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Suppose there is a real signal Amp_time (Amplitude as a function of time) at Sample Rate SampRate. I can transform it into frequency domain with 

fftLength = length(Amp_time);

FT_freq = fft(Amp_time,fftLength); % signal as a function of frequency
FT_freq = FT_freq(1:fftLength/2);  % throw away symmetric part

and the corresponding frequencies:

freq = [0:fftLength-1].*(SampRate/fftLength); 
freq = freq(freq < SampRate/2);   % throw away symmetric part

To transform it back into time domain I simply use

Amp_time2 = ifft(FT_freq,'symmetric');

Question

But how about if I am changing the length of FT_freq?

Suppose I want to reduce length(FT_freq) by averaging FT_freq to a coarser frequency vector (i.e. 10^3 instead of 10^5 frequencies). 

[freq2,FT_freq2] = special_average(freq,FT_freq)   % reducing length of FT_freq

I don't think that it is possible to take the version above (because how can Matlab possibly know to which frequency the signal is corresponding now).

What am I missing?

Do I have to scale the coarse FT_freq2 up again in order to get the right IFFT? (bad in terms of memory, ...)

The reason I wanted to reduce the degrees of freedom of the Fourier Transform is to avoid having matrices with size of order 10^5 x 10^3 (FFTing every tenth of a second of some song) and thereafter to compare freq-spectra/tones.

Lukas
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    Modifying the frequency bins by perhaps averaging elements into a lower length vector will have unintended side effects. This would actually create a completely different signal. If it is your goal to perform some sort of data compression, then adjusting the length of the FFT of the result is probably not the best course of action. – rayryeng Mar 21 '16 at 18:27
  • I wanted to compare different frequency spectra to identify recurring tones. – Lukas Mar 21 '16 at 19:35

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