Low-Pass FIR that keeps all values positive

Question

Let's say, that I want to low-pass the norm (hence positive values) of an analytic signal (related to Hilbert transform). This allows me to perform experiments with the envelope of audio.

For low-passing, I apply a Gaussian blur. It's technically the same as applying multiple time a moving average. The advantages of a Gaussian blur are:

• easy to implement, and enough fast.
• good attenuation in the "stop band" (I just have to iterate more moving average to improve this).
• all values remain positives. This is important, because I use the low-passed norm to perform divisions.

But I'd like to test a low-pass with a narrower transition band.

Could you suggest me a simple solution, for a low-pass FIR (or something "linear phase" anyways) , with a narrower transition band than a straightforward Gaussian blur, and that allows to keep all values positive. For instance, with a sinc based FIR, there are oscillations around the zero line, and I want to avoid that. Adding an offset is out of question, as I want a result that is representative of the amplitude of the audio.

Answer 1

You can't, at least not strictly. Low pass filters with an all positive impulse response will always have a very wide transition band.

However, if you want to do audio envelope detection both Hilbert Transform and low pass filters are the wrong tools for the job: Hilbert Transformers don't work well on broad band signals and low pass filters mess up your transients and onsets.