## 1 Introduction  
Multiplicative Finite Impulse Response (MFIR) filters are a class of filter structures that were originally introduced by Fam in the early 1980s [1]. It was shown that MFIR filters can be used to replace recursive Infinite Impulse Response (IIR) filters with FIR equivalents requiring significantly less hardware than classical FIR architectures that fulfill the same specifications [2]. The replacement of a pole that is implemented in a recursive structure, by a non-recursive FIR, has the advantage that it will always be stable. This is particularly interesting when the original pole is situated close to the unit circle. The MFIR filters are able to realize low-pass, high-pass, band-pass, and notch filters. Although the MFIR structures require approximately the same number of delay elements as the classical FIR implementations, they require, logarithmically, fewer adders and multipliers [1,2]. MFIR filters are based on the identity [3]:  
\[\sum_{i=0}^{2^{r}-1}x^{i}=\prod_{i=0}^{P-1}\left(1+x^{2^{i}}\right). \tag{1}\]  
This identity can be used to approximate both real pole and conjugate pole pair filters. In case \(H\),\(\left(z\right)\) is the transfer function of a stable IIR filter with a real pole, the cascade MFIR filter approximation using Equation (1) yields:  
\[ \begin{split} H_{r}\left(z\right)&=\frac{1}{1- \lambda z^{-1}}=\sum_{i=0}^{\infty}\left(\lambda z^{-1}\right)^{i}\\ H_{r}\left(z\right)&=\sum_{i=0}^{2^{r}-1}\left( \lambda z^{-1}\right)^{i}=\prod_{i=0}^{P-1}\left(1+\left(\lambda z^{-1} \right)^{2^{r}}\right)=\prod_{i=0}^{P-1}M_{i}\left(z\right)=M\left(z\right). \end{split}  \tag{2}\]  
with the obvious definitions for \(M_{i}\left(z\right)\) and \(M\left(z\right)\). In this correspondence, every single \(M_{i}\left(z\right)\) will be called a \"stage\" of the MFIR filter. It has been shown in [1,2] that even for the approximation of poles extremely close to the unit circle, maximum 10 stages (\(P=10\)) are required when a deviation smaller than \(\left|0.01\right|\) dB in the magnitude response between the IIR filter and the MFIR approximation is allowed. The efficiency of the MFIR approximation is immediately clear from Equation (2). Only \(P\) multipliers and adders are required for the MFIR filter while the direct form requires \(2^{P}\) multipliers and adders. An IIR filter with a transfer function \(H_{c}\left(z\right)\) having a conjugate pole pair \(\lambda=re^{+i\theta}\) and \(\lambda^{*}=re^{-i\theta}\) with \(\left|r\right|<1\), can be approximated with a cascade MFIR structure [1,2]:  
\[ \begin{split} H_{c}\left(z\right)&=\frac{1}{\left(1 -\lambda z^{-1}\right)\left(1-\lambda^{*}z^{-1}\right)}\\ &\approx\prod_{i=0}^{P-1}\left(1+\left(\lambda z^{-1}\right)^{2^ {r}}\right)\prod_{i=0}^{P-1}\left(1+\left(\lambda^{*}z^{-1}\right)^{2^{r}} \right)\\ &\approx\prod_{i=0}^{P-1}\left(1+2\left(rz^{-1}\right)^{2^{r}} \cos\left(2^{i}\theta\right)+\left(rz^{-1}\right)^{2^{r+1}}\right)=\prod_{i=0} ^{P-1}M_{i}\left(z\right)=M\left(z\right)\end{split}  \tag{3}\]  
MFIR filters with a linear phase and a desired magnitude response \(\left|H\left(e^{i\omega_{n}}\right)\right|\) (Here \(\omega_{n}\) is the normalized angular frequency: \(\omega_{n}=\omega T_{S}\), and \(T_{S}\) is the sampling period.) can also be realized using the following procedure [1,4]. 1. Design an IIR filter that approximates \(\left|H\left(e^{i\omega_{n}}\right)\right|^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! Consequently, the linear phase approximation of a stable IIR filter \(H_{\text{c}}(z)\) with a real pole at \(\lambda\), can be designed by determining \(\left|H_{\text{c}}^{\cdot}\left(e^{i\alpha_{\text{c}}}\right)\right|=\left|H_{ \text{c}}\left(e^{i\alpha_{\text{c}}}\right)\right|^{\
icefrac{{1}}{{2}}}\) and approximating the real pole \(\lambda^{\prime}\) of \(H_{\text{c}}^{\prime}(z)\) with:  
\[