Is signal reconstruction possible using phase/magnitude only

ramdas

I am studying Fourier Transform and it's inverse. We get phase and magnitude of a signal from it's Fourier transform and reconstruct it back from both together(magnitude of signal +phase of signal)

My question is that is it possible to reconstruct given signal back using it's phase only or magnitude only?

tech99

Gold Member
I am studying Fourier Transform and it's inverse. We get phase and magnitude of a signal from it's Fourier transform and reconstruct it back from both together(magnitude of signal +phase of signal)

My question is that is it possible to reconstruct given signal back using it's phase only or magnitude only?
For a complex waveform, we would need to add each component in its correct amplitude and phase in order to obtain the correct shape. But for sound, the ear does not seem to notice the phase, so the shape of the wave is not important provided the spectral response is correct.

Hechima

There is such a notion as a minimal phase reconstruction in signal processing. Here's an outline of the process:
Suppose you have an analytic signal:
x(t)=A(t)eiφ(t)

Then (a) log of the signal would be:
log(x(t)) = log(|A(t)|)+iφ(t)

Now, what if we just have A(t)?
You can take log(|A(t)|) and call it the real part of an analytic signal, but what about the imaginary part?
It turns out you can take what is called the Hilbert transform of log(|A(t)|) to get a good candidate for the missing imaginary part. Add the real and imaginary parts, then exponentiate to get your reconstruction.

If you have access to MATLAB, you can try this out using the built in 'hilbert' function. If you pass it a time series, it uses the Fast Fourier Transform to make an analytic signal.

mfb

Mentor
For a complex waveform, we would need to add each component in its correct amplitude and phase in order to obtain the correct shape. But for sound, the ear does not seem to notice the phase, so the shape of the wave is not important provided the spectral response is correct.
That is true for short timescales (e.g. if you want to describe a single note played by an instrument at constant amplitude) but it is not true in general.
Consider a 440 Hz wave and a 441 Hz wave at the same amplitude together: a human will interpret this as ~440 Hz sound that oscillates in amplitude once per second. The question "when do we hear sound?" depends on the phases of the two waves.

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