Chapter 10 FIR Digital Filter Design. §10.2.1 Least Integral-Squared Error Design of FIR Filters ...
-
Upload
louisa-martina-fisher -
Category
Documents
-
view
231 -
download
5
Transcript of Chapter 10 FIR Digital Filter Design. §10.2.1 Least Integral-Squared Error Design of FIR Filters ...
Chapter 10
FIR Digital Filter DesignFIR Digital Filter Design
§10.2.1 Least Integral-Squared Error Design of FIR Filters
Let Hd(e jω) denote the desired frequency response
Since Hd(e jω) is a periodic function of ω with a period 2π,it can be expressed as a Fourier series
n
njd
jd enheH ][)(
ndeeHnh njjdd ,)(
2
1][
where
§10.2.1 Least Integral-Squared Error Design of FIR Filters
In general Hd(e jω) is piecewise constant with sharp transitions between bands
In which case {hd[n]} is of infinite length and noncausal
Objective – Find a finite-duration {ht[n]} of length 2M+1 whose DTFT Ht(e jω) approximates the desired DTFT Hd(e jω) in some sence
§10.2.1 Least Integral-Squared Error Design of FIR Filters
Commonly used approximation criterion – Minimize the integral-squared error
deHeH jd
jt
2
)()(2
1
njM
Mnt
jt enheH
][)(
where
§10.2.1 Least Integral-Squared Error Design of FIR Filters
Using Parseval’s relation we can write
M
Mn
M
n Mndddt
dt
nhnhnhnh
nhnh
1
1
222
2
][][][][
][][
It follows from the above that Φis minimum when ht[n]= hd[n] for -M≤n≤M
Best finite-length approximation to ideal infinite-length impulse response in the mean-square sense is obtained by truncation
§10.2.1 Least Integral-Squared Error Design of FIR Filters
A causal FIR filter with an impulse response h[n] can be derived from ht[n] by delaying:
h[n]=ht[n-M] The causal FIR filter h[n] has the same mag
nitude response as ht[n] and its phase response has a linear phase shift of ωM radians with respect to that of ht[n]
§10.2.2 Impulse Responses of Ideal Filters
Ideal lowpass filter –
nn
nnh c
LP ,sin
][
0,sin
0,1][
nn
n
nnh
c
c
LP
Ideal highpass filter –
§10.2.2 Impulse Responses of Ideal Filters
Ideal bandpass filter –
0,
0,)sin()sin(
][12
12
n
nn
nn
n
nhcc
cc
BP
§10.2.2 Impulse Responses of Ideal Filters
Ideal bandstop filter –
0,
0,)sin()sin(
1][
21
12
n
n
nn
nn
nhcc
cc
BS
§10.2.2 Impulse Responses of Ideal Filters
Ideal multiband filter –
Lk
AeH
kk
kj
ML
,...,2,1,
,)(
1
nAAnH Ln
L
ML )sin(
)(][1
1
§10.2.2 Impulse Responses of Ideal Filters
Ideal discrete-time Hilbert transformer –
oddnfor,/2evennfor,0
][
0,0,
)(
nnh
jj
eH
HT
jHT
§10.2.2 Impulse Responses of Ideal Filters
Ideal discrete-time differentiator –
0n,cos
0n,0][
0,)(
nnnh
jeH
DIF
jDIF
§10.2.3 Gibbs Phenomenon Gibbs phenomenon - Oscillatory behavior in
the magnitude responses of causal FIR filters obtained by truncating the impulse response coefficients of ideal filters
§10.2.3 Gibbs Phenomenon As can be seen, as the length of the lowpas
s filter is increased, the number of ripples in both passband and stopband increases, with a corresponding decrease in the ripple widths
Height of the largest ripples remain the same independent of length
Similar oscillatory behavior observed in the magnitude responses of the truncated versions of other types of ideal filters
§10.2.3 Gibbs Phenomenon Gibbs phenomenon can be explained by tre
ating the truncation operation as an windowing operation:
deeHeH jjd
jt )()(
21)( )(
][][][ nwnhnh dt
where Ht(ejω) and Ψ(ejω) are the DTFTs of ht[n] and w[n] , respectively
In the frequency domain
§10.2.3 Gibbs Phenomenon Thus Ht(ejω) is obtained by a periodic continuo
us convolution of Hd(ejω) with Ψ(ejω)
§10.2.3 Gibbs Phenomenon
If Ψ(ejω) is a very narrow pulse centered at ω=0 (ideally a delta function) compared to variations in Hd(ejω) , then Ht(ejω) will approximate Hd(ejω) very closely
Length 2M+1 of w[n] should be very large On the other hand, length 2M+1 of ht[n] shou
ld be as small as possible to reduce computational complexity
§10.2.3 Gibbs Phenomenon
A rectangular window is used to achieve simple truncation:
Presence of oscillatory behavior in Ht(ejω) is basically due to:
– 1) hd[n] nitely long and not absolutely summable, and hence filter is unstable – 2) Rectangular window has an abrupt transition to zero
§10.2.3 Gibbs Phenomenon Oscillatory behavior can be explained by ex
amining the DTFT ΨR(ejω) of hwR[n]:
R(ejω) has a main lobe centered at ω=0
Other ripples are called sidelobes
§10.2.3 Gibbs Phenomenon Main lobe of ΨR(ejω) characterized by its width
4π/(2M+1) defined by first zero crossings on both sides of ω=0
As M increases, width of main lobe decreases as desired
Area under each lobe remains constant while width of each lobe decreases with an increase in M
Ripples in Ht(ejω) around the point of discontinuity occur more closely but with no decrease in amplitude as M increases
§10.2.3 Gibbs Phenomenon
Rectangular window has an abrupt transition to zero outside the range -M≤n≤M, which results in Gibbs phenomenon in Ht(ejω)
Gibbs phenomenon can be reduced either:(1) Using a window that tapers smoothly to
zero at each end, or(2) Providing a smooth transition from passband to stopband in the magnitude specifications
§10.2.4 Fixed Window Functions Using a tapered window causes the height o
f the sidelobes to diminish, with a corresponding increase in the main lobe width resulting in a wider transition at the discontinuity
MnMM
nnw ,)
122
cos(5.05.0][ Hann:
MnMM
nnw ,)
122
cos(46.054.0][ Hamming:
)12
4cos(08.0)
122
cos(5.042.0][ M
nM
nnw
Blackman:
§10.2.4 Fixed Window Functions
Plots of magnitudes of the DTFTs of these windows for M=25 are shown below:
0 0.2 0.4 0.6 0.8 1-100
-80
-60
-40
-20
0
/
Gai
n, d
B
Rectangular window
0 0.2 0.4 0.6 0.8 1-100
-80
-60
-40
-20
0
/
Gai
n, d
B
Hanning window
0 0.2 0.4 0.6 0.8 1-100
-80
-60
-40
-20
0
/
Gai
n, d
B
Hamming window
0 0.2 0.4 0.6 0.8 1-100
-80
-60
-40
-20
0
/
Gai
n, d
BBlackman window
§10.2.4 Fixed Window Functions Magnitude spectrum of each window charac
terized by a main lobe centered at ω= 0 followed by a series of sidelobes with decreasing amplitudes
Parameters predicting the performance of a window in filter design are:
Main lobe width Relative sidelobe level
§10.2.4 Fixed Window Functions
Main lobe width ML - given by the distance between zero crossings on both sides of main lobe
Relative sidelobe level Asl - given by the difference in dB between amplitudes of largest sidelobe and main lobe
§10.2.4 Fixed Window Functions
Observe Thus, Passband and stopband ripples are the same
1)()( )()( cc jt
jt eHeH
5.0)( cjt eH
§10.2.4 Fixed Window Functions
Distance between the locations of the maximum passband deviation and minimum stopband value ML
Width of transition band
= s - p < ML
§10.2.4 Fixed Window Functions
To ensure a fast transition from passband to stopband, window should have a very small main lobe width
To reduce the passband and stopband ripple δ, the area under the sidelobes should be very small
Unfortunately, these two requirements are contradictory
§10.2.4 Fixed Window Functions
In the case of rectangular, Hann, Hamming, and Blackman windows, the value of ripple does not depend on filter length or cutoff frequency c , and is essentially constant
In addition,
c / M
where c is a constant for most practical purposes
§10.2.4 Fixed Window Functions Rectangular window - ML=4/(2M+1)
Asl=13.3 dB, s=20.9 dB, =0.92/M
Hann window - ML=8/(2M+1)
Asl=31.5 dB, s=43.9 dB, =3.11/M
Hamming window - ML=8/(2M+1)
Asl=42.7 dB, s=54.5 dB, =3.32/M,
Blackman window - ML=12/(2M+1)
Asl=58.1 dB, s=75.3 dB, =5.56/M
§10.2.4 Fixed Window Functions
Filter Design Steps -
(1) Set
c =(p + s )/2
(2) Choose window based on specified s
(3) Estimate M using
c / M
§10.2.4 Fixed Window FunctionsFIR Filter Design Example
Lowpass filter of length 51 and c=/2
0 0.2 0.4 0.6 0.8 1
-100
-50
0
/
Gai
n, d
B
Lowpass Filter Designed Using Hann window
0 0.2 0.4 0.6 0.8 1
-100
-50
0
/
Gai
n, d
B
Lowpass Filter Designed Using Hamming window
0 0.2 0.4 0.6 0.8 1
-100
-50
0
/
Gai
n, d
B
Lowpass Filter Designed Using Blackman window
§10.2.4 Fixed Window Functions
FIR Filter Design Example An increase in the main lobe width is associ
ated with an increase in the width of the transition band
A decrease in the sidelobe amplitude results in an increase in the stopband attenuation
§10.2.5 Adjustable Window Functions
Dolph-Chebyshev Window –
MnMMnk
MkT
Mnw
M
kk
,]12
2cos)12
cos(21[12
1][1
)1cosh21cosh(
amplitudelobemainsidelobeofamplitude
1
M
where
1,)coshcosh(1,)coscos(
)( 1
1
xxxx
xT
and
§10.2.5 Adjustable Window Functions
Dolph-Chebyshev window can be designed with any specified relative sidelobe level while the main lobe width adjusted by choosing length appropriately
Filter order is estimated using
)(85.24.16056.2
sN
where is the normalized transition bandwidth, e.g, for a lowpss filter
= s - p
§10.2.5 Adjustable Window Functions
Gain response of a Dolph-Chebyshev window of length 51 and relative sidelobe level of 50 dB is shown below
§10.2.5 Adjustable Window Functions
Properties of Dolph-Chebyshev window: All sidelobes are of equal height Stopband approximation error of filters desig
ned have essentially equiripple behavior For a given window length, it has the smalle
st main lobe width compared to other windows resulting in filters with the smallest transition band
§10.2.5 Adjustable Window Functions
where βis an adjustable parameter and I0(u) is the modified zeroth-order Bessel function of the first kind:
MnMI
MnInw
,
)(})/(1{
][0
20
1
20 ]
!)2/(
[1)(r
r
ru
uI
Kaiser Window –
20
1
20 ]
!)2/(
[1)(r
r
ru
uIIn practice
Note I0(u)>0 for u>0
§10.2.5 Adjustable Window Functions
βcontrols the minimum stopband attenuation of the windowed filter response
βis estimated using
21for,05021for,)21(07886.0)21(5842.0
50for,)7.8(1102.04.0
s
sss
ss
)(285.28
sN
where is the normalized transition bandwidth
Filter order is estimated using
§10.2.5 Adjustable Window Functions
FIR Filter Design Example Specifications: p= , s= , c=40 dB Thus
4.02/)( spc
01.010 20/ ss
3953.31907886.0)19(5842.0 4.0
2886.22)2.0(285.2
32
N
Choose N=24 implying M=1
§10.2.5 Adjustable Window Functions
FIR Filter Design Example Hence ht[n]=sin(0.4n)/ n, -12n12
where w[n] is the n-th coefficient of a length-25 Kaiser window with =3.3953
0 0.2 0.4 0.6 0.8 1-80
-60
-40
-20
0
/
Gai
n, d
B
Kaiser Window
0 0.2 0.4 0.6 0.8 1-80
-60
-40
-20
0
/
Gai
n, d
B
Lowpass filter designed with Kaiser window
§10.2.6 Impulse Responses of FIR Filters with a Smooth Transition
First-order spline passband-to-stopband transition
ps
spc
2/)(
0,)sin()2/sin(2
0,/][
nn
nnn
nnh c
c
LP
§10.2.6 Impulse Responses of FIR Filters with a Smooth Transition
Pth-order spline passband-to-stopband transition
0,)sin(
2/)2/sin(2
0,/][
nn
nPn
Pnn
nh c
c
LP
§10.2.6 Impulse Responses of FIR Filters with a Smooth Transition
Lowpass FIR Filter Design Example
§10.3 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
The FIR filter design techniques discussed so far can be easily implemented on a computer
In addition, there are a number of FIR filter design algorithms that rely on some type of optimization techniques that are used to minimize the error between the desired frequency response and that of the computer-generated filter
§10.3 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
Basic idea behind the computer-based iterative technique
Let H(ejω) denote the frequency response of the digital filter H(z) to be designed approximating the desired frequency response D(ejω), given as a piecewise linear function of ω, in some sense
§10.3 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
Objective - Determine iteratively the coefficients of H(z) so that the difference between H(ejω) and D(ejω) over closed subintervals of 0≤ω≤π is minimized
This difference usually specified as a weighted error function
ε(ω)=W(ejω)[H(ejω)-D(ejω)]where W(ejω) is some user-specified weighting function
§10.3 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
Chebyshev or minimax criterion - Minimizes the peak absolute value of the weighted error:
)(max R
where R is the set of disjoint frequency bands in the range 0≤ω≤π, on which D(ejω) is defined
§10.3 Computer-Aided Design of Equiripple Linear-Phase FIR Filters
The linear-phase FIR filter obtained by minimizing the peak absolute value of
)(max R
is usually called the equiripple FIR filter After ε is minimized, the weighted error funct
ion ε(ω) exhibits an equiripple behavior in the ferquency range R
§10.3.1 the Parks-McClellan Algorithm
The general form of frequency response of a causal linear-phase FIR filter of length 2M+1:
)()( HeeH jMj
where the amplitude response is a real function of ω
)(H
where D(ω) is the desired amplitude response and W(ω) is a positive weighting function
)]()()[()( DHW
Weighted error function is given by
§10.3.1 the Parks-McClellan Algorithm
The Parks-McClellan Algorithm is based on iteratively adjusting the coefficients of until the peak absolute value of ε(ω) is minimized
)(H
baWDH
,
)()()( 0
If peak absolute value of ε(ω) in a band ωa≤ω≤ωb is ε0, then the absolute error satisfies
§10.3.1 the Parks-McClellan Algorithm
For filter design,
is required to satisfy the above desired response with a ripple of ±δp in the passband and a ripple of δs in the stopband
)(H
stopbandthein,0passbandthein,1
)(D
§10.3.1 the Parks-McClellan Algorithm
Thus, weighting function can be chosen either as
stopbandthein,/
passbandthein,1)(
spW
stopbandthein,1passbandthein,/
)( psW
or
§10.3.1 the Parks-McClellan Algorithm
M
k
kkaH0
)cos(][)(
2/)12(
1
)21(cos][)(
M
k
kkbH
MkkMhkaMha 1,][2][,][]0[
2121,]
212[2][ MkkMhkb
where
Type 2 FIR Filter –
where
Type 1 FIR Filter –
§10.3.1 the Parks-McClellan Algorithm
M
k
kkcH1
)sin(][)(
MkkMhkc 1,][2][
2/)12(
1
)21(sin][)(
M
k
kkdH
2121,]
212[2][ MkkMhkd
where
Type 4 FIR Filter –
where
Type 3 FIR Filter –
§10.3.1 the Parks-McClellan Algorithm
Amplitude response for all 4 types of linear- phase FIR filters can be expressed as
)()()( AQH
4Typefor,)2/sin(3Typefor,)sin(2Typefor,)2/cos(1Typefor,1
)(
Q
where
§10.3.1 the Parks-McClellan Algorithm
)cos(][~)(0
kkaAL
k
4Typefor,][~
3Typefor,][~2Typefor,][
~1Typefor,][
][~
kdkckbka
ka
where
and
§10.3.1 the Parks-McClellan Algorithm
4Typefor,2
123Typefor,1
2Typefor,2
121Typefor,
MM
MM
L
are elated to b[k], c[k], and d[k], respectively
,][~
and][~,][~
kdkckb
with
§10.3.1 the Parks-McClellan Algorithm
Modified form of weighted error function
)(/)()(~
)()()(~
QDDQWW
Modified form of weighted error function
)](~
)()[(~
])(
)()()[()(
)]()()()[()(
DAWQ
DAQW
DAQWE
§10.3.1 the Parks-McClellan Algorithm
L
k
DkkaW0
)](~
)cos(][~)[(~
)(
Optimization Problem – Determine which minimize the peak absolute value εof
][~ ka
After has been determined, corresponding coefficients of the original A(ω) are computed from which h[n] are determined
][~ kaover the specified frequency bands ω∈R
§10.3.2 Alternation Theorem The aplitude function A(ω) is the best unique
approximation of obtained by minimizing peak absolute value ω of
)(~ D
)]()()()[()( DAQW
if and only if there exist at least L+2 extremal frequencies, {ωi}, 0≤i≤L+1 in a closed subset R of the frequency range 0≤ω≤π such that ω0<ω1<···ωL
<ωL+1 and ε(ωi)=-ε(ωi+1), |ε(ωi)|=ε for all i
§10.3.1 Design of EquirippleLinear-Phase FIR Filters
Consider a Type 1 FIR filter with an amplitude response A(ω) whose approximation error ε(ω) satisfies the Alternation Theorem
Peaks of E(ω) are at ω=ωi, 0≤i≤L+1 where d E(ω)/dω=0
iddA
ddE
at0
)()(
Since in the passband and stopband, and are piecewise constant,
)(~ D
)(~ W
§10.3.1 Design of EquirippleLinear-Phase FIR Filters
Using cos(ωk)=Tk(cosω), where Tk(x) is the k-th order Chebyshev polynomial
)coscos()( 1 xkxTk
kL
kkA
0
)](cos[)( which is an Lth-order polynomial in cosω
Hence, A(ω) can have at most L-1 local minima and maxima inside specified passband and stopband
A(ω) can be expressed as
§10.3.1 Design of EquirippleLinear-Phase FIR Filters
At bandedges, ω=ωp, and ω=ωs , |ε(ω)| is a maximum, and hence A(ω) has extrem at these points
A(ω) can have extrema at ω=0, and ω=π
Therefore, there are at most L+3 extremal frequencies of ε(ω)
For linear-phase FIR filters with K specified bandedges, there can be at most L+K+1 extremal frequencies
§10.3.1 Design of EquirippleLinear-Phase FIR Filters
The set of equations
10,)1()](~
)()[(~ LiDAW i
ii
)(~
)(~
)(~
)(~
][~
]1[~]0[~
)(~
/)1()cos()cos(1)(
~/)1()cos()cos(1
)(~
/1)cos()cos(1)(
~/1)cos()cos(1
1
1
0
111
1
111
000
L
L
LL
LL
LL
LL
DD
DD
La
aa
WLWL
WLWL
is written in a matrix form
§10.3.1 Design of EquirippleLinear-Phase FIR Filters
The matrix equation can be solved for the unknowns andεif the locations of the L+2 extremal frequencies are known a priori
][~ ka
The Remez exchange algorithm is used to determine the locations of the extremal frequencies
§10.3.2 Remez Exchange Algorithm Step 1: A set of initial values of extremal fre
quencies are either chosen or are available from completion of previous stage
Step 2: Value ofεis computed using
1
0)cos()cos(
1L
nii in
nc
)(~
)1()(
~)(
~
)(~
)(~
)(~
1
11
1
1
0
0
111100
L
LL
LL
Wc
Wc
Wc
DcDcDc
where
§10.3.2 Remez Exchange Algorithm Step 3: Values of A(ω) at ω=ωi are then com
puted using
11,)(~
)(~
)1()( LiD
WA i
i
i
i
Step 4: The polynomial A(ω) is determined by interpolating the above values at the L+2 extremal frequencies using the Lagrange interpolation formula
§10.3.2 Remez Exchange Algorithm
Step 4: The new error function)](
~)()[(
~)( DAWE
is computed at a dense set S(S≥L) of frequencies. In practice S=16L is adequate. Determine the L+2 new extremal frequencies from the values of ε(ω) evaluated at the dense set of frequencies.
Step 5: If the peak values εof ε(ω) are equal in magnitude, algorithm has converged. Otherwise, go back to Step 2.
§10.3.2 Remez Exchange Algorithm
Illustration of algorithm Iteration process is stopped if the difference between the values of the peak absolute errors between two consecutive stages is less than a preset value, e.g., 10-6
§10.3.2 Remez Exchange Algorithm
Example – Approximate the desired y a linear function D(x)=1.1x2-0.1 defined for the range 0≤x≤2 by a linear function a1x+a0 by minimizing the peak value of the absolute error
xaaxx
102
]2.0[1.01.1max
5.1,5.0,0 321 xxx
Stage 1:Choose arbitrarily the initial extremal points
§10.3.2 Remez Exchange Algorithm Solve the three linear equations
3,2,1,)()1(10
xDxaa
3.40.11.0
121111
101
1
0
aa
275.0,65.1,375.0 10 aafor the given extremal points yielding
i.e.,
§10.3.2 Remez Exchange Algorithm Plot of ε1(x)=1.1x2-1.65x+0.275 along with valu
es of error at chosen extremal points shown below
Note: Errors are equal in magnitude and alternate in sign
§10.3.2 Remez Exchange Algorithm Stage 2: Choose extremal points where ε1(x) assumes
its maximum absolute values These are x1=0, x2=0.75, x3=2 New values of unknowns are obtained by sol
ving
3.45188.0
1.0
121175.01
101
1
0
aa
yielding a0=-0.6156, a1=2.2, ε=0.5156
§10.3.2 Remez Exchange Algorithm
Plot of ε2(x)=1.1x2-2.2x+0.5156 along with values of error at chosen extremal points shown below
§10.3.2 Remez Exchange Algorithm Stage 3: Choose extremal points where ε2(x) assumes
its maximum absolute values These are x1=0, x2=1, x3=2 New values of unknowns are obtained by sol
ving
3.40.11.0
121111
101
1
0
aa
yielding a0=-0.65, a1=2.2, ε=0.55
§10.3.2 Remez Exchange Algorithm Plot of ε3(x)=1.1x2-2.2x+0.55 along with value
s of error at chosen extremal points shown below
Algorithm has converged as εis also the maximum value of the absolute error
§10.4 Design of Minimum-PhaseFIR Filters
Linear-phase FIR filters with narrow transition bands are of very high order, and as a result have a very long group delay that is about half the filter order
By relaxing the linear-phase requirement, it is possible to design an FIR filter of lower order thus reducing the overall group delay and the computational cost
§10.4 Design of Minimum-PhaseFIR Filters
A very simple method of minimum-phase FIR filter is described next
Consider an arbitrary FIR transfer function of degree N:
N
n
N
kk
n zhznhzH0 1
1)1(]0[][)(
§10.4 Design of Minimum-PhaseFIR Filters
The mirror-image polynomial to H(z) is given by
The zeros of are thus at z=1/ξk, i.e., are reciprocal to the zeros of H(z) at z=ξk
)(ˆ zH
N
n
N
kk
n
N
zNhznN
zHzzH
0 1
1
1
)/1(][][
)()(ˆ
§10.4 Design of Minimum-PhaseFIR Filters
has zeros exhibiting mirror-image symmetry in the z-plane and is thus a Type 1 linear- phase transfer function of order 2N
)()()(ˆ)()( 1 zHzHzzHzHzG N
Moreover, if H(z) has a zero on the unit circle, will also have a zero on the unit circle at the conjugate reciprocal position
)(ˆ zH
As a result
§10.4 Design of Minimum-PhaseFIR Filters
Thus, unit circle zeros of G(z) occur in pairs On the unit circle we have
Moreover, the amplitude response has double zeros in the frequency range [0,π]
)(G
0)(
)()(2
G
GeH j
§10.4 Design of Minimum-PhaseFIR Filters
Design Procedure – Step 1: Design a Type 1 linear-phase transf
er function F(z) of degree 2N satisfying the specifications:
],[for)(
],0[for1)(1)()(
)()(
sF
pF
s
pF
pF
p
F
F
Note that F(z) has single unit circle zeros
§10.4 Design of Minimum-PhaseFIR Filters
Step 2: Determine the linear-phase transfer function
],[for2)(0
],0[for1)(1)(
)()()()(
sF
s
pF
pF
sF
pF
s
G
G
)()( )( zFzzG NFs
Its amplitude response satisfies
§10.4 Design of Minimum-PhaseFIR Filters
Note that G(z) has double zeros on the unit circle and all other zeros are situated with a mirror-image symmetry
Hence, it can be expressed in the form
G(z)=z-n Hm(z)Hm(z-1)
where Hm(z) is a minimum-phase transfer function containing all zeros of G(z) that are inside the unit circle and one each of the unit circle double zeros
§10.4 Design of Minimum-PhaseFIR Filters
Step 3: Determine Hm(z) from G(z) by applying a spectral factorization
The passband ripple and the stopband ripple of F(z) must be chosen to ensure that the specified passband ripple δp and the stopband rippleδp of Hm(z) are satisfied
)(Fs
)(Fp
§10.4 Design of Minimum-PhaseFIR Filters
It can be shown
s
sFs
s
pFp
12
,11
1 )()(
An estimate of the order N of Hm(z) can be found by first estimating the order of F(z) and then dividing it by 2
If the estimated order of F(z) is an odd integer, it should be increased by 1
§10.5 FIR Digital Filter Design Using MATLAB
Order Estimation – Kaiser’s Formula:
2/)(6.14
)(log20 10
ps
spN
Note: Filter order N is inversely proportional to transition band width (ωp-ωs ) and does not depend on actual location of transition band
§10.5 FIR Digital Filter Design Using MATLAB
Hermann-Rabiner-Chan’s Formula:
2/)(]2/))[(,(),( 2
ps
psspsp FDN
]log[log),(])(log)(log[
log])(log)(log[),(
101021
61052
104
1031022
101
spsp
pp
sppsp
bbFaaa
aaaD
51244.0,01217.11,4278.0,5941.0,00266.0
,4761.0,07114.0,005309.0
21
654
321
bbaaa
aaa
where
with
§10.5 FIR Digital Filter Design Using MATLAB
Formula valid for δp≥δs
For δp<δs , Formula to be used is obtained by interchanging δp and δs
Both formulas provide only an estimate of the required filter order N
Frequency response of FIR filter designed using this estimated order may or may not meet the given specifications
If specifications are not met, increase filter order until they are met
§10.5 FIR Digital Filter Design Using MATLAB
MATLAB code fragments for estimating filter order using Kaiser’s formula
num = - 20*log10(sqrt(dp*ds))-13;
den = 14.6*(Fs - Fp)/FT
N = ceil(num/den); M-file remezord implements Hermann-Rabin
er-Chan’s order estimation formula
§10.5 FIR Digital Filter Design Using MATLAB
For FIR filter design using the Kaiser window, window order is estimated using the M-file kaiserord
The M-file kaiserord can in some cases generate a value of N which is either greater or smaller than the required minimum order
If filter designed using the estimated order N does not meet the specifications, N should either be gradually increased or decreased until the specifications are met
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
The M-file remez can be used to design an equiripple FIR filter using the Parks- McClellan algorithm
Example – Design an equiripple FIR filter with the specifications: Fp=0.8kHz, Fs=1kHz, FT=4kHz, αp=0.5dB, αs=40dB
Here δp=0.0559 and δs=0.01
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
MATLAB code fragments used are
[N, fpts, mag, wt]=
remezord(fedge, mval, dev, FT);
b = remez(N, fpts, mag, wt);
where fedge = [800 1000],
mval = [1 0], dev = [0.0559 0.01],and FT=400
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
The computed gain response with the filter order obtained (N=28) does not the specifications (αp=0.6dB, αs=38.7dB)
Specifications are met with N=30
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Example – Design a linear-phase FIR bandpass filter of order 26 with a passband from 0.3 to 0.5, and stopbands from 0 to 0.25 and from 0.55 to 1
The pertinent input data here areN = 26fpts = [0 0.25 0.3 0.5 0.55 1]mag = [0 0 1 1 0 0]wt = [1 1 1]
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Computed gain response shown below where αp=1dB, αs=18.7dB
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
We redesign the filter with order increased to 110
Computed gain response shown below where αp=0.024dB, αs=51.2dB
Note: Increase in order improves gain response at the expense of increased computational complexity
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
αs can be increased at the expenses of a larger αp by decreasing the relative weight ratio W(ω)=αp/αs Gain response of bandpass filter of order 110 obtained with a weight vector[1 0.1 1]
Now αp=0.076dB, αs=60.86dB
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
As L=13, and thereare 4 band edges, there can be at most L-1+6=18 extrema
Error plot exhibits 17 extrema
Absolute error has same peak value all bands
Plots of absolute error for 1st design
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Absolute error has same peak value in all bands for the 2nd design
Absolute error in passband of 3rd design is 10 times the error in the stopbands
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Example – Design a linear-phase FIR bandpass filter of order 60 with a passband from 0.3 to 0.5, and stopbands from 0 to 0.25 and from 0.6 to 1 with unequal weights
The pertinent input data here areN = 60fpts = [0 0.25 0.3 0.5 0.6 1]mag = [0 0 1 1 0 0]wt = [1 1 0.3]
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Plots of gain response and absolute error shown below
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Response in the second transition band shows a peak with a value higher than that in passband
Result does not contradict alternation theorem
As N = 60, M = 30, and hence, there must be at least M + 2 = 32 extremal frequencies
Plot of absolute error shows the presence of 32 extremal frequencies
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
If gain response of filter designed exhibits a nonmonotonic behavior, it is recommended that either the filter order or the bandedges or the weighting function be adjusted until a satisfactory gain response has been obtained
Gain plot obtained by moving the second stopband edge to 0.55
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
FIR Differentiator Design Examples A lowpass differentiator has a bandlimited frequenc
y response
s
pjDIF
jeH
,00,
)(
where 0≤|ω|≤ωp represents the passband and ωs≤|ω|≤π represents the stopband
For the design phase we choose
W(ω)=1/ω, D(ω)=1, 0≤|ω|≤ωp
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
The M-file remezord cannot be used to estimate the order of an FIR differentiator
Example - Design a full-band (ωp=π) differentiator of order 11
Code fragment to useb = remez(N, fpts, mag, ‘differentiator’);
where N = 11
fpts = [0 1]
mag = [0 pi]
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Plots of magnitude response and absolute error
Absolute error increases with ω as the algorithm result in an equiripple error of the function [A(ω)/ ω-1]
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Example – Design a lowpass differentiator of order 50 with ωp=0.4π, and ωs=0.45π
Code fragment to use
b = remez(N, fpts, mag, ‘differentiator’);
where
N = 50
fpts = [0 0.4 0.45 1]
mag = [0 0.4*pi 0 0]
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Plot of the magnitude response of the lowpass differentiator
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
FIR Hilbert Transformer Design Examples Desired amplitude response of a bandpass Hil
bert transformer is
D(ω)=1, ωL≤|ω|≤ωH
with weighting function
W(ω)=1, ωL≤|ω|≤ωH Impulse response of an ideal Hilbert transform
er satisfies the condition
hHT[n]=0, for n evenwhich can be met by a Type 3 FIR filter
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Example - Design a linear-phase bandpass FIR Hilbert transformer of order 20 with ωL=0.
1π, ωH =0.9π Code fragment to use
b = remez(N, fpts, mag, ‘Hilbert’);where
N = 20fpts = [0.1 0.9]
§10.5.2 Equiripple FIR Digital FilterDesign Using MATLAB
Plots of magnitude response and absolute error
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
The minimum-phase FIR filter design method outlined earlier involves the spectral factorization of a Type 1 linear-phase FIR transfer function G(z) with a non-negative amplitude response in the form
G(z)=z-NHm(z)Hm(z-1)
where Hm(z) contains all zeros of G(z) that are inside the unit circle and one each of the unit circle double zeros
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
Spectral Factorization We next outline the basic idea behind a sim
ple spectral factorization method Without any loss of generality we consider t
he spectral factorization of a 6-th order linear-linear phase FIR transfer function G(z) with a non-negative amplitude response:
G(z)=g3+g2z-1+g1z-2+g0z-3+g1z-4+g2z-5+g3z-6
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
Our objective is to express the above G(z) in the form
G(z)=z-3Hm(z)Hm(z-1)
where
Hm(z)=a0+a1z-1+a2z-2+a3z-3
is the minimum-phase factor of G(z)
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
Expressing G(z) in terms of the coefficients of Hm(z) we get
G(z)=(a0+a1z-1+a2z-2+a3z-3) ×
(a3+a2z-1+a1z-2+a0z-3) Forming the product of the two polynomials
given above and comparing the coefficients of like powers of z-1 the product with that of G(z) given on the previous slide we arrive at 4 equations given in the next slide
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
The above set of equations is then solved iteratively using the Newton-Raphson method
303
31202
3221101
23
22
21
200
aagaaaag
aaaaaagaaaag
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
First, the initial values of ai are chosen to ensure that Hm(z) has all zeros strictly inside the unit circle
Then, the coefficients ai are changed by adding the corrections ei so that the modified values ai +ei satisfy better the set of 4 equalities given in the previous slide
The process is repeated until the iteration converges
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
Substituting ai +ei in the 4 equations given earlier and expanding the products, a set of linear equations are obtained by eliminating all quadratic terms in ei from the expansion
In matrix form, these equations can be written as Ae=b where
03
1032
213201
3210
00
2222
aaaaaaaaaaaaaaaa
A
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
302
31202
3221101
23
22
21
200
3
2
1
0
,
aagaaaag
aaaaaagaaaag
b
eeee
e
0
10
210
3210
3
32
321
3210
00000
0
000000
aaaaaaaaaa
aaa
aaaaaaa
A
and
The matrix A can be expressed
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
The iteration convergence is checked at each step by evaluating the error term
3
0
2
i ie
The error term first decreases monotonically and the iteration is stopped when the error starts increasing
The M-file minphase.m implements the above spectral factorization method
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
Example – Design a minimum-phase lowpass FIR filter with the following specifications: ωp=0.45π, ωs=0.6π,
Rp=2 dB and Rs=26 dB Using Program 10_3.m we arrive at the desi
red filter Plots of zeros of G(z), zeros of Hm(z), and the
gain response of Hm(z) are shown in the next slide
§10.5.3 Minimum-Phase FIR FilterDesign Using MATLAB
§10.5.3 Maximum-Phase FIR FilterDesign Using MATLAB
A maximum-phase spectral factor of a linear-phase FIR filter with an impulse response b of even order with a non- negative zero-phase frequency response can be designed by first computing its minimum-phase spectral factor h and the using the statement
G = fliplr(h)
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Window Generation – Code fragments to usew = blackman(L); w = hamming(L); w = hanning(L);w = chebwin(L, Rs);w = kaiser(L, beta);where window length L is odd
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Example – Kaiser window design for use in a lowpass FIR filter design
Specifications of lowpass filter: ωp=0.4π, ωs=0.45π, αs=50 dB, → δs=0.003162
Code fragments to use[N, Wn, beta, ftype] = kaiserord(fpts, mag,dev);w = kaiser(N+1, beta); where
fpts = [0.3 0.4] mag = [1 0]dev = [0.003162 0.003162]
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Plot of the gain response of the Kaiser window
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
M-files available are fir1 and fir2 fir1 is used to design conventional lowpass,
highpass, bandpass, bandstop and multiband FIR filters
fir2 is used to design FIR filters with arbitrarily shaped magnitude response
In fir1, Hamming window is used as a default if no window is specified
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Example – Design using a Kaiser window a lowpass FIR filter with the specifications: ωp=0.3π, ωs=0.4π, δs=0.003162
Code fragments to use[N, Wn, beta, ftype] = kaiserord(fpts, mag, dev);b = fir1(N, Wn, kaiser(N+1, beta));where
fpts = [0.3 0.4]mag = [1 0]dev = [0.003162 0.003162]
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Plot of gain response
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Example – Design using a Kaiser window a highpass FIR filter with the specifications: ωp=0.55π, ωs=0.4π, δs=0.02
Code fragments to use[N, Wn, beta, ftype] = kaiserord(fpts, mag, dev); b = fir1(N, Wn, ‘ftype’, kaiser(N+1, beta));where
fpts = [0.4 0.55]mag = [0 1]dev = [0.02 0.02]
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Plot of gain response
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Example – Design using a Hamming window an FIR filter of order 100 with three different constant magnitude levels:
0.3 in the frequency range [0, 0.28], 1.0 in the frequency range [0.3, 0.5], and 0.7 in the frequency range [0.52, 1.0]
§10.5.4 Window-Based FIR FilterDesign Using MATLAB
Code fragment to useb = fir2(100, fpts, mval);where fpts = [0 0.28 0.3 0.5 0.52 1];
mval = [0.3 0.3 1.0 1.0 0.7 0.7];
§10.6 Design of ComputationallyEfficient FIR Digital Filters
As indicated earlier, the order N of a linear- phase FIR filter is inversely proportional to the width ∆ω of the transition band
Hence, in the case of an FIR filter with a very sharp transition, the order of the filter is very high
This is particularly critical in designing very narrow-band or very wide-band FIR filters
§10.6 Design of ComputationallyEfficient FIR Digital Filters
The computational complexity of a digital filter is basically determined by the total number of multipliers and adders needed to implement the filter
The direct form implementation of a linear- phase FIR filter of order N requires, in general, multipliers and N two-input adders
21N
§10.6 Design of ComputationallyEfficient FIR Digital Filters
We now outline two methods of realizing computationally efficient linear-phase FIR filters
The basic building block in both methods is an FIR subfilter structure with a periodic impulse response
§10.6.1 The Periodic Filter Section
Consider a Type 1 linear-phase FIR filter F(z) of even degree N:
N
n
nznfzF0
][)(
N
Nnn
nN
N
n
nNN
znfzNf
znfzzFzzE
2/0
2/
0
2/2/
][])2/[1(
][)()(
Its delay-complementary filter E(z) is given by
§10.6.1 The Periodic Filter Section
The transfer function H(z) obtained by replacing z-1 in F(z) with z-L, with L being a positive integer, is given by
N
n
nLL znfzFzH0
][)()(
The order of H(z) is thus NL A direct realization of H(z) is obtained by sim
ply replacing each unit delay in the realization of F(z) with L unit delays
§10.6.1 The Periodic Filter Section
Note: The number of multiplers and delays in the realization of H(z) is the same as those in the realization of F(z)
The transfer function H(z) has a sparse impulse response of length NL+1, with L-1 zero-valued samples inserted between every consecutive pair of impulse response samples of F(z)
§10.6.1 The Periodic Filter Section
The parameter L is called the sparsity factor The relation between the amplitude respons
es of these two filters is given by)()( LFH
It follows from the above that the amplitude response is a period function of ω with a period 2π/L
)(H
§10.6.1 The Periodic Filter Section
One period of is obtained by compressing the amplitude response in the interval [0, 2π] to the interval [0, 2π/L]
)(H
)(F
A transfer function H(z) with a frequency response that is a periodic function of ω with a period 2π/L is called a periodic filter
§10.6.1 The Periodic Filter Section
If F(z) is a lowpass filter with a single pasband and a single stopband, H(z) will be a multiband filter with pasbands and stopbands as shown in the next slide for L=4
12/ L
2/L
§10.6.1 The Periodic Filter Section
§10.6.1 The Periodic Filter Section
Then, the passband and stopband edges of the first band of H(z) are at and respectively
LFp /)( LF
p /)(
The passband and stopband edges of the second band of H(z) are at
and , respectively, and so on as shown on the previous slide
LFp /)2( )(
LFp /)2( )(
Let F(z) be a lowpass filter with passband edge at and stopband edge at ,where
)(Fs
)(Fs
)(Fp
§10.6.1 The Periodic Filter Section
Likewise, the transfer function G(z) by replacing z-1 in E(z) with z-L, is given by
The width of the transition bands of H(z) are , which is 1/L-th of that of F(z)LF
pF
s /)( )()(
N
n
nLNL
LNLL
znfz
zFzzEzG
0
2/
2/
][
)()()(
)()(1)( LFHzG
The amplitude response of G(z) is given by
§10.6.2 Interpolated FIR Filter The overall filter HIFIR(z) is designed as a cas
cade of a linear-phase FIR filter F(zL) and another filter I(z) that suppreses the undesired passbands of the periodic filter section as shown below
The widths of the transition band and the passband of the cascade are 1/L-th of those of F(z)
F(zL) I(z)
periodic filter interpolator
§10.6.2 Interpolated FIR Filter
The cascaded structure is called the interpolated finite impulse response (IFIR) filter, as the missing impulse response samples of the periodic filter section are being interpolated by the filter section I(z), called the interpolator
As the filter F(z) determines approximately the shape of the amplitude response of the IFIR filter, it is called a shaping filter
§10.6.2 Interpolated FIR Filter Design Steps – IFIR specifications: passband edge ωp, stopb
and edge ωs, passband rippleδp, stopband ripple δs
Shaping filter specifications:
passband edge
stopband edge
passband ripple
stopband ripple
pF
p L )(
sF
s L )(
2/)(p
Fp
sF
s )(
§10.6.2 Interpolated FIR Filter The interpolator I(z) has to be designed to pr
eserve the passband of F(zL) in the frequency range [0, ωp] and mask the amplitude response of F(zL) in the frequency range [ωs ,π] , where the periodic subfilter has unwanted passbands and transition bands
This latter region is defined by
2/
1
),2
min(,2L
kss L
kL
kR
§10.6.2 Interpolated FIR Filter
The transition band of the interpolator is the frequency range [ωp , 2π/L-ωs]
Figure below shows the responses of HIFIR(z) and I(z)
§10.6.2 Interpolated FIR Filter Summarinzing, the design specifications for
F(z) and I(z) are as follows:
RI
ILF
LF
Is
Is
pI
pI
p
sF
sF
s
pF
pF
p
for)(
],0[for1)(1],[for)(
],0[for1)(1
)()(
)()(
)()(
)()(
The two linear-phase FIR filters F(z) and I(z) can be designed using the Parks- McClellan method
§10.6.2 Interpolated FIR Filter Example – Filter specifications are as follow
s: ωp=0.15π, ωs=0.2π, δp=0.002, δs=0.001 It follows from the figure in Slide 101 that to
ensure no overlaps between adjacent passbands of F(zL), we should choose L to satisfy the condition
LL
Fs
Fs
)()( 2
§10.6.2 Interpolated FIR Filter
For our example, this reduces to
0.2π< 2π/L- 0.2π Hence, the largest value of L that can be us
ed is L=4, yielding an IFIR structure requiring the least number of multipliers
As a result, the specifications for F(z) and I(z) are as given in the next slide
§10.6.2 Interpolated FIR Filter
The filter orders of F(z) and I(z) obtained using remezord are: Order of F(z)=32 Order of I(z)=43
001.0,001.0
8.0,6.0)()(
)()(
Fs
Fp
Fs
Fp
F(z):
001.0,001.0
3.0,15.0)()(
)()(
Is
Ip
Is
Ip
I(z):
§10.6.2 Interpolated FIR Filter
It can be shown that the filters F(z) and I(z) designed using remez with the above orders do not lead to an IFIR design meeting the minimum stopband attenuation of 60 dB
To meet the stopband specifications, the orders of F(z) and I(z) need to be increased to 33 and 46, respectively
§10.6.2 Interpolated FIR Filter The pertinent gain responses of the redesign
ed IFIR filter are shown below:
172/)133( FR
The number of multipliers needed to implement F(z) and hence, F(z4) is
§10.6.2 Interpolated FIR Filter The number of multipliers needed to implem
ent I(z) is:
242/)146( IR
412417 IFIRR
622/)1122(
The number of multipliers needed to implement the direct single-stage implementation of the FIR filter is
As a result, the total number of multipliers needed to implement HIFIR(z) is
§10.6.3 Frequency-Response Masking Approach
This approach makes use of the relation between a periodic filter H(z)=F(zL) generated from a Type 1 linear-phase FIR filter of even degree N and its delay- complementary filter G(z) given by
)()()( 2/2/ LNN zFzzHzzG
The amplitude responses of F(z), its delay- complentary filter E(z), the periodic filter H(z) and its delay-complentary filter G(z) are shown in the next slide
§10.6.3 Frequency-Response Masking Approach
§10.6.3 Frequency-Response Masking Approach
By selectively masking out the unwanted pasbands of both H(z) and G(z) by cascading each with appropriate masking filters I1(z) and I2(z), respectively, and connecting the resulting cascades in prallel, we can design a large class of FIR filters with sharper transition bands
The overall structure is then realized as indicated in the next slide
§10.6.3 Frequency-Response Masking Approach
Note: The delay block z-NL/2 can be realized by tapping the FIR structure implementing F(zL)
Also, I1(z) and I2(z) can share the same delay-chain if they are realized using the transposed direct form structure
F(zL) I1(z)
z-LN/2 I2 (z)
●
●
§10.6.3 Frequency-Response Masking Approach
The transfer function of the overall structure is given by
)()]([)()()()()()()(
22/
1
21
zIzFzzIzFzIzGzIzHzH
LNLLFM
)()](1[)()()( 21 ILFILFHFM
The corresponding amplitude response is
§10.6.3 Frequency-Response Masking Approach
The overall computational complexity is given by the complexities of F(z), I1(z) and I2(z)
All these three filters have wide transition bands and, in general, require considerably fewer multipliers and adders than that required in a direct design of the
§10.6.3 Frequency-Response Masking Approach
Design Objective – Given the specifications of HFM(z) , determine the specifications of F(z), I1(z) and I2(z) design these 3 filters
Design method – Illustrated for lowpass filter design
Two different situations may arise depending on how the transition band of HFM(z) is created
§10.6.3 Frequency-Response Masking Approach
Case A – Transition band of HFM(z) is from one of the transition bands of H(z)
§10.6.3 Frequency-Response Masking Approach
Bandedges of HFM(z) are related to the bandedges of F(z) as follows:
10,2
,2
)()(
LLL
Fp
s
Fs
p
§10.6.3 Frequency-Response Masking Approach
Case B – Transition band of HFM(z) is from one of the transition bands of G(z)
§10.6.3 Frequency-Response Masking Approach
Bandedges of HFM(z) are related to the bandedges of F(z) as follows:
LL
Fp
s
Fp
p
)()( 2,
2
Example – Specifications for a lowpass filter:ωp=0.4π, ωs=0.402π, δp=0.01 and δs=0.0001
§10.6.3 Frequency-Response Masking Approach
For designing HFM(z) the optimum value of L is in the range
By calculating the total number of multipliers needed to realize F(z), I1(z) and I2(z) for all possible values of L, we arrive at the realization requiring the least number of multipliers obtained for L =16 is 229 which is about 15% of that required in a direct single-stage realization
§10.6.3 Frequency-Response Masking Approach
The gain response of the designed filter is shown below: