Autocorrelation and PSD (Wiener Khinchin Theorem) FFT Amplitude Scaling Problem

I've recently been working on creating on generating a random equiprobable polar NRZ signal (or Polar PAM). I'm attempting to compute its autocorrelation function and then its power spectral density (PSD) through the wiener-khinchin theorem. Theoretically the autocorrelation function of a bipolar NRZ signal is given as:

where A is the amplitude of the signal and T is the bit duration. Assuming A = 5 and T = 1 (meaning a bit rate of 1 bps). The following is the signal of the polar nrz signal using 100 bits and 10 samples per bit.

Below is the biased autocorrelation function (ACF). Rx(0) = 25 which was expected in the theoretical formula. Also the ACF has a triangular shape between tau = [-1 1]. This works fine.

[ACF,tau] = xcorr(obj.bit_stream,'biased');

tau = obj.sampling_period*tau;

figure;

plot(tau,ACF);

With more samples

Lastly, I computed the PSD using MATLAB's fft function. Below is the PSD shown

N = length(ACF);

fshift = (-N/2+0.5:N/2-0.5)*((obj.sampling_rate)/N);

df = fshift(2)-fshift(1);

PSD = abs((fftshift(fft(ACF))));

figure;

plot(fshift,PSD);

title('Graph 1');

xlim([-20 20]);

The shape looks fine and looks even more like a sinc^2 function as I increase the number of bits. The nulls of the sinc^2 function are also correct at f = 1,2.3...n assuming T = 1. Some aliasing did occur but didnt really affect the amplitude, The only problem is the amplitude. According to the theoretical equation above, sinc^2(0) should have a value of A^2 * T = 25.

Please note, I have tried scaling by multiplying the PSD with the following :

• sampling time (dt),
• dt^2
• df (df = fs/N)
• df^2
• (1/N), where N is the length of the ACF,
• 1/(sqrt(N)),
• 1 / (N*fs), where fs is the sampling frequency
• 1/(N^2).
• 1

When I tried scaling with dt, the parseval's power theorem was correct as shown below

N = length(ACF);

fshift = (-N/2+0.5:N/2-0.5)*((obj.sampling_rate)/N);

PSD = obj.sampling_time*abs((fftshift(fft(ACF))));

df = fshift(2)-fshift(1);

P_avg_TD = (sum(abs(obj.bit_stream).^2))*(obj.sampling_period/obj.time(end));

P_avg_FD = sum(PSD)*df;

disp(["The Time Domain Power is",P_avg_TD]);

disp(["The F-Domain Power is ",P_avg_FD]);

Both gave a power value of 25.

So my only concern is the amplitude scaling. I would appreciate any help as I've been working on this matter for a while now.