# Controlling a signals amplitude in the frequency domain [closed]

I want to control the amplitude of a signal I'm creating from a user drawn spectrum by scaling the magnitude values in the frequency domain. Here is my scenario.

• Sample rate $F_s= 44100\textrm{ Hz}$
• FFT size $NFFT = 512$
• Desired waveform frequency: $86.1328125\textrm{ Hz}$ ($44100/512$ so a single cycle)

My user input screen has the magnitudes presented like a bar chart and they are all stored as values with a range of $0.0$ to $1.0$.

How do I scale these in the frequency domain so that the output signal is a $0\textrm{ dB}$ signal?

## closed as unclear what you're asking by Marcus Müller, MBaz, lennon310, A_A, Laurent DuvalJul 3 '17 at 16:41

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Can you please clarify your question? Are you trying to understand what you should scale a "free-hand" sketch of frequency domain coefficients so that the output signal has a p-p amplitude of 1? That would depend on the scaling factor effected by your FFT library too – A_A Jun 30 '17 at 1:40

## 2 Answers

You can use Parseval's theorem for DFT. $$\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N} \sum_{к=0}^{N-1} |X[n]|^2$$ Where $$x[n]$$ - n-th signal sample, $$X[n]$$ - n-th value of the DFT of the signal

• Thanks Andrei. I've encountered this a lot in trying to solve this but I'm not certain I understand how to apply the right hand side. Is it correct to apply it to the the spectrum in rectangular form? I think I understand the basic concept which is that frequency domain spectrum is equivalent to the output signal. I just don't understand the mathematic notation enough to convert it to c++. – cixelsyd Jun 29 '17 at 19:37
• – Andrei Keino Jun 30 '17 at 2:38
• Math notation: en.wikipedia.org/wiki/Summation – Andrei Keino Jun 30 '17 at 3:30
% Parseval's theorem in matlab
close all;
format long;
Fs=40;f=4;Ts=1/Fs;=2;t=0:Ts:T-Ts;N=length(t);
x=2*cos(2*pi*f*t);
fx=fft(x);
figure,
subplot(1,2,1), area(t,abs(x.^2)),title(' Time Domain');
subplot(1,2,2),area(abs(fx)), title(' Frequency Domain');

E1_timedomain=sum(abs(x.^2))
E1_frequdomain=sum(abs(fx.^2))/N

Energy_Of_Signal = sum(x.^2)
Energy_Of_Signal_DFT = sum(abs(fx).^2) / length(fx)

// Parseval's theorem in C++ - pesudocode

vector<double> x; // signal vector
GetSignal(x); // acquire signal
vector<Complex> fx; // DFT of signal vector
GetDFT(x, fx); // get the DFT of signal
double sumX = 0, sumFx = 0;
assert(x.size() == fx.size()); // length of x and fx are equal
for(int i = 0; i < x.size(); i++)
{
sumX += x[i] * x[i];
sumFx += fx[i].real() * fx[i].real() + fx[i].imag() * fx[i].imag(); // or sumFx += fx[i].abs() * fx[i].abs();
}
assert(sumX == sumFx / (double)fx.size()); // Parseval's theorem