# Band-limited signal recovery with finite bit depth

The sampling theorem tells us that a signal with no frequencies above $$f$$ can be completely described by sampling it a rate of $$2f$$. However, the theorem makes no reference to quantization, and so I would think that the result only holds for samples measured with infinite precision.

However, in DSP we never have infinite precision, we instead have a fixed (or at least bounded) bit-depth. So what is the relationship between the bit-depth of each sample and the precision with which we could reconstruct the original signal? Does grossly over-sampling allow us to heavily quantize each sample?

For the purposes of the question, we could even imagine taking countably infinite samples from a signal that is infinite in the time domain, but each sample is itself limited to a resolution of $$n$$ bits.

• have you looked at oversampled sigma-delta quantizers ? – Fat32 Nov 1 '19 at 21:37
• In most practical applications, one first determines what is the acceptable quantization signal-to-noise ratio, and then how many bits per sample are needed. Adding more bits (finer quantization) would be unnoticeable to the final user. – MBaz Nov 1 '19 at 22:21

Quantization is a loss of information. So if you quantize a signal, you will never be able to reconstruct it perfectly.

That being said usually we can, in certain circumstances, assume that quantization is a kind of noise. In these circumstances, assuming an N-bit ideal ADC with a full-scale sine wave as an input.

$$SNR (dB) = 1.76 dB + 6.02N$$

So an ideal 16-bit AC would give you a dynamic range of about 98 dB. The hard question, is whether or not that's enough for your application. Unfortunately we cannot answer that for you.

Now, as to oversampling. If the quantization noise is not correlated, then yes oversampling can help you decrease the quantization noise, and thus increase the dynamic range in a certain frequency band. The rule-of-thumb is that oversampling by a factor of 4, then filtering, then downsampling by 4 is equivalent to adding another ADC bit, i.e. adding 6.02 dB to your dynamic range.

However, sigma-delta ADCs also perform quantization noise-shaping which allows them to vastly increase their dynamic range in a certain frequency band. Combined with oversampling, this allows them to implement 24-bit and 32-bit ADCs.