MATLAB: Analog QAM with Stereo Signals

Hello Guys,

I hope you are enjoying there. Today, I am having a lot of work esp. final projects for semester end. My college is off from next week due to Diwali Vacation. So, having a lot of work…

Anyway, I am taking some rest. So, wanna get back to MATLAB.

We are starting with DSB-SC scheme for modulation and demodulation. And then we will discuss Analog Qudrature Amplitude modulation scheme for sending two different signals at the same frequency band modulated with sinusoid carrier. For one signal to be modulated, we are using cosine and, for another is modulated by Sine. And we will also write a code in MATLAB to do the analysis.

Let’s look at the simple modulation scheme.

DSB-SC

Figure 1. Schematic representations of the amplitude modulation and demodulation schemes: (a)

modulator and (b) demodulator.

Here n is time domain representing variable (continuous time).  And Acosw_o­n is carrier wave. And mixer multiplies these two signals in time domain and generates the output y[n].

Looking the time domain representation of signals…

Figure 2: (a) Message signal = m(t) = x[n]. (b) Carrier Signal = c(t) = c[n] =Acosw_on. (c) Modulated Signal = y[n]

From Figure 2, Modulated signal is having carrier varing in the amplitude with reference to the message signal. This scheme is also called DSB-SC (Double Side Band Suppressed Carrier).

DSB-Sc in frequency domain is shown in following figure.

Figure 3. |X(F)| = Magnitude spectrum of message signal and |Y(F)| is Magnitude Spectrum of modulated signal

From the frequency domain representation, message signal is of bandwidth F_M and frequency of carrier is F_c. The Modulated signal is shifted to + and - F_c. And the bandwidth of modulated signal is doubled i.e. 2F_M than the modulating signal.

To demodulate the signal at the receiver end, we will multiply the received signal i.e. y[n] with locally generated carrier (can be extracted from received signal) and then Low Pass filtering of the signal will give the original signal back… This way we can send a signal over the channel i.e. wireless or wired.

Analog QAM

It is possible to modulated one signal by sine wave and other by cosine and add them together to get the modulated signal. By using this, we can save the requirement of different frequency band for other signal. This is also called Analog Quadrature Amplitude modulation (Analog QAM).

Let’s look at the model of QAM.

Figure 4. Schematic representations of the quadrature amplitude modulation

Here, two different signals x₁[n] and x₂[n] are being modulated by Acosw_on and Asinw_on (i.e. 90^o shifted of Acosw_on) respectively. So, the output of two mixer will be Ax₁[n]cosw_on and Ax₂[n]sinw_on. Now the next stage is to add these two signals to get the modulated signal y[n].

Figure 5. Carrier wave (a) Asinw_ot (b) Acosw_ot

Figure 6. Message Signal (a) x₂[n] (b) x1[n]

Figure 7. Modulated Results (a) Ax₂[n]sinw_on (b) Ax₁[n]cosw_on

Figure 8. Combined signal for transmission i.e. final modulated signal = y[n]

Writing the equation of y[n],

                y[n] = Ax₁[n] cos(ω_on) + Ax₂[n] sin(ω_on)

Note that the two carrier signals have the same carrier frequency ωo but have a phase difference of 90o. In general, the carrier A cos(ωon) is called the in-phase component and the carrier A in(ωon) is called the quadrature component. The spectrum Y (ejω) of the composite signal y[n] is now given by,

Y (e^jω) = A/2 {X₁(e^j(w-wo)) + X₁(^j(w+wo))} + A/2j {X₂(e^j(w-wo)) - X₂(^j(w+wo))}

This signal y[n] is transmitted through channel and it is received by the receiver.

To recover the original modulating signals, the composite signal is multiplied by both the in-phase (cosw_on) and the quadrature (sinw_on) components of the carrier separately resulting in two signals:

Figure 9. Demodulator of QAM

The demodulation process requires the carrier locally generated. Here also, the process is same as DSB-SC demodulation. But in upper path, the carrier which is multiplying with signal is cosw_on and for lower path it is sinw_on. So, at the output of mixer is something like,

                        r₁[n] = y[n] cos(ω_on) and r₂[n] = y[n] sin(ω_on)

So,

r₁ [n] = A/2 x₁[n] + A/2 x₁[n] cos (2w_on) + A/2 x₂[n] sin (2w_on) and

r2[n] = A/2 x₂[n] + A/2 x₁[n] sin (2w_on) - A/2 x₂[n] cos (2w_on).

Now to get both the signals back, we will pass these signals with low-pass filter with cut-off frequency = w_m. So the output of low-pass filter will be A/2 x₁[n] and A/2 x₂[n] respectively. And hence, we will get both the signals at the receiver end by transmitting in the same frequency band.

Now this is the time to implement this scheme in MATLAB.

MATLAB Implementation

For doing this, we will use an audio file (in wav format). This file have two audio component i.e. left and right audio. So, we will get two different signal like x₁[n]=left signal and x₂[n]=right signal. Now, we will modulate these signals with sinusoid signals of freq w_o = 16kHz. But t do that, first we need to understand the problem with matlab to do modulation. Matlab works with signal in terms of matrix. All matrix are discrete quantities. So, to get the perfect (not perfect but somewhat more) glimpse of analog signal we will up-sample the signal by 10. And then we will modulate it add it and transmit it. To understand the modulation and demodulation only we are not considering the noise (In practice, the noise will always be present). Now, at the other end, received signal is same as transmitted signal because, noise is not considered. So, to get the original signal back, we will multiply the signal with in-phase and quadrature-phase component of w_o frequency. Then for low-pass filter we will design Butterworth low-pass filter having pass band frequency 4kHz and stop band = 10kHz. The next thing is to down sample the signal to remove the extra samples which we have added at the modulation time. Now we will get the left and right signal back. To get the original audio signal back i.e. stereo signal, we will structure the matrix of received left and right file. And then we will listen the audio.

Following are the figures, which will help to learn in better way with graphical representation.

Figure 10. Stereo signal with left and Right channel signal and their zooming version

Figure 11. Left signal and Up-Sampled with interpolation (Zooming Version)

Figure 12. Frequency Response of Cosine wave

Figure 13. Frequency Response of Sine wave

Figure 14. Left and Right wave with carrier and modulated wave

Figure 15. Separately Modulated wave and their addition i.e. final transmitted wave

Figure 16. Frequency Spectrum of Message signal, Carrier and Modulated wave.

Figure 17. Spectra of Received signal and their multiplication with in-phase and quadrature component of frequency w_o.(Here for rec1 and rec2 the higher freq component is overwriting. i.e. actually higher freq. component is at 3.2kHz. But due to insufficient FFT samples it is showing ar 1.2kHz)

  

Figure 18. Butterworth Low-Pass Filter Response (F_p=4kHz and F_s = 10kHz)

Figure 19. Frequency Spectrum of Received signal, Multiplied received signal with carrier overlaying the bandpass response and low pass filtered signal.

Figure 20. Received Signal 1 = Left (Upper one), Received signal 2 = Right (Middle One) and Generated Stereo Signal (Lower one)

The code for this program is

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%% Project
% Aim : To implement the analog Quadrature amlitude modultaion scheme on stereo.
% Here we approch, modulating two signals i.e. left and right channel signal of stereo
% signal, with same frequency of sinusoid. But one of the signal will get
% modulated by sine and other with cosine. As the sine and cosine are of
% pi/2 phase, the corelation (i.e. integreation over a period) is zero. So
% to use this phenomena, both the modulated signal is added with
% each-other and then we will transmit them. Here to give the glimps of
% analog signal we are using audio in wav file format, which is loss-less
% format among the all audio compression standard. Though, now-a-days the
% audio becomes digital. So one more thing to do is upsample the signal, so
% that it will look like upsampled(20-20kHz upsampled by 10. so 2-2kHz)
% analog singnal. And then we will modulate both the singal by 16kHz
% sinusiod wave. Then add it. And transmitting the modulated singal.

% On the other hand, we receive the signal which is sent through the
% channel, maybe wireless or wired depeding upon the choice w.r.t economy or
% environment(beheaviour). So there will be some noise received at the
% receiver end. But for simplicity, to understand the modulation and
% demodulation process, assuming the nosie on the receiver end is zero i.e.
% ideal condition. Now, To demodulate the signal, we pass the received
% signal in two parallel circuits. In which, one mixes the sine wave and
% other mixes the cosine wave by mixer. Here also assumed that the carreir
% at the receiver side is in sync. with the received signal carreir. We can
% extract carrier from the received signal also. Now the next thing is to
% get the left and right channel signal back by passing the output of
% mixer into low-pass filter at the receiver end. Here we are using
% ButterWorth Low-Pass Filter. Then we will downsample the signals and we
% will get both the singal. If we are modulating stereo, then at the other
% end we need to get the stereo sound back by formating the two signals at
% the received end. 

%% Detalis about Author 
% Author : Sagar Patel
% Email ID : sagarpatel.9556@gmail.com

%% Initilizing and Reading the data
clear all;
clc;
close all;
clf;
[signal fs]=wavread('sample.wav');
Time=length(signal)/fs;  % In Seconds
%% Left and Right Audio Signals
s_l=signal(:,1)';
s_r=signal(:,2)';
%% Plotting the Data
figure(1);
t=0:1/fs:Time-(1/fs);
subplot(321);
plot(t',signal);
title('Stereo Signal');
xlabel('Time (seconds)');ylabel('Amplitude');
subplot(322);
plot((50000:51000)/fs,signal(50000:51000,:));
title('Stereo Signal (ZOOM UP) <50000-51000>');
xlabel('Time (seconds)');ylabel('Amplitude');
subplot(323);
plot(t',s_l);
title('Left Signal');
xlabel('Time (seconds)');ylabel('Amplitude');
subplot(324);
plot((50000:51000)/fs,s_l(50000:51000));
title('Left Signal (ZOOM UP) <50000-51000>');
xlabel('Time (seconds)');ylabel('Amplitude');
subplot(325);
plot(t',s_r);
title('Right channel Signal');
xlabel('Time (seconds)');ylabel('Amplitude');
subplot(326);
plot((50000:51000)/fs,s_r(50000:51000));
title('Right Signal (ZOOM UP) <50000-51000>');
xlabel('Time (seconds)');ylabel('Amplitude');
%% Listening the Sound
display('Lets listen to the stereo Sound ');
pause;
sound(signal,fs);
display('Stereo Sound');
pause;
display('Lets listen to left channel signal ');
pause;
sound([s_l',zeros(length(s_l),1)],fs);
display('Left sound');
pause;
display('Lets listen to right channel signal ');
pause;
sound([zeros(length(s_r),1),s_r'],fs);
display('Right sound');
pause;
%% Up Sampling to Modulate the Signal
L = 10;         % Up-Sampling Co-Efficient
x1 = s_l;
x2 = s_r;
% Generating the interpolated output sequence
y1 = interp(x1,L);
y2 = interp(x2,L);
% Ploting the input (left) and the output (upsampled left) sequences
figure(2);
subplot(2,1,1);
n=10000:10100;
stem(n,x1(n(1):n(length(n))));
title('Left Sequence ');
xlabel('Time index n'); ylabel('Amplitude');
subplot(2,1,2);
m = L*n(1):(n(length(n))*L)-1;
stem(m,y1(m(1):m(length(m))));
title('Up Sampled Left Sequence ');
xlabel('Time index n'); ylabel('Amplitude');
s_l_up = y1;
s_r_up = y2;
%% Generating a carrier
fc = 16000;
A = 1;
t = 0:1/fs:(length(s_l_up)-1)/fs;
carrier = A*cos(2*pi*fc*t);
carrier2 = A*sin(2*pi*fc*t);
figure(3);
subplot(221);
plot(t,carrier);
title('Carrier Wave - Cosine (16 kHz)');
xlabel('Time (seconds)');
ylabel('Amplitude (seconds)');
subplot(222);
plot(t(100:150),carrier(100:150));
title('Carrier Wave - Cosine ');
xlabel('Time (seconds)');
ylabel('Amplitude (seconds)');
subplot(223);
plot(t,carrier2);
title('Carrier Wave - Sine (16 kHz)');
xlabel('Time (seconds)');
ylabel('Amplitude (seconds)');
subplot(224);
plot(t(100:150),carrier2(100:150));
title('Carrier Wave - Sine ');
xlabel('Time (seconds)');
ylabel('Amplitude (seconds)');

figure(4);
freqz(carrier,1,512,fs);
title('Frequency Response of Carrier - Cosine');
figure(5);
freqz(carrier2,1,512,fs);
title('Frequency Response of Carrier - Sine');
%% Modulation the Snare signal with Carrier
mod1 = s_l_up.* carrier;
mod2 = s_r_up.* carrier2;
mod = mod1 + mod2;
%% Plotting in Time domain
figure(6);
subplot(311);
plot(s_l_up(104000:105000));
hold on;
plot(s_r_up(104000:105000),'-g');
title('Left(B) and Right(G) in time domain');
subplot(312);
plot(carrier(104000:105000));
title('Carrier in time domain');
subplot(313);
plot(mod1(104000:105000));
hold on;
plot(mod2(104000:105000),'-g');
title('Modulated - Left(B) Right(G) - in time domain');

figure(7);
subplot(311);
plot(mod1(104000:105000));
title('Left Modulated in time domain');
subplot(312);
plot(mod2(104000:105000));
title('Right Modulated in time domain');
subplot(313);
plot(mod(104000:105000));
title('Modulated signal in time domain');
%% Plotting the FFT of Signals
figure(8);
NFFT = 65536*2*2;
wd = [-pi:2*pi/(NFFT-1):pi]; %digital frequency in rad/s
fd = [-pi:2*pi/(NFFT-1):pi]/(2*pi); %digital frequency in hz
Fa = fd * fs; %analog freq in hz

subplot(321);
plot(Fa,abs(fftshift(fft(s_l_up,NFFT))));
title('FFT of Up-Sampled of Left signal');
subplot(322);
plot(Fa,abs(fftshift(fft(s_r_up,NFFT))));
title('FFT of Up-Sampled of Right signal');
subplot(323);
plot(Fa,abs(fftshift(fft(carrier,NFFT))));
title('FFT of Carrier Signal - Cosine');
subplot(324);
plot(Fa,abs(fftshift(fft(carrier2,NFFT))));
title('FFT of Carrier Signal - Sine');
subplot(325);
plot(Fa,abs(fftshift(fft(mod1,NFFT))));
title('FFT of modulated signal');
subplot(326);
plot(Fa,abs(fftshift(fft(mod2,NFFT))));
title('FFT of modulated signal');
%% Demodulation
rec=mod;
%Here we can generate carrier locally or by extracting from signal
%So using the transmitted carreir to co-relate
rec1 = rec.*carrier;
rec2 = rec.*carrier2;
%% Plotting the FFt of Demodulated rec1 and rec2
figure(9);
subplot(311);
plot(Fa,abs(fftshift(fft(rec,NFFT))));
title('Received Signal Spectrum');
subplot(312);
plot(Fa,abs(fftshift(fft(rec1,NFFT))));
title('rec1 Signal Spectrum');
subplot(313);
plot(Fa,abs(fftshift(fft(rec2,NFFT))));
title('rec2 Signal Spectrum');

%% Generating a Low Pass Filter
F_Pass= 4000;    %PassBand Freq = 4kHz
F_Stop= 10000;   %StopBand Freq = 10kHz

F_s=fs;

fd_p = F_Pass / F_s;
fd_s = F_Stop / F_s;

Rp = 0.5;
Rs = 30;
[Order, Cut_off] = BUTTORD(fd_p, fd_s, Rp, Rs);
[num,den] = BUTTER(Order,Cut_off);

%% Demodulating the signal by extracting the carrier
clear mod1 mod2;
rec_1_Filtrd = filter(num,den,rec1);
rec_2_Filtrd = filter(num,den,rec2);
figure(10);
[H,F] = FREQZ(num,den,512,F_s*2);
plot(F,abs(H));
title('Frequency Respose of LowPass Filter');
%% Plotting the FFT
figure(11);
subplot(311);
plot(Fa,abs(fftshift(fft(rec,NFFT))));
title('Received Signal Spectrum');
subplot(312);
plot(Fa,abs(fftshift(fft(rec1,NFFT))));
hold on;
f_vary = [-flipud(F); F];
h_vary = [flipud(abs(H)); abs(H)];
plot(f_vary,2000*h_vary,'r');
title('Received 1 Signal Spectrum and LowPass Response');
subplot(313);
plot(Fa,abs(fftshift(fft(rec_1_Filtrd,NFFT))));
title('Low Pass Filtered Signal Spectrum');
%% Downsampling Data
lft_dwn = rec_1_Filtrd(1:L:end);
rht_dwn = rec_2_Filtrd(1:L:end);
%% Generating Stereo
final = [lft_dwn' rht_dwn'];
%% Plotting Received Downsampled signals in Time domain
figure(12);
t=0:1/fs:Time-(1/fs);
subplot(311);
plot(t,lft_dwn);
title('Left Signal in Time Domain');
xlabel('Time (Seconds)');ylabel('Amplitude');
subplot(312);
plot(t,rht_dwn);
title('Right Signal in Time Domain');
xlabel('Time (Seconds)');ylabel('Amplitude');
subplot(313);
plot(t',final);
title('Stereo Signal (Left = B and Right = G) in Time Domain');
xlabel('Time (Seconds)');ylabel('Amplitude');
%% Plotting Error Signal (We have not included AWGN. So Error = 0)
E_l = s_l - lft_dwn;
E_r = s_r - rht_dwn;
E_str = signal - final;
figure(13);
t=0:1/fs:Time-(1/fs);
subplot(311);
plot(t,E_l);
title('Error in Left Signal in Time Domain');
xlabel('Time (Seconds)');ylabel('Amplitude');
subplot(312);
plot(t,E_r);
title('Error in Right Signal in Time Domain');
xlabel('Time (Seconds)');ylabel('Amplitude');
subplot(313);
plot(t',E_str);
title('Error in Stereo Signal (Left = B and Right = G) in Time Domain');
xlabel('Time (Seconds)');ylabel('Amplitude');
%% Listening to the Saperate channels
display('Left Received and Down-sampled Signal ');
pause;
sound([lft_dwn' zeros(length(lft_dwn),1)],fs);
display('Left Signal');
pause;
display('Right Received and Down-sampled Signal ');
pause;
sound([zeros(length(rht_dwn),1) rht_dwn'],fs);
display('Right Signal');
pause;
%% Playing a stereo Signal
display('Received Stereo Signal ');
pause;
sound(final,fs);
display('Received Signal');
pause;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Guys, I hope that you will run this code and I am pretty sure that you will enjoy…

If you would like to do some innovation like adding an additive white Gaussian noise, changing the filter type i.e. chebysev, Elliptic, Bessel,etc.

Thank you buddy… Bye…