Gordana Jovanovic Dolecek
This article explores some tips and tricks on how to decrease the number of additions per output sample (APOS) in a cascaded-integrator-comb (CIC) multistage decimation filter, ensuring the smallest number of APOS for a given decimation factor; the desired minimum worst-case attenuation (WCA) in dB; and the desired passband edge frequency ${\omega}_{p}$. We assume that the decimation factor M is a product of prime factors. In the last stage, we propose modifying the optimal multistage structure with the minimum number of APOS presented in the literature.
Two cases are investigated depending on if the decimation factor at the last stage is equal to two or not. In both cases, we present tips and tricks to decrease the number of APOS. Finally, we compare the proposed approach and the original optimal multistage structure for ${\omega}_{p} = {\pi} / {2}{M}$ and for different values of M and WCA.
Recently, in [1], the authors proposed a CIC multistage structure with the minimum number of APOS for a given minimum WCA in dB and the desired passband edge frequency ${\omega}_{p}$. The decimation factor is presented as a product of prime factors ${M}_{1},{\ldots},{M}_{N}$, where N is the number of stages, and ${M}_{i}$ is the decimation factor at ith stage, ${i} = {1},{\ldots},{N}$. The parameters of the structure are obtained by optimization. The principal idea in [1] is to use a multirate identity to move combs at the ith stage after decimation by ${M}_{i}$ and cancel combs and integrators at the ${(}{i} + {1}{)}$-th stage. In the last stage, combs are moved only after the decimation by ${M}_{N}$. The transfer function of the filter in the ith stage is written in a generalized form as \[{H}_{i}{\left({z}\right)} = {\left({{1}{-}{z}^{{-}{1}}}\right)}^{{L}_{i}},{i} = {1},{\ldots},{N}{.} \tag{1} \]
Similarly, the filter after the last stage is denoted as ${H}_{{N} + {1}}{(}{z}{)}$ \[{H}_{{N} + {1}}{\left({z}\right)} = {\left({{1}{-}{z}^{{-}{1}}}\right)}^{{L}_{{N} + {1}}}{.} \tag{2} \]
The parameters ${L}_{i}$, ${i} = {1},{\ldots},{N} + {1}$ are defined by the orders of the CIC filter, as shown next, in an equation from [1]. \[{L}_{i} = \begin{cases}\begin{array}{lll}{{-}{K}_{1}}&{\text{for}}&{i} = {1} \\ {K}_{i}{-}{K}_{{i} + {1}}&{\text{for}}&{{1}\,{<}\,{i}\leq{N}}\\{{K}_{N}}&{\text{for}}&{{i} = {N} + {1}}\end{array}\end{cases}{.} \tag{3} \]
The values ${L}_{i}\,{<}\,{0}$ and ${L}_{i}\,{>}\,{0}$ denote the multiplicity of integrators and combs, respectively, while ${L}_{i} = {0}$ means their absence [1].
The work in [1] presents the optimal solution for the optimal distributions of decimation factors ${M}_{i}$, integrators, and combs for a given M, WCA, and ${\omega}_{p}$, ensuring the smallest number of APOS. Figure 1(a) presents the obtained multistage CIC structure with N-stages, decimation factors ${M}_{i}$, and parameters ${L}_{i}$, ${i} = {1},{\ldots},{N} + {1}$, with the minimum number of APOS. Figure 1(b) shows the multistage CIC structure with the decimation factors ${M}_{i}$ and parameters ${K}_{i}$.
Figure 1. Structures and one example, taken from [1]. (a) The proposed general structure in [1]. (b) The initial CIC multistage structure in [1]. (c) An example in [1]: M = 12, Mi = [2, 3], Li = [−4, −2, −3, 9], Ki = [4, 6, 9]. (Source: Taken from [1].)
The method is illustrated with the example from [1] for ${M} = {12}$, ${\text{WCA}} = {79.1}{\text{ dB}}$, ${M}_{i} = {\left[{3},{2},{2}\right]}$, ${L}_{i} = {\left[{-}{4},{-}{2},{-}{3},{9}\right]}$, ${K}_{i} = {\left[{4},{6},{9}\right]}$, and ${\omega}_{p} = {0.5}{\pi} / {M}$. The number of APOS is equal to 71. Figure 1(c) illustrates this example.
In the continuation, we present tips and tricks to decrease the number of APOS in the structure [1] without alternating the desired values of WCA and ${\omega}_{p}$. We consider two cases, depending on if the decimation at the last stage is equal to two or not.
We propose to modify the last stage of the initial CIC structure from [1]. To this end, Figure 2(a) presents the last stage of the initial multistage CIC structure before moving the combs to a lower rate. We propose to split the ${K}_{N}$ combs at the last stage into two parts. The ${K}_{{N}{-}{1}}$ combs are presented in a recursive form (CIC) filters, while the rest of the combs of the order ${K}_{N}{-}{K}_{{N}{-}{1}}$ are presented in a nonrecursive form, as shown in the following: \begin{align*} \left[{\frac{{1}{-}{z}^{{-}{2}}}{{1}{-}{z}^{{-}{1}}}}\right]^{{K}_{N}} & = {\left[{\frac{{1}{-}{z}^{{-}{2}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{{K}_{{N}{-}{1}}}\,{\times}\,{\left[{\frac{{1}{-}{z}^{{-}{2}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{{K}_{N}{-}{K}_{{N}{-}{1}}} \\ & = {\left[{\frac{{1}{-}{z}^{{-}{2}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{{K}_{{N}{-}{1}}}\,{\times}\,{\left[{{1} + {z}^{{-}{1}}}\right]}^{{K}_{N}{-}{K}_{{N}{-}{1}}}{.} \tag{4} \end{align*}
Figure 2. The proposed structure for Case 1. (a) The initial last stage. (b) The modified last stage. (c) The final last stage. (d) The structure in Example 2.
Figure 2(b) presents the modified last stage according to (4).
By canceling ${K}_{{N}{-}{1}}$ combs and integrators and moving ${K}_{{N}{-}{1}}$ combs after the decimation by two, we get the structure shown in Figure 2(c).
In the obtained modified structure, we have \[{R}_{i} = \begin{cases}\begin{array}{lll}{{L}_{i}}&{\text{for}}&{{1}\leq{i}\,{<}\,{N}{-}{1}}\\{{K}_{N}{-}{K}_{{N}{-}{1}}}&{\text{for}}&{{i} = {N}}\\{{K}_{{N}{-}{1}}}&{\text{for}}&{{i} = {N} + {1}}\end{array}\end{cases}, \tag{5} \] where ${L}_{i}$ is given in (3).
Observe that in the modified structure, the order of the filter, after the last decimation, is equal to ${R}_{{N} + {1}} = {K}_{{N}{-}{1}}$, while in the original structure, it is equal to ${L}_{{N} + {1}} = {K}_{N}$.
As a result, the number of APOS is decreased by ${K}_{N}{-}{K}_{{N}{-}{1}}$ in comparison with the original structure from [1]. \[{\Delta}{\text{APOS}} = {L}_{{N} + {1}}{-}{R}_{{N} + {1}} = {K}_{N}{-}{K}_{{N}{-}{1}}{.} \tag{6} \]
Taking the example in Figure 1(c), with parameters ${M}_{i} = {\left[{3},{2},{2}\right]}$, ${L}_{i} = {\left[{-}{4},{-}{2},{-}{3},{9}\right]}$, ${K}_{i} = {\left[{4},{6},{9}\right]}$, and the number of APOS equal to 71, we get the modified structure with the parameters ${R}_{i} = {\left[{-}{4},{-}{2},{3},{6}\right]}$ and the number of APOS equal to 68. The number of APOS is decreased by ${K}_{N}{-}{K}_{{N}{-}{1}} = {9}{-}{6} = {\bf{3}}$. Figure 2(d) presents the modified structure.
We can observe that in the last stage of the modified structure, there are ${R}_{N}$ combs in a nonrecursive form, which can be easily presented using a polyphase decomposition [2], [3]. \[{\left({1} + {z}^{{-}{1}}\right)}^{{R}_{N}} = {P}_{0}{\left({z}^{2}\right)} + {z}^{{-}{1}}{P}_{1}{\left({z}^{2}\right)} \tag{7} \] where ${P}_{0}{\left({z}^{2}\right)}$ and ${P}_{1}{\left({z}^{2}\right)}$ are the polyphase components.
The polyphase components are moved to a lower rate, i.e., after the decimation by two, and the last stage is obtained, as shown in Figure 3(a).
Figure 3. Polyphase decomposition. (a) The general structure. (b) Example 3.
Denoting the number of adders of the polyphase components ${P}_{0}{(}{z}{)}$ and ${P}_{1}{(}{z}{)}$ as ${N}_{0}$ and ${N}_{1}$, respectively, and taking into account that we need the ${M}_{{N}{-}{1}} = {1}$ adder to add the polyphase components, we get the decrease in the number of APOS in comparison with the modified structure without the polyphase decomposition as \[{\Delta}{\text{APOS}} = {R}_{N}\,{\times}\,{2}{-}{(}{N}_{0} + {N}_{1} + {M}_{{N}{-}{1}}{)}{.} \tag{8} \]
We present the polyphase decomposition by taking the example from Figure 2(d). We have \[{\left({1} + {z}^{{-}{1}}\right)}^{3} = {1} + {3}{z}^{{-}{1}} + {3}{z}^{{-}{2}} + {z}^{{-}{3}}{.}\]
The polyphase components are \[{P}_{0}{{\left({z}\right)}} = {1} + {\left({2}^{0} + {2}^{1}\right)}{z}^{{-}{1}}\] \[{P}_{1}{{\left({z}\right)}} = {\left({2}^{0} + {2}^{1}\right)} + {z}^{{-}{1}}{.}\]
Both components require two adders, i.e., ${N}_{0} = {N}_{1} = {2}$.
The polyphase structure is shown in Figure 3(b).
The decrease of the number of APOS is equal to \begin{align*}{\Delta}{\text{APOS}} & = {R}_{N}\,{\times}\,{2}{-}{\left({N}_{0} + {N}_{1} + {1}\right)} \\ & = {3}\,{\times}\,{2}{-}\left({{2} + {2} + {1}}\right) = {1}{.} \end{align*}
The total number of APOS in the modified structure with the polyphase decomposition equals ${68}{-}{1} = {\bf{67}}$. In comparison with the original number of APOS [1], it is four APOS less.
Similarly, as in Case 1, we will modify the last stage by presenting ${K}_{N}$ combs in the last stage, as a ${K}_{{N}{-}{2}}$ cascade of CIC filters and two combs in a nonrecursive form, as shown in Figure 4(a). \[{\left[{\frac{{1}{-}{z}^{{-}{M}_{N}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{{K}_{N}} = {\left[{\frac{{1}{-}{z}^{{-}{M}_{N}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{{K}_{N}{-}{2}}\,{\times}\,{\left[{\frac{{1}{-}{z}^{{-}{M}_{N}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{2}{.} \tag{9} \]
Figure 4. The last stages of the original structure [1] and the proposed modified structure for the decimation factor differ from the two. (a) The initial modified last stage. (b) The final last stage with a recursive comb. (c) The final last stage is the polyphase decomposition. (d) An example of the proposed last stage for M = 15, Mi = [3, 5], and Ki = [3, 7]. (Source: Taken from [1].)
The comb filter of order 2 in (9) is presented in a nonrecursive form. \begin{align*}{\left[{\frac{{1}{-}{z}^{{-}{M}_{N}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{2} = & {1} + {2}{z}^{{-}{1}} + {3}{z}^{{-}{2}} + {\cdots} \\ & + {M}_{N}{z}^{{M}_{N}{-}{1}} + \cdots + {z}^{{-}{2}{(}{M}_{N}{-}{1}{)}} \tag{10} \end{align*}
Note that all coefficients in (10) are integers ${1},{\ldots},{M}_{N}$.
After canceling ${K}_{{N}{-}{1}}$ combs and ${K}_{{N}{-}{1}}$ integrators, of the total ${K}_{{N}{-}{2}}$ integrators, we get the filter of the order ${R}_{N} = {K}_{{N}{-}{1}}{-}{K}_{N} + {2}$. Applying the multirate identity, the ${K}_{{N}{-}{2}}$ combs are moved after decimation by ${M}_{N}$. Finally, using (9), we get the last stage of the multirate structure as given in Figure 4(b).
We have \[{R}_{i} = \begin{cases}\begin{array}{lll}{{L}_{i}}&{\text{for}}&{{1}\leq{i}\leq{N}{-}{1}}\\{{K}_{{N}{-}{1}}{-}{K}_{N} + {2}}&{\text{for}}&{{i} = {N}}\\{{K}_{N}{-}{2}}&{\text{for}}&{{i} = {N} + {1}}\end{array}\end{cases}{.} \tag{11} \]
The nonrecursive comb (10) is presented with the polyphase components. \begin{align*}&{1} + {2}{z}^{{-}{1}} + {3}{z}^{{-}{2}} + \cdots + {M}_{N}{z}^{{-}{(}{M}_{N}{-}{1}{)}} \\ & \quad + \cdots + {z}^{{-}{2}\left({{M}_{N}{-}{1}}\right)} = \mathop{\sum}\limits_{{k} = {0}}\limits^{{M}_{N}{-}{1}}{z}^{{-}{k}}{P}_{k}{(}{z}^{{M}_{N}}{),} \tag{12} \end{align*}
After moving the polyphase components to a lower rate, we get the structure of the last stage, as shown in Figure 4(c).
The decrease of the number of APOS in the modified structure in comparison to the original structure from Figure 1(c) is equal to \begin{align*}&{\Delta}{\text{APOS}} = {\left({L}_{N}\,{\times}\,{M}_{N} + {L}_{{N} + {1}}\right)} \\ & \quad {-}{\left({R}_{N}\,{\times}\,{M}_{N} + {R}_{{N} + {1}} + {M}_{N}{-}{1} + \mathop{\sum}\limits_{{i} = {1}}\limits^{{M}_{N}{-}{1}}{N}_{i}\right)} \tag{13} \end{align*} where ${N}_{i}$ is the number of adders in the polyphase component ${P}_{i}{(}{z}{)}$; ${L}_{N}$ and ${L}_{{N} + {1}}$ are given in (3); and ${R}_{N}$ and ${R}_{{N} + {1}}$ are given in (11).
We consider an example from [1] with the parameters: ${M} = {15}$, ${M}_{i} = {\left[{5},{3}\right]}$, ${K}_{i} = \left[{{4},{9}}\right]$, ${L}_{i} = {\left[{-}{4},{-}{5},{9}\right]}$. The number of the APOS is 84.
We have ${R}_{i} = {\left[{-}{4},{-}{3},{7}\right]}$ in the modified structure. The transfer function of nonrecursive combs is given as \begin{align*}{\left[{\frac{{1}{-}{z}^{{-}{3}}}{{1}{-}{z}^{{-}{1}}}}\right]}^{2} & = {\left({{1} + {z}^{{-}{1}} + {z}^{{-}{2}}}\right)}^{2} \\ & = {1} + {2}{z}^{{-}{1}} + {3}{z}^{{-}{2}} + {2}{z}^{{-}{3}} + {z}^{{-}{4}} \\ & = \mathop{\sum}\limits_{{i} = {0}}\limits^{2}{z}^{{-}{i}}{P}_{i}{\left({z}^{3}\right)}{.} \end{align*}
The polyphase components are \[{P}_{0}{\left({z}\right)} = {1} + {2}^{1}{z}^{{-}{1}}\] \[{P}_{1}{\left({z}\right)} = {2}^{1} + {z}^{{-}{1}}\] \[{P}_{2}{\left({z}\right)} = {2}^{1} + {2}^{0}{.}\]
The decrease in the number of APOS from (13) is \begin{align*}{\Delta}{\text{APOS}} & = \left({{5}\,{\times}\,{3} + {9}}\right){-}{\left({3}\,{\times}\,{3} + {7} + {2} + {3}\right)} \\ & = {24}{-}{21} = {3}{.}\end{align*}
The number of APOS in the modified structure is equal to \begin{align*}{\text{APOS}} & = {4}\,{\times}\,{15} + {3}\,{\times}\,{3} + {7} + {2} + {3} \\ & = {\bf{81}}{.} \end{align*}
The modified structure is given in Figure 4(d).
Table 1 compares examples provided in [1, Table 4] for ${\omega}_{p} = {0.5}{\pi} / {M}$ with the proposed approach. The better values are presented in bold.
Table 1. Decimation factor M; the WCA in dB; the number of APOS; and the design parameters, Mi, Ki, Li, Ri, and ∼p = 0.5π/M.
This article presents an update of the multistage CIC decimation structure recently published in the literature. The structure provides the minimum number of APOS obtained by optimization for a given decimation factor, WCA, and the passband edge frequency. We noticed that it is possible to modify only the last stage of this structure to get a decreased number of APOS. We considered two cases depending on whether the final decimation factor equals two. In the first case, as a difference from the original structure, the combs decimated by two are presented in recursive and nonrecursive forms. The integrators of the recursive combs cancel combs at that stage, while the combs of the recursive combs are moved to a lower rate. As a result, the number of APOS is decreased.
Additionally, applying the polyphase decomposition to the nonrecursive comb, the number of APOS is further decreased since the polyphase components are moved after the decimation by two. In the second case, in which the decimation at the last stage is different from two, the comb at the last stage is also presented in nonrecursive and recursive forms. As a difference from Case 1, the combs in a nonrecursive form always have the order of two. As a result, the polyphase decomposition with the simple polyphase components applied to the nonrecursive combs may be used, decreasing the original number of APOS.
Gordana Jovanovic Dolecek (gordana@ieee.org) received her Ph.D. degree from the Faculty of Electrical Engineering, University of Sarajevo, Bosnia and Herzegovina. She is a full professor at the Institute INAOE, Department of Electronics, Tonantzintla, Puebla, Mexico 72740, which she joined in 1995. Her research interests include digital signal processing and digital communications. In 2012, she received the Science and Technology Puebla State Award for her research work in electronics. She is an associate editor for IEEE Transactions on Circuits and Systems II: Express Briefs; IEEE Transactions on Circuits and Systems I: Regular Papers; IEEE Circuits and Systems Magazine; and IET Signal Processing. She is a member of the Mexican Academy of Sciences and National System of Researchers (SNI) Mexico and a Life Senior Member of IEEE.
[1] A. Dudarin, M. Vucic, and G. Molnar, “Decimation filters with minimum number of additions per output sample,” Electron. Lett., vol. 58, no. 6, pp. 246–248, Mar. 2022, doi: 10.1049/ell2.12415.
[2] G. J. Dolecek, Multirate Systems: Design and Applications. Hershey, PA, USA: Idea Group Publishing, 2001.
[3] F. J. Harris, Multirate Signal Processing for Communication Systems, 2nd ed. Gistrup, Denmark: River Publisher, 2021.
Digital Object Identifier 10.1109/MSP.2022.3216720