Oversampled analog to
digital conversion uses a higher sampling
rate necessary to combat the effects of
quantization noise. Consider an analog to
digital conversion scheme, where the input signal
is band limited, we can raw sample the signal
with a sampling period T_s and as long as T_s complies with the
sampling theorem, we don't have to
worry about aliasing, and then we quantize the samples
into the digital domain. The resulting signal
can be seen as the sum of the true sample values plus a sequence of noise samples that are
created by the quantizer. The standard model is that the
noise sequence is an i.i.d process and it is independent
of the underlying signal. This is a very strong
assumption that is rarely true in practice, but for the time being
let's assume that it works. If this is true, the power spectral density of
the noise is constant over the entire frequency band and its magnitude is equal to
delta square over 12, where delta is the
quantization step if we use a uniform quantizer. The key observation that leads to oversampled analog to
digital conversion is that the digital spectrum
after sampling has a support that is proportional
to the sampling period. So if we sample fast with a
very small sampling period, the support of the resulting
spectrum will be narrow, whereas if we sample with
a large sampling period, the support will be large. This is true even if we consider an input as a stochastic process, and it will apply to the support of the
power spectral density. So graphically, it looks like so. Imagine this as a signal
sampled at the critical rate, so using the sampling frequency dictated by the sampling theorem, we know in this case
that the support of the power spectral
density will be full band, so for minus pi to pi,
and it looks like so. Here we have plotted as well, the power spectral
density of the noise. Now, let's suppose
that we're sampling at twice the minimum
sampling rate. At that point the new
sampling period will be the previous sampling
period divided by 2. This will shrink the support of the power spectral density of the input to minus pi
over 2 and pi over 2. So we're contracting
the signal that used to be full band to a
half band signal. We can continue with this process by increasing the
sampling frequency. So at three times the
minimal sampling frequency, the support will be one- third, at four times, the support
will be one-forth. You also see that the amplitude of the power spectral
density increases linearly with the oversampling factor as shown in the formula
a couple of slides ago. The key observation is that even if we increase
the sampling rate, the power spectral density of the noise does not
change because we have assumed that the noise is independent of the
underlying sample. You can immediately see the
fallacy in this reasoning, when you sample really fast, the samples would
be very very close to each other and
so the error that the quantizer introduces for each sample will be
strongly correlated. So it will be very unlikely, that the power
spectral density is independent of
sampling frequency. But for moderate
oversampling factors, this is a reasonable
approximation. Now of course we can
use a low-pass filter to eliminate the out-of-band
quantization noise, the noise the falls outside
of the minus pi over capital N to pi over N interval, which is where the
signal resides. Then once we have eliminated the out-of-band
noise by filtering, we down-sample the signal
to bring it back to the nominal sampling frequency dictated by the sampling theorem. In so doing, we scale down
the signal according to the down-sampling formula and we scale down the
power of the noise. So in the end you have the same power spectral
density as you would have had if you had sampled
at the critical rate, but the noise power
has been divided by a factor of N. The final
scheme is like so, you have your band
limited signal here. You have a sampler
that operates at n times the minimal sampling
frequency or conversely, whose sampling period is 1 over n times the maximum
sampling period. You have a quantizer, and then you have
a low-pass filter with cutoff frequency pi over N followed by a down
sampler by n. The signal to noise ratio at the output of this chain is n
times the signal to noise ratio that you
would have had with the simple straightforward
sampling and quantization. So that means that you gain about 3dBs per doubling of
the sampling frequency. However as we said, the key assumption of
noise independence doesn't really hold for very
high up-sampling factors. We can quickly look at how Oversample A/D conversion works entirely in the time domain
just to get some intuition. The oversampling
factor of N basically produces N samples where we
would have needed just one. Now, if we assume that the quantization noise is uncorrelated between
this extra samples, the low-pass filtering
operation is simply a local average that by averaging neighboring samples with
uncorrelated noise will reduce the overall noise
for this group of samples. Then by down-sampling by N, we're discarding the
redundant information introduced by oversampling.