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Say, I'm amplifying a signal using a device with gain bandwidth, ΔG Hz, which is unknown.

My source signal which is being amplified is known to have a bandwidth of X Hz, and the amplified output signal has a measured bandwidth of Y Hz.

Is it possible to process the input and output signal to find the gain bandwidth?

I was thinking that the output spectrum is likely to be the convolution of the gain profile and input spectrum, so maybe deconvolution would work?

In theory, could I measure input and output spectra and then try to numerically deconvolve, or would I need to first approximate the spectral shape (e.g. Lorentz, Gaussian etc.)?

Thanks for any tips!

Wostwl
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2 Answers2

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I'm assuming that your filter is linear and time invariant, then we know that in the frequency domain

$Y(\omega)=G(\omega)X(\omega)$

and thus conceptually:

$G(\omega)=Y(\omega)/X(\omega)$.

The (first) hard part is that we know that $X$ is band-limited so you'll only be able to evaluate $G(\omega)$ for those frequencies that lie within the frequency extent of your input signal. So you won't be able to fully characterize the filter if it passes signal that are not present in your input, test, signal. There is nothing (that I know of) that you can do about estimating $G(\omega)$ for frequencies where $X(\omega)=0$.

If you didn't have any noise in the system, within the bandwidth of $X$ you could just do the division and be done with it.

To account for the noise, I'd use Wiener Deconvolution which, in a nutshell, gives the estimate:

\begin{equation} G(\omega) = \frac{ X^*(\omega) Y(\omega)}{ \lVert X(\omega) \rVert^2 +\sigma^2 } \end{equation}

but requires an estimate of the power of the noise floor, $\sigma^2$ in your system.

Dave
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  • I think that if noise is a problem then deconvolution won't work. The SNR needs to be high enough that noise won't contribute otherwise the deconvolution errors will be high. – user6972 Sep 06 '13 at 17:51
  • The Wiener filter is optimal (mean-squared error sense) assuming that the noise is additive and independent of the signals; without additional details/assumptions what more can you do? – Dave Sep 06 '13 at 18:56
  • I'm pointing out that when the goal is to reconstruct a black box's filter/gain function if the OP also has a noise problem too then the likelihood of success is low, especially with a bandwidth limited input. Adding another filter to the process will also make estimating the response more complex. – user6972 Sep 06 '13 at 21:28
  • Thanks for the interesting comments. The process I'm thinking about is actually a nonlinear optical process: an input signal experiences gain causing the output signal to exhibit a feature at a different wavelength.

    Given this is a nonlinear process which depends on the intensity of the input signal, I suspect the above linear model is invalid?

     – Wostwl Sep 07 '13 at 11:46
  • Right, this is invalid for nonlinear systems. – Dave Sep 07 '13 at 18:35
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Yes and no. Typically your input signal bandwidth needs to be larger than your expected bandwidth that you are trying to measure otherwise you won't find the edges. A complex input waveform can make it difficult to characterize the system properly. Deconvolution can be very error prone and time consuming to try to invent the filter response that gives you back your original input. And the end result is only a filter that gives you back what you put into it, not necessarily a complete picture of the true filter.

This is usually done by introducing a transient impulse function and then directly measuring the output response. The frequency response of the system T(f) will be its Fourier transform of its impulse response.

EDIT: In the non-linear case this is convolution in the frequency domain instead of the time domain. It is often easier to characterize the non-linear element in terms of a polynomial using a Volterra series than try to create a model using deconvolution which requires a lot of guessing and iteration. Once your model responds like the real system then you can explore responses to the input using frequency convolution. Here is an example of estimating the Volterra function in an optics application (see the references too for more info). Perhaps the device you're using has already been characterized?

It might be helpful for us to know your end goal and if you can control the input source characteristics. Otherwise we have to answer with generalities.

user6972
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