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The paper analyzes the effects of roundoff noise on Multiplicative Finite Impulse Response (MFIR) filters used to approximate the behavior of pole filters. General expressions to calculate the signal to roundoff noise ratio of a cascade structure of Finite Impulse Response (FIR) filters are obtained and applied on the special case of MFIR filters. The analysis is based on fixedpoint implementations, which are most common in digital signal processing algorithms implemented in FieldProgrammable GateArray (FPGA) technology. Three well known scaling methods,
Multiplicative Finite Impulse Response (MFIR) filters are a class of filter structures that were originally introduced by Fam in the early 1980s [
MFIR filters are based on the identity [
This identity can be used to approximate both real pole and conjugate pole pair filters. In case
An IIR filter with a transfer function
MFIR filters with a linear phase and a desired magnitude response
Design an IIR filter that approximates
Approximate the poles of the IIR filter with the MFIR structure.
Cascade to every zero in the resulting MFIR filter its reciprocal with respect to the unit circle.
Despite the advantage of the logarithmically more efficient use of multipliers and adders, MFIR filters have not been popular. Indeed, the large number of delay elements required to approximate the behavior of an IIR filter was considered prohibitively expensive. This made them impractical for implementation on standard DSP platforms with fixed memory maps. Advances in Very Large Scale Integration VLSI technology in general, and FieldProgrammable GateArray (FPGA) architectures in particular, make it necessary to reevaluate MFIR filters and the technical barriers to their widespread use. Several applications and implementations of MFIR filters in modern FPGA fabrics [
It is clear from Equations (2–5) that MFIR filters are basically a cascade of simple sparse FIR filters. An example of a cascade of three MFIR stages realizing an MFIR filter approximating a real pole is given in
General architecture of an MFIR filter approximating a real pole (three stages).
Every stage in an MFIR filter has the same structure as the other stages but the number of delay elements differs. Therefore, it is obvious to design an optimized component that implements a general MFIR stage per MFIR filter approximation,
As floatingpoint arithmetic is only recently available in FPGAs, only fixedpoint arithmetic is taken into consideration. Unfortunately, every fixedpoint addition or multiplication requires a widening of the bus width, which has to be avoided; this implies that rounding must be applied to keep the data path widths manageable and implementable.
In this text, rounding is defined as the process whereby the width of the data path after a multiplication or an addition is reduced to the original width of the data path before the multiplication or addition. This is done by taking the most significant bits, conventional rounding of the result, and using saturation if necessary.
Scaling is defined as the process that changes the filter coefficients in order to increase the SNR based on the fulfillment of a specified criterion (as will be defined in
In order to create general MFIR stage components, the data path bitwidth at the input and at the output of each stage are kept constant. Inside the stage, the bit width is appropriately incremented to avoid accumulation of roundoff errors. Consequently, in practice at the end of each stage, a rounding block will bring the output bit width back to the original input bitwidth. It is, however, not excluded that a change in bus width between the stages would yield better results. However, there are so many possible combinations that it is almost impossible to investigate the behavior of all these possible implementations.
In this correspondence, the Signal to Noise Ratio SNR degradation effect of the consecutive roundings and scalings in the cascaded MFIR structures is analyzed and conclusions on optimal ordering of the stages are drawn. In
Although it will not always be explicitly mentioned, in this correspondence, SNR refers to the Signal to roundoff Noise Ratio,
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 fixedpoint implementation, the rounding process is of vital importance.
A. Fam developed, in [
Moreover, it is suggested in [
The objective of the paper is to determine how much the SNR of a signal is deteriorated by the noise (due to roundoff errors) in the overall MFIR structure. Although the roundoff noise performance of FIR and IIR cascade structures has been studied intensively over the past decades, [
It will be shown that the ordering has a large impact on the SNR. However,
The study is based on the following assumptions. Each multiply and accumulate action in a stage is modeled as an infinite precision multiplier, followed by a summation node. After the summation node, rounding is performed and consequently roundoff noise is added to the system. In the present paper, it is assumed that the rounding process uses conventional rounding and saturation. For the real pole approximation, the conjugate pole pair cascade approximation and their respective linear phase approximations, each stage has one single noise source at the output of the stage. It is assumed that
every sample of the noise source is uncorrelated with the previous sample,
all noise sources are uncorrelated,
the noise sources are uncorrelated with the input signal,
every noise source is a time discrete stationary zero mean white random process with output variance
The following definitions are used throughout the text.
Every stage without rounding or scaling is indicated by
A number of transfer functions are defined in
The filter cascade without rounding or scaling and its transfer functions.
For the roundoff noise analysis, it is assumed there is a roundoff noise source at the output of each stage. The transfer function from the filter input to the output of the stage with transfer function
The transfer function from the output of the stage with transfer function
The rounded and scaled filter cascade and its transfer functions.
The “worked out” rounded and scaled filter cascade and its transfer functions.
Consequently, the following holds:
In analogy to [
For L_{2} bound scaling, the scaling factors are determined by:
For infinity bound scaling, L_{∞}, the scaling factors are determined by:
For absolute bound scaling, the scaling factors are determined by:
Contrary to the absolute bound scaling approach, when using infinity bound scaling or L_{2} bound scaling, overflow is still possible. Although all scaling methods prevent overflow according to a certain criterion, none of them will force all multiplier coefficients to be smaller than (or equal to) one. This implies that in a practical implementation it can happen that the bits used to represent a multiplier coefficient are not sufficient. The problem can be solved in several ways [
The present paper uses a method that has minimum impact on the overall accuracy. More precisely, the multiplier values of the stages where the coefficients are larger than 1 are divided by a power of 2 (using shifting) to force all coefficients of this stage to be smaller than 1. This division by the power of 2 is undone in the output signal of the stage,
As the sequential ordering of the stages has a large impact on the scaling factors and the noise performance of the filter, the roundoff output noise variance as a function of the ordering must be calculated and a method to determine the optimal sequential ordering must be found.
The roundoff output noise variance is the summation of the noise sources inside the filter cascade, with suitable weightings and filtering. This variance is affected by the ordering of the stages of the filter structure. The variance of the noise of each rounding operation equals
Under the assumptions given in
The PSD of the noise generated at the output of the total filter structure, by the noise source of a (scaled and rounded) stage
Using Equations (7) and (25) in Equation (24) yields:
In Equations (26) and (27), the scaling is taken into account and the unscaled stage equations can now be used in
In case of L_{2} bound scaling (
If infinity bound scaling, L_{∞}, is considered (
for every
In case of absolute bound scaling the equivalent of Equation (25) is given by:
In case this ratio is significantly larger than 1 for every
The roundoff noise performance of a filter may not be correctly evaluated by only analyzing the roundoff noise. A more reliable result is obtained by calculating the Signal to roundoff noise ratio (SNR). The signal to noise ratios (when using the previously discussed scaling methods) are investigated in this section.
The discrete input signal of the filter is indicated by
In case the input signal is a wide sense stationary random signal with uniform PDF and variance
In case of L_{2} or L_{∞} bound scaling, Equation (34) can be written as (using Equation (10) or Equation (13) as appropriate):
In case of absolute bound scaling, the SNR is given by:
In Equations (37) and (38)
In general for an arbitrary
As mentioned in
Even in case of an optimized ordering for infinity bound scaling or absolute bound scaling, L_{2} bound scaling will always have the best SNR. Note that SNR in this text is signal to roundoff noise ratio and not the overall SNR of the filter. However, an optimized ordering for absolute bound scaling can have a better SNR than a nonoptimized ordering for infinity bound scaling and vice versa. In case of L_{∞} bound scaling, the optimal SNR must be determined by finding an ordering that minimizes
Notice that the equations derived in this section are generally valid for any filter cascade (with one noise source at the output of each filter stage) and not only for MFIR structures.
In this section, the general theory developed in
In case
The ordering where the first stage is stage
The unscaled and scaled MFIR filter cascade transfer functions for reverse ordering.
The ordering where stage
Notice that in case of reverse ordering
Although there are
The equations derived in
Scaling factors for Multiplicative Finite Impulse Response (MFIR) structures.
Forward ordering  Reverse ordering  

L_{2} bound 


Infinity bound 


Absolute bound 


Length of the impulse responses of the partial MFIR transfer functions.
Forward ordering  
Real pole  0 → 
Real pole linear phase  0 → 
Complexconjugate pole pair  0 →

Complexconjugate pole pair linear phase  0 →

Real pole  0 → 
Real pole linear phase  0 → 
Complexconjugate pole pair  0 → 
Complexconjugate pole pair linear phase  0 → 
In case of absolute bound scaling, the length
It is shown in [
In case of forward ordering, Equations (39–41) become:
In case of reverse ordering, Equations (39–41) become:
Here,
In order to keep the text more readable, the figures of the several calculation results of Equations (55–60) are grouped together at the end of the section.
Overview of the figures of the SNR calculations.
Approximation of  Scaling type  Ordering  Range  Figure 

Real pole  L_{2}  Forward and Reverse   

Real Pole  Abs and Inf.  Forward and Reverse  
Real Pole Linear Phase  L_{2}  Forward and Reverse  
Real Pole Linear Phase  Abs and Inf.  Forward and Reverse  
Compl. Conj. Pole pair  L_{2}  Forward and Reverse  
Compl. Conj. Pole pair  Inf  Forward and Reverse  
Compl. Conj. Pole pair  Abs  Forward and Reverse  
Compl. Conj. Pole pair  Inf  Reverse  
Compl. Conj. Pole pair  L_{2}, Inf, Abs  Forward and Reverse  
Compl. Conj. Pole pair, Lin Phase  L_{2}, Inf, Abs  Forward  
Compl. Conj. Pole pair, Lin Phase  L_{2}, Inf, Abs  Reverse  
Compl. Conj. Pole pair, Lin Phase  Inf  Reverse 
Here,
As explained in
In case of absolute or infinity bound scaling, the MFIR filter has a better signal to noise performance than the approximated real pole filter for 
In case 
The more 
There is no objective comparison possible between this MFIR approximation and the IIR filter, since the IIR filter is not a linear phase filter and the MFIR filter approximates the
The ordering has no impact when L_{2} bound scaling is used and the noise performance with L_{2} bound scaling is always better than absolute and infinity bound scaling. Infinity bound scaling and absolute bound scaling have the same performance. Reverse ordering yields again better results than forward ordering. However, the maximum difference between the two orderings is rather small (3.3 dB).
Compared to the nonlinear phase approximation, the difference between L_{2} bound scaling and the other scaling methods are somewhat larger.
All pole magnitudes between 0.8 and 0.99 in steps of 0.01 have been calculated but are not all shown here. The results shown in the figures are however representative for all combinations of scaling methods, pole magnitudes and angles that were calculated.
After extensive analysis of the data, the following conclusions can be drawn for the MFIR approximation of a complexconjugate pole pair filter realized using the cascade structure. The roundoff noise performance of an MFIR filter approximating a complexconjugate pole pair filter
is significantly better (up to 20 dB) than the noise performance of its corresponding IIR filter when the approximated poles are situated in the neighborhood of the real axis;
is far less pole angle
is up to 2.5 dB better for infinity bound scaling than for absolute bound scaling (using reverse ordering);
is always better for L_{2} bound scaling than for the other scaling methods (obeys Equation (44));
is pole magnitude dependent, but not that much as the corresponding IIR filter;
is fairly insensitive to an extra MFIR filter stage (typically 1 dB);
is very sensitive to the stage ordering for absolute and infinity bound scaling;
is ordering independent in case of L_{2} bound scaling;
is in general better in reverse ordering than in forward ordering, except for pole angles in the neighborhood of
is for pole angles in the neighborhood of
can be up to 6 dB worse than the corresponding IIR filter for pole angles in the neighborhood of
In the normal range of pole magnitudes that are considered for MFIR approximations (
Compared with the nonlinear phase approximation, the
Unfortunately in the neighborhood of the pole angles
In
It is clear from Equations (62) and (63) that forward ordering (
cos (2
Even in case of forward ordering, the SNR performance for these angles is not good. Indeed the stages with larger
An approach to model roundoff noise in general cascade filter structures has been studied. This roundoff noise depends on the used scaling method and on the ordering of the stages. These general results are used to study and optimize the roundoff noise behavior of MFIR filters.
It has been shown that the roundoff noise performances of the MFIR pole approximations indeed depend on the used scaling method. L_{2} bound scaling results in the best performance, followed by infinity and absolute bound scaling.
In general, it can be concluded that in the region of interest (approximating pole behaviors with magnitudes
The analysis presented in [
when approximating a complexconjugate pole pair having a pole angle in the neighborhood of π/2 and the cascade structure has been used;
when approximating the squared magnitude response of a complexconjugate pole pair filter in case the pole angles are situated in the neighborhood of π/4, π/2 and 3π/4 and the linear phase cascade structure has been used.
Further research is required to determine if alternative orderings can be found which would yield a better noise performance. Note that the orderings considered here are not the only possible orderings. For best performance, one must determine the “declining amplification” order for every pole pair that is approximated. At the moment, no systematic approach has been found, so trial and error is required. Research will have to prove if an optimal ordering can be calculated. If not, a heuristic approach as in [
The authors declare no conflict of interest.