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Problem:

I have the impulse response of a matched filter(therefore its phase and magnitude response. See figure below) of a filter, and I need to implement its response using only off-the-shelf components i.e low-pass filter, delay lines, multiplexers, etc.. I don't require extreme precision but I'd like to approximate its response as exact as possible.

Filter Response

Possible Solutions:

  • Cascade low-pass and high pass filters to obtain the overall shape of the magnitude response. Use delay lines to implement two taps to give the magnitude response its "sinusoidal" nature as shown by the above figure.
  • Use a multiplexers/splitters with delays, phase shifters, and attenuators with values according to the filter's frequency and phase response.

I am extremely new to signal processing so I don't know much of what is out there. So please, if anyone could guide me or give me an insight into how to achieve my goal, I'd really appreciate it.

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    $\begingroup$ So this is originally an analog filter? Is it a standard form (low pass/high pass/ band pass etc)? Is it linear at least? Your plot is kinda noisy we can't really see... Also I imagine red is phase and blue is amplitude but could you confirm? $\endgroup$ – Florent Jul 25 '17 at 23:22
  • $\begingroup$ I highly doubt that this is a filter at all... is this a sensor response? a channel response? How did you generate this so called FIR response? $\endgroup$ – Fat32 Jul 26 '17 at 16:12
  • $\begingroup$ Yes the red curve is the phase and the blue one is the amplitude response. @Fat32 It is a filter. The way I obtained the tap coefficients of the filter was by reversing in time data of a single pulse, which acts as a matching filter if convoluted with the unfiltered data. Then by definition the time-reversed data is the impulse response of my desired filter. I am new to this so let me know if I need to elaborate more. $\endgroup$ – Agustin Pacheco Jul 26 '17 at 21:03
  • $\begingroup$ ahaa! Yes this is the impulse response of a matched filter. Ok so you are right. Such a meaningless impulse response can be generated as a matched filter which is indeed a signal's waveform reversed. So you want to detect something with the matched filter? Why do you want to do that using analog technology? $\endgroup$ – Fat32 Jul 26 '17 at 21:15
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    $\begingroup$ ok good but that requires more microwave engineering than DSP, I'm sure you'r aware of it. Coming to the approximation, analog approximation techiques is the least topic I could be talking of... sorry. You would need very high orders and comlplex transfer function to realize though. Unless you state explicitly what degree of approximation is acceptable. $\endgroup$ – Fat32 Jul 26 '17 at 22:06
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FIR filters typically have a sinc^x type impulse response spectrum or something that looks like it, in some but not all cases. The sinc^x shape is mainly seen when the impulse response h(n), if plotted on cartesian coordinates, look like an even function time shifted where the coefficients are placed symmetrical about the origin. This is a simplistic answer based on a particular type of FIR filter such as a Type I or II. You have not given enough information to give a more specific answer.

So your analog filter would have to look like a sinc^x or some type of comb filter with lobes falling off faster than a typical comb filter. But again this is a simplified answer.

In short it is theoretically impossible to do what you want exactly, or even approximately, but I can't tell you that you cannot somehow come up with a crude approximation, say using delay lines

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The plot shown has a frequency axis horizontally, so this is not an impulse response (which is the magnitude and phase of an impulse vs time), but is the Frequency Response.

From the comments it sounds like the OP would like to optimally detect very narrow time pulses (UWB?). Designing a static matched filter for this that would offer any benefit would have considerable challenges: a 0.1 ns pulse in time is a Sinc function in frequency with the first null of the main lobe out at 10 GHz! There will be significant phase dispersion and multipath propagation delay affects over this ultra-wide bandwidth extending over multiple GHz that will most certainly change with time, so unless this experiment is set up in a fixed and controlled environment such an "off-the-shelf components" based filter will likely not be able to offer any improvement.

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