David Shiung, Jeng-Ji Huang, Ya-Yin Yang
Designing filters with perfect frequency responses (i.e., flat passbands, sharp transition bands, highly suppressed stopbands, and linear phase responses) is always the ultimate goal of any digital signal processing (DSP) practitioner. High-order finite impulse response (FIR) filters may meet these requirements when we put no constraint on implementation complexity. In contrast to FIR filters, infinite impulse response (IIR) filters, owing to their recursive structures, provide an efficient way for high-performance filtering at reduced complexity. However, also due to their recursive structure, IIR filters inherently have nonlinear phase responses, and this does restrain their applicability. In this article, we propose two tricks regarding cascading a prototype IIR filter with a few shaping all-pass filters (APFs) for an almost linear phase response over its passband. After performing a delicate design on the prototype and shaping filters, we approach perfect filtering with reduced complexity.
Over the past decades, we have witnessed the power of DSP in various fields of applications, e.g., wireless communication [1], [2], seismology [3], and biomedical sciences [4]. Digital filtering undoubtedly plays an important role in realizing these fancy applications. When filtering with a linear phase response, the intended signal merely experiences a constant group delay and preserves its waveform. This is a vital feature for many applications, e.g., the denoising of electrocardiography (ECG) records [4] and seismologic signals [3].
Traditionally, high-performance filtering with linear phase responses is achievable by high-order FIR filters. This, in turn, increases the system complexity (the number of adders and multipliers), although there are techniques to cut the system complexity in half by folding the symmetric filter coefficients [5]. The filter cascade technique can be used for designing filters with reduced complexity, e.g., the composite filter in [6] and interpolated FIR filters [5], [7]. A comprehensive survey regarding the design techniques of FIR filters is presented in [8]. Among these design techniques, the works in [9] and [10] extend the idea of interpolated FIR filters by first designing a bandpass filter and then modifying it by shaping the model filter by using some masking filters. This technique is an attractive candidate for obtaining a filter with a sharp transition band. The results show supremacy over other designs in terms of filtering performance and implementation complexity. However, there still exists room for further improvement. In contrast to FIR filters, IIR filters do achieve high-performance filtering with low system complexity due to their recursive structures. But also due to their recursive structures, IIR filters are unable to provide linear phase responses. In this article, we provide our solution to the problem of perfect filtering with reduced complexity. Our solution is realized through cascading a prototype IIR filter with a few shaping APFs for an almost linear phase response over the passband. The proposed composite filter can be used to replace any ordinary FIR filter with fixed filter coefficients. We then approach perfect filtering with reduced complexity.
APFs have wide applications in the fields of DSP and communication [11], [12], [13]. Ideally, an APF has a constant magnitude response over the whole frequency bands. The novelty of this article is that a cascade of some IIR filters can produce a composite filter with an almost linear phase response over the filter passband. In particular, the filtering performance of this composite filter can perform quite close to a high-order (thus, high-complexity) FIR filter. This composite filter provides a way to perfect filtering by using limited complexity. We first introduce the design of a composite low-pass filter (LPF) through first designing a prototype filter that meets the design specifications for the magnitude response. Then, the prototype filter is cascaded with a few shaping APFs to rebuild an almost linear phase response over the filter passband. This idea is then extended to designing a high-pass filter (HPF). The example filter shows that the intended signal waveform is preserved, while the unwanted signal is highly suppressed.
The transfer function of a feasible APF suitable for our composite filter is of the form [14] \[{H}_{\text{ap}}\left({z}\right) = \frac{{z}^{{-}{1}}{-}{a}^{\ast}}{{1}{-}{az}^{{-}{1}}} \tag{1} \] where ${a} = {re}^{{j}{\theta}}$ is the complex pole of ${H}_{\text{ap}}\left({z}\right)$ and ${0}\leq{\theta}\lt{2}{\pi}$. For stability, we need ${0}\leq{r}\lt{1}$. By substituting ${z} = {e}^{{j}{\omega}}$ into (1), the frequency response of the APF can be written as \begin{align*}{H}_{\text{ap}}{(}{\omega}{)} & = \frac{{e}^{{-}{j}{\omega}}{-}{a}^{\ast}}{{1}{-}{ae}^{{-}{j}{\omega}}} \\ & = {e}^{{-}{j}{\omega}}\frac{{1}{-}{a}^{\ast}{e}^{{j}{\omega}}}{{1}{-}{ae}^{{-}{j}{\omega}}}{.} \tag{2} \end{align*}
We can confirm that ${H}_{\text{ap}}\left({\omega}\right)$ has a unity magnitude that is independent of ${\omega}$. Reorganizing (2), the phase function of ${H}_{\text{ap}}\left({\omega}\right)$ is \begin{align*}{∡}{H}_{\text{ap}}\left({\omega}\right) = & {-}{\omega}{-}{2}\,{\tan}^{{-}{1}} \\ & \left({\frac{{r}\sin{(}{\omega}{-}{\theta}{)}}{{1}{-}{r}\cos{(}{\omega}{-}{\theta}{)}}}\right){.} \tag{3} \end{align*}
The group delay and phase delay associated with a system with frequency response ${H}\left({\omega}\right)$ are defined as \[{\tau}_{\text{gr}}\left({\omega}\right){≜}{-}\frac{{d}{∡}{H}\left({\omega}\right)}{{d}{\omega}} \tag{4} \] and \[{\tau}_{\text{ph}}\left({\omega}\right){≜}{-}\frac{{∡}{H}\left({\omega}\right)}{\omega} \tag{5} \] respectively. The group delay and phase delay have an important implication for an APF. If a narrow-band sequence ${x}{(}{n}{)} = s(n)\cos({\omega}_{0}n)$ is passed through an APF, the filter output $y(n)$ becomes [14] \begin{align*}{y}{(}{n}{)} = &{s}{(}{n}{-}{\tau}_{\text{gr}}\left({{\omega}_{0}}\right){)}\cos \\ & \left({{\omega}_{0}\left({{n}{-}{\tau}_{\text{ph}}\left({{\omega}_{0}}\right)}\right)}\right){.} \tag{6} \end{align*}
By definition, the group delay of ${H}_{\text{ap}}{(}{\omega}{)}$ can be presented as \begin{align*}{\text{grd}}\left[{{H}_{\text{ap}}\left({\omega}\right)}\right] & = \frac{{1}{-}{r}^{2}}{{1} + {r}^{2}{-}{2}{r}\cos{(}{\omega}{-}{\theta}{)}} \\ & = \frac{{1}{-}{r}^{2}}{{\left|{{1}{-}{re}^{{j}{\theta}}{e}^{{-}{j}{\omega}}}\right|}^{2}} \tag{7} \end{align*}
Since ${0}\leq{r}\lt{1}$ for a stable APF, by (7), we confirm that the group delay is always positive for all frequency bands.
Consider the design of a real-coefficient APF. If the filter order is one, we can set the angle of pole ${\theta}$ to zero or ${\pi}$. If the filter order is two, its transfer function can be the multiplication of that of two order 1 APFs with complex conjugate poles $a$ and ${a}^{\ast}$, respectively. This is because a real-coefficient equation has paired conjugate roots. Thus, the transfer function of a real-coefficient shaping filter can be defined as \begin{align*} & {H}_{S}\left({z}\right){≜} \\ & \quad\begin{cases}{\begin{array}{ll}{\frac{{z}^{{-}{1}}{-}{r}}{{1}{-}{rz}^{{-}{1}}},}&{\theta} = {0} \\ \frac{{z}^{{-}{1}}{-}{a}^{\ast}}{{1}{-}{az}^{{-}{1}}}\cdot\frac{{z}^{{-}{1}}{-}{a}}{{1}{-}{a}^{*}{z}^{{-}{1}}},&{0}\lt{\theta}\lt{\pi} \\ \frac{{z}^{{-}{1}} + {r}}{{1} + {rz}^{{-}{1}}},&{{\theta} = {\pi}}\end{array}}{.}\end{cases} \tag{8} \end{align*}
Note that the shaping filter defined in (8) still has a frequency magnitude response independent of ${\omega}$. Extending the result in (7) for the transfer function defined in (8), we have \begin{align*}&{\text{grd}}\left[{H}_{S}\left({\omega}\right)\right] = \\ &\quad\begin{cases}{\begin{array}{l} \frac{{1}{-}{r}^{2}}{{1} + {r}^{2}{-}{2}{r}\cos{(}{\omega}{)}},{\theta} = {0} \\ \frac{{1}{-}{r}^{2}}{{1} + {r}^{2}{-}{2}{r}\cos{(}{\omega}{-}{\theta}{)}} + \\ \,\,\,\frac{{1}{-}{r}^{2}}{{1} + {r}^{2}{-}{2}{r}\cos{(}{\omega} + {\theta}{)}},{0}\lt{\theta}\lt{\pi} \\ \frac{{1}{-}{r}^{2}}{{1} + {r}^{2} + {2}{r}\cos{(}{\omega}{)}},{\theta} = {\pi}\end{array}}{.}\end{cases} \tag{9} \end{align*}
By (7) and (9), we know that the group delay contributed by a real-coefficient APF is always positive. From (4), we also know that rebuilding the frequency phase response of a composite filter is equivalent to rebuilding its group delay. Four shaping filters with parameters $\left({{r},{\theta}}\right) = \left({0.9,0}\right),\left({{0}{.}{9},{0}{.}{2}{\pi}}\right),\left({{0}{.}{8},{0}{.}{3}{\pi}}\right)$, and $\left({{0}{.}{9},{\pi}}\right)$ are shown in Figure 1, where the frequency is normalized to ${\omega} = {\pi}$ (radians/sample). Here, ${\theta}$ corresponds to the peak of the group delay curve, while $r$ controls its shape. The choices of $r$ and ${\theta}$ provide degrees of freedom in shaping the group delay of the composite filter. Obviously, our design freedom increases when more shaping APFs are used.
Figure 1. The group delays of the shaping APFs for $\left({{r},{\theta}}\right) = \left({0.9,0}\right){, }\left({{0}{.}{9},{0}{.}{2}{\pi}}\right),\left({{0}{.}{8},{0}{.}{3}{\pi}}\right)$, and $\left({{0}{.}{9},{\pi}}\right)$, respectively.
The central idea of the cascade technique for filter design is to build a high-performance filter by cascading a number of low-performance filters [5]. This technique can be used to sharpen the transition band or to suppress the stopband of the prototype filter [6], [15], [16], [17]. In this article, we focus on cascading a prototype filter with ${M}$ shaping filters for an almost linear phase response. The relationship of the input and output sequences for the composite filter is presented in Figure 2. The ${z}$-domain input and output sequences are denoted by $X(z)$ and $Y(z)$, respectively. In particular, the prototype filter with a transfer function ${H}_{P}\left({z}\right)$ is an IIR filter for meeting the specifications regarding the magnitude response. A family of ${M}$ shaping APFs with transfer functions ${H}_{S,m}\left({z}\right),{m} = {1},\ldots,{M}$ is for remodeling the phase response of the composite filter. The aggregate transfer function of the composite filter is \[{H}_{C}\left({z}\right) = \frac{Y(z)}{X(z)} = {H}_{P}\left({z}\right)\mathop{\Pi}\limits_{{m} = {1}}^{M}{H}_{S,m}{(}{z}{)}{.} \tag{10} \]
Figure 2. The input and output relationship when cascading the prototype filter with $M$ shaping APFs.
Clearly, the frequency magnitude response of the composite filter obtained by replacing ${z} = {e}^{{j}{\omega}}$ into (10) is identical to that of the prototype filter; i.e., $\left|{{H}_{C}\left({\omega}\right)}\right| = \left|{{H}_{P}\left({\omega}\right)}\right|$. The frequency phase response of the composite filter can be related with that of the prototype filter and the ${M}$ shaping APFs by \[{∡}{H}_{C}\left({\omega}\right) = {∡}{H}_{P}\left({\omega}\right) + \mathop{\sum}\limits_{{m} = {1}}^{M}{∡}{H}_{S,m}\left({\omega}\right){.} \tag{11} \]
Taking the negative derivative with respect to ${\omega}$ at both sides of (11), we relate the relationship of the composite filter with its component filters as follows: \begin{align*}{\text{grd}}\left[{{H}_{C}\left({\omega}\right)}\right] & = {\text{grd}}\left[{{H}_{P}\left({\omega}\right)}\right] \\ & \quad + \mathop{\sum}\limits_{{m} = {1}}^{M}{\text{grd}}{[}{H}_{S,m}\left({\omega}\right){]}{.} \tag{12} \end{align*}
One trick for designing a filter with a near perfect frequency response is cascading a number of simple APFs with a prototype filter. The other trick is using a Chebyshev type 2 filter as the prototype filter for efficient compensation of the group delay. We elaborate on these two points in the following.
In the context, we use the terms constant group delay and linear phase interchangeably for ease of explanation. Although a FIR filter can have a linear frequency phase response over the whole frequency bands, it is unnecessary for a band-limited signal. Actually, the filter phase response can be relaxed to be linear only within the passband. This is because the other bands are already suppressed by the prototype filter.
Consider the problem of designing a low-pass composite filter where its phase response is linear within the passband edge frequency ${\omega}_{p}$. Figure 3 explains the idea of compensating the group delay of the prototype filter within the frequency ${\omega}_{p}$. The group delay margin for compensation is in green. In essence, this is an optimization problem for finding the best parameters ${r}$ and ${\theta}$ for the M shaping filters. Clearly, as the number of shaping filters increases, we expect a better fill for the margin. In addition, the geometry of the compensated margin also impacts the error caused by compensation. This idea is further verified in the design examples, and the results are promising when ${M}\geq{3}$ for the considered design specifications. The mean of the synthesized group delay of the composite filter over the passband ${[}{0},{\omega}_{p}{]}$ can be written as \[{m}_{\text{GD}} = \frac{1}{{\omega}_{p}}\mathop{\int}\nolimits_{0}\nolimits^{{\omega}_{p}}{\text{grd}}\left[{{H}_{C}{(}{\omega}{)}}\right]{\text{d}}{\omega}{.} \tag{13} \]
Figure 3. Compensating the group delay of a prototype filter within ${\omega}_{p}$ by using a number of shaping APFs.
The flatness of the synthesized group delay over the passband can be defined as the root mean square (RMS) group delay error of ${H}_{C}{(}{\omega}{):}$ \[{\Delta}_{\text{GD}}{≜}\sqrt{\frac{1}{{\omega}_{p}}\mathop{\int}\nolimits_{0}\nolimits^{{\omega}_{p}}{{\left\{{{\text{grd}}\left[{{H}_{C}{(}{\omega}{)}}\right]{-}{m}_{\text{GD}}}\right\}}^{2}}}{.} \tag{14} \]
Here, ${\Delta}_{\text{GD}}$ can be used in the filter design specifications to qualify to what degree the composite filter can perform.
The design specifications include the following:
The prototype filter is for meeting the frequency magnitude response of the design specifications, while the shaping APFs are for the linear phase response. To facilitate the design, we divide the design specifications into two groups. The specifications in the first group, including constraints 1–4, are for designing the prototype filter; the specifications in the second group, including constraints 5–7, are for designing the shaping filters. Some popular IIR filters, e.g., Butterworth filters, elliptic filters, least-pth-norm filters, and Chebyshev filters, all are candidates for the prototype filter. We just arbitrarily choose the least-pth-norm filter and Chebyshev type 2 filter as the candidate prototype filters. We can use some software package, e.g., MATLAB, to design a prototype filter satisfying specifications 1–4. The frequency magnitude responses are given in Figure 4. The filter orders are eight and 12 for the candidate least-pth-norm IIR filter and Chebyshev type 2 filter, respectively. We also arbitrarily choose a least-squares FIR filter of filter order 200 as a baseline for comparison. Note that all three filters meet specifications 1–4, and the baseline FIR filter is far more complex than the other two IIR filters, considering its high filter order. We can easily improve the stopband suppression for the two IIR filters by cascading them with a complementary comb filter [6], [15], [16], [17]. The frequency magnitude responses of the prototype filters are identical to those of the corresponding composite filters.
Figure 4. A comparison of the frequency magnitude responses for a least-squares FIR filter, least-pth-norm IIR filter, and Chebyshev type 2 IIR filter.
The group delays of the baseline FIR filter and the other two prototype filters are provided in Figure 5. We can see that the group delays of the two prototype filters are monotonously increasing over the passband $\left[{{0},{\omega}_{p}}\right]$. In particular, the least-pth-norm IIR filter has a large group delay margin to be filled by the shaping APFs as compared with that of the Chebyshev type 2 filter. The baseline FIR filter undoubtedly has a constant group delay throughout the whole frequency bands, due to its symmetric filter coefficients. But we show in the example that the linear phase over the filter passband is enough to preserve the intended signal waveform.
Figure 5. A comparison of the group delays for a least-squares FIR filter, least-pth-norm IIR filter, and Chebyshev type 2 IIR filter.
Provided that we know the filter orders of the two candidate prototype filters, the problem of designing a composite filter is then reduced to finding the optimal parameters ${(}{r}_{1},\ldots,{r}_{M},{\theta}_{1},\ldots,{\theta}_{M}{)}$ for a cascade of M shaping APFs satisfying specifications 5–7. The objective function of the optimization problem is \begin{align*} & {\min}_{{r}_{1},\ldots,{r}_{M},{\theta}_{1},\ldots,{\theta}_{M}} \\ & {\Delta}_{\text{GD}}{(}{r}_{1},\ldots,_{M}{r}_{M},{\theta}_{1},\ldots,{\theta}_{M}{)} \tag{15} \end{align*} subject to \begin{align*}{1} & \leq{M}\leq{3} \tag{16} \\ {0} & \leq{r}_{1},\ldots{r}_{M}\leq{1} \tag{17} \\ {0} & \leq{\theta}_{1},\ldots,{\theta}_{M}\leq{\pi}{.} \tag{18} \end{align*}
Equation (15) ensures that the RMS group delay error is minimized over the solution space, that is, (16)–(18). Note that M is a positive integer, and the variables ${(}{r}_{1},\ldots,{r}_{M},{\theta}_{1},\ldots,{\theta}_{M}{)}$ are real numbers. In addition, one shaping filter contributes an additional filter order of two to the composite filter. Thus, specification 6 is satisfied if the composite filter is constrained by (16). Note that (15) is not a linear function, and we cannot use linear programming to solve the problem. To simplify the problem, we set M to one, two, and three, respectively, and search through the reduced solution space constrained by (17) and (18). The solution space can be first sliced using a coarse grid for finding candidate solutions. Then, the solution space around the candidate solutions is sliced using a fine grid. The process is repeated a few rounds, as in the work in [18]. This method is especially useful for well-behaved functions, and the optimal solution can be obtained in a few rounds.
Figure 6 shows the RMS group delay errors of using the two candidate prototype filters when ${M} = {1},\ldots,{4}$. We find that choosing the Chebyshev type 2 filter as the prototype filter always achieves a lower RMS group delay error than that of the least-pth-norm filter. This is because the margin of the group delay to be compensated is smaller for the Chebyshev type 2 filter than that of the least-pth-norm filter. Thus, we get ${M} = {3}$, and $\left({{r}_{1},{r}_{2},{r}_{3},{\theta}_{1},{\theta}_{2},{\theta}_{3}}\right) = {(}{0}{.}{8820},{0}{.}{8869},$${0}{.}{8892},{0}{.}{0514},{0}{.}{1553},{0}{.}{2625}{)}$. The corresponding RMS group delay error is 0.1196. Using these numerical results, we accomplish the design of the composite filter and obtain the frequency phase response in Figure 7. As compared with the baseline FIR filter, we find that the composite filter can achieve an almost linear phase response over the filter passband $\left[{{0},{\omega}_{p}}\right]$.
Figure 6. A comparison of the RMS group delay errors by using different prototype filters.
Figure 7. A comparison of the frequency phase responses for a least-squares FIR filter and our composite filter. The prototype filter is a Chebyshev type 2 IIR filter.
Figure 8 illustrates the group delays of the composite filter and those of its component filters. In this case, the group delay of the composite filter over the passband is around 58.5, which is smaller than that of the baseline FIR filter. The component filter parameters of the low-pass composite filter are tabulated in Table 1.
Figure 8. The group delays of the composite filter and associated shaping filters.
Table 1. The component filter parameters of the low-pass composite filter.
Table 2 provides a comparison of the complexity among five LPFs, which are 1) our composite LPF, 2) a least-squares FIR filter of order 200, 3) an equiripple FIR filter of order 134, 4) a FIR filter of order 182 designed by the window design method, and 5) a narrow transition-band FIR filter designed in [9]. From (8), we know that one shaping APF needs four multipliers and four adders. With an order 12 prototype filter, the composite filter needs a total of ${36}{(}{4}\,{\times}\,{3} + {12}\,{\times}\,{2} = {36}{)}$ multipliers and ${36}{(}{4}\,{\times}\,{3} + {12}\,{\times}\,{2} = {36}{)}$ adders. The other three low-pass FIR filters (i.e., cases 2, 3, and 4) are designed using MATLAB. All five of them meet the same design specifications, e.g., passband/stopband edge frequencies, passband peak-to-peak ripple, suppression in the stopband, and so on, as outlined in the preceding. Clearly, the composite filter is significantly simpler than the other four FIR filters. Note that we fold the FIR filter architectures for cases 2, 3, and 4 by utilizing the symmetric filter coefficients presented in [5] for reduced complexity. Also shown in Table 2 are a comparison of the timing complexity and the group delay of the five designs. Cases 1–4 all have constant filter coefficients, and there is no need to update them constantly. Thus, the timing control and timing complexity for all fives designs are low. From Figure 8, we know that the group delay for the signals inside the passband of the composite filter is around 58.5, which is smaller than for the other four designs.
Table 2. A comparison of the complexity among five LPFs.
Table 2 is based on the central idea of using the minimum filter order for each design so as to satisfy the same design specifications. DSP practitioners then leverage the constraints on the implementation platform and freely choose among the feasible designs. The philosophy of our comparison is fairly common and widely used in commercial software packages, e.g., the Filter Design and Analysis Tool in MATLAB, although we, indeed, can set all the filter orders to be fixed and compare their frequency responses. In addition, our comparisons belong to the architecture level. This means we can bypass circuit-level concerns, and it is a fair comparison.
Figure 9(a) shows an input sequence $x(n)$ consisting of three narrow-band pulses of sinusoids. The pulses are given as follows: \begin{align*}{x}_{1}{(}{n}{)} & = {w}{(}{n}{)}\cos{(}{\omega}_{1}{n}{)} \tag{19a} \\ {x}_{2}{(}{n}{)} & = {w}{(}{n}{)}\cos\left({{\omega}_{2}{n}{-}\frac{\pi}{2}}\right) \tag{19b} \\ {x}_{3}{(}{n}{)} & = {w}{(}{n}{)}\cos\left({{\omega}_{3}{n} + \frac{\pi}{5}}\right) \tag{19c} \end{align*}
Figure 9. The input sequence for verifying example filters: the (a) waveform of signal $x[n]$ and (b) corresponding discrete-time Fourier transform magnitude ${|}{X}{(}{\omega}{)}{|}$.
where ${\omega}_{1} = {0}{.}{07}{\pi}$ (radians/sample), ${\omega}_{2} = {0}{.}{02}{\pi}$ (radians/sample), ${\omega}_{3} = {0}{.}{4}{\pi}$ (radians/sample), and $w(n)$ is a Hamming window; ${\omega}_{1}$ and ${\omega}_{2}$ are located at the filter passband, while ${\omega}_{3}$ is at the stopband. The Hamming window of length ${L} + {1}$ is defined as [5], [14] \begin{align*}&{w}{(}{n}{)} = \\ &\quad\begin{cases}{\begin{array}{ll}{{0}{.}{54}{-}{0}{.}{46}\cos\left({\frac{{2}{\pi}{n}}{L}}\right),}&{{0}\leq{n}\leq{L},} \\ {0,}&{\text{otherwise}}\end{array}}{.}\end{cases} \tag{20} \end{align*}
The complete input sequence is defined as \begin{align*}{x}{(}{n}{)} & = {x}_{1}{(}{n}{)} + {x}_{2}{(}{n}{-}{L}{-}{1}{)} \\ & \quad + {x}_{3}{(}{n}{-}{2}{L}{-}{2}{),}\,{n}\geq{0}{.} \tag{21} \end{align*}
In fact, $x(n)$ can be regarded as a simulation of an ECG record from a field trial. Parts ${x}_{1}(n)$ and ${x}_{2}(n)$ denote the intended signals, while ${x}_{3}(n)$ is an interfering signal. A composite filter with an almost linear phase response does preserve the intended waveform and remove the interfering signal. This can be vital for correct diagnosis. The discrete-time Fourier transform (DTFT) of windowed sinusoids equals the convolution of the DTFT of the window with that of an infinitely long sinusoid. A windowed sinusoid thus has a spread of spectrum, depending on the window length, around the center frequency of the sinusoid. The corresponding DTFT magnitude $\mid{X}{(}{\omega}{)}\mid$ when ${L} = {60}$ is in Figure 9(b). Note that the filter passband is full of a wideband signal, with an accompanied out-of-band signal around ${\omega} = {\omega}_{3} = {0}{.}{4}{\pi}$ (radians/sample).
Figure 10 compares the output sequences filtered by 1) a Chebyshev type 2 filter, 2) the composite filter, and 3) a least-squares FIR filter. All three filters meet the filter magnitude specifications. The output sequences for each filter are denoted as ${y}_{1}(n)$, ${y}_{2}(n)$, and ${y}_{3}(n)$, respectively. The Chebyshev type 2 filter is actually the prototype filter of the composite filter; it is of filter order 12. The composite filter has three shaping APFs and is of filter order 18. The least-squares FIR filter has a linear phase response over the whole frequency bands and is of filter order 200; it is used as a benchmark for performance comparison. Comparing Figure 10(a) with Figure 9(a), we can see that the Chebyshev type 2 filter is unable to maintain an undistorted waveform for the wideband signal, due to its nonlinear phase response. Comparing ${y}_{2}(n)$ with ${y}_{3}(n)$, we see that both filters do suppress the out-of-band signals while preserving the in-band signals with high fidelity. The output sequences for both filters are similar except that the baseline FIR filter produces an additional delay of approximately 42 samples ${(}{100}{-}{58} = {42}{)}$. This is due to the composite filter having a lower in-band group delay as compared with that of the least-squares FIR filter.
Figure 10. A comparison of the output sequences filtered by a (a) Chebyshev type 2 IIR filter, (b) composite filter, and (c) least-squares FIR filter.
The pole-zero diagram of the composite filter for ${M} = {3}$ is presented in Figure 11. There is a total of six ${(}{2}\,{\times}\,{M}{)}$ pole-zero pairs scattered over the band ${[}{-}{0}{.}{1}{\pi},{0}{.}{1}{\pi}{]}$ as expected. All the poles are within the unit circle, and the composite filter is stable in any case.
Figure 11. A pole-zero diagram of the composite filter.
The procedures for designing a high-pass composite filter are similar to those for a low-pass composite filter. Assume that the filter specifications are
We can arbitrarily choose an IIR filter that meets specifications 1–4 as the prototype filter. For example, we choose a Butterworth IIR filter and a Chebyshev type 2 IIR filter as the candidate filters, which results in a filter order of 20 and 10, respectively. The objective function of the optimization problem can be formulated as \begin{align*} & {\min}_{{r}_{1},\ldots,{r}_{M},{\theta}_{1},\ldots,{\theta}_{M}} \\ & {\Delta}_{\text{GD}}{(}{r}_{1},\ldots,{r}_{M},{\theta}_{1},\ldots,{\theta}_{M}{)} \tag{22} \end{align*} subject to \begin{align*}{1} & \leq{M}\leq{4} \tag{23} \\ {0} & \leq{r}_{1},\ldots{r}_{M}\leq{1} \tag{24} \\ {0} & \leq{\theta}_{1},\ldots,{\theta}_{M}\leq{\pi}{.} \tag{25} \end{align*}
For the two candidate prototype filters, specification 6 is met by the constraint (23).
The frequency magnitude responses of the two candidate prototype filters are shown in Figure 12. Also shown in Figure 12 for comparison is an equiripple FIR filter of filter order 82. The frequency magnitude responses of the composite filters are the same as those of their corresponding prototype filters.
Figure 12. A comparison of the frequency magnitude responses for an equiripple FIR filter, a Butterworth IIR filter, and a Chebyshev type 2 IIR filter.
Figure 13 demonstrates the group delays of the three filters. The Chebyshev type 2 IIR filter has a strictly decreasing group delay over the passband ${[}{\omega}_{p},{\pi}{]}$ and is easier to be compensated by the shaping APFs than the Butterworth IIR filter for a given number of shaping APFs.
Figure 13. A comparison of the group delays for an equiripple FIR filter, a Butterworth IIR filter, and a Chebyshev type 2 IIR filter.
Figure 14 compares the RMS group delay errors for the two candidate prototype filters. We see that the RMS group delay error of the Chebyshev type 2 filter is lower than that of the Butterworth filter. Cascading the Chebyshev type 2 filter with ${M} = {4}$ shaping APFs results in an RMS group delay error of 0.2191, and the resultant filter order is only 18. In contrast to the baseline FIR filter of filter order 82, the composite filter shows a significant reduction in complexity. However, when using the Butterworth IIR filter as the prototype filter, the RMS group error for ${M} = {4}$ is 0.7421, which is unable to meet the design specifications.
Figure 14. A comparison of the RMS group delay errors for two candidate prototype filters.
We already obtained ${M} = {4},\,{(}{r}_{1},{r}_{2},{r}_{3},{r}_{4},{\theta}_{1},{\theta}_{2},{\theta}_{3},{\theta}_{4}{)} = {(}{0}{.}{8928},\,{0}{.}{9175},\,{0}{.}{9038},\,{0}{.}{8845},\,{3}{.}{0135},\,{2}{.}{822},\,{2}{.}{92},\,{3}{.}{1005}{)}$. Figure 15 describes how the group delay of the composite filter is synthesized by the five component filters (one prototype filter and four shaping filters). The in-band group delay of the composite filter is around 66. We confirm that the composite HPF is stable in any case. The component filter parameters of the high-pass composite filter are tabulated in Table 3. This completes the design of a high-pass composite filter.
Figure 15. The synthesis of the group delay of the composite filter.
Table 3. The component filter parameters of the high-pass composite filter.
Table 4 provides a comparison of the complexity among four HPFs, which are 1) the composite HPF, 2) an equiripple FIR filter of order 82, (3) a generalized equiripple FIR filter of order 104, and 4) a FIR filter of order 146 designed by the window design method. With an order 10 prototype filter, the composite filter needs a total of ${36}{(}{4}\,{\times}\,{4} + {10}\,{\times}\,{2} = {36}{)}$ multipliers and a total of ${36}{(}{4}\,{\times}\,{4} + {10}\,{\times}\,{2} = {36}{)}$ adders. The three high-pass FIR filters are all designed using MATLAB, and all meet the same design specifications as the composite filter. Clearly, the composite filter is the simplest among the four filters. Notice that we already simplified the three FIR filters by using the folded architectures presented in [5].
Table 4. A comparison of the complexity among four HPFs.
If we want to design a bandpass or band-stop composite filter, the design procedures are the same as those of the two example filters. We find that Chebyshev type 2 filters inherently have relatively lower group delay margins for compensation than the least-pth-norm and Butterworth filters. It is beneficial to select a Chebyshev type 2 filter as the prototype filter when designing a composite filter.
The proposed composite filter can be used to replace any ordinary FIR filter with fixed filter coefficients. This means our composite filter is not suited for adaptive filters that are constantly changing filter coefficients according to some optimization algorithms. This is because the recursive structures of IIR filters inherently have a good memory for samples. Nevertheless, a recursive structure does not necessarily result in a slow operating speed from a very large-scale integration (VLSI) hardware perspective. In fact, the peak operating speed of a filter is constrained by its critical path. The peak operating speed of an IIR filter can be increased by cutting down the critical path by using, e.g., pipelining or retiming techniques, and so can that of the composite filter [19].
This article presented two tricks for approaching a perfect filter (i.e., flat passband, sharp transition band, highly suppressed stopband, and linear phase) with reduced complexity. This goal was realized through first designing a prototype filter to meet the design specifications regarding the frequency magnitude response. The phase function of the prototype filter was then remodeled by a cascade of delicately designed shaping filters. After cascading the prototype IIR filter with the shaping APFs, we obtained a composite filter with an almost linear phase response over the filter passband. Two example filters with highly reduced complexity were demonstrated. We found that the Chebyshev type 2 filter is an appealing candidate for the prototype filter of the composite filter. Our composite filter shows quite similar filtering performance as the baseline FIR filter of significantly higher complexity. The composite filter provides a way to approach perfect filtering using limited complexity and is especially useful for replacing any ordinary FIR filter with fixed filter coefficients. Further VLSI implementations focusing on operating speed, hardware cost, and so on can be the next steps to further investigate the benefit of the proposed composite filter.
David Shiung (davids@cc.ncue.edu.tw) received his Ph.D. degree from National Taiwan University, Taipei, 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, in 2004. He is a professor at National Taiwan Normal University, Taipei 106, Taiwan. His research interests include 5G, LoRaWAN, and vehicular ad hoc networks. He is a Member of IEEE.
Ya-Yin Yang (ivyyang64@gmail.com) received her Ph.D. degree in electrical engineering from National Taiwan University, Taipei, in 2009. She is currently an assistant researcher with the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan 701, Taiwan. Her research interests include channel estimation, radio resource allocation, and interference cancellation for wireless communication systems.
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Digital Object Identifier 10.1109/MSP.2023.3290772