# FFT processing gain

I am working on radar received signal which is $15000\times 2008$ matrix. Here each row corresponds to one chirp or one signal, this way there are 15000 signals. The difference among them is just the time difference and doppler shift.

If I plot the PSD of one signal(sweep) C(1,:) with MATLAB's psd I get a distinct peak ,

If I take the PSD of ten signals, i.e ten signals combined in one row [C(1,:) C(2,:)...C(10,:)] then take the psd, the plot is similar the only difference is that the gain increasing, I assume it is the processing gain, also as I increase the number of signals and take the psd the gain increases even more

I want to know what is the reason of the gain in the plots as the lenth of the signal is increasing, the fft length is not altered. I Would like to know any mathematics proving this.

fs=1e6;
C=Chirp;
x=[C(1,:) C(2,:) C(3,:) C(4,:) C(5,:) C(6,:) C(7,:) C(8,:) C(9,:) C(10,:)];
Pxx = periodogram(x);
hpsd = dspdata.psd(Pxx,'Fs',Fs); % Create a psd data object.
plot(hpsd);


In the code for the periodogram this occurs:

if (n > nfft)
Pxx = 0;
rr = rem (length (x), nfft);
if (rr)
x = [x(:); zeros(nfft-rr, 1)];
endif
x = sum (reshape (x, nfft, []), 2);
endif


Where nfft is the length of the fft and n is the length of your vector x.

Note how if n> nfft (as in your case) the vector is reshaped and summed together.