# How to understand the sum of all “fourier frequencies”?

I got the data of an acceleration sensor to analyze. It consists of special terms of 30 Hz, 60 Hz and 120 Hz. In the following you can see in the first plot the 60 Hz data and in the second one the spectrum:

The plotted values are the results of a FFT. My question now is: How to understand the spectrum? The 60 Hz plot is clear. A FFT was operated and all the values were extracted which correspond to the 60 Hz signal. This is (or should be) the upper plot. The spectrum is allegedly the value across the whole frequency range - this I do not quite understand. How can I make use of this or how can it physically be interpreted? Is it finally kind of a total amplitude? Is it the origin signal (I don't think so..)? Is it even possible without an inverse transformation? Or is it just what it is: The sum of all (fourier) frequencies without any higher meaning? I'm not sure how to name the signals/peaks in the frequency domain.

• fft gives the contribution of each frequency element and larger the peak, more the contribution. A 60 Hz signal will have a peak at the 60Hz bin of FFT data. – DSP Novice Mar 12 at 4:47

Let me put a practical answer with the following Matlab / Octave Code :

L = 2*1000;     % signal sample count
n = 0:L-1;      % discrete-time index

Fs = 44100;     % sampling frequency
am = [1, 0.5, 2, 0.5, 0.3, 0.6, 0.1, 0.2];                % magnitudes of 8 components
fm = [882, 2646, 4410, 6615, 8820, 10000, 13230, 15876];  % frequencies

% Time domain signal :
% x = a1*sin(2*pi*f1*n) + a2*sin(2*pi*f2*n) + ... + am*sin(2*pi*fm*n)
x = sin((2*pi/Fs)* n'*fm)*(am');

% Frequency domain signal :
X = abs( fft( x, L ) );

% Display :
figure,subplot( 2,1,1 );
plot( n/Fs , x );
title(' signal as a sum of sinusoidal frequencies ' );
xlabel(' time [s]');
subplot( 2,1,2 );
plot(linspace( 0 , Fs/2 , L/2 ) , (2/L)*X(1:L/2) );
title(' Fourier (frequency) spectrum magnitude of the        signal' );
xlabel(' Frequencies from 0 to Fs/2 [Hz]' );
ylabel(' FFT Magnitude ');


The output will be :

The output speaks for itself. The upper plot is the time-domain signal as a superposition of (M=8) different sinusoidal signals each with a particular frequency and amplitude. The lower plot shows its frequency spectrum magnitude which displays a peak per frequency component in $$x[n]$$. The peak's amplitude is proportional to the time-domain component amplitude.

• Thanks, but I think this is more a generel response with regard to "What is a fourier analysis"? My primary question is: What is the "spectrum" or, meanwhile, I think the term "sum" is better suited. The data consists of the fourier components of 30, 60 and 120 Hz. And in addition there is the "so-called" spectrum which summarizes all frequency components. How can I make use of that? – Ben Mar 12 at 6:33
• @Ben but: "how to understand the sum of all Fourier frequencies" is exactly that: Fourier analysis. What is a "spectrum": the result of what you get when you do Fourier analysis. I really don't think Fat32's answer is missing anything you've asked, but I might be mistaken? – Marcus Müller May 27 at 10:01