signal - Semantic Scholar
SIGNAL
PROCESSING Signal Processing
60 (1997)
127-130
Computationally efficient weighted least-squares design of FIR filters satisfying prescribed magnitude and phase specifications Sunder S. Kidambi* Analog Devices Inc., Ray Stata Technology Center, 804 Woburn Street, Wilmington, MA 01887, USA Received
1 February
1996; revised
10 March 1997
Abstract An efficient method for the design of FIR filters satisfying prescribed magnitude and phase specifications is described. The method involves formulating an objective function in a quadratic form. The filter coefficients are obtained by solving a system of linear equations which involves either a Toeplitz matrix or a Toeplitz-plus-Hankel matrix. The computational complexity associated with such systems is only O(N*). 0 1997 Elsevier Science B.V. Zusammenfassung Eine wirksame Methode, urn FIR Filter unter vorgeschriebenen Amplituden- und Phasenspezifikationen zu entwickeln, wird beschrieben. Die Methode beinbaltet die Formulierung einer quadratischen Objektfunktion. Die Filterkoetiienten erhIilt man durch L&en von linearen Gleichungen, wobei die Matrix entweder eine Toeplitz- oder eine Toeplitz-plus-Hankel-Struktur besitzt. Der Rechenaufwand, den solche Gleichungssysteme benBtigen, hat eine Gri%enordmmg von O(N*). 0 1997 Elsevier Science B.V.
Nous d&ivons dans cet article une mCthode efficiente de conception de filtres FIR satisfaisant des spicifications d’amplitude et de phase. Cette methode implique la formulation d’une fonction objectif de forme quadratique. Les coefficients du filtre sont obtenus par &olution d’un systtme d’bquations lineaires correspondant g une matrice de type Toeplitz ou Toeplitzplus-Hankel. La complexit en calcul associ6e g de tels systkmes est seulement O(N*). 0 1997 Elsevier Science B.V. Keywords:
FIR filter; Least squares; Efficient
1. Introduction For narrow transition-band specifications, linearphase finite impulse response (FIR) filters produce large group delays. On the other hand, minimum* Tel.: 617937 1195; fax: 617937 e-mail: [email protected].
1022;
phase FIR filters can be designed to achieve a lower group-delay, but these do not provide a constant group-delay for the frequencies of interest. In such applications, there is a need to design filters having
an approximately constant group-delay which is less than that produced by linear-phase filters. In many phase equalization problems, filters with constant magnitude and nonlinear phase characteristics are
0161%1684/97/$17.00 @ 1997 Elsevier Science B.V. All rights resen led. PII SO165- I684(97)00066-2
S.S. Kidambi / Signal Processing 60 (1997)
128
needed. In all these situations, FIR filters can be designed by approximating both magnitude and phase specifications. Weighted least-squares (WLS) techniques have been successfully used in the design of FIR filters satisfying prescribed magnitude and phase specifications having equiripple magnitude characteristics [ 1,3]. In these techniques, a suitable frequency-dependent weighting function that yields an equiripple design is determined using iterative methods. In all these techniques, computational complexity involved in obtaining the filter coefficients is 0(N3). In this correspondence, we present an efficient WLS method for the design of FIR filters meeting prescribed magnitude and phase specifications. It is shown that the filter coefficients can be obtained by solving a sytem of linear equations involving a Toeplitz-symmetric and positive-definite matrix. The solution of such a system of linear equations entails only O(N2) complexity. In the design of allpass filters with a symmetric response about one-quarter the sampling frequency, the coefficients are obtained by solving a system of linear equations that involves a Toeplitz-plus-Hankel matrix. Again, the complexity of such a system of equations is 0(N2). Evidently, our method has an order of magnitude lower computational complexity than the techniques of [ 1,3]. Our method, however, can be used in conjunction with the iterative methods of [ 1,3] to obtain the suitable frequency-dependent weighting function that yields an equiripple design.
127-130
where h=[h(O)h(l)h(2)
. . . h(N - l)]T,
c(o) = [l cos(w) cos(2w) . . . cos((N - l)~)]~, s(w) = [0 sin(o) sin(2o) . . . sin((N - l)~)]~. The phase response of the filter is given by 4(W)=-tan-’
$$$ (
(2)
>
and the group-delay is given by (3) The desired frequency response D(ej”) having an amplitude response M(o) and a phase response p(w) is given by D(e’O)=M(o)e’p(o)=MR(0)+jMI(O)
(4)
and (5) is the desired group-delay response. The mean-square error between D(ej”) and H(ejO) can be expressed as Emse= 2 W(o,)]D(ejWI) - H(ejOl)]’ I=1
2. Error function formulation and minimization The frequency response of an FIR digital filter with N taps specified by a real-valued impulse response h(n) is given by N-l H(ejw)
=
C
h(n)e-jnw
n=O N-l
= C
WI(w) + hTo4))21~
(6)
where W(o) is a nonnegative frequency-dependent weighting fnnction and M is the number of points at which D(ej”) is sampled. In minimizing E,,,, we set aE,,,,,/ah = 0 to obtain a system of linear equations Qh = d, where
N-l
h(n) cos(no) - j C
n=O
=
+
hTC(o)-
h(n) sin(no)
n=O
jhTs(co),
(1)
M d= c I=1
W(~l)(MR(~f)C(WI)
-
Mr(W)dW)),
(8)
S.S. Kiaizmbi/ Signal Processing 60 (1997)
where Q(w) = c(o)cT(o) and Q(o) = s(o)sT(o). Since Ms( w ) and Mr( o) are both zero in the stopband of the of the desired characteristics, the above equation for d can be written as
127-130
129
about x/2. It follows that [6] h(y
-,)=h(y
+.)
= 0
when n is odd.
(14)
where MP is the number of sample points in the passband of D(ejO). The entries of Q and d are given as
It must be mentioned that the number of filter taps N is odd. In designing such allpass filters, Q and d are given as in (7) and (9) except that the rows and columns corresponding to the zero-valued coefficients are deleted.
Q(n,m)=&W,)cos((n /=I
2.2. Symmetric phase characteristics
0)
=2
-mkJr),
W(wrW(W)cos(nW
+ p(wr))
(10)
(11)
When b(o) in (13) is symmetric about x/2, the following conditions hold [6]:
I=1
for 0 < n, m < N - 1. It can be seen that Q is real, Toeplitz-symmetric and positive definite and hence only the Iirst row (or column) of entries has to be evaluated. Consequently, the system of linear equations can be solved by the computationally efficient and robust Levinson method [4]. It can also be noted that the entries of Q are independent of D(+). For the case of allpass phase equalizers, the desired characteristic can be expressed as
h(y-n)=h(y+n)
whenniseven,
h(&$-n)=-h(y+n)
whennisodd.
(15) From (1 ), the allpass phase equalizer can be characterized by ej((N-l)12)~~(ejw)
o(ej”)
= ejP(w) = cos(~(w)) +j sin@(w)),
w E [O,nl,
(12)
u =Ca(n)cos(2nw)
V
+ jCb(n)sin((2n
where p(o) is the desired phase response. Again, a mean-square error Emse can be minimized and system of linear equations can be obtained. If, however, p(w) is antisymmetric or symmetric with respect to x/2, the computational complexity of Q and d can be reduced significantly since only half the number of coefficients need be determined. Below, we shall consider the design of two allpass phase equalizers presented in [6] using our method.
- l)o),
n=l
n=O
(16) where
n = 0, a(n) = --
n=1,2 ,..., U,
,
2. I. Antisymmetric phase characteristics The desired phase characteristic can be written as p(o)=
--
N-l 2
b(n)=2h 0+&w),
(13)
where the first term on the right-hand side is the linear phase term and b(o) is a function of o antisymmetric
NG
- 2n+l
,
n=1,2
,..., V.
( Consequently (16) can be written as ej((N-i)/2)oH(ejo)
=ar;(o)
+jbrqo),
(17)
S.S. Ki&mbi/ Signal Processing 60 (1997) 127-130
130
where u = [a(O) a( 1) . . . a( U)]T, b=[b(l)b(2)
. . . b(V)]T,
2((w)= [l cos(2w) . . . cos(2Uo)]T, $0) = [sin(w) sin(3w) . . . sin((2V - l)o)lT. Since the coefficients associated with the real and imaginary parts of (16) differ in number, they are.computed separately. The mean-square error associated with the real part can be written as E, = 5
W(ol)(cos(fi(or))
- aTP(or))2 do.
(18)
I=1
By setting aE,liYa = 0, we get &z = &,,, where 0, = 5 IWMW)~T(W) I=1
(19)
and a, = 5
W(w,)cos(fi(w,))@(o/).
An efficient WLS method for the design of FIR filters satisfying prescribed magnitude and phase specifications is described. By incorporating symmetry and antisymmetry constraints of the impulse response, linear-phase FIR filters can easily be designed thus making our method general for the design of FIR filters.
References
Similarly, the mean-square error associated with the imaginary part can be written as W(wr)(sin(j?(w,)) - bT$(~l)).
(21)
I=1
Again, by setting dEbjab = 0, we get &b = & where & is given by
(22) I=1
and & = 5
3. Conclusion
(20)
l=l
Eb = 5
Hankel matrix. Such a system of linear equations can be efficiently solved using the methods presented in [5,2]. These methods have a computational complexity of O(N2). In order to design a filter having an equiripple magnitude response, appropriate weighting function, W(o), has to be used in the minimization of the mean-square error. Since there are no analytical methods to obtain the appropriate W(o), iterative techniques presented in [ 1,3] can be used.
W(o,) sin(&ol))&wl).
(23)
k=O
It has been shown in [7] that matrices & and &, can be decomposed into a sum of a Toeplitz and a
[I] C.-Y. Chi, S.-L. Chiou, A new WLS Chebyshev approximation method for the design of FIR digital filters with arbitrary complex frequency response, Signal Processing 29 (1992) 335-347. [2] I. Gohberg, I. Koltracht, Efficient algorithm for Toeplitz-plusHankel matrices, Integ. Equations Oper Theory 12 (1989) 136-142. [3] Y.C. Lim, J.H. Lee, C.K. Chen, R.H. Yang, A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design, IEEE Trans. Signal Process. 40 (1992) 551-558. [4] S.L. Marple Jr., Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, NJ, 1987. [5] GA. Merchant, T.W. Parks, Efficient solution of a Toeplitzplus-Hankel coefficient matrix system of equations, IEEE Trans. Acoust. Speech Signal Process. ASSP-30 (1982) 40-44 [6] K. Steiglitz, Design of FIR digital phase networks, IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1981) 171-176. [7] S. Sunder, An’efficient weighted least-squares design of linearphase nonrecursive filters, IEEE Trans. Circuits Systems 42 (1995) 359-361.
PROCESSING Signal Processing
60 (1997)
127-130
Computationally efficient weighted least-squares design of FIR filters satisfying prescribed magnitude and phase specifications Sunder S. Kidambi* Analog Devices Inc., Ray Stata Technology Center, 804 Woburn Street, Wilmington, MA 01887, USA Received
1 February
1996; revised
10 March 1997
Abstract An efficient method for the design of FIR filters satisfying prescribed magnitude and phase specifications is described. The method involves formulating an objective function in a quadratic form. The filter coefficients are obtained by solving a system of linear equations which involves either a Toeplitz matrix or a Toeplitz-plus-Hankel matrix. The computational complexity associated with such systems is only O(N*). 0 1997 Elsevier Science B.V. Zusammenfassung Eine wirksame Methode, urn FIR Filter unter vorgeschriebenen Amplituden- und Phasenspezifikationen zu entwickeln, wird beschrieben. Die Methode beinbaltet die Formulierung einer quadratischen Objektfunktion. Die Filterkoetiienten erhIilt man durch L&en von linearen Gleichungen, wobei die Matrix entweder eine Toeplitz- oder eine Toeplitz-plus-Hankel-Struktur besitzt. Der Rechenaufwand, den solche Gleichungssysteme benBtigen, hat eine Gri%enordmmg von O(N*). 0 1997 Elsevier Science B.V.
Nous d&ivons dans cet article une mCthode efficiente de conception de filtres FIR satisfaisant des spicifications d’amplitude et de phase. Cette methode implique la formulation d’une fonction objectif de forme quadratique. Les coefficients du filtre sont obtenus par &olution d’un systtme d’bquations lineaires correspondant g une matrice de type Toeplitz ou Toeplitzplus-Hankel. La complexit en calcul associ6e g de tels systkmes est seulement O(N*). 0 1997 Elsevier Science B.V. Keywords:
FIR filter; Least squares; Efficient
1. Introduction For narrow transition-band specifications, linearphase finite impulse response (FIR) filters produce large group delays. On the other hand, minimum* Tel.: 617937 1195; fax: 617937 e-mail: [email protected].
1022;
phase FIR filters can be designed to achieve a lower group-delay, but these do not provide a constant group-delay for the frequencies of interest. In such applications, there is a need to design filters having
an approximately constant group-delay which is less than that produced by linear-phase filters. In many phase equalization problems, filters with constant magnitude and nonlinear phase characteristics are
0161%1684/97/$17.00 @ 1997 Elsevier Science B.V. All rights resen led. PII SO165- I684(97)00066-2
S.S. Kidambi / Signal Processing 60 (1997)
128
needed. In all these situations, FIR filters can be designed by approximating both magnitude and phase specifications. Weighted least-squares (WLS) techniques have been successfully used in the design of FIR filters satisfying prescribed magnitude and phase specifications having equiripple magnitude characteristics [ 1,3]. In these techniques, a suitable frequency-dependent weighting function that yields an equiripple design is determined using iterative methods. In all these techniques, computational complexity involved in obtaining the filter coefficients is 0(N3). In this correspondence, we present an efficient WLS method for the design of FIR filters meeting prescribed magnitude and phase specifications. It is shown that the filter coefficients can be obtained by solving a sytem of linear equations involving a Toeplitz-symmetric and positive-definite matrix. The solution of such a system of linear equations entails only O(N2) complexity. In the design of allpass filters with a symmetric response about one-quarter the sampling frequency, the coefficients are obtained by solving a system of linear equations that involves a Toeplitz-plus-Hankel matrix. Again, the complexity of such a system of equations is 0(N2). Evidently, our method has an order of magnitude lower computational complexity than the techniques of [ 1,3]. Our method, however, can be used in conjunction with the iterative methods of [ 1,3] to obtain the suitable frequency-dependent weighting function that yields an equiripple design.
127-130
where h=[h(O)h(l)h(2)
. . . h(N - l)]T,
c(o) = [l cos(w) cos(2w) . . . cos((N - l)~)]~, s(w) = [0 sin(o) sin(2o) . . . sin((N - l)~)]~. The phase response of the filter is given by 4(W)=-tan-’
$$$ (
(2)
>
and the group-delay is given by (3) The desired frequency response D(ej”) having an amplitude response M(o) and a phase response p(w) is given by D(e’O)=M(o)e’p(o)=MR(0)+jMI(O)
(4)
and (5) is the desired group-delay response. The mean-square error between D(ej”) and H(ejO) can be expressed as Emse= 2 W(o,)]D(ejWI) - H(ejOl)]’ I=1
2. Error function formulation and minimization The frequency response of an FIR digital filter with N taps specified by a real-valued impulse response h(n) is given by N-l H(ejw)
=
C
h(n)e-jnw
n=O N-l
= C
WI(w) + hTo4))21~
(6)
where W(o) is a nonnegative frequency-dependent weighting fnnction and M is the number of points at which D(ej”) is sampled. In minimizing E,,,, we set aE,,,,,/ah = 0 to obtain a system of linear equations Qh = d, where
N-l
h(n) cos(no) - j C
n=O
=
+
hTC(o)-
h(n) sin(no)
n=O
jhTs(co),
(1)
M d= c I=1
W(~l)(MR(~f)C(WI)
-
Mr(W)dW)),
(8)
S.S. Kiaizmbi/ Signal Processing 60 (1997)
where Q(w) = c(o)cT(o) and Q(o) = s(o)sT(o). Since Ms( w ) and Mr( o) are both zero in the stopband of the of the desired characteristics, the above equation for d can be written as
127-130
129
about x/2. It follows that [6] h(y
-,)=h(y
+.)
= 0
when n is odd.
(14)
where MP is the number of sample points in the passband of D(ejO). The entries of Q and d are given as
It must be mentioned that the number of filter taps N is odd. In designing such allpass filters, Q and d are given as in (7) and (9) except that the rows and columns corresponding to the zero-valued coefficients are deleted.
Q(n,m)=&W,)cos((n /=I
2.2. Symmetric phase characteristics
0)
=2
-mkJr),
W(wrW(W)cos(nW
+ p(wr))
(10)
(11)
When b(o) in (13) is symmetric about x/2, the following conditions hold [6]:
I=1
for 0 < n, m < N - 1. It can be seen that Q is real, Toeplitz-symmetric and positive definite and hence only the Iirst row (or column) of entries has to be evaluated. Consequently, the system of linear equations can be solved by the computationally efficient and robust Levinson method [4]. It can also be noted that the entries of Q are independent of D(+). For the case of allpass phase equalizers, the desired characteristic can be expressed as
h(y-n)=h(y+n)
whenniseven,
h(&$-n)=-h(y+n)
whennisodd.
(15) From (1 ), the allpass phase equalizer can be characterized by ej((N-l)12)~~(ejw)
o(ej”)
= ejP(w) = cos(~(w)) +j sin@(w)),
w E [O,nl,
(12)
u =Ca(n)cos(2nw)
V
+ jCb(n)sin((2n
where p(o) is the desired phase response. Again, a mean-square error Emse can be minimized and system of linear equations can be obtained. If, however, p(w) is antisymmetric or symmetric with respect to x/2, the computational complexity of Q and d can be reduced significantly since only half the number of coefficients need be determined. Below, we shall consider the design of two allpass phase equalizers presented in [6] using our method.
- l)o),
n=l
n=O
(16) where
n = 0, a(n) = --
n=1,2 ,..., U,
,
2. I. Antisymmetric phase characteristics The desired phase characteristic can be written as p(o)=
--
N-l 2
b(n)=2h 0+&w),
(13)
where the first term on the right-hand side is the linear phase term and b(o) is a function of o antisymmetric
NG
- 2n+l
,
n=1,2
,..., V.
( Consequently (16) can be written as ej((N-i)/2)oH(ejo)
=ar;(o)
+jbrqo),
(17)
S.S. Ki&mbi/ Signal Processing 60 (1997) 127-130
130
where u = [a(O) a( 1) . . . a( U)]T, b=[b(l)b(2)
. . . b(V)]T,
2((w)= [l cos(2w) . . . cos(2Uo)]T, $0) = [sin(w) sin(3w) . . . sin((2V - l)o)lT. Since the coefficients associated with the real and imaginary parts of (16) differ in number, they are.computed separately. The mean-square error associated with the real part can be written as E, = 5
W(ol)(cos(fi(or))
- aTP(or))2 do.
(18)
I=1
By setting aE,liYa = 0, we get &z = &,,, where 0, = 5 IWMW)~T(W) I=1
(19)
and a, = 5
W(w,)cos(fi(w,))@(o/).
An efficient WLS method for the design of FIR filters satisfying prescribed magnitude and phase specifications is described. By incorporating symmetry and antisymmetry constraints of the impulse response, linear-phase FIR filters can easily be designed thus making our method general for the design of FIR filters.
References
Similarly, the mean-square error associated with the imaginary part can be written as W(wr)(sin(j?(w,)) - bT$(~l)).
(21)
I=1
Again, by setting dEbjab = 0, we get &b = & where & is given by
(22) I=1
and & = 5
3. Conclusion
(20)
l=l
Eb = 5
Hankel matrix. Such a system of linear equations can be efficiently solved using the methods presented in [5,2]. These methods have a computational complexity of O(N2). In order to design a filter having an equiripple magnitude response, appropriate weighting function, W(o), has to be used in the minimization of the mean-square error. Since there are no analytical methods to obtain the appropriate W(o), iterative techniques presented in [ 1,3] can be used.
W(o,) sin(&ol))&wl).
(23)
k=O
It has been shown in [7] that matrices & and &, can be decomposed into a sum of a Toeplitz and a
[I] C.-Y. Chi, S.-L. Chiou, A new WLS Chebyshev approximation method for the design of FIR digital filters with arbitrary complex frequency response, Signal Processing 29 (1992) 335-347. [2] I. Gohberg, I. Koltracht, Efficient algorithm for Toeplitz-plusHankel matrices, Integ. Equations Oper Theory 12 (1989) 136-142. [3] Y.C. Lim, J.H. Lee, C.K. Chen, R.H. Yang, A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design, IEEE Trans. Signal Process. 40 (1992) 551-558. [4] S.L. Marple Jr., Digital Spectral Analysis with Applications, Prentice-Hall, Englewood Cliffs, NJ, 1987. [5] GA. Merchant, T.W. Parks, Efficient solution of a Toeplitzplus-Hankel coefficient matrix system of equations, IEEE Trans. Acoust. Speech Signal Process. ASSP-30 (1982) 40-44 [6] K. Steiglitz, Design of FIR digital phase networks, IEEE Trans. Acoust. Speech Signal Process. ASSP-29 (1981) 171-176. [7] S. Sunder, An’efficient weighted least-squares design of linearphase nonrecursive filters, IEEE Trans. Circuits Systems 42 (1995) 359-361.
