Synthesis of Wideband Linear-Phase FIR Filters with a Piecewise ...
Synthesis of Wideband Linear-Phase FIR Filters with a Piecewise-Polynomial-Sinusoidal Impulse Response Raija Lehto and Tapio Saram¨aki
Olli Vainio
Institute of Signal Processing Tampere University of Technology Tampere, Finland Email: [email protected], [email protected]
Institute of Digital and Computer Systems Tampere University of Technology Tampere, Finland Email: [email protected]
Abstract— A method is presented to synthesize wideband linear-phase FIR filters with a piecewise-polynomial-sinusoidal impulse response. The proposed method is based on merging the earlier synthesis scheme proposed by the authors with the method proposed by Chu and Burrus and by modifying it by using an arbitrary number of separately generated center coefficients instead of only one used in the method by Chu-Burrus. The desired impulse response is created by using a parallel connection of several filter branches and by adding an arbitrary number of center coefficients to form it. The arithmetic complexity of these filters is proportional to the number of branches, the common polynomial order for each branch, the number of separate center coefficients, and the number of complex multipliers. The filter coefficients are optimized by using linear programming. An example shows the benefits of these filters with regard to reducing the number of coefficients as well as the arithmetic complexity.
I. I NTRODUCTION The synthesis scheme proposed in this paper is based on the previous work done by the authors [1] and on merging it with and on modifying the method proposed by Chu and Burrus [2–3] for the synthesis scheme of wideband FIR filters. Chu and Burrus first design a so-called 2N -order envelope filter with a piecewise-polynomial impulse response, for which the design scheme of the authors [1] is very efficient. Secondly, the coefficient values of this envelope filter are modified by multiplying them with sin[(n − N )ωc ] and by adding at least one center coefficient, which leads to a wideband FIR filter. The transition-band center is approximately located at ω = ωc . The method uses a recursive implementation approach and it involves complex-valued arithmetics. Also other approaches have been developed to reduce the arithmetic complexity of FIR filters, especially in wideband cases with a narrow transition-band. One of the most efficient techniques is the frequency response masking technique (FRM). This technique was originally introduced by Lim and improved by Saram¨aki and Lim [4] by using the Remez multiple exchange algorithm. Because of the importance of center coefficients in FIR filters, the approach presented in this paper to synthesize wideband piecewise-polynomial-sinusoidal linear-phase FIR filters uses an arbitrary number of separately generated center coefficients instead of only one used by Chu and Burrus [2– 1-4244-0921-7/07 $25.00 © 2007 IEEE.
3]. The implementation structures proposed by the authors earlier can be used with a slight modification; in fact, the implementation structure is modified to be complex and a conventional FIR filter is added in order to form the center of the impulse response. Linear programming is used to optimize the unknowns in the overall filter. The arithmetic complexity of these filters is proportional to the number of impulse-response pieces, the overall polynomial order, the number of separately generated center coefficients and complex multipliers. II. P ROPOSED S YNTHESIS S CHEME FOR T YPE 1 W IDEBAND FIR F ILTERS This section shows how the original Chu-Burrus [2–3] approach, to synthesize wideband linear phase filters based on the use of piecewise-polynomial-sinusoidal impulse responses, can be significantly improved. First, the original approach is briefly reviewed for the causal case. Secondly, it is shown how to improve the Chu-Burrus synthesis technique in two ways, namely by merging it with the method proposed by the authors in [1] and by adding an arbitrary number of center coefficients separately by using an additional transfer function. A. Startup Idea in the Chu-Burrus Approach This idea arises from a windowing technique for Type 1 filters of order 2N [5], where the corresponding overall causal transfer function can be expressed as F (z) = F1 (z) + F2 (z), where F1 (z) =
2N
W (n) sin[ωc (n − N )]z −n ,
(1)
(2)
n=0
with W (n) = w(n − N )/[π(n − N )] and
(3)
ωc −N z . (4) π w(n − N ) in (3) is a window function satisfying w(N ) = 1, w(n) = 0 for |n| > N + 1 and w(2N − n) = w(n) for n = 0, 1, . . . , N .
2052
F2 (z) =
There is a need to use the sub-transfer function, F2 (z), because the value of the center coefficient in F1 (z) becomes automatically zero due to the multiplication by sin[ωc (n − N )]. A way to generate the piecewise-polynomial-sinusoidal transfer function F1 (z) is to consider the following transfer function: G(z) = =
2N
2N
W (n)z −n (ej(n−N )ωc )
n=0
W (n)[cos((n − N )ωc ) + j sin((n − N )ωc )]z −n .
n=0
(5)
From this it follows that F1 (z) can be expressed as F1 (z) = Im{ = Im{
2N
2N
W (n)z −n (ej(n−N )ωc )}
n=0
W (n)z
(6) −N
(z
e
) }.
B. Proposed Linear-Phase Filter Classes for Type 1 Wideband FIR filter First, the overall transfer function, denoted by H(z), is constructed as described in our earlier method [1], i.e., M parallel branches are connected and delayed with z −Nm in order to keep the center of the symmetry at the same location for all the subimpulse responses. Secondly, the subresponses are modulated with a sinusoidal function as described in the previous subsection. Third, an arbitrary number of separately generated center coefficients is added as follows. b z −Nm Hm (z) + z −N H(z),
(7)
m=1
where the integers Nm in the delay terms z −Nm satisfy N1 = 0 and Nm+1 > Nm for m = 1, 2, . . . , M − 1,
(8)
and Hm (z) is a 2(N − Nm )th-order transfer function with the impulse response given for n = 1, 2, . . . , N − Nm by hm (n) =
L r=0
(10)
and
In the Chu-Burrus approach, a wideband FIR filter is obtained by first generating the envelope filter with W (n) for n = 0, 1, . . . , 2N as the impulse-response coefficients. This impulse response is designed to become piecewise polynomial. The coefficient values of this envelope filter are modified by multiplying them with sin[(n − N )ωc ], which gives F1 (z). The transition-band center of the resulting overall filter with the transfer function F1 (z) + F2 (z) is approximately located at ω = ωc . Because of the importance of center coefficients of an FIR filter, it is worth taking an arbitrary number of center coefficients separately, instead of only one center coefficient. This leads to our earlier design method [1] on how to create the overall transfer function for a linear-phase FIR filter.
M
Xm = [Nm , Nm+1 − 1] for m = 1, 2, . . . , M − 1
−1 jωc n
n=0
H(z) =
th-order direct-form In addition, H(z) is a conventional 2N transfer function with the additional impulse response coeffi = N − c + 1 with c = (T /2) and T is cients, h(n) and N the number of the additional coefficients at the center of the filter. means rounding upwards. The delay terms in (7) are used to shift the center of the symmetry at the desired location. First, the center impulse response of each term occurs at n = N . The impulse-response coefficients of the envelope transfer function, which were mentioned in the previous subsection and are indirectly included in the summation of (7), are multiplied with sin[(n − N )ωc )]. The arrival at 2N th-order Type 1 overall transfer function is guaranteed by these facts. In order to indicate that the overall filter has a piecewisepolynomial-sinusoidal impulse response the time interval n ∈ [0, N ] is divided into the following M subintervals:
r a(L) m (r)n × sin[ωc (n − (N − Nm ))] . (9)
XM = [NM , N ]
(11)
The following facts should be pointed out. First, X1 = [0, N2 − 1] because N1 = 0, Secondly, the overall impulse response can be studied up to n = N because of the even symmetry around this point. The impulse response on Xm can be expressed as h(n) =
M
hm (n),
(12)
m=1
where hm (n) =
m L (L) ak (r)(n − Nm )r × sin((n − N )ωc ) k=1
r=0
(13)
for m = 1, 2, · · · , M − 1, and M L (L) ak (r)(n − NM )r k=1 r=0 × sin((n − N )ωc ) + h (n − N ),
hM (n) =
(14)
which equals the overall impulse response and where h (n − N ) is a conventional direct-form Type 1 filter with nonzero coefficients for n = N − c + 1, . . . , N , in which c = T /2 and T is the number of separately generated additional center coefficients. Slices Nm s should be chosen so that |N2 − N1 | = |N3 − N2 | = · · · = |NM − NM−1 |, where N1 = 0 and M is the number of subintervals in the overall impulse response. Based on the above equations, in each Xm for m = 1, 2, . . . , M , a separate piecewise-polynomial-sinusoidal impulse response can be generated. In addition, in the XM , there are additional center coefficients, which are of great importance for fine-tuning the overall filter to meet the given criteria. Given the filter criteria as well as the design parameters ωc , M , N , L, Nm ’s, and the number of center coefficients in cluded in H(z), the overall problem is solvable by using linear
2053
e-j(N-L)wc
programming, instead of using ad-hoc nonlinear programming as used in the original Chu-Burrus approach.
e -j(N-L+1)wc
e j(L+1)/2 wc αL
+
e jwc
Z-
1
Z -1
In this section, an efficient implementation structure for the overall transfer function is shown based on the structure proposed in [1] as given by (7) and (9). The proposed structure is modified by an additional branch for the separately generated center coefficients, which do not follow the polynomials, and the branches are multiplied by additional complex terms. The subtransfer function, Hm (z), as given by (7), is expressed in the form: N −1 Hm (z) ≡h(0) (N )z −N + h(0) (n)[z −n + z −(2N −n) ],
1 Z-
e-j(N-3)wc 1 Z-
e -j(N-2)wc
+
β2 e -jNwc
e jwc
-
α3
+
α2
+
Z-
Fig. 1.
Z -1
Z-N
e jwc Z-
1
ejwc e jwc
+ -
(16)
Z -1
1 Z-
Z -1
IN
e jwc
e j2wc
+
e -j(N-1)wc
(15) where N is an integer corresponding to Nm −N in the previous section and h(0) (n) = pL (n) with an Lth-order modulated polynomial given by
α4
+ β4
1 Z-
n=0
a(L) (r)nr .
1 Z-
βL+1
1 Z-
p(L) (n) = sin(ωc (n − N )) ×
e jwc
+
III. A N I MPLEMENTATION S TRUCTURE FOR T YPE 1 F ILTER
L
αL+1
+
α1
Z -1
1 Z-
N -1
ej(N+1)wc
Im(OUT)
Basic implementation block for Type 1 filters.
r=0
The following polynomials are determined recursively for k = 1, 2, . . . , L as p(L−k) (n) = p(L−k+1) (n + 1) − p(L−k+1) (n) = (L−k) a(L−k) (r)nr , sin(ωc (n − N )) ×
(17)
r=0
where a
(L−k)
(L−k+1)
(r) =
j=r+1
j (L−k+1) (r). a r
(18)
The filter coefficients, αk s and β2k s, are related to the polynomials as given by (16), (17) and (18) through αk = p(L−k) (0) = a(L−k) (0), k = 1, 2, ..., L + 1,
(19)
and β2k = −p(L−2k+1) (N + 1 − k) − p(L−2k+1) (N − k), k = 1, 2, . . . , L/2. (20) Additionally, αL+1 = p(0) (0), whereas for odd L, βL+1 = −2p(0) . An efficient implementation block for the above transfer function is shown in Fig. 1 for odd L. For even L the uppermost parts are absent. Based on the above structure as well as our earlier implementation structure for Type 1 filters [1], a similar structure is developed for each of the branch filters, Hm (z). To generate the corresponding coefficients, αk s and β2k s, the impulse response given by (9) is used. (m) (m) The resulting αk s and β2k s are denoted by αk and β2k . An efficient implementation structure for the overall filter is shown in Fig. 2, where Tm = Nm for m = 1, 2, . . . , M − 1, TM1 = N − NM and TM2 = N − NM + 1, and H(z) is a conventional direct-form FIR filter for the separately b generated center coefficients with delay element z −N with
= N − c + 1. Furthermore, the β2k s are the sums of the N (m) corresponding β2k s. In order to keep the overall diagram (m) simple, the coefficients αk s have been used twice. In the practical implementation, the overall number of multipliers can be reduced by first adding [subtracting] the inputs of the left(m) (m) (m) hand side αk s and the right-hand side αk s [−αk s] and (m) then by multiplying the result by αk s. In Fig. 2 the complex multipliers are only drawn before the filter coefficients in feedforward loops because of the available space, i.e., the purpose is that the data do not become complex until after multiplication. IV. A D ESIGN E XAMPLE In this section, the properties and efficiency of the proposed filter classes for Type 1 FIR filters over FRM-based FIR filters are shown by means of an example in a wideband application. Consider a very narrow-transitionband design of linearphase FIR filters for ωp = 0.400π, ωs = 0.402π, ωc = 0.401 and with passband ripple δp = 0.01, stopband attenuation δs = 0.001 (60 dB). For the proposed approach, the given criteria are met by the filter of order 3381, M = 10, L = 2, the number of separate center coefficients, T = 7, N1 = 0, N2 = 400, N3 = 900, N4 = 1300, N5 = 1500, N6 = 1600, N7 = 1638, N8 = 1665, N9 = 1675, and N10 = 1683. This filter thus consists of ten slices with the polynomial degree L = 2 in each slice. In this case, the number of unknowns required both in the optimization and in the implementation is only 34. The number of multipliers is 35 + 51 in practical implementation. The magnitude and impulse response for the optimized filter are shown in Fig. 3. The corresponding FRM-based FIR filter in a three stage design [4] requires 94 multipliers, and the overall filter order
2054
βL+1
e j wc e-j(N-L)wc
e- jwc
α L+1 (1)
1 Z-
1 Z-
L+1
ejwc
α(1) L+1
−α(M) L+1
e jwc Z-
αL
(1)
e- jwc
e-j (N-L+1)wc
Z -1
α
α(M) L
Z -1
β4 - jwc
e
α(M) 4
jwc
e
Z -1
Z -1
α(M) 4
α(1) 4
e jwc
-j (N-2)wc
α3
(1)
α(M) 3
e- jwc
Z -1
−α(1) 3
−α(M3)
Z -1
α(1) 2
α(M) 2
- jwc
e
1 Z-
α2 e j (TM1 +1)wc
(M)
1 Z-
α(M) 1
e- jT1wc
T Z- 1
Z -1
Z -1
β2
e -j (N-1)wc
α(1) 1
e jwc 1 Z-
α4
(1)
e- jNwc
1
−α(1) L
−α(M) L
1 Z-
e-j (N-3)wc
e
Z -1
(M)
−α(M) 1
Z -1
Z - TM2
Z - TM1
e jwc
Z -1
α(1) 2
ejwc e jTM2 wc
−α(1) 1
Z -1
e j(N+1)wc
Z -T 1
Im
IN OUT
H(z)
Z -N
Fig. 2.
Implementation structure of Type 1 filter.
is 3196 compared to our design which requires 86 multipliers in a practical implementation, and a slightly increased filter order, 3381.
0 Magnitude in dB
−20 −40 −60 −80 −100 −120 0
0.1π
0.2π
0.3π
0.4π 0.5π 0.6π 0.7π Angular frequency ω
0.8π
0.9π
π
V. C ONCLUSION In this paper, we have shown a straightforward approach to synthesize linear-phase FIR filters with a piecewisepolynomial-sinusoidal impulse response for wideband specifications. An efficient implementation structure was shown for Type 1 FIR filters. An example has been given in order to show the benefits of the proposed structure. It was also found that the synthesis of the filter is the most efficient with the polynomial degree two. ACKNOWLEDGMENT This work was supported by the Academy of Finland, Project No. 213462 (Finnish Centre of Excellence Program (2006 - 2011)). R EFERENCES
Overall impulse response 0.6 0.4
0.5
0.2
Amplitude
0.4
0
0.3
1650
1700
1750
0.2 0.1 0 −0.1 0
Fig. 3.
500
1000
1500 2000 n samples
2500
3000
A design example of proposed linear-phase FIR filters.
[1] Lehto, R., Saram¨aki, T. and Vainio, O., “Synthesis of narrowband linearphase FIR filters with a piecewise-polynomial impulse response, ” Proc. IEEE Int. Symp. on Circuits and Systems, ISCAS 2005, Kobe, Japan, pp. 2012–2015, May 2005. [2] S. Chu and S. Burrus, “Efficient recursive realizations of FIR filters, Part I: The filter structures,” Circuits, Systems and Signal Processing, vol. 3, number 1, pp. 2–20, 1984. [3] S. Chu and S. Burrus, “Efficient recursive realizations of FIR filters, Part II: Design and applications,” Circuits, Systems and Signal Processing, vol. 3, number 1, pp. 21–57, 1984. [4] T. Saram¨aki, Y. C. Lim, “Use of the Remez Algorithm for Designing FRM based FIR Filters,” Circuits Systems Signal Processing, Birkhauser Boston, Vol. 22, No. 2, pp. 77-97, 2003. [5] T. Saram¨aki, “Finite Impulse Response Filter Design,” in Digital Signal Processing Handbook, S. K. Mitra, J. F. Kaiser, Ed., pp. 155–272, John Wiley and Sons, NY, 1993.
2055
Olli Vainio
Institute of Signal Processing Tampere University of Technology Tampere, Finland Email: [email protected], [email protected]
Institute of Digital and Computer Systems Tampere University of Technology Tampere, Finland Email: [email protected]
Abstract— A method is presented to synthesize wideband linear-phase FIR filters with a piecewise-polynomial-sinusoidal impulse response. The proposed method is based on merging the earlier synthesis scheme proposed by the authors with the method proposed by Chu and Burrus and by modifying it by using an arbitrary number of separately generated center coefficients instead of only one used in the method by Chu-Burrus. The desired impulse response is created by using a parallel connection of several filter branches and by adding an arbitrary number of center coefficients to form it. The arithmetic complexity of these filters is proportional to the number of branches, the common polynomial order for each branch, the number of separate center coefficients, and the number of complex multipliers. The filter coefficients are optimized by using linear programming. An example shows the benefits of these filters with regard to reducing the number of coefficients as well as the arithmetic complexity.
I. I NTRODUCTION The synthesis scheme proposed in this paper is based on the previous work done by the authors [1] and on merging it with and on modifying the method proposed by Chu and Burrus [2–3] for the synthesis scheme of wideband FIR filters. Chu and Burrus first design a so-called 2N -order envelope filter with a piecewise-polynomial impulse response, for which the design scheme of the authors [1] is very efficient. Secondly, the coefficient values of this envelope filter are modified by multiplying them with sin[(n − N )ωc ] and by adding at least one center coefficient, which leads to a wideband FIR filter. The transition-band center is approximately located at ω = ωc . The method uses a recursive implementation approach and it involves complex-valued arithmetics. Also other approaches have been developed to reduce the arithmetic complexity of FIR filters, especially in wideband cases with a narrow transition-band. One of the most efficient techniques is the frequency response masking technique (FRM). This technique was originally introduced by Lim and improved by Saram¨aki and Lim [4] by using the Remez multiple exchange algorithm. Because of the importance of center coefficients in FIR filters, the approach presented in this paper to synthesize wideband piecewise-polynomial-sinusoidal linear-phase FIR filters uses an arbitrary number of separately generated center coefficients instead of only one used by Chu and Burrus [2– 1-4244-0921-7/07 $25.00 © 2007 IEEE.
3]. The implementation structures proposed by the authors earlier can be used with a slight modification; in fact, the implementation structure is modified to be complex and a conventional FIR filter is added in order to form the center of the impulse response. Linear programming is used to optimize the unknowns in the overall filter. The arithmetic complexity of these filters is proportional to the number of impulse-response pieces, the overall polynomial order, the number of separately generated center coefficients and complex multipliers. II. P ROPOSED S YNTHESIS S CHEME FOR T YPE 1 W IDEBAND FIR F ILTERS This section shows how the original Chu-Burrus [2–3] approach, to synthesize wideband linear phase filters based on the use of piecewise-polynomial-sinusoidal impulse responses, can be significantly improved. First, the original approach is briefly reviewed for the causal case. Secondly, it is shown how to improve the Chu-Burrus synthesis technique in two ways, namely by merging it with the method proposed by the authors in [1] and by adding an arbitrary number of center coefficients separately by using an additional transfer function. A. Startup Idea in the Chu-Burrus Approach This idea arises from a windowing technique for Type 1 filters of order 2N [5], where the corresponding overall causal transfer function can be expressed as F (z) = F1 (z) + F2 (z), where F1 (z) =
2N
W (n) sin[ωc (n − N )]z −n ,
(1)
(2)
n=0
with W (n) = w(n − N )/[π(n − N )] and
(3)
ωc −N z . (4) π w(n − N ) in (3) is a window function satisfying w(N ) = 1, w(n) = 0 for |n| > N + 1 and w(2N − n) = w(n) for n = 0, 1, . . . , N .
2052
F2 (z) =
There is a need to use the sub-transfer function, F2 (z), because the value of the center coefficient in F1 (z) becomes automatically zero due to the multiplication by sin[ωc (n − N )]. A way to generate the piecewise-polynomial-sinusoidal transfer function F1 (z) is to consider the following transfer function: G(z) = =
2N
2N
W (n)z −n (ej(n−N )ωc )
n=0
W (n)[cos((n − N )ωc ) + j sin((n − N )ωc )]z −n .
n=0
(5)
From this it follows that F1 (z) can be expressed as F1 (z) = Im{ = Im{
2N
2N
W (n)z −n (ej(n−N )ωc )}
n=0
W (n)z
(6) −N
(z
e
) }.
B. Proposed Linear-Phase Filter Classes for Type 1 Wideband FIR filter First, the overall transfer function, denoted by H(z), is constructed as described in our earlier method [1], i.e., M parallel branches are connected and delayed with z −Nm in order to keep the center of the symmetry at the same location for all the subimpulse responses. Secondly, the subresponses are modulated with a sinusoidal function as described in the previous subsection. Third, an arbitrary number of separately generated center coefficients is added as follows. b z −Nm Hm (z) + z −N H(z),
(7)
m=1
where the integers Nm in the delay terms z −Nm satisfy N1 = 0 and Nm+1 > Nm for m = 1, 2, . . . , M − 1,
(8)
and Hm (z) is a 2(N − Nm )th-order transfer function with the impulse response given for n = 1, 2, . . . , N − Nm by hm (n) =
L r=0
(10)
and
In the Chu-Burrus approach, a wideband FIR filter is obtained by first generating the envelope filter with W (n) for n = 0, 1, . . . , 2N as the impulse-response coefficients. This impulse response is designed to become piecewise polynomial. The coefficient values of this envelope filter are modified by multiplying them with sin[(n − N )ωc ], which gives F1 (z). The transition-band center of the resulting overall filter with the transfer function F1 (z) + F2 (z) is approximately located at ω = ωc . Because of the importance of center coefficients of an FIR filter, it is worth taking an arbitrary number of center coefficients separately, instead of only one center coefficient. This leads to our earlier design method [1] on how to create the overall transfer function for a linear-phase FIR filter.
M
Xm = [Nm , Nm+1 − 1] for m = 1, 2, . . . , M − 1
−1 jωc n
n=0
H(z) =
th-order direct-form In addition, H(z) is a conventional 2N transfer function with the additional impulse response coeffi = N − c + 1 with c = (T /2) and T is cients, h(n) and N the number of the additional coefficients at the center of the filter. means rounding upwards. The delay terms in (7) are used to shift the center of the symmetry at the desired location. First, the center impulse response of each term occurs at n = N . The impulse-response coefficients of the envelope transfer function, which were mentioned in the previous subsection and are indirectly included in the summation of (7), are multiplied with sin[(n − N )ωc )]. The arrival at 2N th-order Type 1 overall transfer function is guaranteed by these facts. In order to indicate that the overall filter has a piecewisepolynomial-sinusoidal impulse response the time interval n ∈ [0, N ] is divided into the following M subintervals:
r a(L) m (r)n × sin[ωc (n − (N − Nm ))] . (9)
XM = [NM , N ]
(11)
The following facts should be pointed out. First, X1 = [0, N2 − 1] because N1 = 0, Secondly, the overall impulse response can be studied up to n = N because of the even symmetry around this point. The impulse response on Xm can be expressed as h(n) =
M
hm (n),
(12)
m=1
where hm (n) =
m L (L) ak (r)(n − Nm )r × sin((n − N )ωc ) k=1
r=0
(13)
for m = 1, 2, · · · , M − 1, and M L (L) ak (r)(n − NM )r k=1 r=0 × sin((n − N )ωc ) + h (n − N ),
hM (n) =
(14)
which equals the overall impulse response and where h (n − N ) is a conventional direct-form Type 1 filter with nonzero coefficients for n = N − c + 1, . . . , N , in which c = T /2 and T is the number of separately generated additional center coefficients. Slices Nm s should be chosen so that |N2 − N1 | = |N3 − N2 | = · · · = |NM − NM−1 |, where N1 = 0 and M is the number of subintervals in the overall impulse response. Based on the above equations, in each Xm for m = 1, 2, . . . , M , a separate piecewise-polynomial-sinusoidal impulse response can be generated. In addition, in the XM , there are additional center coefficients, which are of great importance for fine-tuning the overall filter to meet the given criteria. Given the filter criteria as well as the design parameters ωc , M , N , L, Nm ’s, and the number of center coefficients in cluded in H(z), the overall problem is solvable by using linear
2053
e-j(N-L)wc
programming, instead of using ad-hoc nonlinear programming as used in the original Chu-Burrus approach.
e -j(N-L+1)wc
e j(L+1)/2 wc αL
+
e jwc
Z-
1
Z -1
In this section, an efficient implementation structure for the overall transfer function is shown based on the structure proposed in [1] as given by (7) and (9). The proposed structure is modified by an additional branch for the separately generated center coefficients, which do not follow the polynomials, and the branches are multiplied by additional complex terms. The subtransfer function, Hm (z), as given by (7), is expressed in the form: N −1 Hm (z) ≡h(0) (N )z −N + h(0) (n)[z −n + z −(2N −n) ],
1 Z-
e-j(N-3)wc 1 Z-
e -j(N-2)wc
+
β2 e -jNwc
e jwc
-
α3
+
α2
+
Z-
Fig. 1.
Z -1
Z-N
e jwc Z-
1
ejwc e jwc
+ -
(16)
Z -1
1 Z-
Z -1
IN
e jwc
e j2wc
+
e -j(N-1)wc
(15) where N is an integer corresponding to Nm −N in the previous section and h(0) (n) = pL (n) with an Lth-order modulated polynomial given by
α4
+ β4
1 Z-
n=0
a(L) (r)nr .
1 Z-
βL+1
1 Z-
p(L) (n) = sin(ωc (n − N )) ×
e jwc
+
III. A N I MPLEMENTATION S TRUCTURE FOR T YPE 1 F ILTER
L
αL+1
+
α1
Z -1
1 Z-
N -1
ej(N+1)wc
Im(OUT)
Basic implementation block for Type 1 filters.
r=0
The following polynomials are determined recursively for k = 1, 2, . . . , L as p(L−k) (n) = p(L−k+1) (n + 1) − p(L−k+1) (n) = (L−k) a(L−k) (r)nr , sin(ωc (n − N )) ×
(17)
r=0
where a
(L−k)
(L−k+1)
(r) =
j=r+1
j (L−k+1) (r). a r
(18)
The filter coefficients, αk s and β2k s, are related to the polynomials as given by (16), (17) and (18) through αk = p(L−k) (0) = a(L−k) (0), k = 1, 2, ..., L + 1,
(19)
and β2k = −p(L−2k+1) (N + 1 − k) − p(L−2k+1) (N − k), k = 1, 2, . . . , L/2. (20) Additionally, αL+1 = p(0) (0), whereas for odd L, βL+1 = −2p(0) . An efficient implementation block for the above transfer function is shown in Fig. 1 for odd L. For even L the uppermost parts are absent. Based on the above structure as well as our earlier implementation structure for Type 1 filters [1], a similar structure is developed for each of the branch filters, Hm (z). To generate the corresponding coefficients, αk s and β2k s, the impulse response given by (9) is used. (m) (m) The resulting αk s and β2k s are denoted by αk and β2k . An efficient implementation structure for the overall filter is shown in Fig. 2, where Tm = Nm for m = 1, 2, . . . , M − 1, TM1 = N − NM and TM2 = N − NM + 1, and H(z) is a conventional direct-form FIR filter for the separately b generated center coefficients with delay element z −N with
= N − c + 1. Furthermore, the β2k s are the sums of the N (m) corresponding β2k s. In order to keep the overall diagram (m) simple, the coefficients αk s have been used twice. In the practical implementation, the overall number of multipliers can be reduced by first adding [subtracting] the inputs of the left(m) (m) (m) hand side αk s and the right-hand side αk s [−αk s] and (m) then by multiplying the result by αk s. In Fig. 2 the complex multipliers are only drawn before the filter coefficients in feedforward loops because of the available space, i.e., the purpose is that the data do not become complex until after multiplication. IV. A D ESIGN E XAMPLE In this section, the properties and efficiency of the proposed filter classes for Type 1 FIR filters over FRM-based FIR filters are shown by means of an example in a wideband application. Consider a very narrow-transitionband design of linearphase FIR filters for ωp = 0.400π, ωs = 0.402π, ωc = 0.401 and with passband ripple δp = 0.01, stopband attenuation δs = 0.001 (60 dB). For the proposed approach, the given criteria are met by the filter of order 3381, M = 10, L = 2, the number of separate center coefficients, T = 7, N1 = 0, N2 = 400, N3 = 900, N4 = 1300, N5 = 1500, N6 = 1600, N7 = 1638, N8 = 1665, N9 = 1675, and N10 = 1683. This filter thus consists of ten slices with the polynomial degree L = 2 in each slice. In this case, the number of unknowns required both in the optimization and in the implementation is only 34. The number of multipliers is 35 + 51 in practical implementation. The magnitude and impulse response for the optimized filter are shown in Fig. 3. The corresponding FRM-based FIR filter in a three stage design [4] requires 94 multipliers, and the overall filter order
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βL+1
e j wc e-j(N-L)wc
e- jwc
α L+1 (1)
1 Z-
1 Z-
L+1
ejwc
α(1) L+1
−α(M) L+1
e jwc Z-
αL
(1)
e- jwc
e-j (N-L+1)wc
Z -1
α
α(M) L
Z -1
β4 - jwc
e
α(M) 4
jwc
e
Z -1
Z -1
α(M) 4
α(1) 4
e jwc
-j (N-2)wc
α3
(1)
α(M) 3
e- jwc
Z -1
−α(1) 3
−α(M3)
Z -1
α(1) 2
α(M) 2
- jwc
e
1 Z-
α2 e j (TM1 +1)wc
(M)
1 Z-
α(M) 1
e- jT1wc
T Z- 1
Z -1
Z -1
β2
e -j (N-1)wc
α(1) 1
e jwc 1 Z-
α4
(1)
e- jNwc
1
−α(1) L
−α(M) L
1 Z-
e-j (N-3)wc
e
Z -1
(M)
−α(M) 1
Z -1
Z - TM2
Z - TM1
e jwc
Z -1
α(1) 2
ejwc e jTM2 wc
−α(1) 1
Z -1
e j(N+1)wc
Z -T 1
Im
IN OUT
H(z)
Z -N
Fig. 2.
Implementation structure of Type 1 filter.
is 3196 compared to our design which requires 86 multipliers in a practical implementation, and a slightly increased filter order, 3381.
0 Magnitude in dB
−20 −40 −60 −80 −100 −120 0
0.1π
0.2π
0.3π
0.4π 0.5π 0.6π 0.7π Angular frequency ω
0.8π
0.9π
π
V. C ONCLUSION In this paper, we have shown a straightforward approach to synthesize linear-phase FIR filters with a piecewisepolynomial-sinusoidal impulse response for wideband specifications. An efficient implementation structure was shown for Type 1 FIR filters. An example has been given in order to show the benefits of the proposed structure. It was also found that the synthesis of the filter is the most efficient with the polynomial degree two. ACKNOWLEDGMENT This work was supported by the Academy of Finland, Project No. 213462 (Finnish Centre of Excellence Program (2006 - 2011)). R EFERENCES
Overall impulse response 0.6 0.4
0.5
0.2
Amplitude
0.4
0
0.3
1650
1700
1750
0.2 0.1 0 −0.1 0
Fig. 3.
500
1000
1500 2000 n samples
2500
3000
A design example of proposed linear-phase FIR filters.
[1] Lehto, R., Saram¨aki, T. and Vainio, O., “Synthesis of narrowband linearphase FIR filters with a piecewise-polynomial impulse response, ” Proc. IEEE Int. Symp. on Circuits and Systems, ISCAS 2005, Kobe, Japan, pp. 2012–2015, May 2005. [2] S. Chu and S. Burrus, “Efficient recursive realizations of FIR filters, Part I: The filter structures,” Circuits, Systems and Signal Processing, vol. 3, number 1, pp. 2–20, 1984. [3] S. Chu and S. Burrus, “Efficient recursive realizations of FIR filters, Part II: Design and applications,” Circuits, Systems and Signal Processing, vol. 3, number 1, pp. 21–57, 1984. [4] T. Saram¨aki, Y. C. Lim, “Use of the Remez Algorithm for Designing FRM based FIR Filters,” Circuits Systems Signal Processing, Birkhauser Boston, Vol. 22, No. 2, pp. 77-97, 2003. [5] T. Saram¨aki, “Finite Impulse Response Filter Design,” in Digital Signal Processing Handbook, S. K. Mitra, J. F. Kaiser, Ed., pp. 155–272, John Wiley and Sons, NY, 1993.
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