How Can I Compensate an Integrator Filter in the Digital Domain?

[divB]

Hi,

I have a signal $$f(t)$$ with nyquist rate W, i.e. the maximum frequency is W/2.

This signal is filtered with an integrator (simulated in Simulink) the following way:

$$f_I(t) = \int_t^{t+\frac{1}{W}} f(t)\,dt$$

In words: I integrate the signal for a period of 1/W, then the integrator is reset. It is obvious that the signal won't be the same afterwards; however I only integrate for 1/nyquistrate long. So I think it should be possible to compensate this filter in digital domain.

But: How? This is a lowpass first order. I tried to get the transfer function which should be something like (1+z^-1) and filter with the reverse, i.e. 1/(1+z^-1). But this filter is unstable and the results therefore unuseble.

Does anybody know how I could compensate this filter in digital domain (when I have the nyquist samples of $$f_I(t)$$)?

Regards, divB


PS: The whole thing should be equivalent to oversample $$f(t)$$ with a factor of e.g. 100, that is, the sampling rate is W*100; and afterwards summing up 100 consecutive samples in digital domain ...

 

[mmwave]

Dear divB,

Thank you for sharing your signal processing challenge with us. It seems like you are trying to compensate for the effects of an integrator filter on your signal f(t). This type of filter is commonly used in signal processing to perform smoothing or averaging on a signal. In order to compensate for this filter in the digital domain, you will need to use a technique called deconvolution.

Deconvolution is a mathematical process that can be used to reverse the effects of a filter on a signal. In your case, you will need to use the transfer function of the integrator filter to deconvolve your signal f_I(t) and recover the original signal f(t). The transfer function of an integrator filter is indeed (1+z^-1), but in order to use it for deconvolution, you will need to take the inverse of this function, which is 1/(1+z^-1). This function is known as the deconvolution filter.

However, as you have mentioned, this filter can be unstable. This is because the integrator filter introduces a pole at z=1, which can cause instability when used in deconvolution. To overcome this, you can use a technique called regularization, which involves adding a small value to the denominator of the deconvolution filter to avoid the pole at z=1. This will stabilize the filter and allow you to successfully deconvolve your signal.

Another approach you can try is using a different type of filter, such as a moving average filter, to compensate for the integrator filter. This type of filter can also be used to perform smoothing on a signal and is more stable for deconvolution.

I hope this helps you in your signal processing efforts. Please let me know if you have any further questions.