## 2 The Round-Off Noise Model  
### Introduction  
In order to avoid overflow of the signal data due to the successive multiplications of the signal with the filter coefficients, rounding (as defined above) will have to be performed. Rounding can be seen as a quantization action on the signal. Each rounding action is treated as a random process with uniform probability density function, producing white noise that is uncorrelated with the signal and other quantization sources in the filter. As in this correspondence the implementation of the MFIR filters is investigated for fixed-point implementation, the rounding process is of vital importance. [PERSON] developed, in [1], the \"pure multiplicity property\" which forms the basis of the investigation of the relationship between the noise variance at the output (relative to the round-offnoise variance) and the ordering of the different stages of the MFIR filter. This is done for an implementation without scaling (which is not very realistic for fixed point implementations) and an implementation with L\({}_{2}\) scaling. The analysis in [1] is performed for MFIR structures that approximate a real pole or a conjugate pole pair using the forward lattice structure. The approximation of a conjugate pole pair in cascade is not analyzed as \"it does not have the pure multiplicity property\" [1]. However, in the present research, the forward lattice MFIR implementation is not considered because of its (unpractical) large hardware impact [2]. Consequently, the study of the noise behavior of the complex-conjugate pole pair cascade MFIR implementation and the linear phase MFIR implementations are completely new. Moreover, it is suggested in [1] that the analysis of the round-off noise for the MFIR structures that do not have the \"pure multiplicity property\" should be done in the style of [8] or [9]. In this text, the method of [8] will be used for the real pole approximation as well as for the complex-conjugate pole pair cascade approximation. All three well known scaling methods (L\({}_{2}\) bound, infinity bound and absolute bound) [8] will be considered. The noise performance is evaluated by first calculating the noise variance at the output due to the noise variance of the round-off error sources. However, a good evaluation of the round-off noise performance can only be made when the actual signal to round-off noise ratio (SNR) is considered [10]. With scaling, the output signal is also scaled, implying the actual signal to noise ratio at the output and not only the noise variance is considered for all possible scaling cases. The objective of the paper is to determine how much the SNR of a signal is deteriorated by the noise (due to round-off errors) in the overall MFIR structure. Although the round-off noise performance of FIR and IIR cascade structures has been studied intensively over the past decades, [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18] no directly applicable expression of the SNR degradation in cascade structures could be found. Therefore, the theory will first be developed in a general manner. More precisely, the stages in the cascade will not be considered as MFIR stages with transfer function \(M_{{}_{i}}\left(z\right),\) but as general filter stages with transfer function \(H_{{}_{k}}\left(z\right).\) The index \(k\) will be used as an index for the general stages in the cascade. Note that \(k\) is not the same as the index \(i\) that is used to indicate the stages of the MFIR structure _and_ the power of the multiplier coefficients of the MFIR stages (as in Equations (2) and (3)). It will be shown that the ordering has a large impact on the SNR. However, \(P\) stages of an MFIR structure can be ordered in \(P!\) possible orderings, making an exhaustive search for the optimal ordering unpractical. Unlike typical cascade structures studied in [15; 16; 17; 18], the transfer functions of the stages of the MFIR structure are fixed by the expressions given in Equations (2-5), _i.e._, in every stage the grouping of the zeros is fixed by the MFIR approach.