# Reason for decrease in the PSD gain with increase in the number of FFT points

what is the reason for that? The gain should increase ideally I guess.

This is the code I tried. As I increase the length of F by zero padding,the gain of PSD reduces.

A=["48","03","0C","F6","00","F5","40","02","04","02","94","F4","D0","F5","2C","03","50","01","38","F4","24","F7","14","04","80","00","F4","F3","5C","F8","C4","04","28","FF","84","F3","08","F9","6C","04","EC","FD","CC","F2","2C","FA","E4","04","44","FC","C4","F2","1C","FC","6C","05","E4","FB","08","F3","40","FD","60","05","5C","FA","20","F3","70","FE","C8","04","A4","F8","04","F3","1C","FF","F0","03","84","F7","D4","F3","84","00","DC","03","2C","F7","00","F5","18","02","24","03","BC","F5","88","F5","08","03","78","02","20","F5","40","F6","34","03","D0","00","08","F4","5C","F7","0C","04","DC","FF","98","F3","7C","F8","C8","04","F8","FE","48","F3","F0","F9","34","05","70","FD","C8","F2","40","FB","40","05","24","FC","C0","F2","DC","FB","F4","04","18","FB","E4","F2","70","FD","D4","04","A8","F9","54","F3","F4","FE","FC","04","68","F8","50","F3","0C","00","34","04","50","F7","80","F3","BC","00","54","03","00","F6","8C","F4","48","02","60","03","CC","F5","A0","F5","14","03","D8","01","44","F4","28","F7","D0","03","70","00","44","F4","E0","F7","70","04","D4","FF","6C","F3","A8","F8","E8","04","9C","FE","3C","F3","38","FA","2C","05","FC","FC","EC","F2","38","FB","0C","05","20","FC","C4","F2","80","FC","24","05","40","FA","BC","F2","14","FE","1C","05","38","F9","10","F3","F8","FE","94","04","38","F8","D0","F3","80","00","0C","04","70","F7","68","F4","24","01","F0","02","CC","F5","BC","F4","CC","01","FC","01","18","F5","14","F6","70","03","6C","01","68","F4","60","F7","24","04","BC","00","0C","F4","3C","F8","A8","04","80","FF","9C","F3","50","F9","C4","04","98","FD","8C","F2","78","FA","0C","05","98","FC","E8","F2","E8","FB","20","05","80","FB","2C","F3","B4","FD","B0","05","80","FA","F4","F2","58","FE","D8","04","C0","F8","24","F3","4C","FF","CC","03","78","F7","1C","F4","B4","00","DC","03","14","F7","A0","F4","E8","01","1C","03","D8","F5","6C","F5","30","03","A0","01","0C","F4","FC","F5","5C","03","F0","00","14","F4","50","F7","E0","03","C8","FF","DC","F3","C8","F8","AC","04","AC","FE","60","F3","0C","FA","A8","05","F0","FD","44","F3","40","FB","54","05","88","FC","04","F3","04","FC","2C","05","CC","FA","80","F2","3C","FD","00","05","CC","F9","5C","F3","D8","FE","6C","04","5C","F8","B4","F3","44","00","6C","04","AC","F7","3C","F4","00","01","90","03","5C","F6","CC","F4","FC","01","7C","02","50","F5","C4","F5","E8","02","74","01","94","F4","B0","F6","C4","03","AC","00","E0","F3","FC","F7","38","04","70","FF","64","F3","E4","F8","E4","04","90","FE","1C","F3","1C","FA","08","05","FC","FC","EC","F2","4C","FB","F0","04","F0","FB","3C","F3","F4","FC","9C","05","C0","FA","04","F3","40","FE","E8","04","48","F9","54","F3","44","FF","80","04","BC","F7","6C","F3","5C","00","EC","03","0C","F7","30","F4","1C","01","04","03","30","F6","60","F5","C8","02","C4","02","EC","F4","04","F6","5C","03","60","01","4C","F4","E8","F6","DC","03","D8","FF","98","F3","3C","F8","54","04","18","FF","74","F3","4C","F9","E4","04","28","FE","38","F3","74","FA","0C","05","E0","FC","28","F3","30","FC","7C","05","58","FB","6C","F2","0C","FD","98","05","4C","FA","18","F3","50","FE","64","04","BC","F8","68","F3","8C","FF","9C","04","20","F8","40","F4","CC","00","98","03","E0","F6","EC","F4","DC","01","D4","02","B0","F5","50","F5","E8","02","2C","02","AC","F4","08","F6","28","03","0C","01","28","F4","A0","F7","20","04","E4","FF","94","F3","8C","F8","6C","04","78","FE","78","F3","14","FA","E0","04","74","FD","60","F2","9C","FA","3C","05","30","FC","9C","F2","40","FC","E0","04","48","FB","28","F3","90","FD","0C","05","BC","F9","1C","F3","FC","FE","DC","04","A0","F8","D8","F3","3C","00","18","04","54","F7","58","F4","2C","01","80","03","2C","F6","80","F4","0C","02","C0","02","88","F5","A0","F5","E8","02","68","01","94","F4","EC","F6","1C","04","E8","00","FC","F3","E0","F7","8C","04","9C","FF","7C","F3","FC","F8","BC","04","0C","FE","E4","F2","D8","F9","0C","05","70","FD","C8","F2","EC","FA","20","05","CC","FB","98","F2","B8","FC","68","05","A0","FA","D0","F2","E8","FD","F4","04","24","F9","3C","F3","7C","FF","A4","04","18","F8","AC","F3","1C","00","A4","03","2C","F7","7C","F4","5C","01","84","03","78","F6","78","F5","54","02","6C","02","58","F5","EC","F5","3C","03","78","01","04","F4","FC","F6","10","04","24","00","94","F3","CC","F7","34","04","0C","FF","64","F3","84","F9","0C","05","0C","FE","10","F3","BC","FA","34","05","B4","FC","44","F3","F8","FB","04","05","EC","FA","88","F2","18","FD","34","05","00","FA","C4","F2","50","FE","D8","04","F0","F8","8C","F3","9C","FF","50","04","F8","F7","44","F4","1C","01","C4","03","7C","F6","94","F4","B8","01","84","02","6C","F5","8C","F5","CC","02","28","02","BC","F4","54","F6","D4","03","5C","01","68","F4","8C","F7","04","04","D0","FF","D8","F3","B8","F8","A4","04","74","FE","EC","F2","B8","F9","1C","05","70","FD","AC","F2","D8","FA","2C","05","1C","FC","A4","F2","AC","FC","9C","05","2C","FB","D8","F2","7C","FD","20","05","A0","F9","38","F3","08","FF","5C","04","F0","F7","64","F3","3C","00","E0","03","54","F7","40","F4","B8","00","40","03","B0","F6","40","F5","08","02","B0","02","3C","F5","70","F5","0C","03","F8","01","24","F5","F8","F6"];
B=A';
C1 = reshape(A,2,[])';
C=[C1(:,2) C1(:,1)];
D=join(C);   %512*1
E=regexprep(D,'[^\w'']','');  %remove spaces
F=hex2dec(E);   %hex2dec

N = length(F);
xdft = fft(F);
xdft = xdft(1:N/2+1);
psdx = (1/(Fs*N)) * abs(xdft).^2;
psdx(2:end-1) = 2*psdx(2:end-1);
freq = 0:Fs/length(F):Fs/2;
figure
plot(freq,20*log10(psdx))
grid on
title('Periodogram Using FFT')
xlabel('Frequency (Hz)')
ylabel('Power/Frequency (dB/Hz)')

• Possible duplicate of MATLAB: $\tt fft$ and $\tt ifft$ scaling – Marcus Müller Jul 18 '17 at 9:07
• i read the info from the link,i am unable to link to my question,when i double the no of points in fft there shud be a 3db gain in psd,but in matlab it shows an exact 3db loss – Akshay Rathod Jul 18 '17 at 10:37
• Can you please show us how you are forming F ? – Peter K. Jul 18 '17 at 11:17
• @AkshayRathod no, there's nothing that says you should have a "gain". Mathematically justify that claim, and then check against the documentation of the fft function you're using. – Marcus Müller Jul 18 '17 at 16:00
• @MarcusMüller the formula for fft gain is 10log(M/2)..where M is the no of points..for details pls visit this page,it has somewhat given the answer. designnews.com/aerospace/where-does-fft-process-gain-come/… – Akshay Rathod Jul 19 '17 at 5:44

$$\sum_{n=0}^{N-1} | x[n] |^2 = \frac{1}{N} \sum_{k=0}^{N-1} | X[k] |^2$$
when you increase $N$ by zero padding, you increase the denominator on the right, which reduces the levels.