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My question may be a bit naive, as I don't know this stuff well, but bear with me.

I'm looking a encoding a time-varying signal x(t) into a pulse-stream s(t) with a Sigma-Delta modulator, and producing a reconstruction y(t) from s(t) using a low-pass filter. What I'm wondering is what method I should use to design/tune this low-pass filter to optimize reconstruction - defined, for example, by sum_over_t((x(t)-y(t))^2). Is this something with a clear solution, or is it one of those things with a bunch of solutions depending on your requirements/application?

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  • \$\begingroup\$ Your least-square-sum criterion needs another parameter to account for the time-delay from x(t) to y(t). \$\endgroup\$ – glen_geek Jan 23 '17 at 16:33
  • \$\begingroup\$ I'm trying to minimize both delay and signal distortion. It seems least-squares meets this objective. \$\endgroup\$ – Peter Jan 24 '17 at 12:46
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There will be at least three contributors to the error signal as you define it

a) in band distortion of the passband signal, caused by the filter cutting off too early

b) inadequate filtering of the out of band noise that the converter adds to the signal

c) in band distortion and noise added to the signal by the converter itself

A sigma delta converter produces out of band noise, that rises at a rate defined by the order of the converter. You want the reconstruction filter to have at least the same order as the converter, preferrably one order higher. Having it higher order will render far out noise irrelevant, and only close to band edge noise will be significant. As converter noise rises slowly, there's little need to use elliptic filters, conventional all pole filters will be just fine.

Assuming your converter is adequate for the job (which means the noise added does indeed fall outside your bandwidth of interest), then a filter that passes the passband, and rolls off faster than the converter noise rises, will also do the job.

If your converter is inadequate, then no amount of filtering will rescue it.

Your detailed definitions of x(t) and y(t), sample rates, any other filtering, might affect the detail of an optimum filter, but passing the passband while stopping the stopband is about all one can say without that detail.

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It will depend on the design of your sigma-delta modulator. The sigma-delta modulator shapes the quantization noise of the resultant digital signal (s(t)) in a way that weights the noise towards the higher frequencies. You then oversample the signal with the modulator to move the noise out of the signal bandwidth of interest. You can then reconstruct the original signal by simply filtering out this quantization noise.

The particular details of the reconstruction filter you use will depend on:

a) The bandwidth of the signal y(t) that is of interest to you. This will determine its passband.

b) The shaping of the quantization noise by the particular modulator you use and the amount of oversampling employed. This will determine how aggressive your filter needs to be to remove enough of the quantization noise.

c) How much quantization noise you can accept leaking into the reconstructed signal.

In practice, most people would just oversample with a sufficiently good modulator to ensure they can use a simple single pole reconstruction filter. This makes the filter implementation simple, and also has the advantage that it won't introduce any phase distortion to the reconstruction.

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  • \$\begingroup\$ Any sigma delta converter worth its salt uses a higher order than can be filtered adequately with a single pole. Even a single pole filter introduces phase distortion to the reconstruction, have you ever tried passing a square wave through an RC lowpass filter? \$\endgroup\$ – Neil_UK Jan 23 '17 at 14:16
  • \$\begingroup\$ The order of the modulator is not related to the order of the reconstruction filter. The order of the modulator sets the aggressiveness of the noise shaping function, but you can shift the entire noise spectrum up and down (relative to the signal of interest) by changing the oversampling ratio. The combination of these two design decisions sets the requirements for the reconstruction filter. In many commercial DACs they are indeed designed to work with a single pole filter. \$\endgroup\$ – Jon Jan 23 '17 at 14:43
  • \$\begingroup\$ The order of the converter is related to the ratio of the sampling frequency to the input passband, and the degree of attenuation of quantisation noise that's required. The order of the minimal reconstruction filter is related to the converter order, though you can use other filters that are not minimal, or do not perform as well. Generally, we use the most efficient filter we can, which results in it being one order higher than the converter. Would you like a list of my published patents in the field of sigma delta converters? \$\endgroup\$ – Neil_UK Jan 23 '17 at 14:49

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