Signal Processing Toolbox 6.12

FIR Gaussian Pulse-shaping Filter Design

This demo shows the design of the Gaussian pulse-shaping FIR filter and the parameters influencing this design. The FIR Gaussian pulse-shaping filter design is done by truncating a sampled version of the continuous-time impulse response of the Gaussian filter which is given by:

The parameter 'a' is related to 3-dB bandwidth-symbol time product (B*Ts) of the Gaussian filter as given by:

There are two approximation errors in the design: a truncation error and a sampling error. The truncation error is due to a finite-time (FIR) approximation of the theoretically infinite impulse response of the ideal Gaussian filter. The sampling error (aliasing) is due to the fact that a Gaussian frequency response is not really band-limited in a strict sense (i.e. the energy of the Gaussian signal beyond a certain frequency is not exactly zero). This can be noted from the transfer function of the continuous-time Gaussian filter, which is given as below:

As f increases, the frequency response tends to zero, but never is exactly zero, which means that it cannot be sampled without some aliasing occurring.

Contents

Continuous-time Gaussian Filter

To design a continuous-time Gaussian filter, let us define the symbol time (Ts) to be 1 micro-second and the number of symbols between the start of the impulse response and its peak (NT) to be 3. From the equations above, we can see that the impulse response and the frequency response of the Gaussian filter depend on the parameter 'a' which is related to the 3 dB bandwidth-symbol time product. To study the effect of this parameter on the Gaussian FIR filter design, we will define various values of 'a' in terms of Ts and compute the corresponding bandwidths. Then, we will plot the impulse response for each 'a' and the magnitude response for each bandwidth.

Ts = 1e-6; % Symbol time (sec)
NT = 3; % Number of symbol periods between beginning and peak of impulse res
ponse

a = Ts*[.5, .75, 1, 2];
B = sqrt(log(2)/2)./(a);

t = linspace(-NT*Ts,NT*Ts,1000)';
hg = zeros(length(t),length(a));
for k = 1:length(a),
    hg(:,k) = sqrt(pi)/a(k)*exp(-(pi*t/a(k)).^2);
end

plot(t/Ts,hg)
title('Impulse response of a continuous-time Gaussian filter for various ban
dwidths');
xlabel('Normalized time (t/Ts)')
ylabel('Amplitude')
legend(sprintf('a = %g*Ts',a(1)/Ts),sprintf('a = %g*Ts',a(2)/Ts),sprintf('a 
= %g*Ts',a(3)/Ts),sprintf('a = %g*Ts',a(4)/Ts))

Note that the impulse responses are normalized to the symbol time.

Frequency Response for Continuous-time Gaussian Filter

We will compute and plot the frequency response for continuous-time Gaussian filters with different bandwidths. In the graph below, the 3-dB cutoff is indicated by the red circles ('o') on the magnitude response curve. Note that 3-dB bandwidth is between DC and B.

f = linspace(0,32e6,10000)';
Hideal = zeros(length(f),length(a));
for k = 1:length(a),
    Hideal(:,k) = exp(-a(k)^2*f.^2);
end

plot(f,20*log10(Hideal))
title('Ideal magnitude response for a continuous-time Gaussian filter for va
rious bandwidths');
legend(sprintf('B = %g',B(1)),sprintf('B = %g',B(2)),sprintf('B = %g',B(3)),
sprintf('B = %g',B(4)))
hold on
for k = 1:length(a),
    plot(B,20*log10(exp(-a.^2.*B.^2)),'ro')
end

axis([0 5*max(B) -50 5])
xlabel('Frequency (Hz)')
ylabel('Magnitude (dB)')

FIR approximation of the Gaussian Filter

We will design the FIR Gaussian filter using the FDESIGN.PULSESHAPING function. The FDESIGN.PULSESHAPING function takes the oversampling factor (i.e. the number of samples per symbol), the 3-dB bandwidth-symbol time product and number of symbol periods between the start of the filter impulse response and the end as inputs.

The oversampling factor (OF) determines the sampling frequency and the filter length and hence, plays a significant role in the Gaussian FIR filter design. The approximation errors in the design can be reduced with an appropriate choice of oversampling factor. We illustrate this by comparing the Gaussian FIR filters designed with two different oversampling factors.

First, we will consider oversampling factor of 16 to design the discrete Gaussian filter.

OF = 16; % Oversampling factor (samples/symbol)
d = fdesign.pulseshaping(OF,'Gaussian','Nsym,BT',2*NT);
for k = 1:length(a),
    d.BT = B(k)*Ts;
    h(k) = design(d);
end
[iz,t] = impz(h);
t = (t-t(end)/2)/Ts;
figure('color','white')
stem(t,iz)
title('Impulse response of the Gaussian FIR filter for various bandwidths, O
F=16');
xlabel('Normalized time (t/Ts)')
ylabel('Amplitude')
legend(sprintf('a = %g*Ts',a(1)/Ts),sprintf('a = %g*Ts',a(2)/Ts),sprintf('a 
= %g*Ts',a(3)/Ts),sprintf('a = %g*Ts',a(4)/Ts))

Frequency Response for FIR Gaussian Filter (oversampling factor=16)

We will calculate the frequency response for the Gaussian FIR filter with an oversampling factor of 16 and we will compare it with the ideal frequency response (i.e. frequency response of a continuous-time Gaussian filter).

Fs = OF/Ts;
hfvt = fvtool(h, 'FrequencyRange', 'Specify freq. vector', ...
    'FrequencyVector',f,'Fs',Fs,'color','white');
title({'Ideal mag. responses (dashed lines) and corresponding FIR appro
ximations (solid lines)' ;' for various values of B, OF=16'})
legend(hfvt, sprintf('B = %g',B(1)),sprintf('B = %g',B(2)),...
    sprintf('B = %g',B(3)),sprintf('B = %g',B(4)), ...
    'Location','NorthEast')
hold on;
plot(f*Ts,20*log10(Hideal),'--')
axis([0 32 -350 5])

Notice that the first two FIR filters exhibit aliasing error and the last two FIR filters exhibit the truncation error. Aliasing occurs when the sampling frequency is not greater than the Nyquist frequency. In case of the first two filters, the bandwidth is large enough that the oversampling factor does not separate the spectral replicas enough to avoid aliasing. The amount of aliasing is not very significant however.

On the other hand, the last two FIR filters show the FIR approximation limitation before any aliasing can occur. The magnitude responses of these two filters reach a floor before they can overlap with the spectral replicas.

Significance of the Oversampling Factor

The aliasing and truncation errors vary according to the oversampling factor. If the oversampling factor is reduced, these errors will be more severe, since this reduces the sampling frequency (thereby moving the replicas closer) and also reduces the filter lengths (increasing the error in the FIR approximation).

For example, if we select the oversampling factor of 4, we will see that all the FIR filters exhibit aliasing error as the sampling frequency is not large enough to avoid the overlapping of the spectral replicas in case of any FIR filter.

OF = 4; % Oversampling factor (samples/symbol)
d.SamplesPerSymbol = OF;
for k = 1:length(a),
    d.BT = B(k)*Ts;
    h(k) = design(d);
end
[iz,t] = impz(h);
t = (t-t(end)/2)/Ts;
figure('color','white')
stem(t,iz)
title('Impulse response of the Gaussian FIR filter for various bandwidths, O
F=4');
xlabel('Normalized time (t/Ts)')
ylabel('Amplitude')
legend(sprintf('a = %g*Ts',a(1)/Ts),sprintf('a = %g*Ts',a(2)/Ts),sprintf('a 
= %g*Ts',a(3)/Ts),sprintf('a = %g*Ts',a(4)/Ts))

Frequency Response for FIR Gaussian Filter (oversampling factor=4)

We will plot and study the frequency response for the Gaussian FIR filter designed with oversampling factor of 4. A smaller oversampling factor means smaller sampling frequency. As a result, this sampling frequency is not enough to avoid the spectral overlap and all the FIR approximation filters exhibit aliasing in this case.

Fs = OF/Ts;
hfvt = fvtool(h, 'FrequencyRange', 'Specify freq. vector', ...
    'FrequencyVector',f,'Fs',Fs,'color','white');
title({'Ideal mag. responses (dashed lines) and corresponding FIR appro
ximations (solid lines)' ;' for various values of B, OF=4'})
legend(hfvt, sprintf('B = %g',B(1)),sprintf('B = %g',B(2)),...
    sprintf('B = %g',B(3)),sprintf('B = %g',B(4)), ...
    'Location','NorthEast')
hold on;
plot(f*Ts,20*log10(Hideal),'--')
axis([0 32 -350 5])