IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 6, JUNE 2000 149
Closed-Form Design and Efficient Implementation of
Generalized Maximally Flat Half-Band FIR Filters
Soo-Chang Pei and Peng-Hua Wang
Abstract—In this letter, a closed-form expression for the
im-pulse response of the generalized half-band (HB) maximally flat (MF) FIR filters is obtained by solving the linear equations of the MF conditions. Based on the resultant impulse responses, an effi-cient implementation structure is derived. The dynamic range of the multipliers of the new structure is shown to be greatly reduced in comparison to the one of the direct form impulse response.
Index Terms—Half-band filter, maximal flatness.
I. INTRODUCTION
H
ALF-BAND (HB) filters are often used for decimation, interpolation, and multirate systems. An th order linear phase FIR filter is called HB if its impulse response has the following propertyand
where the desired group delay is , and is a constant. In other words, one of the polyphase components of the HB FIR filter is just a delay . In [1], the definition of the HB FIR filters is recently generalized. According to the generalized def-inition, the desired group delay is not restricted to be a half of the filter order, and the resulting HB filter can have nonsym-metric impulse response (i.e., nonlinear phase response).
In [2], the authors summarized the design methods of the HB filters and the Hilbert transformers with linear phase. A closed-form design of the nonlinear phase HB maximally flat (MF) FIR filters was recently proposed in [3] using the Cheby-shev polynomial. Although the final impulse response was not expressed for direct form structure, the transfer function repre-sented as the Bernstein polynomial form was analytically ob-tained. In [4], a closed-form expression for the transfer function of linear phase HB MF FIR filter was derived. Two implemen-tations were presented. It is interesting that one of the two struc-tures needs no multipliers. That is, a multiplierless structure for the linear phase MF HB FIR filter was obtained in [4].
In this letter, a new closed-form expression for the impulse response of the generalized HB MF FIR filters is proposed. The impulse response is solved directly from the linear equations of the MF conditions. An efficient new implementation struc-ture is derived based on the resultant impulse responses. The dynamic range of the multipliers of the new structure is shown to be greatly reduced in comparison with the dynamic range of
Manuscript received December 12, 1999. The associate editor coordinating the review of this paper and approving it for publication was Prof. R. Shenoy.
The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. (e-mail: [email protected]).
Publisher Item Identifier S 1070-9908(00)05110-5.
the direct form impulse response by an example. The multipliers used are also fewer in the proposed structure.
II. CLOSED-FORMIMPULSERESPONSE OFGENERALIZEDHB MF FIR FILTERS
Suppose the th order ( ) generalized HB FIR filter is characterized by the transfer function of
(1)
where the odd integer is the symmetry center. The desired frequency response is defined by
for
for (2)
where and denote the passband and the stopband, respec-tively. The filter is said to be maximally flat if the frequency re-sponse satisfies the following properties
for
(3) and
for
(4) where and represent the degree of flatness at 0 and , respectively. Since the desired HB frequency response is symmetric about is assigned to be equal to in this paper.
Substituting (1) and (2) for (3) and (4) and simplifying the equations, we obtain the following equations:
(5)
and
(6)
for . It is easy to show that
(7) 1070–9908/00$10.00 © 2000 IEEE
150 IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 6, JUNE 2000
(a) (b)
Fig. 1. Design of the tenth-order maximally flat HB FIR filter with the desired group delayL = 3. (a) Impulse response. (b) Magnitude response.
(a) (b)
Fig. 2. Design of the tenth-order maximally flat HB FIR filter with the desired group delayL = 5. (a) Impulse response. (b) Magnitude response.
and (5) and (6) are reduced as
(8)
for . Since (8) is a Vandermonde system [5], it
is of full rank. We conclude that the quantities of , , and are related as
(9) Equation (8) can be solved analytically by carrying out the Cramer’s rule [6]. The resultant closed-form impulse responses are obtained and expressed by
(10)
for .
Fig. 1(a) and (b) shows the impulse responses and the magni-tude response of the tenth-order MF HB FIR filter with 3. The impulse response is not symmetric and consequently, the filter does not have linear phase response. Fig. 2(a) and (b) shows the impulse responses and the magnitude response of the tenth-order MF HB FIR filter with 5. The impulse response is symmetric about the desired group delay 5. Thus, the phase response of the designed filter is linear.
III. EFFICIENTIMPLEMENTATION OFGENERALIZEDHB MF FIR FILTERS
Although the filter can be implemented in direct form with the impulse response solved in Section II, a new efficient
struc-Fig. 3. Efficient implementation structure for theNth-order generalized MF HB FIR filter, whereM = N=2, and L is the desired group delay.
ture can be derived. Since 1/2, the transfer function ex-pressed by (1) can be written as
(11) where
and .
If we express in the following form:
for . It is easy to show that
(12)
Substituting for (13), the weighting coefficient is simplified as
(13)
for and 1. Based on the property that
can be expressed by the following cumulative product:
then we can derive an efficient implementation structure as shown in Fig. 3.
The main advantage of the new structure is that the dynamic range of the weighting coefficients is greatly reduced. For ex-ample, according to (10), the impulse response of the tenth-order MF HB FIR filter with 3 is
where the ratio of the largest amplitudes of coefficients to the
least one is . If the filter is implemented by
an FIR lattice structure, the reflection coefficients are
PEI AND WANG: IMPLEMENTATION OF HALF-BAND FIR FILTERS 151
where the ratio of the largest amplitudes of coefficients to the
least one is . However, the weighting
coeffi-cients of the proposed structure are
and the ratio of the largest amplitudes of multiplier to the least
one is 9. Another advantage of the structure in
Fig. 3 is there is one multiplier less than the direct form imple-mentation. There are multipliers for the direct form structure, and is needed for the proposed one.
IV. CONCLUSION
In this paper, a closed-form expression for the impulse re-sponse of the generalized HB MF FIR filters is derived. We solve the impulse response directly from the linear equations of the MF conditions by using the Carmer’s rule. The impulse response form is rather simple. Design examples are presented.
A new efficient implementation structure is obtained based on the derived impulse response. The dynamic range of the multi-pliers of the new structure can be greatly reduced in comparison to the one of the direct form impulse response.
REFERENCES
[1] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993.
[2] H. W. Schüßler and P. Steffen, “Halfband filters and Hilbert trans-formers,” Circuits Syst. Signal Process., vol. 17, pp. 137–164, Oct. 1998.
[3] S. Samadi, A. Nishihara, and H. Iwakura, “Generalized half-band max-imally flat FIR filters,” in Proc. IEEE Int. Symp. Circuits and Systems, Orlando, FL, May 1999.
[4] S. Samadi, H. Iwakura, and A. Nishihara, “Mutiplierless and hierarchical structure for maximally flat half-band FIR filters,” IEEE Trans. Circuits
Syst. II, vol. 46, pp. 1225–1230, Sept. 1999.
[5] G. H. Golub and C. F. Van Loan, Matrix Computing. Baltimore, MD: The Johns Hopkins Univ. Press, 1989.
[6] B. Noble and J. W. Daniel, Applied Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1988.
