,Mitra 29
Zero Locations of Linear-
Phase FIR Transfer Functions
? Now consider first an FIR filter with an
anti-symmetric impulse response,
? Its transfer function can be written as
? By making a change of variable,
we get
][][ nNhnh ???
??
?
?
?
? ???? N
n
nN
n
n znNhznhzH
00
][][)(
)(][][ 1
00
??
?
??
?
? ?????? ?? zHzzmhznNh NN
m
mNN
n
n
nNm ??
Copyright ? 2001,S,K,Mitra 30
Zero Locations of Linear-
Phase FIR Transfer Functions
? Hence,the transfer function H(z) of an FIR
filter with an antisymmetric impulse
response satisfies the condition
? A real-coefficient polynomial H(z)
satisfying the above condition is called a
anti-mirror-image polynomial (AIP)
)()( 1???? zHzzH N
Copyright ? 2001,S,K,Mitra 31
Zero Locations of Linear-
Phase FIR Transfer Functions
? It follows from the relation
that if is a zero of H(z),so is
? Moreover,for an FIR filter with a real
impulse response,the zeros of H(z) occur in
complex conjugate pairs
? Hence,a zero at is associated with a
zero at
)()( 1???? zHzzH N
oz ??
oz ??
oz ?/1?
*oz ??
Copyright ? 2001,S,K,Mitra 32
Zero Locations of Linear-
Phase FIR Transfer Functions
? Thus,a complex zero that is not on the unit
circle is associated with a set of 4 zeros given
by
? A zero on the unit circle appear as a pair
as its reciprocal is also its complex conjugate
,?jrez ?? ?jr ez ?? 1
?jez ??
Copyright ? 2001,S,K,Mitra 33
Zero Locations of Linear-
Phase FIR Transfer Functions
? Since a zero at is its own reciprocal,it
can appear only singly
? Now a Type 2 FIR filter satisfies
with degree N odd
? Hence
implying,i.e.,H(z) must have a
zero at
1??z
)()( 1??? zHzzH N
1??z
)()()()( 1111 ??????? ? HHH N
01 ?? )(H
Copyright ? 2001,S,K,Mitra 34
Zero Locations of Linear-
Phase FIR Transfer Functions
? Likewise,a Type 3 or 4 FIR filter satisfies
? Thus
implying that H(z) must have a zero at z = 1
? On the other hand,only the Type 3 FIR
filter is restricted to have a zero at
since here the degree N is even and hence,
)()( 1???? zHzzH N
)()()()( 1111 HHH N ???? ?
1??z
)()()()( 1111 ???????? ? HHH N
Copyright ? 2001,S,K,Mitra 35
Zero Locations of Linear-
Phase FIR Transfer Functions
? Typical zero locations shown below
1? 1
Type 2 Type 1
1? 1
1? 1
Type 4 Type 3
1? 1
Copyright ? 2001,S,K,Mitra 36
Zero Locations of Linear-
Phase FIR Transfer Functions
? Summarizing
Type 1 FIR filter,Either an even number or
no zeros at z = 1 and
Type 2 FIR filter,Either an even number or
no zeros at z = 1,and an odd number of zeros
at
Type 3 FIR filter,An odd number of zeros at
z = 1 and
1??z
1??z
1??z
Copyright ? 2001,S,K,Mitra 37
Zero Locations of Linear-
Phase FIR Transfer Functions
Type 4 FIR filter,An odd number of zeros
at z = 1,and either an even number or no
zeros at
? The presence of zeros at leads to the
following limitations on the use of these
linear-phase transfer functions for designing
frequency-selective filters
1??z
1??z
Copyright ? 2001,S,K,Mitra 38
Zero Locations of Linear-
Phase FIR Transfer Functions
? A Type 2 FIR filter cannot be used to
design a highpass filter since it always has a
zero
? A Type 3 FIR filter has zeros at both z = 1
and,and hence cannot be used to
design either a lowpass or a highpass or a
bandstop filter
1??z
1??z
Copyright ? 2001,S,K,Mitra 39
Zero Locations of Linear-
Phase FIR Transfer Functions
? A Type 4 FIR filter is not appropriate to
design a lowpass filter due to the presence
of a zero at z = 1
? A Type 1 FIR filter has no such restrictions
and can be used to design almost any type
of filter
Copyright ? 2001,S,K,Mitra 40
Bounded Real Transfer
Functions
? A causal stable real-coefficient transfer
function H(z) is defined as a bounded real
(BR) transfer function if
? Let x[n] and y[n] denote,respectively,the
input and output of a digital filter
characterized by a BR transfer function H(z)
with and denoting their
DTFTs
)( wjeX )( wjeY
1|)(| ?wjeH for all values of w
Copyright ? 2001,S,K,Mitra 41
Bounded Real Transfer
Functions
? Then the condition implies that
? Integrating the above from to ?,and
applying Parseval’s relation we get
1|)(| ?wjeH
22 )()( ww ? jj eXeY
??
?
???
?
???
?
nn
nxny 22 ][][
??
Copyright ? 2001,S,K,Mitra 42
Bounded Real Transfer
Functions
? Thus,for all finite-energy inputs,the output
energy is less than or equal to the input
energy implying that a digital filter
characterized by a BR transfer function can
be viewed as a passive structure
? If,then the output energy is
equal to the input energy,and such a digital
filter is therefore a lossless system
1|)(| ?wjeH
Copyright ? 2001,S,K,Mitra 43
Bounded Real Transfer
Functions
? A causal stable real-coefficient transfer
function H(z) with is thus
called a lossless bounded real (LBR)
transfer function
? The BR and LBR transfer functions are the
keys to the realization of digital filters with
low coefficient sensitivity
1|)(| ?wjeH
Linear-Phase FIR Transfer
Functions
? It is nearly impossible to design a linear-
phase IIR transfer function
? It is always possible to design an FIR
transfer function with an exact linear-phase
response
? Consider a causal FIR transfer function H(z)
of length N+1,i.e.,of order N,
? ? ?? Nn nznhzH 0 ][)(
Copyright ? 2001,S,K,Mitra 2
Linear-Phase FIR Transfer
Functions
? The above transfer function has a linear
phase,if its impulse response h[n] is either
symmetric,i.e.,
or is antisymmetric,i.e.,
NnnNhnh ???? 0],[][
NnnNhnh ????? 0],[][
Copyright ? 2001,S,K,Mitra 3
Linear-Phase FIR Transfer
Functions
? Since the length of the impulse response can
be either even or odd,we can define four
types of linear-phase FIR transfer
functions
? For an antisymmetric FIR filter of odd
length,i.e.,N even
h[N/2] = 0
? We examine next the each of the 4 cases
Copyright ? 2001,S,K,Mitra 4
Linear-Phase FIR Transfer
Functions
Type 1,N = 8 Type 2,N = 7
Type 3,N = 8 Type 4,N = 7
Copyright ? 2001,S,K,Mitra 5
Linear-Phase FIR Transfer
Functions
Type 1,Symmetric Impulse Response with
Odd Length
? In this case,the degree N is even
? Assume N = 8 for simplicity
? The transfer function H(z) is given by
1 2 3( ) [ 0 ] [ 1 ] [ 2 ] [ 3 ]H z h h z h z h z? ? ?? ? ? ?
87654 87654 ????? ????? zhzhzhzhzh ][][][][][
Copyright ? 2001,S,K,Mitra 6
Linear-Phase FIR Transfer
Functions
? Because of symmetry,we have h[0] = h[8],
h[1] = h[7],h[2] = h[6],and h[3] = h[5]
? Thus,we can write 8 1 7( ) [ 0 ] ( 1 ) [ 1 ] ( )H z h z h z z? ? ?? ? ? ?
45362 432 ????? ????? zhzzhzzh ][)]([)]([
)]([)]([{ 33444 10 ??? ???? zzhzzhz
]}[)]([)]([ 432 122 hzzhzzh ????? ??
Copyright ? 2001,S,K,Mitra 7
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
then given by
? The quantity inside the braces is a real
function of w,and can assume positive or
negative values in the range ??w?0
)3c o s (]1[2)4c o s (]0[2{)( 4 w?w? w?w hheeH jj
]}4[)c o s (]3[2)2c o s (]2[2 hhh ?w?w?
Copyright ? 2001,S,K,Mitra 8
Linear-Phase FIR Transfer
Functions
? The phase function here is given by
where b is either 0 or ?,and hence,it is a
linear function of w in the generalized sense
? The group delay is given by
indicating a constant group delay of 4 samples
b?w??w? 4)(
4)( )( ???w? ww?dd
Copyright ? 2001,S,K,Mitra 9
Linear-Phase FIR Transfer
Functions
? In the general case for Type 1 FIR filters,
the frequency response is of the form
where the amplitude response,also
called the zero-phase response,is of the
form
)()( 2/ w? w?w HeeH jNj~
)(wH~
)(wH~ ? w???
?
2/
1 22
)c o s (][2][
N
n
NN nnhh
Copyright ? 2001,S,K,Mitra 10
Linear-Phase FIR Transfer
Functions
? Example - Consider
which is seen to be a slightly modified version
of a length-7 moving-average FIR filter
? The above transfer function has a symmetric
impulse response and therefore a linear phase
response
][)( 6215432121610 ?????? ??????? zzzzzzzH
Copyright ? 2001,S,K,Mitra 11
Linear-Phase FIR Transfer
Functions
? A plot of the magnitude response of
along with that of the 7-point moving-
average filter is shown below
)(zH 0
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
w / ?
M
a
gni
t
ude
m odi f i e d f i l t e r
m ovi ng- a ve r a ge
Copyright ? 2001,S,K,Mitra 12
Linear-Phase FIR Transfer
Functions
? Note the improved magnitude response
obtained by simply changing the first and the
last impulse response coefficients of a
moving-average (MA) filter
? It can be shown that we an express
which is seen to be a cascade of a 2-point MA
filter with a 6-point MA filter
? Thus,has a double zero at,i.e.,
(w = ?)
)()()( 54321611210 11 ?????? ???????? zzzzzzzH
1??z)(zH 0
Copyright ? 2001,S,K,Mitra 13
Linear-Phase FIR Transfer
Functions
Type 2,Symmetric Impulse Response with
Even Length
? In this case,the degree N is odd
? Assume N = 7 for simplicity
? The transfer function is of the form
321 3210 ??? ???? zhzhzhhzH ][][][][)(
7654 7654 ???? ???? zhzhzhzh ][][][][
Copyright ? 2001,S,K,Mitra 14
Linear-Phase FIR Transfer
Functions
? Making use of the symmetry of the impulse
response coefficients,the transfer function
can be written as
)]([)]([)( 617 110 ??? ???? zzhzhzH
)]([)]([ 4352 32 ???? ???? zzhzzh
)]([)]([{ ///// 2525272727 10 ??? ???? zzhzzhz
)}]([)]([ //// 21212323 32 ?? ???? zzhzzh
Copyright ? 2001,S,K,Mitra 15
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
given by
? As before,the quantity inside the braces is a
real function of w,and can assume positive
or negative values in the range
)c o s(]1[2)c o s(]0[2{)( 25272/7 www?w ?? hheeH jj
)}c o s(]3[2)c o s(]2[2 223 ww ?? hh
??w?0
Copyright ? 2001,S,K,Mitra 16
Linear-Phase FIR Transfer
Functions
? Here the phase function is given by
where again b is either 0 or ?
? As a result,the phase is also a linear function
of w in the generalized sense
? The corresponding group delay is
indicating a group delay of samples
b?w??w? 27)(
27
27)( ?w?
Copyright ? 2001,S,K,Mitra 17
Linear-Phase FIR Transfer
Functions
? The expression for the frequency response
in the general case for Type 2 FIR filters is
of the form
where the amplitude response is given by
)(wH~ ? ?w??
?
?
?2/)1(
1 2
1
2
1 ))(c o s(][2 N
n
N nnh
)()( 2/ w? w?w HeeH jNj~
Copyright ? 2001,S,K,Mitra 18
Linear-Phase FIR Transfer
Functions
Type 3,Anti-symmetric Impulse Response
with Odd Length
? In this case,the degree N is even
? Assume N = 8 for simplicity
? Applying the symmetry condition we get
)]([)]([{)( 33444 10 ??? ???? zzhzzhzzH
)}]([)]([ 122 32 ?? ???? zzhzzh
Copyright ? 2001,S,K,Mitra 19
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
given by
? It also exhibits a generalized phase response
given by
where b is either 0 or ?
)3s i n (]1[2)4s i n (]0[2{)( 2/4 w?w? ??w?w hheeeH jjj
)}s i n (]3[2)2s i n (]2[2 w?w? hh
b??w??w? ?24)(
Copyright ? 2001,S,K,Mitra 20
Linear-Phase FIR Transfer
Functions
? The group delay here is
indicating a constant group delay of 4 samples
? In the general case
where the amplitude response is of the form
4)( ?w?
)()( 2/ w? w?w HjeeH jNj~
)(wH~ ? w??
?
2/
1 2
)s in (][2
N
n
N nnh
Copyright ? 2001,S,K,Mitra 21
Linear-Phase FIR Transfer
Functions
Type 4,Anti-symmetric Impulse Response
with Even Length
? In this case,the degree N is odd
? Assume N = 7 for simplicity
? Applying the symmetry condition we get
)]([)]([{)( ///// 2525272727 10 ??? ???? zzhzzhzzH
)}]([)]([ //// 21212323 32 ?? ???? zzhzzh
Copyright ? 2001,S,K,Mitra 22
Linear-Phase FIR Transfer
Functions
? The corresponding frequency response is
given by
? It again exhibits a generalized phase
response given by
where b is either 0 or ?
)si n (]1[2)si n (]0[2{)( 25272/2/7 ww??w?w ?? hheeeH jjj
)}si n (]3[2)si n (]2[2 223 ww ?? hh
b??w??w? ?227)(
Copyright ? 2001,S,K,Mitra 23
Linear-Phase FIR Transfer
Functions
? The group delay is constant and is given by
? In the general case we have
where now the amplitude response is of the
form
27)( ?w?
)()( 2/ w? w?w HjeeH jNj~
)(wH~ ? ?w??
?
?
?2/)1(
1 2
1
2
1 ))(si n (][2 N
n
N nnh
Copyright ? 2001,S,K,Mitra 24
Linear-Phase FIR Transfer
Functions
General Form of Frequency Response
? In each of the four types of linear-phase FIR
filters,the frequency response is of the form
? The amplitude response for each of
the four types of linear-phase FIR filters can
become negative over certain frequency
ranges,typically in the stopband
)()( 2/ w? bw?w HeeeH jjNj~ ~
)(wH
Copyright ? 2001,S,K,Mitra 25
Linear-Phase FIR Transfer
Functions
? The magnitude and phase responses of the
linear-phase FIR are given by
? The group delay in each case is
)(|)(| w?w HeH j
?
?
?
?
?
?w? )(
0)(f o r,
0)(f o r,
2
2
?w??b??
?wb??
w
w
H
H
N
N~
~
~
2)( N?w?
Copyright ? 2001,S,K,Mitra 26
Linear-Phase FIR Transfer
Functions
? Note that,even though the group delay is
constant,since in general is not a
constant,the output waveform is not a
replica of the input waveform
? An FIR filter with a frequency response that
is a real function of w is often called a zero-
phase filter
? Such a filter must have a noncausal impulse
response
|)(| wjeH
Copyright ? 2001,S,K,Mitra 27
Zero Locations of Linear-
Phase FIR Transfer Functions
? Consider first an FIR filter with a symmetric
impulse response,
? Its transfer function can be written as
? By making a change of variable,
we can write
??
?
?
?
? ??? N
n
nN
n
n znNhznhzH
00
][][)(
][][ nNhnh ??
nNm ??
???
?
?
?
??
?
? ??? N
m
mNN
m
mNN
n
n zmhzzmhznNh
000
][][][
Copyright ? 2001,S,K,Mitra 28
Zero Locations of Linear-
Phase FIR Transfer Functions
? But,
? Hence for an FIR filter with a symmetric
impulse response of length N+1 we have
? A real-coefficient polynomial H(z) satisfying
the above condition is called a mirror-image
polynomial (MIP)
)()( 1??? zHzzH N
)(][ 10 ?? ?? zHzmhNm m
Copyright ? 2001,S,K,Mitra 29
Zero Locations of Linear-
Phase FIR Transfer Functions
? Now consider first an FIR filter with an
anti-symmetric impulse response,
? Its transfer function can be written as
? By making a change of variable,
we get
][][ nNhnh ???
??
?
?
?
? ???? N
n
nN
n
n znNhznhzH
00
][][)(
)(][][ 1
00
??
?
??
?
? ?????? ?? zHzzmhznNh NN
m
mNN
n
n
nNm ??
Copyright ? 2001,S,K,Mitra 30
Zero Locations of Linear-
Phase FIR Transfer Functions
? Hence,the transfer function H(z) of an FIR
filter with an antisymmetric impulse
response satisfies the condition
? A real-coefficient polynomial H(z)
satisfying the above condition is called a
anti-mirror-image polynomial (AIP)
)()( 1???? zHzzH N
Copyright ? 2001,S,K,Mitra 31
Zero Locations of Linear-
Phase FIR Transfer Functions
? It follows from the relation
that if is a zero of H(z),so is
? Moreover,for an FIR filter with a real
impulse response,the zeros of H(z) occur in
complex conjugate pairs
? Hence,a zero at is associated with a
zero at
)()( 1???? zHzzH N
oz ??
oz ??
oz ?/1?
*oz ??
Copyright ? 2001,S,K,Mitra 32
Zero Locations of Linear-
Phase FIR Transfer Functions
? Thus,a complex zero that is not on the unit
circle is associated with a set of 4 zeros given
by
? A zero on the unit circle appear as a pair
as its reciprocal is also its complex conjugate
,?jrez ?? ?jr ez ?? 1
?jez ??
Copyright ? 2001,S,K,Mitra 33
Zero Locations of Linear-
Phase FIR Transfer Functions
? Since a zero at is its own reciprocal,it
can appear only singly
? Now a Type 2 FIR filter satisfies
with degree N odd
? Hence
implying,i.e.,H(z) must have a
zero at
1??z
)()( 1??? zHzzH N
1??z
)()()()( 1111 ??????? ? HHH N
01 ?? )(H
Copyright ? 2001,S,K,Mitra 34
Zero Locations of Linear-
Phase FIR Transfer Functions
? Likewise,a Type 3 or 4 FIR filter satisfies
? Thus
implying that H(z) must have a zero at z = 1
? On the other hand,only the Type 3 FIR
filter is restricted to have a zero at
since here the degree N is even and hence,
)()( 1???? zHzzH N
)()()()( 1111 HHH N ???? ?
1??z
)()()()( 1111 ???????? ? HHH N
Copyright ? 2001,S,K,Mitra 35
Zero Locations of Linear-
Phase FIR Transfer Functions
? Typical zero locations shown below
1? 1
Type 2 Type 1
1? 1
1? 1
Type 4 Type 3
1? 1
Copyright ? 2001,S,K,Mitra 36
Zero Locations of Linear-
Phase FIR Transfer Functions
? Summarizing
Type 1 FIR filter,Either an even number or
no zeros at z = 1 and
Type 2 FIR filter,Either an even number or
no zeros at z = 1,and an odd number of zeros
at
Type 3 FIR filter,An odd number of zeros at
z = 1 and
1??z
1??z
1??z
Copyright ? 2001,S,K,Mitra 37
Zero Locations of Linear-
Phase FIR Transfer Functions
Type 4 FIR filter,An odd number of zeros
at z = 1,and either an even number or no
zeros at
? The presence of zeros at leads to the
following limitations on the use of these
linear-phase transfer functions for designing
frequency-selective filters
1??z
1??z
Copyright ? 2001,S,K,Mitra 38
Zero Locations of Linear-
Phase FIR Transfer Functions
? A Type 2 FIR filter cannot be used to
design a highpass filter since it always has a
zero
? A Type 3 FIR filter has zeros at both z = 1
and,and hence cannot be used to
design either a lowpass or a highpass or a
bandstop filter
1??z
1??z
Copyright ? 2001,S,K,Mitra 39
Zero Locations of Linear-
Phase FIR Transfer Functions
? A Type 4 FIR filter is not appropriate to
design a lowpass filter due to the presence
of a zero at z = 1
? A Type 1 FIR filter has no such restrictions
and can be used to design almost any type
of filter
Copyright ? 2001,S,K,Mitra 40
Bounded Real Transfer
Functions
? A causal stable real-coefficient transfer
function H(z) is defined as a bounded real
(BR) transfer function if
? Let x[n] and y[n] denote,respectively,the
input and output of a digital filter
characterized by a BR transfer function H(z)
with and denoting their
DTFTs
)( wjeX )( wjeY
1|)(| ?wjeH for all values of w
Copyright ? 2001,S,K,Mitra 41
Bounded Real Transfer
Functions
? Then the condition implies that
? Integrating the above from to ?,and
applying Parseval’s relation we get
1|)(| ?wjeH
22 )()( ww ? jj eXeY
??
?
???
?
???
?
nn
nxny 22 ][][
??
Copyright ? 2001,S,K,Mitra 42
Bounded Real Transfer
Functions
? Thus,for all finite-energy inputs,the output
energy is less than or equal to the input
energy implying that a digital filter
characterized by a BR transfer function can
be viewed as a passive structure
? If,then the output energy is
equal to the input energy,and such a digital
filter is therefore a lossless system
1|)(| ?wjeH
Copyright ? 2001,S,K,Mitra 43
Bounded Real Transfer
Functions
? A causal stable real-coefficient transfer
function H(z) with is thus
called a lossless bounded real (LBR)
transfer function
? The BR and LBR transfer functions are the
keys to the realization of digital filters with
low coefficient sensitivity
1|)(| ?wjeH
