Oversample analog to digital conversion uses a higher sampling rate

than 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 the sampling period TS.

And as long as TS complies with the sampling theorem we don't have to worry

about aliasing.

And then we quantize the samples into the digital domain.

The resultant 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 oversample analog to digital conversion.

Is that the digital spectrum after sampling has a support

that is inversely 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 considered 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 is 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 from minus pi to pi.

And it looks like so.

And 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.

And 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.

And 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 fourth.

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.

And so it would be very unlikely that the power spectral density is independent

of sampling frequency.

But for moderate oversampling factors, this is a reasonable approximation.

And now of course we can use a low pass filter.

To eliminate the out-of-band quantization noise, the noise that

falls outside of the minus pi over capital N to pi over capital N interval.

Which is where the signal resides.

Then, once we have eliminated the out-of-band noise by filtering.

We downsample the signal to bring it back to the nominal sampling frequency

dictated by the sampling theorem.

And 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've had

if you had sampled at the critical rate.

But the noise power has been divided by a factor of capital 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 cut off

frequency pi over capital 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 a 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 upsampling factors.

We can quickly look at how over sample [INAUDIBLE] conversion

works entirely in the time domain just to get some intuition.

The over sampling factor of capital N basically produces

N samples where we would've needed just one.

Now if we assume that the quantization noise is uncorrelated between these extra

samples.

The low pass filtering operation is simply a local average.

That by average in neighboring samples with uncorrelated noise

will reduce the overall noise for this group of samples.

And then by downsampling by capital N,

we're discarding the redundant information introduced by oversampling.

Another application where we can trade sampling rate for

precision is in the conversion from the digital to analog domain.

Here we're not concerned anymore with quantization errors.

We are actually trying to use very cheap interpolation filters to produce

an analog signal.

This is particularly relevant in audio applications.

And in fact, Oversampled D/A is the reason why we can have very cheap,

and yet very high quality, audio consumer electronics today.

So let's see how this works.

We know that in the ideal case when we move from the digital to the analog

domain, we should use sinc interpolation.

Which in practice corresponds to using an analog low-pass filter with a very,

very sharp, almost ideal, characteristic.

Remember, this is the continuous time spectrum here,

which is obtained from the discrete time spectrum.

By simply selecting the base band component of the spectrum.

Via the Fourier transform of the sinc interpolator that we

know to be [INAUDIBLE] in the frequency domain.

So graphically, if this is your digital spectrum, you know that

in the continuous time domain, you will have all these repetitions.

This is the positive frequency axis, right.

And the sinc interpolator acts as an ideal low pass filter.

That selects just the base band component and

kills all the repetitions at high frequency.

Now you understand for instance that in audio applications.

These high-frequency repetitions of the spectrum would sound extremely disturbing.

And even if these copies are beyond the limit of human hearing,

they're really undesirable.

Because they would overload the amplification stages of any music

system like your stereo for instance.

Now the problem is that using such a sharp low pass filter in

digital to analog conversion.

Means that you have to have an analog low pass filter with a very,

very sharp characteristic.

Now analog low pass filters are very expensive to manufacture with this

precision.

And also, the more precise the filter is, the more delicate it becomes.

In the sense that, if you use analog components.

You have a natural aging of the components of the capacitors, for instance,

that are used in the filter.

That will change the characteristic of the filter over time, or

if the temperature changes, and so on and so forth.

So if you want a very good digital to analog conversion,

it seems that you have to invest a lot in the interpolation filter.

And this indeed was the situation when the first

digital audio devices came into the market in the 80s.