Difference Between Built-in Periodogram and Self-Calculated Periodogram

Hello,

I have the following code block. Im just trying to calculate the periodogram of the signal 'x' with built-in periodogram function and without built-in periodogram function. I got the same pattern but the amplitudes are not equal. What am I missing there? Thank you for your help.

clc
clear
close all

j = sqrt(-1);
n = 1:1:256;

noise_mean = 0;
noise_deviation = sqrt(5);
noise = noise_deviation.*randn(1,256) + noise_mean;

x = 20*exp(j*2*pi*(0.15)*n) + 30*exp(j*2*pi*(0.20)*n) + noise;


N = length(x); % N point FFT
for k = 1:N 
    Sx(k) = 0;
    for n = 1:N
        Sx(k) = Sx(k)+x(n)*exp(-1i*2*pi*(k-1)*(n-1)/N);
    end
end

Sx = (1/N)*abs(Sx).^2;

figure
plot(10*log10(Sx)) , title('Periodogram without Built-in');

figure
plot(10*log10(periodogram(x))), title('Built-in Periodogram');

Answer 1

[Sulaymon Eshkabilov]

The only thing that you are missing that you ahve not specified FFT window size when you've employed MATLAB's builtin fcn:

clc
clear
close all

j = sqrt(-1);
n = 1:1:256;

noise_mean = 0;
noise_deviation = sqrt(5);
noise = noise_deviation.*randn(1,256) + noise_mean;

x = 20*exp(j*2*pi*(0.15)*n) + 30*exp(j*2*pi*(0.20)*n) + noise;


N = length(x); % N point FFT
for k = 1:N 
    Sx(k) = 0;
    for n = 1:N
        Sx(k) = Sx(k)+x(n)*exp(-1i*2*pi*(k-1)*(n-1)/(N));
    end
end

Sx = (1/N)*(abs(Sx).^2);

figure
plot(10*log10(Sx)) , title('Periodogram without Built-in');

figure
plot(10*log10(periodogram(x, [], N, 1))), title('Built-in Periodogram');   % FFT Window size is specified 

figure
plot(10*log10(Sx), 'b-', 'linewidth', 2.5, 'DisplayName', 'Manual Calc') , hold on
plot(10*log10(periodogram(x,[], N, 1)), 'DisplayName', 'Built-in Periodigram'), title('Built-in Periodogram vs. Manual Calc');
legend('toggle'), grid on

[tinkyminky93]

Can you explain the syntax sir? What does [], N, 1 corresponds to?

[Sulaymon Eshkabilov]

It is quite straightforward. N is Discrete Fourier Transform window size (also called resolution), which you set equal to the length of your generated signal (N = length(x)), and 1 corresponds to the sampling frequency. In your example, Fs is 1, becuase n = 1:1:256. [] is coming from the syntax of periodigram() fcn becuase of the other chosen parameters.