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Pulse Shaping Filter Design

This example shows how to design pulse shaping filters. Pulse shaping filters are used at the heart of many modern data transmission systems (e.g. mobile phones, HDTV) to keep a signal in an allotted bandwidth, maximize its data transmission rate, and minimize transmission errors. Raised cosine filters form a well-established solution to this filter design problem. In this example, we review why and explore less known alternatives that provide significant implementation cost savings.

Ideal Pulse Shaping Filter

The ideal pulse shaping filter has two properties:

  • A high stopband attenuation to reduce the interchannel interference as much as possible.

  • Minimized intersymbol interferences (ISI) to achieve a bit error rate as low as possible. The first Nyquist criterion states that in order to achieve a ISI-free transmission, the impulse response of the shaping filter should have zero crossings at multiples of the symbol period.

A time-domain sinc pulse meets these requirements since its frequency response is a brick wall but this filter is not realizeable. We can however approximate it by sampling the impulse response of the ideal continuous filter. The sampling rate must be at least twice the symbol rate of the message to transmit. That is, the filter must interpolate the data by at least a factor of two and often more to simplify the analog circuitry.

In its simplest system configuration, a pulse shaping interpolator at the transmitter is associated with a simple downsampler at the receiver.

Hrxd = mfilt.firdecim(8,1); % Downsampler

Raised Cosine Filter Design

A raised cosine filter is typically used to shape and oversample a symbol stream before modulation/transmission. The rolloff factor, R, determines the width of the transition band. Practical digital communication systems use a rolloff factor between 0.1 and 0.5. A minimum stopband attenuation of 60 to 80 dB is also desirable to suppress interchannel interference. In this example, we work with an oversampling or interpolation factor of 8. The filter specifications are given below.

L = 8;                  % Interpolation factor
R = 0.25;               % Roll-off factor
Ast = 60;               % Minimum stopband attenuation
f = fdesign.interpolator(L,'Raised Cosine',L,'Ast,Beta',Ast,R);
Htxrc = design(f,'window');
order(Htxrc)
ans =

   272

The raised-cosine filter designed above is obtained by truncating the analytical impulse response and it is not optimal in any sense. In fact, a filter order as high as 272 is necessary to attain a minimum stopband attenuation of 60 dB.

FIR Nyquist Filter Design

Nyquist filters can replace raised cosine filters for a fraction of the cost because they have an optimal equiripple response. The same stopband attenuation and transition width can be obtained with a much lower order. The transition width requirement can be deduced from the roll-off and interpolation factors, as follows:

TW = R/(L/2);          % Transition Bandwidth
d = fdesign.interpolator(L,'nyquist',L,'TW,Ast',TW,Ast);
Htxnyq = design(d,'equiripple');

The magnitude response of this filter and the raised cosine filter above have the same transition width and minimum stopband attenuation but the filter order of the equiripple Nyquist design has been reduced to 106.

order(Htxnyq)
ans =

   106

Multistage Halfband Filter Design

An even more efficient design is obtained by cascading 3 halfband filters. The main advantage of multistage over single stage designs is that longer (i.e. more expensive) filters can be operated at lower sample rates while shorter filters are operated at higher sample rates. Halfband filters can be designed using FIR and IIR design techniques.

d.Astop = 65;
Htxhbfir = design(d,'multistage',...
    'HalfBandDesignMethod','equiripple','Nstages',3);

FIR designs have an additional advantage in that every other coefficient is equal to zero. IIR designs can achieve quasi-linear phase and they offer a greater cost savings while achieving extremely low passband ripples.

Htxhbiir = design(d,'multistage',...
    'HalfBandDesignMethod','iirlinphase','Nstages',3);

Performance and Cost Analysis

The magnitude response of the four designs described above shows minimum stopband attenuation over 60 dB which reduces the interchannel interference to satisfactory levels.

Hrc1 = cascade(Htxrc,Hrxd);
Hnyq1 = cascade(Htxnyq,Hrxd);
Hhbfir1 = cascade(Htxhbfir,Hrxd);
Hhbiir1 = cascade(Htxhbiir,Hrxd);
hfvt = fvtool([Hrc1,Hnyq1,Hhbfir1,Hhbiir1], ...
    'NormalizeMagnitudeto1','on','Color','white');
legend(hfvt,'Raised Cosine','FIR Nyquist',' Multistage FIR Halfband',...
    'Multistage IIR Linear Phase Halfband');

Additionally, the bit error rate (BER) introduced by the four design is very similar. The function pulseshapedemogetBER.mpulseshapedemogetBER.m measures the BER. It assumes a 16-QAM modulation scheme and an additive white Gaussian noise channel. Notice that it requires Communications System Toolbox™ functions to run.

------------------------------------------------------------------------
                         Bit Error Rate (BER)
------------------------------------------------------------------------
                      10-dB SNR      15-dB SNR     20-dB SNR
Raised Cosine            0.0542         0.0035             0
FIR-Nyquist              0.0550         0.0037             0
FIR-Halfband             0.0549         0.0038             0
IIR-Halfband             0.0591         0.0045             0

While the performance of the four designs is almost identical, their implementation cost varies greatly, as show in the table below.

C1 = cost(Hrc1);
C2 = cost(Hnyq1);
C3 = cost(Hhbfir1);
C4 = cost(Hhbiir1);
------------------------------------------------------------------------
                    Implementation Cost Comparison
------------------------------------------------------------------------
               Multipliers         Adders  Mult/InSample   Add/InSample
Raised Cosine          272            265            272            265
FIR-Nyquist             94             87             94             87
FIR-Halfband            32             29             60             53
IIR-Halfband            12             24             22             44

As can be seen, alternatives to the raised cosine filter provide significant savings both in terms of hardware and operations per sample. They range from over 60% savings for the FIR Nyquist design to about 80% savings for the multistage FIR halfband design and 90% for the multistage IIR halfband design.

Using a Matched Filter at the Receiver

Sometimes the filtering is split between the transmitter and receiver. The data stream is upsampled and filtered at the transmitter and then the transmitted signal is filtered and downsampled by a matched filter at the receiver. This approach is very popular because, for a given processing power, using two square root raised cosine filters (one in the transmitter and one in the receiver) provides better stopband attenuation than using a raised cosine filter in the transmitter and a downsampler in the receiver.

SQRT Raised Cosine Filter Design

In theory, the cascade of two square root raised cosine filters is equivalent to a single normal raised cosine filter. However, the limited impulse response of practical square root raised cosine filters causes a slight difference between the responses of two cascaded square root raised cosine filters and of one raised cosine filter.

f1 = fdesign.interpolator(L,'Square Root Raised Cosine',L,'N,Beta',56,R);
f2 = fdesign.decimator(L,'Square Root Raised Cosine',L,'N,Beta',56,R);
Htxsqrc = design(f1,'window');  % Tx Filter
Hrxsqrc = design(f2,'window');  % Rx Filter
Hrc2 = cascade(Htxsqrc,Hrxsqrc);

Minimum Phase FIR Halfband Design

Minimum phase FIR halfband filters can be substituted for square root raised cosine filters.

d3 = fdesign.interpolator(2,'halfband','N,TW',15,0.25);
Hhb1min = design(d3,'equiripple','MinPhase',true);

setspecs(d3,'N,TW',7,0.5);
Hhb2min = design(d3,'equiripple','MinPhase',true);

setspecs(d3,'N,TW',3,0.78);
Hhb3min = design(d3,'equiripple','MinPhase',true);

In such a case, a minimum phase Nyquist filter is used at the transmiter while its maximum phase counterpart is used for filtering at the receiver.

Htxmin = cascade(Hhb1min,Hhb2min,Hhb3min); % Tx Filter
Hhb1max = mfilt.firdecim(2,fliplr(Hhb1min.Numerator/2));
Hhb2max = mfilt.firdecim(2,fliplr(Hhb2min.Numerator/2));
Hhb3max = mfilt.firdecim(2,fliplr(Hhb3min.Numerator/2));
Hrxmax = cascade(Hhb3max,Hhb2max,Hhb1max); % Rx Filter

The convolution of the minimum phase and maximum phase filters produces a Nyquist filter.

Hhbfir2 = cascade(Htxmin,Hrxmax);

Performance Analysis and Cost Analysis

The following code verifies that the cascaded filters provide a total stopband attenuation of 60 dB.

set(hfvt,'Filters',[Hrc2,Hhbfir2]);
legend(hfvt,'SQRT Raised Cosine','Multistage FIR Halfband');

Additionally, using a matched filter instead of a downsampler at the receiver improves the BER substantially for relatively low SNRs. The results in the table below were measured using the function pulseshapedemogetBER.mpulseshapedemogetBER.m.

------------------------------------------------------------------------
                         Bit Error Rate (BER)
------------------------------------------------------------------------
                      10-dB SNR      15-dB SNR     20-dB SNR
SQRT Raised Cosine    4.100e-05      3.333e-06     6.667e-06
Matched FIR-Halfband  2.000e-05              0             0

For the implementation cost, a multistage minimum and maximum phase FIR halfband design provides over 50% savings compared to the square root raised cosine design:

C5 = cost(Hrc2);
C6 = cost(Hhbfir2);
------------------------------------------------------------------------
                    Implementation Cost Comparison
------------------------------------------------------------------------
                     Multipliers  Adders  Mult/InSample  Add/InSample
SQRT Raised Cosine           114     105            114           105
Matched FIR-Halfband          56      47             54            39.12

Summary

Raised cosine and square root raised cosine filters are widely used in data transmission systems. Despite their popularity, we have shown in this example that they are not optimal in any sense. In many applications they could be advantageously replaced by alternative designs that are more cost efficient.

Appendix

The following helper functions are used in this example.

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