# How to obtain impulse response from a plot of magnitude of 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?

• 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. – Matt L. Jan 26 '18 at 15:35

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.

Without further constraints on the system, it's not possible to obtain the impulse response from the magnitude alone. If the system is LTI and has a real impulse response however, then there are procedures to reconstruct the impulse response from phase or magnitude alone. Monson Hayes has published a rich literature on oing so.

In your case, you probably assume that the phase is zero ? Then it's possible to reconstrcut the impulse response $h[n]$ from the inverse Fourier transform operation: $$h[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} H(e^{j\omega}) e^{j \omega n} d\omega$$

• 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. – Matt L. Jan 26 '18 at 15:33
• @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. – Fat32 Jan 26 '18 at 16:34
• @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"... – Fat32 Jan 26 '18 at 16:50