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RandomProcess & Quantization

Determining if given random processes are stationary processes(WSS) or not with statistical analysis MATLAB.

Converting an analog signal to digital to transfer & receive it in MATLAB.

Random Process

$x_1(t) = cos(2\pi t+\phi); \phi \approx U[-\pi,\pi]$

$x_2(t) = cos(2\pi t+\phi); \phi \approx U[-\pi/4,\pi/4]$

Let's start by sampling from the random processes mentioned above at a frequency of $F_s = 100 Hz$. Next, we generate three sets of samples for $\phi$, containing 100, 1000, and 10000 samples respectively. Finally, here are the results for $x_1$ and $x_2$ using these different sample sizes.

Mean of $x_1$ Mean of $x_2$

Here are the exact values of the mean calculated manually.

$E[x_1] = 0$

$E[x_2] = 2[sin(2\pi t+\pi/4)+cos(2\pi t+\pi/4)]/\pi$

By increasing the number of samples in $x_1$, we observe a convergence of its mean towards zero, indicating that the mean stationarity condition is fulfilled. However, the mean of $x_2$ varies with time, thus violating the condition of mean stationarity. Consequently, $x_2$ cannot be considered a stationary random process.

Sample Size Autocorrelation function of $x_1$ Autocorrelation function of $x_2$
100
1000

Here are the exact values of the autocorrelation function calculated manually.

$R_{x_1} = cos(2\pi(t_1-t_2))/2 $

$R_{x_2} = cos(2\pi(t_1+t_2))/\pi + cos(2\pi(t_1-t_2))/2$

By analyzing the diagrams from the standpoint of a constant time difference ($\tau = t_1 - t_2$), we can conclude that the autocorrelation function of $x_1$ is solely determined by $\tau$, thus meeting the criteria for a stationary random process.

However, in the case of $x_2$, the autocorrelation function yields different values depending on $\tau$.

Quantization

In this section, the $g(t)$ signal will undergo a uniform quantization process to convert it into a digital signal. Once received, the digital signal will be converted back into an analog signal.

$g(t) = 4 + sin(2\pi t) + cos(\pi t) + cos(\pi t/2) + tan(\pi t/6)$

Sampling

To begin, an analog signal $g(t)$ was generated consisting of 15,000 points. Subsequently, the signal was sampled at a frequency of $fs = 300 Hz$. The obtained results are as follows.

Analog Signal Sampled Signal

Applying Quantization Levels

To perform uniform quantization, the range of the sampled signal amplitude was divided into 33 levels. The process was implemented using the following code.

N = 33;
lines = linspace(min(g_sample),max(g_sample),N);
quantization = zeros(1,length(g_sample));
hold on
for k=1:length(lines)
    yline(lines(k));
end
for i=1:length(quantization)
    for j=1:length(lines)-1
        mean_two_line = (lines(j+1)+lines(j))/2;
        if(g_sample(i) >= lines(j) && g_sample(i) <= lines(j+1))
            quantization(i) = mean_two_line;
        end
    end
end
Quantization levels Quantized Signal

Digitalizing

This section describes the procedure of transmitting signals from the transmitter to the receiver using PAM (Pulse Amplitude Modulation) principles. For each quantized point (Digit or Symbol), the base pulse is multiplied by a specific range before transmission.

Base Pulse Modulation Result Modulation Result (Zoomed)

Receiving the Digital Signal

The receiver will receive the signal along with noise, which is assumed to follow a normal random process. The Signal-to-Noise Ratio (SNR) is assumed to be 2dB.

noise = normrnd(0,0.48,[1,length(modulate_signal)]);

signal_with_noise = modulate_signal + noise;

Here is the received signal with the aforementioned noise.

Decoding

A policy has been implemented to convert the pulses into their corresponding digits, with the knowledge that each digit is sent in 1 second. The basic pulse is multiplied in the received pulse string for every second. By calculating their mutual energy and considering the energy of the basic pulse, the amplitude of each of these pulses is determined.

for i=1:length(bits)
    noise_pulse = signal_with_noise(((i-1)*1000)+1:i*1000).*pulse;
    energy_each_pulse(i) = sum(noise_pulse) / energy_pulse;
end

Now, the energy_each_pulse values are compared with the two-dimensional array obtained from the Digitizing section in order to derive the quantization level.

for i=1:length(energy_each_pulse)
    nearest_odd_number = energy_each_pulse(i);
    is_odd = mod(nearest_odd_number,2) < 1;
    nearest_odd_number = floor(nearest_odd_number);
    nearest_odd_number(is_odd) = nearest_odd_number(is_odd)+1;
    energy_each_pulse(i) = nearest_odd_number;
    for j=1:length(lines)-1
        if (energy_each_pulse(i) == TwoD_array(1,j))
            decode_signal(i) = TwoD_array(2,j);
        end
    end
end

Course Description