# Documentation

## IIR Polyphase Filter Design

This example shows how to design IIR polyphase filters.

IIR polyphase filters present several interesting properties: they require a very small number of multipliers to implement, they are inherently stable, have low roundoff noise sensitivity and no limit cycles. Furthermore, it is possible to achieve almost linear phase designs.

Butterworth, Chebyshev I and II, and elliptic IIR filters can be designed using allpass subfilters as building blocks for the following response types: lowpass, highpass, bandpass and bandstop. Butterworth and elliptic IIR filters can also be designed with a halfband, Hilbert transform, multirate halfband and multirate dyadic halfband response type.

Cost Efficiency

A way of measuring a filter's computational cost is to determine how many multiplications need to be computed (on average) per input sample (MPIS). Consider a MPIS count case study: FIR vs IIR for the following filter specifications:

```Fs = 9.6e3; % Sampling frequency: 9.6kHz Fc = 300; % Cutoff frequency: 300Hz TW = 9.6; % Transition width: 9.6Hz Ap = 3e-3; % Maximum passband ripple: 0.003 dB Ast= 80; % Minimum stopband attenuation: 80 dB Fp = Fc-TW/2; Fst= Fc+TW/2; LowpassDesign = fdesign.lowpass(Fp,Fst,Ap,Ast,Fs); ```

Elliptic filters provide the lowest order IIR filter of all minimum order IIR filters that meet the specifications (optimal minimax solution). The classic elliptic design can be implemented with a direct-form II, second-order sections in a Biquad structure:

```IIRFilt = design(LowpassDesign,'ellip','SystemObject',true) ```
```IIRFilt = System: dsp.BiquadFilter Properties: Structure: 'Direct form II' SOSMatrixSource: 'Property' SOSMatrix: [8x6 double] ScaleValues: [9x1 double] InitialConditions: 0 OptimizeUnityScaleValues: true ```

This filter requires 32 MPIS.

```cost(IIRFilt) ```
```ans = NumCoefficients: 32 NumStates: 16 MultiplicationsPerInputSample: 32 AdditionsPerInputSample: 32 ```

In most cases, if we are reducing the bandwidth of a signal, we should also reduce its sampling rate to improve its computational cost. Rather than using the conventional approach that the stopband frequency must be set at 1/M, where M is the decimation factor, Nyquist filters when used for decimation set the cutoff frequency at 1/M. This allows for some aliasing to be introduced. However, this aliasing occurs in the transition region of the filter only, a region in which the signal is being distorted anyway.

```M = (Fs/2)/Fc; NyquistDecimDesign = fdesign.decimator(M,'Nyquist'); setspecs(NyquistDecimDesign,M,TW,Ast,Fs); MultistageFIRDecim = design(NyquistDecimDesign,'multistage',... 'HalfbandDesignMethod','equiripple','SystemObject',true) ```
```MultistageFIRDecim = System: dsp.FilterCascade Properties: Stage1: [1x1 dsp.FIRDecimator] Stage2: [1x1 dsp.FIRDecimator] Stage3: [1x1 dsp.FIRDecimator] Stage4: [1x1 dsp.FIRDecimator] ```

A way of obtaining efficient FIR designs is through the use of multirate multistage techniques. This design results in four FIR halfband filters in cascade. Halfband filters are extremely efficient because every other coefficient is zero.

```cost(MultistageFIRDecim) ```
```ans = NumCoefficients: 314 NumStates: 612 MultiplicationsPerInputSample: 23.8125 AdditionsPerInputSample: 22.8750 ```

This method achieves computational costs lower than that of a direct-form II, second-order sections IIR elliptic filter since it requires only 23.8125 MPIS on average compared to 32 MPIS for the classic elliptic IIR design.

"Modern" IIR designs can take advantage of the same sort of "tricks" we can play with FIR filters resulting in extremely efficient designs.

```CoupledAPFilt = design(LowpassDesign,'ellip',... 'FilterStructure','cascadeallpass','SystemObject',true) ```
```CoupledAPFilt = System: dsp.CoupledAllpassFilter Properties: Structure: 'Minimum multiplier' PureDelayBranch: false AllpassCoefficients1: {4x1 cell} AllpassCoefficients2: {5x1 cell} Gain1: '1' Gain2: '1' ```

The polyphase elliptic implementation is composed of two allpass IIR subfilters. Notice that this design does take advantage of multirate techniques.

```cost(CoupledAPFilt) ```
```ans = NumCoefficients: 18 NumStates: 34 MultiplicationsPerInputSample: 18 AdditionsPerInputSample: 35 ```

The cost of this structure is only 18 MPIS. Although single-rate, this structure has already a lower cost than the multistage multirate FIR filter.

Because the specified cutoff frequency of the filter is the inverse of a power of two, we can use multirate multistage techniques based on IIR halfband filters as we previously did in the FIR case.

```MultistageIIRDecim = design(NyquistDecimDesign,'multistage',... 'HalfbandDesignMethod','ellip') ```
``` MultistageIIRDecim = FilterStructure: Cascade Stage(1): IIR Polyphase Decimator Stage(2): IIR Polyphase Decimator Stage(3): IIR Polyphase Decimator Stage(4): IIR Polyphase Decimator PersistentMemory: false ```

This latest design results in four IIR halfband filters in cascade.

```cost(MultistageIIRDecim) ```
``` ans = Number of Multipliers : 17 Number of Adders : 34 Number of States : 25 Multiplications per Input Sample : 2.5 Additions per Input Sample : 5 ```

The multirate multistage design reaches an incredibly low 2.5 MPIS. Starting with the classic optimal elliptic design that required 32 MPIS, we were able to first reduce the computational cost to 18 MPIS by using a single-rate combination of cascade-allpass subfilters and then to only 2.5 MPIS by fully leveraging the benefits of multirate halfband IIR filters.

If we overlay the magnitude responses of the FIR and IIR multirate multistages filters, the two filters look very similar and both meet the specifications.

```fvFig = fvtool(MultistageFIRDecim,MultistageIIRDecim,'Color','white'); legend(fvFig, 'Multirate/Multistage FIR Polyphase', ... 'Multirate/Multistage IIR Polyphase') fvtool(MultistageFIRDecim,MultistageIIRDecim,'Color','white'); title('Passband Magnitude Response (dB)') axis([0 0.325 -0.0016 0.0016]) ```

Close inspection actually shows the passband ripples of the IIR filter to be far superior to that of the FIR filter. So computational cost savings don't come at the price of a degraded magnitude response.

Quasi-Linear Phase Halfband and Dyadic Halfband Designs

By modifying the structure used to implement each IIR halfband filter, it is possible to achieve almost linear phase designs using IIR filters. This comes at the expense of a slight increase in computational cost due to the constrain on the phase (reduction in the degrees of freedom in the design).

```NyquistDecimDesign = fdesign.decimator(8,'Nyquist',8,300,80,9.6e3); FIRLinearPhaseFilt = design(NyquistDecimDesign,'multistage',... 'HalfbandDesignMethod','equiripple','SystemObject',true); IIRLinearPhaseFilt = design(NyquistDecimDesign,'multistage',... 'HalfbandDesignMethod','iirlinphase'); fvFig = fvtool(FIRLinearPhaseFilt,IIRLinearPhaseFilt,'Color','white',... 'Analysis','grpdelay'); axis([0 0.6 85 125]) title('Passband Group delay') legend(fvFig, 'Linear-Phase FIR', 'Quasi-Linear Phase IIR') ```

Nevertheless, these designs achieve very good phase characteristics and are still more efficient than comparable trully linear phase FIR halfband designs.

```cost(FIRLinearPhaseFilt) ```
```ans = NumCoefficients: 43 NumStates: 74 MultiplicationsPerInputSample: 9.8750 AdditionsPerInputSample: 9 ```
```cost(IIRLinearPhaseFilt) ```
``` ans = Number of Multipliers : 16 Number of Adders : 32 Number of States : 38 Multiplications per Input Sample : 3.75 Additions per Input Sample : 7.5 ```

Fixed-Point Robustness

Polyphase IIR filters can be implemented in different ways. We have already encountered single-rate and multirate cascade allpass in previous sections. Now take a Hilbert transformer for example. A quasi linear-phase IIR Hilbert filter with a transition width of 96Hz and a maximum passband ripple of 0.1 dB can be implemented as a cascade wave digital filter using only 10 MPIS compared to 133 MPIS for an FIR equivalent:

```HilbertDesign = fdesign.hilbert('TW,Ap',96,.1,Fs); HilbertIIRFilt = design(HilbertDesign,'iirlinphase',... 'FilterStructure','cascadewdfallpass','SystemObject',true) cost(HilbertIIRFilt) ```
```HilbertIIRFilt = System: dsp.CoupledAllpassFilter Properties: Structure: 'Wave Digital Filter' PureDelayBranch: true WDFCoefficients2: {6x1 cell} Delay: 16 Gain1: '1' Gain2: '0+1i' ans = NumCoefficients: 10 NumStates: 33 MultiplicationsPerInputSample: 10 AdditionsPerInputSample: 25 ```

Wave digital filters have been proven to be very robust even when poles are close to the unit circle. They are inherently stable, have low roundoff noise properties and are free of limit cycles. To convert our IIR Hilbert filter to a fixed-point representation, we can use the realizemdl command and the Fixed-Point Tool to do the floating-point to fixed-point conversion of the Simulink model:

```realizemdl(HilbertIIRFilt) ```

Summary

IIR filters have traditionally been considered much more efficient than their FIR counterparts in the sense that they require a much smaller number of coefficients in order to meet a given set of specifications.

Modern FIR filter design tools utilizing multirate/polyphase techniques have bridged the gap while providing linear-phase response along with good sensitivity to quantization effects and the absence of stability and limit cycles problems when implemented in fixed-point.

However, IIR polyphase filters enjoy most of the advantages that FIR filters have and require a very small number of multipliers to implement.