David Shiung, Jeng-Ji Huang
High-performance filtering (a flat passband, a sharp transition band, and a highly suppressed stopband) is always the ultimate goal of any digital signal processing (DSP) practitioner. However, high-performance filtering is traditionally the synonym of high implementation complexity. In this article, we propose an approach to reach high-performance filtering with lower implementation complexity. This approach designs a composite low-pass filter (LPF) by cascading a number of simple filters of diverse magnitude responses. In particular, this composite filter consists of prototype infinite impulse response (IIR) filters together with a mixture of shaping running-sum filters and their variations. These component shaping filters contain no multiplier at all. The numeric example shows that the filtering performance of our composite filter outperforms that of the conventional IIR filters (e.g., Chebyshev type 1 and elliptic filters) of higher complexity.
Over the past decades, we have already witnessed the wide applicability of DSP in versatile fields, e.g., biomedical engineering [1], wireless communication [2], [3], and audio processing [4]. Digital filtering undoubtedly plays a key role in realizing these fancy applications. High-performance filtering is highly desired by DSP practitioners, but it almost always accompanies high system complexity [5], [6]. Nevertheless, high system complexity can be reduced without sacrificing performance by the use of a cascade technique [7].
Consider designing a composite LPF with its passband edge at frequency ${\omega}_{p}$. Our solution to the problem of perfect filtering with reduced complexity is realized through cascading a prototype filter with a shaping filter. The central idea for building a composite filter is shown in Figure 1. Figure 1(a) shows the design of a prototype filter [with magnitude response $\mid{H}_{P}{(}{e}^{{j}{\omega}}{)}\mid{],}$ while Figure 1(b) shows the design of a shaping filter [with magnitude response $\mid{H}_{S}{(}{e}^{{j}{\omega}}{)}\mid{]}{.}$ In this figure, the prototype filter consists of M component filters, while the shaping filter consists of N component filters with various frequency magnitude responses. In particular, the shaping filter consists of running-sum filters [8] and their variant counterparts, i.e., interlaced running-sum filters and complementary comb filters. The prototype filter consists of elliptic and Chebyshev type 1 filters with filter order ${K}\prime = {1},{2}{.}$ The resulting magnitude response of the composite filter $\mid{H}_{C}{(}{e}^{{j}{\omega}}{)}\mid$ is in Figure 1(c), which is the product of magnitude responses of the shaping and the prototype filters. We normalize the dc response of all filters to unity throughout the whole article for easy illustration. The frequency response of the composite filter is as follows: \[{H}_{C}\left({{e}^{{j}{\omega}}}\right) = \mathop{\mbox{\Large $$}}\limits_{{m} = {1}}\limits^{M}{H}_{P,m}\left({{e}^{{j}{\omega}}}\right)\mathop{\mbox{\Large $$}}\limits_{{n} = {1}}\limits^{N}{H}_{S,n}\left({{e}^{{j}{\omega}}}\right){.}, \tag{1} \]
Figure 1. Cascading component prototype filters (a) with component shaping filters (b) for a high-performance composite filter (c).
${H}_{P,m}\left({{e}^{{j}{\omega}}}\right),{m} = {1},\ldots,{M},$ denote the frequency responses of the component prototype filters, while ${H}_{S,n}\left({{e}^{{j}{\omega}}}\right),$ ${n} = {1},\ldots,{N},$ are for the component shaping filters. Rewriting the magnitude of ${H}_{C}\left({{e}^{{j}{\omega}}}\right)$ in decibels, one has: \begin{align*}\begin{gathered}{{\left|{{H}_{C}\left({{e}^{{j}{\omega}}}\right)}\right|}_{dB} = \mathop{\mbox{\Large $\sum$}}\limits_{{m} = {1}}\limits^{M}{\left|{{H}_{P,m}({e}^{{j}{\omega}})}\right|}_{dB}}\\{ + \mathop{\mbox{\Large $\sum$}}\limits_{{n} = {1}}\limits^{N}{\left|{{H}_{S,n}({e}^{{j}{\omega}})}\right|}_{dB}{.}}\end{gathered}, \tag{2} \end{align*}
We can apply (2) by first designing the passband edge frequency of the prototype filter (and all its component filters) ${{\omega}\prime}_{p}$ slightly over ${\omega}_{p},$ and then shape it by the following shaping filter. In particular, the maximum magnitude response of all of the component prototype filters with ${K}\prime = {2}$ is deliberately designed at frequency ${\omega}_{p},$ so as to provide magnitude margin for shaping. The design procedure is similar to tile mosaic and finally a near-perfect composite filter is obtained.
The central idea of cascade technique for filter design is to cascade a number of low-performance filters for high-performance filtering [9]. This idea is realized through either linear-phase (e.g., [6] and [10]) or nonlinear-phase composite filters (e.g., [5] and [7]). These works mainly use simple shaping filters, e.g., comb filters and complementary comb filters, in cascade with a prototype filter. By (2), we see that this idea does reduce system complexity because parts of ${\left|{{H}_{C}\left({{e}^{{j}{\omega}}}\right)}\right|}_{dB}$ are achieved by the simple shaping filter. Inspired by the success of these preliminary designs regarding cascading two or more subfilters for high-performance filtering, in this article we fully investigate the potential of the cascade technique by introducing the technique of filter diversity. Here the terminology filter diversity means cascading of subfilters with various filter types and orders. In contrast with cascade technique, interpolated finite impulse response (FIR) filters use an interpolation technique to approach this goal [9], [11]. These approaches are compared in the numerical example.
We use two tricks for designing the composite LPF. The first trick is to use the passband ripple of the shaping filter to compensate for the passband ripple of the prototype filter at frequency ${\omega}_{p}{.}$ This trick produces a flat passband for the composite filter. The second trick is to diversify the constituents of the prototype and shaping filters. This ensures a sharp transition band and a highly suppressed stopband of the composite filter. These two tricks are illustrated in Figure 1.
Notice that the component shaping filters are diverse in shaping capabilities and contain no multiplier at all. In addition, low-order IIR filters, e.g., elliptic filters and Chebyshev type 1 filters, sometimes can be further simplified due to their symmetric or specific filter coefficients. These two points significantly reduce the overall system complexity and finally we obtain a high-performance LPF with reduced complexity.
For the prototype filter, we intentionally designed the passband edge frequency of all its component filters at ${{\omega}\prime}_{p},$ where ${{\omega}\prime}_{p}{>}{\omega}_{p}{.}$ In addition, we chose a passband ripple, e.g., 12 dB, for all the component filters if applicable [i.e., ${H}_{P,2}({e}^{{j}{\omega}})$ and ${H}_{P,3}({e}^{{j}{\omega}})$ in Figure 1]. Note that ${H}_{P,1}({e}^{{j}{\omega}})$ has a downward passband ripple. A total of three kinds of component filters can be used for building the prototype filter, since both the elliptic and Chebyshev type 1 filters with ${K}\prime = {1}$ have exactly the same magnitude response. The component prototype filters can be designed using some commercial software packages, e.g., Matlab, and their parameters, e.g., passband ripple, should be optimized as detailed in the following paragraphs.
Figure 2 shows the relationship between ${{\omega}\prime}_{p}$ and ${\omega}_{p}$ for both the elliptic and Chebyshev type 1 filters of filter order ${K}\prime = {2}{.}$ Their magnitude responses are shown in Figure 1(a). Interestingly this relationship remains unchanged no matter how large the passband ripple is. From Figure 2, we know that for a composite filter with a passband edge frequency ${\omega}_{p} = {0}{.}{1}{\pi}$ (rad./sample), ${{\omega}\prime}_{p}$ should be set to 0.1398π (rad./sample).
Figure 2. Relationship between ${{\omega}\prime}_{p}$ and ${\omega}_{p}$ for both the elliptic and Chebyshev type 1 filters of filter order ${K}\prime = {2}{.}$
Determining the number of component prototype filters provided the order of prototype filter is K, where ${K} = {1},{2},{3},\ldots{.}$ Note that K is the sum of filter order of all component prototype filters. Let the number of order-1 and order-2 component prototype filters be ${k}_{1}$ and ${k}_{2},$ respecctively. ${k}_{1}$ and ${k}_{2}$ are nonnegative integers. We can relate: \[{k}_{1} + {2}{k}_{2} = {K}{.}, \tag{3} \]
A practical solution for maximizing the magnitude margin for shaping is: \[\left({{k}_{1},{k}_{2}}\right) = \left({{K}{-}{2}\left\lfloor{\frac{K}{2}}\right\rfloor,\left\lfloor{\frac{K}{2}}\right\rfloor}\right){.}, \tag{4} \]
The notation $\left\lfloor{\cdot}\right\rfloor$ denotes the floor function, which rounds toward negative infinity. For example, when ${K} = {4},$ we have: $\left({{k}_{1},{k}_{2}}\right) = {(}{0},{2}{)}{.}$
We diversify the component shaping filters through using running-sum filters, interlaced running-sum filters, and complementary comb filters of various filter orders. These three types of FIR filters contain no multiplier and are stable in any condition.
The system function of an ${order} {-} {L}_{1}$ running-sum filter is [8]: \[{H}_{RS}\left({z}\right) = {1} + {z}^{{-}{1}} + {z}^{{-}{2}} + {z}^{{-}{3}} + \cdots + {z}^{{-}{L}_{1}}, \tag{5} \] where ${L}_{1} = {0},{1},{2},{3},\ldots{.}$ By the sum formula of geometric series, (5) can be simplified as: \[{H}_{RS}\left({z}\right) = \frac{{1}{-}{z}^{{-}{(}{L}_{1} + {1}{)}}}{{1}{-}{z}^{{-}{1}}}{.}, \tag{6} \]
The frequency response of a running-sum filter is obtained by substituting ${z} = {e}^{{j}{\omega}}$ into (5) or (6). The first null of a running-sum filter ${\omega}_{RS}$ can be obtained by setting (6) to zero; that is, \begin{align*}{\omega}_{RS} = \left\{{\begin{array}{ll}{\frac{{2}{\pi}}{{L}_{1} + {1}},}&{{L}_{1} = {1},{2},{3},\ldots}\\{\text{NA},}&{{L}_{1} = {0}}\end{array}{.}}\right., \tag{7} \end{align*}
To prevent the passband of the prototype filter from being nulled by the shaping filter, one has to set ${\omega}_{RS}{>}{\omega}_{p}{.}$ This implies ${L}_{1}$ should be restricted to the feasible range ${L}_{1} = {1},{2},{3},\ldots,\left\lfloor{{2}{\pi}{/}{\omega}_{p}}\right\rfloor{-}{1}{.}$
We define the system function of an interlaced running-sum filter as: \[{H}_{IRS}\left({z}\right) = {1} + {z}^{{-}{2}} + {z}^{{-}{4}} + {z}^{{-}{6}} + \cdots + {z}^{{-}{L}_{2}}, \tag{8} \] where ${L}_{2} = {0},{2},{4},{6},\ldots{.}$ This filter can be regarded as a modified version of a running-sum filter, with filter coefficients interlaced by 0 and 1. (8) can be further simplified as: \[{H}_{IRS}\left({z}\right) = \frac{{1}{-}{z}^{{-}{(}{L}_{2} + {2}{)}}}{{1}{-}{z}^{{-}{2}}}{.}, \tag{9} \]
Similarly, we obtain the first null of an interlaced running-sum filter: \begin{align*}{\omega}_{IRS} = \left\{{\begin{array}{ll}{\frac{{2}{\pi}}{{L}_{2} + {2}},}&{{L}_{2} = {2},{4},{6},\ldots}\\{\text{NA},}&{{L}_{2} = {0}}\end{array}}\right.{.}, \tag{10} \end{align*}
For a feasible choice of ${L}_{2},$ one has to consider the constraint ${\omega}_{IRS}{>}{\omega}_{p}{.}$ This implies ${L}_{2}$ should be restricted to the range: ${L}_{2} = {2},{4},{6},\ldots,\left\lfloor{{2}{\pi}{/}{\omega}_{p}}\right\rfloor{-}{2}{.}$
The system function of an order- ${L}_{3} {-} $ complementary comb function is [6]: \[{H}_{CCF}\left({z}\right) = {1} + {z}^{{-}{L}_{3}}, \tag{11} \] where ${L}_{3} = {0},{1},{2},{3},\ldots{.}$ It can also be regarded as a variation of running-sum filter. The first null of a complementary comb filter is: \begin{align*}{\omega}_{CCF} = \left\{{\begin{array}{ll}{\frac{\pi}{{L}_{3}},}&{{L}_{3} = {1},{2},{3},\ldots}\\{\text{NA},}&{{L}_{3} = {0}}\end{array}{.}}\right., \tag{12} \end{align*}
We still have to take into account the constraint ${\omega}_{CCF}{>}{\omega}_{p}{.}$ This further constrains ${L}_{3}$ to the range ${L}_{3} = {1},{2},{3},\ldots,\left\lfloor{{\pi}{/}{\omega}_{p}}\right\rfloor{.}$
Since the shaping filter consists of a series of subfilters with diverse filter orders, it is important to know their attenuations at ${\omega}_{p}{.}$ The attenuation of magnitude response of filter order L, at frequency ${\omega}_{p},$ with respect to its dc response can be represented as: \[{\Delta}_{\text{Type},L} = {20}{\log}_{10}\frac{{\left|{{H}_{Type}\left({{e}^{{j}{\omega}}}\right)}\right|}_{{\omega} = {0}}}{{\left|{{H}_{Type}\left({{e}^{{j}{\omega}}}\right)}\right|}_{{\omega} = {\omega}_{p}}}{.}, \tag{13} \]
To simplify the notation, Type can be one of “<I>”, “<II>”, or “<III>”, which corresponds to the running-sum, the interlaced running-sum, and the complementary comb filters, respectively. For example, ${\Delta}_{{<}{I}{>},{2}}$ denotes the attenuation for a running-sum filter of filter order-2. We can collect the attenuations of all feasible filter orders and reorganize them into a vector by ${\mathbf{\Delta}}_{{<}{I}{>}} = $$\left({{\Delta}_{{<}{I}{>},{1}},{\Delta}_{{<}{I}{>},{2}},\ldots,{\Delta}_{{<}{I}{>},\left\lfloor{{2}{\pi}{/}{\omega}_{p}}\right\rfloor{-}{1}}}\right){.}$ We denote the number of subfilters for composing the shaping filter in a vector form by ${\mathbf{k}}_{{<}{I}{>}} = \left({{k}_{{<}{I}{>},{1}},{k}_{{<}{I}{>},{2}},\ldots,{k}_{{<}{I}{>},\left\lfloor{{2}{\pi}{/}{\omega}_{p}}\right\rfloor{-}{1}}}\right){.}$ These notations can be extended to that for the interlaced running-sum filter and complementary comb filter with a modification on the suffixes of ${\mathbf{\Delta}}_{{<}{I}{>}}$ and ${\mathbf{k}}_{{<}{I}{>}}{.}$
In the following section, we demonstrate an example to show the design details and how they perform.
The design specifications and constraints on complexity include: 1) passband edge frequency ${\omega}_{p} = {0}{.}{1}{\pi}$ (rad./sample), 2) maximum passband peak-to-peak ripple 2.78 dB, 3) stopband edge frequency ${\omega}_{s} = {0}{.}{2}{\pi}$ (rad./sample), 4) prototype filter order ${K} = {4},$ and 5) maximum number of component shaping filters of each type is 3. There is no constraint on the suppression of stopband.
Since the prototype filter order is ${K} = {4},$ by (4) we have $\left({{k}_{1},{k}_{2}}\right) = {(}{0},{2}{).}$ This implies that the composite filters contain two order-2 component prototype filters and no order-1 filter. Let one component prototype filter be a Chebyshev type 1 filter, and the other one be an elliptic filter due to diversity. Both filters have an identical passband edge frequency ${{\omega}\prime}_{p} = {0}{.}{1398}{\pi}$ (rad./sample) and identical passband ripples, i.e., ${R}_{p1} = {R}_{p2} = {R}_{p}{/}{2},$ where ${R}_{p}$ denotes the magnitude of passband ripple of the prototype filter. The design specifications are now transformed into a mathematical optimization problem, whose objective function is: \[\mathop{\min}\limits_{{R}_{p},{\mathbf{k}}_{{<}{I}{>}},{\mathbf{k}}_{{<}{II}{>}},{\mathbf{k}}_{{<}{III}{>}}}{\mathop{\int}\nolimits_{{\omega}_{s}}\nolimits^{\pi}{\left|{{H}_{C}({e}^{{j}{\omega}})}\right|}}^{2}{d}{\omega},, \tag{14} \] subject to \[{0}\leq{R}_{p}\leq{40}, \tag{15} \] \begin{align*}\begin{gathered}{{R}_{p}{-}{\epsilon}\leq{\mathbf{k}}_{{<}{I}{>}}{\mathbf{\Delta}}_{{<}{I}{>}}{}^{t} + {\mathbf{k}}_{{<}{II}{>}}{\mathbf{\Delta}}_{{<}{II}{>}}{}^{t}} \\{ + {\mathbf{k}}_{{<}{III}{>}}{\mathbf{\Delta}}_{{<}{III}{>}}{}^{t}\leq{R}_{p} + {\epsilon},} \end{gathered}, \tag{16} \end{align*} \begin{align*}\begin{gathered}{{0}\leq\mathop{\max}\limits_{[0,{\omega}_{p}]}{\left|{{H}_{C}({e}^{{j}{\omega}})}\right|}_{dB}} \\{{-}\mathop{\min}\limits_{[0,{\omega}_{p}]}{\left|{{H}_{C}({e}^{{j}{\omega}})}\right|}_{dB}\leq{2}{.}{78},} \end{gathered}, \tag{17} \end{align*} \[{0}\leq\mathop{\mbox{\Large $\kern-0.7em\sum$}}\limits_{{\kern-0.3emL}_{1}\kern-0.3em = {\kern-0.3em1}}\limits^{\left\lfloor{\frac{{2}{\pi}}{{\omega}_{p}}}\right\rfloor{-}{1}}{\kern-0.9emk}_{{<}{I}{>},{L}_{1}}\kern-0.9em\leq{\kern-0.9em3}{\kern-0.9em,}, \tag{18} \] \[{0}\leq\mathop{\mbox{\Large $\kern-0.8em\sum$}}\limits_{{L}_{2} = {2},{even}}\limits^{\left\lfloor{\frac{{2}{\pi}}{{\omega}_{p}}}\right\rfloor{-}{2}}{\kern-1.0emk}_{{<}{II}{>},{L}_{2}}\kern-1.0em\leq{\kern-1.0em3}{\kern-1.0em,}, \tag{19} \] \[{0}\leq\mathop{\mbox{\Large $\sum$}}\limits_{{L}_{3} = {1}}\limits^{\left\lfloor{\frac{\pi}{{\omega}_{p}}}\right\rfloor}{k}_{{<}{III}{>},{L}_{3}}\leq{3}{.}, \tag{20} \]
In (15), the search range of ${R}_{p}$ extends from 0 to 40, so that the optimal solution is well inside the region. (17) constrains the passband peak-to-peak ripple by 2.78. The problem outlined in (14) to (20) is a mixed-integer programming problem. Unfortunately, the solution space is too large to use brute-force search. In addition, it is nonlinear and no off-the-shelf software package can be used. Nevertheless, the computational complexity of the problem can be reduced if we start from (16) and (18) to (20). We estimate ${R}_{p}$ by using successive zoom-in method [12] and associate the estimate of ${R}_{p}$ with candidates of ${\mathbf{k}}_{{<}{I}{>}},$ ${\mathbf{k}}_{{<}{II}{>}},$ and ${\mathbf{k}}_{{<}{III}{>}}{.}$ The candidate solutions are further examined by (17) and finally by (14). The successive zoom-in method is useful when the search range is continuous and the objective function is well-posed. The method first slices the search range of ${R}_{p}$ into coarse grids and then slices the neighborhood of the candidate solution using fine grids. Usually, the optimal solution converges quickly and can be obtained in a few rounds. The solution set is obtained as follows: ${\mathbf{k}}_{{<}{I}{>}} = \left({0,0,0,0,0,0,0,1,1,1,0,0,0,0,}\right.$$\left.{{0},{0},{0}}\right),$${\mathbf{k}}_{{<}{II}{>}} = \left({{0},{0},{2},{1},{0},{0},{0},{0}}\right),$ ${\mathbf{k}}_{{<}{III}{>}} = \left({0,0,2,1,0,0,0,0,0}\right),{R}_{p} = {24}{.}$ The system function of the composite filter can be written as: \begin{align*}\begin{gathered}{{H}_{C}\left({z}\right) = \frac{{1} + {1}{.}{919}{z}^{{-}{1}} + {z}^{{-}{2}}}{{1}{-}{1}{.}{827}{z}^{{-}{1}} + {0}{.}{924}{z}^{{-}{2}}}} \\{\times\frac{{1} + {2}{z}^{{-}{1}} + {z}^{{-}{2}}}{{1}{-}{1}{.}{827}{z}^{{-}{1}} + {0}{.}{924}{z}^{{-}{2}}}\cdot\frac{{1}{-}{z}^{{-}{9}}}{{1}{-}{z}^{{-}{1}}}} \\{\times\frac{{1}{-}{z}^{{-}{10}}}{{1}{-}{z}^{{-}{1}}}\cdot\frac{{1}{-}{z}^{{-}{11}}}{{1}{-}{z}^{{-}{1}}}\cdot{\left({\frac{{1}{-}{z}^{{-}{8}}}{{1}{-}{z}^{{-}{2}}}}\right)}^{2}} \\{\times\frac{{1}{-}{z}^{{-}{10}}}{{1}{-}{z}^{{-}{2}}}\cdot{(}{1} + {z}^{{-}{3}}{)}^{2}\cdot{(}{1} + {z}^{{-}{4}}{).}} \end{gathered}, \tag{21} \end{align*}
Note, (21) is already reorganized for brevity. We verify the poles of the two component prototype filters are inside the unit circle and our composite LPF is stable in any condition.
Figure 3 shows the frequency magnitude responses of the two composite prototype filters and that of the shaping filter. Also shown in Figure 3 is the frequency magnitude response of the composite filter for comparison. They all perform as expected.
Figure 3. Frequency magnitude responses of the composite filter and its three component filters (two component prototype filters and one shaping filter).
Figure 4 compares the frequency magnitude responses of our composite filter and the other three filters. Two of the three filters are conventional IIR filters of order-6, while the last one is a recent design in [7] with no filter diversity. They all meet the design specifications. Note, by (21) the order of the composite filter is $70$ ${(}{70} = {2} + {2} + {9} + {10} + {11} + {8}\times{2} + $${10} + {3}\times{2} + {4}{).}$ We find that our composite filter outperforms the other three filters in both the stopband and transition-band responses.
Figure 4. Comparison of the frequency magnitude responses of the composite filter with that of the other three filters.
Table 1 shows a comparison of the five filters on complexity. Let the component prototype and shaping filters be in direct-form II [13]. From (6), (9), (11), and (21) we know the composite filter needs a total of five multipliers and $23$ ${(}{4}\times{2} + {6}\times{2} + {3}\times{1} = {23}{)}$ adders. Note multiplication by 2 involves only bit shifting. Also shown in Table 1 for comparison is a linear-phase interpolated FIR filter and the work in [7]. Although our composite filter needs more adders than the other three filters, less multiplier is highly desirable for very large-scale integration (VLSI) implementations [14].
Table 1. Comparison of complexity among the five filters.
Like all conventional filter design methods that lever filtering performance under constrained complexity, our composite filter is of no exception. For an LPF with a wide passband, e.g., ${\omega}_{p} = {0}{.}{45}{\pi}$ (rad./sample) and ${\omega}_{s} = {0}{.}{55}{\pi}$ (rad./sample), one has ${L}_{1} = {1},{2},{3},$ ${L}_{1} = {2},$ and ${L}_{1} = {1},{2}{.}$ There are barely enough candidates to design the component shaping filters. Our composite filter then consistently has a passband peak-to-peak ripple around 1∼4 dB depending on different optimization criterions and performs comparably with conventional IIR filters of order-$6$. If some degree of passband ripple is permitted, the composite filter is still an appealing choice with reduced complexity.
In this article, we realize a high-performance composite LPF by cascading a prototype filter with a shaping filter, both of which consist of a number of component filters. The first trick for designing the composite LPF is to cancel the deliberately produced passband ripple of the prototype filter by the ripple of the shaping filter. This trick produces a flat passband for the composite filter. The second trick is to diversify the constituents of the prototype and shaping filters. This ensures a sharp transition band and a highly suppressed stopband for the composite filter. Diversifying the component filters by using versatile filters, e.g., order-2 Chebyshev type 1 filter, running-sum filter, and the variations, does reduce filter complexity in the architecture level and improve overall filtering performance. Further VLSI implementations focusing on timing performance, hardware cost, power consumption, etc., can be the next step to further elaborate the low-complexity benefit of the proposed composite filter.
David Shiung (davids@cc.ncue.edu.tw) received his Ph.D. degree from National Taiwan University, Taipei, Taiwan in 2002. He is an associate professor at National Changhua University of Education, Changhua 500, Taiwan. His research interests include signal processing for wireless communication and astronomical imaging. He is a Member of IEEE.
Jeng-Ji Huang (hjj2005@ntnu.edu.tw) received his Ph.D. degree from National Taiwan University, Taipei, Taiwan in 2004. He is a professor at National Taiwan Normal University, Taipei 106, Taiwan. His research interests include 5G, LoRaWAN, and VANET. He is a Member of IEEE.
[1] E. Ozpolat, B. Karakaya, T. Kaya, and A. Gulten, “FPGA-based digital filter design for biomedical signal,” in Proc. XII Int. Conf. Perspective Technol. Methods MEMS Des. (MEMSTECH), Lviv, Ukraine, Apr. 2016, pp. 70–73, doi: 10.1109/MEMSTECH.2016.7507523.
[2] I. Pirnog, I. Marcu, A. M. C. Drãgulinescu, and C. Oprea, “Digital filters for sigma-delta modulation in wireless communications,” in Proc. Int. Semicond. Conf. (CAS), Sinaia, Romania, Oct. 2019, pp. 111–114, doi: 10.1109/SMICND.2019.8923999.
[3] S. M. Perera et al., “Wideband N-beam arrays using low-complexity algorithms and mixed-signal integrated circuits,” IEEE J. Sel. Topics Signal Process., vol. 12, no. 2, pp. 368–382, May 2018, doi: 10.1109/JSTSP.2018.2822940.
[4] J. Jiang, “Audio processing with channel filtering using DSP techniques,” in Proc. IEEE 8th Annu. Comput. Commun. Workshop Conf. (CCWC), Las Vegas, NV, USA, Jan. 2018, pp. 545–550, doi: 10.1109/CCWC.2018.8301696.
[5] D. Shiung, Y.-Y. Yang, and C.-S. Yang, “Cascading tricks for designing composite filters with sharp transition bands,” IEEE Signal Process. Mag., vol. 33, no. 1, pp. 151–162, Jan. 2016, doi: 10.1109/MSP.2015.2477420.
[6] D. Shiung, Y.-Y. Yang, and C.-S. Yang, “Improving FIR filters by using cascade techniques,” IEEE Signal Process. Mag., vol. 33, no. 3, pp. 108–114, May 2016, doi: 10.1109/MSP.2016.2519919.
[7] D. Shiung, “A trick for designing composite filters with sharp transition bands and highly suppressed stopbands,” IEEE Signal Process. Mag., vol. 39, no. 5, pp. 70–76, Sep. 2022, doi: 10.1109/MSP.2022.3165960.
[8] J. H. McClellan, R. W. Schafer, and M. A. Yoder, DSP First, 2nd ed. New York, NY, USA: Pearson, 2015.
[9] R. G. Lyons, Understanding Digital Signal Processing, 3rd ed. Boston, MA, USA: Person, 2011.
[10] W.-S. Lu and T. Hinamoto, “Design of least-squares and minimax composite filters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 65, no. 3, pp. 982–991, Mar. 2018, doi: 10.1109/TCSI.2017.2772345.
[11] Y. Neuvo, D. Cheng-Yu, and S. K. Mitra, “Interpolated finite impulse response filters,” IEEE Trans. Acoust., Speech, Signal Process., vol. 32, no. 3, pp. 563–570, Jun. 1984, doi: 10.1109/TASSP.1984.1164348.
[12] D. Shiung, P.-H. Hsieh, and Y.-Y. Yang, “Parallels between wireless communication and astronomical observation,” in Proc. IEEE 29th Annu. Int. Symp. Pers. Indoor Mobile Radio Commun. (PIMRC), Bologna, Italy, Sep. 2018, pp. 1–6, doi: 10.1109/PIMRC.2018.8580926.
[13] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. New York, NY, USA: Pearson, 2010.
[14] K. K. Parhi, VLSI Digital Signal Processing Systems: Design and Implementation, 1st ed. New York, NY, USA: Wiley, 1999.
Digital Object Identifier 10.1109/MSP.2023.3247903