how to retrieve original signal from power spectrum

Question

I have calculated power spectrum of signal. the steps are:

• FFT of time signal
• square of absolute value of FFT/length of signal i.e power spectrum

Now I want to convert it into time domain. What steps should I follow.

Answer 1

The easiest way to see the non-uniqueness of the power (or amplitude) spectrum for describing the time domain signal is that both white noise and the delta function in the time domain have the same power (or amplitude) spectrum - a constant - in the frequency domain.

Answer 2

The reconstruction of the original signal from the frequency-domain requires both the magnitude and the phase information. So, as you compute the power spectrum and keep only the magnitude, you no longer have all the required information to uniquely reconstruct the original signal.

In other words, we can find examples where different signals have the exact same power spectrum. In that case retrieving which one of those different signals was the original one would thus not be possible.

As a simple illustration, let's say the original signal x is:

x = [0.862209 0.43418 0.216947544 0.14497645];

For sake of argument, let's consider some other signal y, which I've specially crafted for the purpose of this example as:

y = [-0.252234 -0.0835824 -0.826926341 -0.495571572];

As show in the following plots, those two signals might appear completely unrelated:

They do however share the same power spectrum:

f = [0:N-1]/N;
Xf = fft(x,N);
Yf = fft(y,N);
hold off; plot(f, Xf.*conj(Xf)/N, 'b');
hold on;  plot(f, Yf.*conj(Yf)/N, 'r:');
xlabel('Normalized frequency');
legend('Px', 'Py')
title('Power spectrum');

As a result, someone who only sees the power spectrum and doesn't know that you started with x, could very well guess that you instead started with y.

That said, the fact that those signals have the same power spectrum could tell you that those signals aren't as unrelated as you might think. In fact those signals also share the same autocorrelation function in the time-domain:

Rx = xcorr(x);
Ry = xcorr(y);
t = [0:length(Rx)-1] - length(x) + 1;
hold off; stem(t, Rx, 'bo');
hold on;  stem(t, Ry, 'rx');
legend('Rxx', 'Ryy');
xlabel('lag');
title('Autocorrelation');

This is to be expected since the autocorrelation can be obtained by computing the inverse transform (with ifft) of the power spectrum. This, however, is about as much as you can recover in the time domain. Any signal with this autocorrelation function would be as good a guess as any for the original signal. If you are very motivated you could attempt to solve the set of non-linear equations that are obtained from the definition of the autocorrelation and obtain a list of possibles signals. That would still not be sufficient to tell which one was the original, and as you noticed when comparing my example x and y, there wouldn't be a whole lot to make of it.