Noise Shaping of \$ \Sigma \Delta \$ Converters

Question

I am currently reading the book Continuous-Time Sigma-Delta A/D Conversion to get myself a better understanding of \$ \Sigma \Delta\$ converters. Unfortunately I got stuck at the point where noise-shaping is explained. The chapter started clearly, saying that for studying the effects of noise, the quantizer can be approximated by a linear model, which then leaves us with a linearized \$ \Sigma \Delta\$ modulator as shown at the bottom.

With this model we want to achieve two different transfer functions, one for \$x(n)\$ and another for \$e(n)\$.

$$ Y(z)=STF(z)X(z)+NTF(z)E(z) $$

STF ... Signal Transfer Function, NTF ... Noise Transfer Function

Since the noise shall occur at high frequencies and the signal at low frequencies, it seems clear that the STF should be a low-pass and the NTF a high-pass filter.

Now the author states that this leads to the following concrete transfer functions:

$$ STF(z)=\frac{1}{\frac{1}{H(z)k} + 1} \\ NTF(z)=\frac{1}{H(z)k + 1} $$

No further explanation. How is it possible to conclude that the STF and NTF need to look like that? Furthermore, how can we conclude that the simplest \$H(z)\$ is an integrator?

Answer 1

Noise shaping refers to the spectral shaping of the quantization noise. It is assumed that the input signal of the quantizer is sufficiently "busy" so that it can be modeled as a linear gain element with some additive noise.

The ultimate goal is to get an output signal that represents the input signal as closely as possible. The modulator shown in your post is the most common type, a lowpass delta-sigma modulator, it works best for low-frequency input signals.

Looking at $$ Y(z)=STF(z)X(z)+NTF(z)E(z) $$ we see that for an accurate representation of the input signal X(z) the STF should be about 1 and the NTF should be as small as possible.

To fulfill the first requirement STF ~ 1 we can look at $$ STF(z)=\frac{1}{\frac{1}{H(z)k} + 1} $$ and quickly see that we only need to have an \$H(z)\$ that is much larger than 1 for small frequencies. An integrator will be a perfect fit, however other filter types are possible as well.

For the second condition, quantization noise suppression at low frequencies, the expression $$ NTF(z)=\frac{1}{H(z)k + 1} $$ shows, that again \$H(z) \gg 1\$ for low frequencies is required. Again this can be achieved with an integrator.