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Magnitude of the frequency response

I have a plot of a magnitude response of a frequency response, how can I get the impulse response using only this plot and nothing else?

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  • $\begingroup$ Your plot shows the frequency response $H(e^{j\omega})$, not only the magnitude. The frequency response happens to be real-valued. So you can just take the inverse DTFT. $\endgroup$
    – Matt L.
    Jan 26, 2018 at 15:35

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You state in your question that you have a "magnitude response". In that case, you cannot reconstruct the time-domain signal corresponding to the impulse response, because phase information is missing. In the plot, however, the vertical axis is indexed by $H(e^{j\omega})$ and not $|H(e^{j\omega})|$, so I suppose you meant "frequency response".

You can relate a system's frequency response to its respective impulse response via inverse DTFT:

$$H(e^{j\omega}) \xrightarrow{\mathscr{F^{-1}}} h(n)$$

Note that you have a typical ideal low-pass filter. That has a well-known DTFT inverse.

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Without further constraints on the system, it's not possible (in genral) to obtain the impulse response of the system from the frequency response magnitude alone. However, if the LTI system has a real impulse response, then there are procedures to reconstruct its (up to a scale factor) from phase or magnitude of frequency response alone. Monson Hayes has published papers on doing so...

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    $\begingroup$ How is LTI and real-valued sufficient for deriving the phase from the magnitude? I can always concatenate an allpass filter with the system so there can't be a way to uniquely determine the phase from the magnitude if nothing else is known other than that the system is a real-valued LTI system. $\endgroup$
    – Matt L.
    Jan 26, 2018 at 15:33
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    $\begingroup$ @MattL. You can find the details here... link The paper is about reconstructing a signal but then the signal is the impulse response here, that's why I stated LTI. $\endgroup$
    – Fat32
    Jan 26, 2018 at 16:34
  • $\begingroup$ @MattL. btw for signal reconstruction from phase or magnitude alone, one more constraint is also needed. As it's also stated as "exact signal recovery except a scaling factor"... $\endgroup$
    – Fat32
    Jan 26, 2018 at 16:50

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