A COMPUTER PROGRAM FOR DESIGNING LINEAR PHASE FIR DIGITAL FILTERS BY EIGEN-APPROACH
Soo-Chang Pei and Jong-Jy Shyu Departmenr: of Electrical Engineering
National Taiwan University Taipei, Taiwan, Rep. o f China ABSTRACT
This paper presents a computer program for designing a large class of FIR linear phase digital filters by eigen-approach. The unique advantage of this approach over the McClellan-Parks algorithm is that it is general enough to incorporate both time and frequency domain constraints, this program not only can design standard bandpass filters, differentiators, Hilbert transformers, and also suitable for time-constrained FIR filter design such a s Nyquist filters, partial response FIR filters, Mth band filters and higher order differentiators. The simplicity and generality of this eigen-design algorithm make this program useful for a wide variety of design applications.
INTRODUCTION
In the past fifteen years, the most popular program for designing linear phase FIR digital filters is that develop- ed by McClellan and Parks [l]. It can design multi-bandpass filters, differenti- ators and Hilbert transformers etc. very efficiently. But the McClellan-Parks algorithm are emphasized on finding the optimal frequency response without any time constraint on the filter coefficients, and it is very difficult to incorporate both the time- and frequency-domain constraints to design some important filters in communication applications such a s pulse shaping filters, multirate Mth band filters and Nyquist filters with zero intersymbol interference [21.
Recently Vaidyanathan and Nguyen have introduced the eigenfilter approach for designing linear phase FIR digital filters [31. By minimizing a quadratic measure of the error in the frequency band, an eigenvector of appropriate matrix is computed to get the filter coefficients. The unique advantage of this approach over the McClellan-Parks algorithm is that it is general enough to incorporate both time and frequency domain constraints: also the design time
is comparable to that of McClellan-Parks algorithm. We have extended this approach to design multiple band-pass filters, higher order differentiators [41 and Hilbert transformers [ 5 ] by eigenfilter, this method is not only simple and fast, but also optimal in the least square sense C31. Comparison with the McClellan- Parks algorithm for minimax equiripple filters shows that both are optimal in the sense of different minimum norms of the error function, but much better performance is obtained with our approach in most o f the frequency band except in the narrowband region near the cutoff edge C41 C51.
This paper presents a general purpose computer program for designing a wide class of FIR linear phase digital filters by eigen-approach. This program can design Nyquist filters, Mth band filters, class 1 partial frequency response filters [SI and higher order differentiators in addition to those McClellan-Parks program can design. It takes the interactive form such that the users give the input data according to the hint, and if the input data is not correct it will give warning and demand another input again. Moreover, the program also allows to get the quantized filter coefficients and the quantized error, if necessary. The simplicity and generality of this eigen- design algorithm make this program useful for a wide variety of design applications.
DESIGN ALGORITHM
A typical FIR digital filter is characterized by a transfer function H(Z), i.e.
n = O
where N is the length of the filter impulse response sequence [ h(n)}
,
its frequency response is presented byc l 1
where "(w) is a real valued function
367
which is the value o f the amplitude response, and L is equal to 0 or 1. The least-squares approach to this filter design problem is to formulate an object function
E
1
/cCD(w)-H'(w)l2dw (3) where D(w) is the desired frequency response we wish to approximate and c represents the band region O<w<x,
but excluding the transition band. Now we wish to reformulate Eq.(3) such that it has the following formwhere t is the vector transponse opera- tion, A is a column vector containing the coefficients of H'(w), and Q is a real, symmetric and positive-definite matrix depending upon the design requirement. Once matrix Q is found, the eigenvector corresponding to the smallest eigenvalue of matrix Q is the desired solution. Due to the limited space, the design detail of multi-band filters, Nyquist filters, Hilbert transforms and differentiators can be referred to the References 3,4,5.
E = A~.Q.A (4)
PROGRAM DESCRIPTION
This program is written in interac- tive form such that the users only input data according to the program's instruc- tion, sometimes it will give the users the warning statements to revise the input data set; The eigen-filter design . algorithm consists of an input section, generation of matrix Q for eigenformura- tion, compute the eigenvector A correspond- ing to the smallest eigenvalue of matrix Q, obtain the filter impulse response h(n) from eigenvector A , and program output the filter's coefficients with/ without quantization, the four subroutines QG1, QG2, QG3 and QG4 are used separately to calculate the elements q(n,m) of matrix Q according to the chosen type of filter.
The input which describes the filter
a) Multiple passbandlstopband (ITYPE=l): Subroutine QG1
.
b) Nyquist filter (ITYPE=2), Mth-band filter (ITYPE=3) and Class 1 partial response filter (ITYPE=4): Subrou- tine Q G 2 .
c ) Hilbert transformer (ITYPE=5): Subroutine Q G 3 .
d) LowIHigh order differentiator (ITYPE=6): Subroutine QG4.
2) The filter length: 35NL1128.
3) The frequency bands, specified by upper and lower cutoff frequency (EDGE array) up to a maximum of 10 frequency bands: l<IBAND_<10.
4) The desired frequency response (DES array) in each band.
specifications consists of the following.
1 ) The type of the filter (ITYPE):
5) A positive weight constant (WE1 array) in each band.
6) The intersymbol duration needs to be specified first for the cases of Nyquist, M-th band and Class 1 partial response filters: ITYPE = 2 , 3 , 4 and 2 5 MIN.
The program output will list the filter's impulse response coefficients (H array, 32 bit precision on VAX 11/780 computer), also it includes the option to quantize the impulse response with fewer number of bits (4sIBIT116: including sign bit) and prints out the quantized impulse response coefficients (HE array), the quantization error of the filters coeffi- cient is stored in E array for the user's reference ( E(. ) HE(. )
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H(. ) 1.DESIGN EXAMPLES
This section presents some specific examples designed by this package; In Fig. 1 , we show a three-passband filter with length N=57 and 3 different filter gain passband responses. Fig.2 is a FIR Nyquist filter with length N.39, fp-0.1 and M = 4 , where M is the intersymbol duration, in this figure we also plotted its impulse response in which every other 4th filter coefficient except the central one is zero due to its time constraints.
Fig.3(a) shows a length N=32, full- band Hilbert transformer with cutoff frequencies fpl=0.04, fp2=0.5, in this example we compare the error curves of the eigen-approach (solid line) and McClellan-Parks algorithm (dotted line) in Fig.3(b), the passband ripple is much smaller in the eigen-approach than that in McClellan-Parks algorithm.
A fifth-order full-band differentiator with length N=32 is designed, compare Fig.Q(a) with Fig.4(b) the performance of the eigen-method is much smoother and less deviation than the McClellan-Parks algorithm.
For the computation of the elements q(n,m) in matrix Q, since Q is a real, symmetric and positive-definite matrix, i.e. q(n,m)=q(m,n), then only half the elements of q(n,m) need to be calculated for computation savings. The design time of the eigenfilter approach is mostly spent in the computation o f the smallest eigenvector, however this computation can be done efficiently. The design time for the eigenfilter approach is comparable to that of the McClellan-Parks algorithms for filter length less than 100. By extensive computer simulations, we should avoid to choose the reference frequency wo at the band edge in our eigenfilter design C41. A better choice is to choose the reference frequency at the center of
CONCLUSIONS c 1 1 c21 C31 c41 C51 C61
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J.H. McClellan, T.W. Parks and L.R. Rabiner. "A comDuter program for
IIR Nyquist filter with zero inter- 0 . 3
Circuits Syst. Vol.CAS-29, pp.23-34, 0 . 2 - symbol interference and its frequency
response approximation," IEEE Trans. Jan. 1982.
P.P. Vaidyanathan and T.Q. Nguyen,
"Eigenfilters: a new approach to ' O .
least squares FIR filter design and applications including Nyquist fil- ters", IEEE Trans. Circuits and Systems, Vol.CAS-34, pp.ll-23, Jan.
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