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How does SDR resolution translate to audio resolution when using an FM demodulation algorithm?

For example: using a 8-bit/2.048 Mhz IQ stream, what is the maximum audio sample rate and bit depth that can be generated from a +/-75kHz mono WFM signal with 20kHz audio?

These are the steps of a basic FM conversion algorithm. My question is, what effect does each step have on the bit depth of the result of the step:

  • Step 1 - Decimate down to the relevant bandwidth (150khz). Doing so reduces the sample rate but increases the bit depth.

  • Step 2 - Take the arctan of the resulting IQ samples. The combination of the I and Q parts further increases the bit depth of the current data.

  • Step 3 - Use a differentiator to get the final audio result. I am not sure how this affects the resolution, but since no data is combined, I suspect there would be no change.

  • Step 4 - Further decimate the audio to desired sample rate (48khz presumably). Would result in further increase in bit depth, but not by much.

In each step, combinations of data result in more data possibilities, eg:

  • Average 2 and 3 to get 2.5
  • Arctan(I/Q) provides N^2 possibilities, though not all are unique

Wikipedia has an explanation of the effect of oversampling on bit depth. But I am not sure about the arctan and differentiator processes.

Thank you for your expertise

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  • $\begingroup$ Note: this question has also been posted to the Signal Processing SE. $\endgroup$ – a CVn May 13 '15 at 8:14
  • $\begingroup$ Want me to migrate that question here, so you can merge, @MichaelKjörling? $\endgroup$ – Peter K. May 13 '15 at 8:44
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    $\begingroup$ I think it's probably more appropriate on the DSP SE. I apologize for the violation. It's just frustrating to not have the question answered adequately and it seemed it needed a bigger audience. $\endgroup$ – NewEndian May 13 '15 at 12:27
  • $\begingroup$ OK. Generally, just post one place. If you don't get an adequate response, see if the mods on the site you posted it on have a better suggestion for where to ask it. They can usually migrate it there. $\endgroup$ – Peter K. May 14 '15 at 7:30
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If all you want to know is what limit is placed on fidelity by the sample format of the SDR, there is a much simpler answer. Sample rate limits the frequency components the digital signal can represent, and the bit depth relates to quantization noise. If we can sample the entire bandwidth of the RF channel, and the quantization noise isn't a significant component of all the noise, then the digitization process isn't significantly degrading the fidelity of the demodulated signal.

At 2048 kHz, the sample rate in the example you gave is more than sufficient to cover a 200 kHz wide FM channel. So we are good on that front by a wide margin.

At 8-bit samples, the quantization noise is more than it would be with a more precise sample. But as long as the quantization noise is several decibels below the analog noise floor (which is itself a sum of RF noise and electrical noise in the receiver, among other things), then digitization adds only negligible noise to the signal.

Oversampling will not help you reduce quantization noise unless your SDR has an analog filter to reduce the bandwidth of the signal at the ADC's input. Indeed, the 2048 kHz bandwidth of the IQ samples is far in excess of the 200 kHz of an WFM channel. But if you aren't doing the filtering before reaching the ADC, that excess bandwidth contains information about other stations and noise outside the channel rather than extra information about the channel you wish to receive.

If you did have an analog filter, then you are oversampled by a factor of at least 8x. Every 4x oversampling effectively adds one bit, so you could reasonably convert your incoming 8-bit samples to 10-bit samples. With quiet electronics and proper application of noise shaping and dithering in the ADC you might even do a little better.

Practically speaking, you'd probably convert your samples to 16 bits at this point since that's convenient for most processors and it guarantees rounding error in the subsequent calculations are buried in the noise floor.

At this point, if the quantization noise is still not a few decibels below the analog noise floor, all the processing that follows is irrelevant. The signal has irreversibly been degraded. Once noise becomes mixed with signal, there's nothing that can be done to unmix it.

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    $\begingroup$ Just because FM is not digital doesn't mean there isn't a relationship. Just like, in audio, sample rate is directly related to frequency range and bit depth is directly related to dynamic range. FM changes that relationship, but as you stated, parameters too low decrease the frequency range and dynamic range of the program material and parameters too high do not increase quality. So what is the actual relationship? $\endgroup$ – NewEndian May 12 '15 at 18:38
  • $\begingroup$ @NewEndian How can there be a relationship to a thing which doesn't exist? FM does not have a sample rate or resolution. Sure, you could sample the analog audio coming out of an FM receiver, but your selection of sample rate and resolution is arbitrary. Likewise when implementing a digital FM demodulation you can pick whatever sample rate and resolution you want for the output. Why do you think there must be some relationship besides the one you pick in your implementation? $\endgroup$ – Phil Frost - W8II May 12 '15 at 22:47
  • $\begingroup$ There is a direct and very real relationship between sample parameters and audio. The maximum frequency that can be represented by a sample rate is one half the sample rate (according to the Nyquist theorem). The quantization noise is related to bit depth at 6dB per bit. Furthermore, I am not asking about the conversion from analog to digital. I am asking about digital to digital conversion. The algorithm cannot produce program content that didn't get recorded, so my question is, how do the digital parameters of the SDR relate to the digital parameters of the FM algorithm results. $\endgroup$ – NewEndian May 13 '15 at 2:41
  • $\begingroup$ @NewEndian You are not wrong that there is a relationship between sample format and quality. These apply whether the thing being sampled is an audio signal or an RF signal. But there is no relationship between input and output sample format beyond the relationship imposed by the implementation of the demodulator. The demodulator could decimate its sample rate and output 1 sample per second. Or it could interpolate and output 50M samples/second. It could use 8-bit numbers throughout. Or it could convert those input samples to 64 bit numbers and perform the calculations in that format. $\endgroup$ – Phil Frost - W8II May 13 '15 at 13:46
  • $\begingroup$ Interpolating would not reconstruct data that was not recorded. You can similarly interpolate audio samples but the higher frequency content would not be restored. Unless you're saying that 1024-bit (and higher to infinity) calculations would yield improvements (however small) over 256-bit calculations, there is a limit to what any given SDR sample rate can provide in output sample rate and bit depth after conversion. $\endgroup$ – NewEndian May 13 '15 at 13:56

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