A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

**Language:** Matlab

**Processor:** Not Relevant

**Submitted by kadhiem Ayob on Jan 14 2012**

Licensed under a Creative Commons Attribution 3.0 Unported License

A popular method for upsampling is zero insertion. A regular zero interleaving of a signal results in an extra copy in its frequency domain. If this copy is removed by a filter that keeps original signal then the filter in effect creates interpolation points. If the copy is not removed efficiently then this implies less efficient interpolation in time domain.

Decimation can be done on a signal by discarding samples provided aliasing is avoided by removing frequencies above half of final sampling frequency.

If both upsampling and downsampling are planned then a single filter after zero insertion and before decimation may be enough.

The following code can check filter performance for interpolation and decimation. Additionally it checks the effect of moving signal band or filter envelope.

The code uses test parameters but you can choose your own input, your filter, your sampling rate, your required interpolation/decimation rates and your input signal centre frequency.

%%%%%%%%%%%%%%%%%% inputs for model %%%%%%%%%%%%%%%%

clear all; close all;

%example bandlimited impulse input & parameters

x = filter(fir1(70,.1),1,[1 zeros(1,2^15-1)]);

Fs = 120; %MHz original sample rate

h = fir1(30,.3); %filter used for upsampling

up = 3; %Interpolation factor

dn = 2; %Decimation factor

Fc = 12; %MHz band centre (-Fs/2 ~ +Fs/2)

Fch = 0; %MHz filter centre (-Fs*up/2 ~ +Fs*up/2)

%move signal to its centre

x = x.*exp(j*2*pi*(0:length(x)-1)*Fc/Fs);

%shift filter

h = h.*exp(j*2*pi*(0:length(h)-1)*Fch/(Fs*up));

%%%%%%%%%%%%%%%%%%%%% model %%%%%%%%%%%%%%%%%%%%%%

%check signal in upsampled domain

x_up = zeros(1,length(x)*up);

x_up(1:up:end) = x;

[P, F] = pwelch(x_up, [], 0, 2^16, Fs*up,'twosided');

F = F - max(F)/2;

P = fftshift(P);

y(find(P == 0)) = -100; %avoid log of zero

y(find(P ~= 0)) = 10*log10(P(find(P ~= 0)));

P_dB = y - 10*log10(max(P)); %normalise

%check filter response in upsampled domain

H = fftshift(20*log10(abs(fft(h,2^16))));

subplot(2,1,1);

hold;grid;

plot(F, P_dB,'.-');

plot(F,H,'m--');

axis([min(F)-1 max(F)+1 -80 1]);

legend('upsampled signal','upsampling filter');

%check signal in downsampled domain

x_f = filter(h,1,x_up);

x_dn = x_f(1:dn:end);

[P, F] = pwelch(x_dn, [], 0, 2^16, Fs*up/dn,'twosided');

F = F - max(F)/2;

P = fftshift(P);

y(find(P == 0)) = -100; %avoid log of zero

y(find(P ~= 0)) = 10*log10(P(find(P ~= 0)));

P_dB = y - 10*log10(max(P)); %normalise

subplot(2,1,2)

plot(F,P_dB,'r.-');

grid;

axis([min(F)-1 max(F)+1 -80 1]);

legend('downsampled signal');

clear all; close all;

%example bandlimited impulse input & parameters

x = filter(fir1(70,.1),1,[1 zeros(1,2^15-1)]);

Fs = 120; %MHz original sample rate

h = fir1(30,.3); %filter used for upsampling

up = 3; %Interpolation factor

dn = 2; %Decimation factor

Fc = 12; %MHz band centre (-Fs/2 ~ +Fs/2)

Fch = 0; %MHz filter centre (-Fs*up/2 ~ +Fs*up/2)

%move signal to its centre

x = x.*exp(j*2*pi*(0:length(x)-1)*Fc/Fs);

%shift filter

h = h.*exp(j*2*pi*(0:length(h)-1)*Fch/(Fs*up));

%%%%%%%%%%%%%%%%%%%%% model %%%%%%%%%%%%%%%%%%%%%%

%check signal in upsampled domain

x_up = zeros(1,length(x)*up);

x_up(1:up:end) = x;

[P, F] = pwelch(x_up, [], 0, 2^16, Fs*up,'twosided');

F = F - max(F)/2;

P = fftshift(P);

y(find(P == 0)) = -100; %avoid log of zero

y(find(P ~= 0)) = 10*log10(P(find(P ~= 0)));

P_dB = y - 10*log10(max(P)); %normalise

%check filter response in upsampled domain

H = fftshift(20*log10(abs(fft(h,2^16))));

subplot(2,1,1);

hold;grid;

plot(F, P_dB,'.-');

plot(F,H,'m--');

axis([min(F)-1 max(F)+1 -80 1]);

legend('upsampled signal','upsampling filter');

%check signal in downsampled domain

x_f = filter(h,1,x_up);

x_dn = x_f(1:dn:end);

[P, F] = pwelch(x_dn, [], 0, 2^16, Fs*up/dn,'twosided');

F = F - max(F)/2;

P = fftshift(P);

y(find(P == 0)) = -100; %avoid log of zero

y(find(P ~= 0)) = 10*log10(P(find(P ~= 0)));

P_dB = y - 10*log10(max(P)); %normalise

subplot(2,1,2)

plot(F,P_dB,'r.-');

grid;

axis([min(F)-1 max(F)+1 -80 1]);

legend('downsampled signal');

Experienced FPGA Engineer, focussed on DSP functionality within FPGAs

Comments

1/17/2012

2/12/2012

I ran your code. In the top panel of Figure 1 there's supposed to be a solid blue curve and a dashed magenta curve. But only the magenta curve is displayed in the top panel. In the bottom panel of Figure 1 there's supposed to be a solid red curve, but the bottom panel contains no curve at all. Am I doing something wrong?

[-Rick-]

2/12/2012

Seems like copy paste issue of those characters like 'r.-'

You may need to edit them or remove the dot.

Regards

kadhiem