Signal Processing Magic (2) -- Sigma Delta ADC

Sigma Delta ADC is a commonly used architecture for converting analog signal to digital. The conception of ADC is quite simple: let us say we have 4 bits with bit3 representing 1volt, bit2 for 0.5 v, bit1 for 0.25v and bit0 for 0.125v, I can quantize an analog input in volt to these 4 bits. For example, if the input is 1.2 volts, then it becomes [bit3 bit2 bit1 bit0] = [1010] = 1.25v. The quantization error is equal to 1.2-1.25 = -0.05v. Quantization noise comes from quantization error. A straightforward thought is to use binary search to convert analog input to digital. This is the basic idea behind SAR (Successive Approximation Register) ADC. 

Sigma Delta ADC, our main topic in this article, is constructed differently. Assuming x(n) is the input and y(n) is the output of sigma delta modulator, y(n) has a much higher sampling rate than x(n) and y(n) is also only 1 bit. So in term of y(n), sigma delta can be seen as a trade off between sampling rate and bitwidth. y(n) will then pass through lowpass filter and decimator to produce the multiple-bit ADC output we commonly work with.

Now let us do some math to figure out what really happens in sigma delta. First we calculate the transfer function between v(n) and u(n)

\[z^{-1}(U(z)+V(z))=V(z)\]
which gives
\[H(z)=\frac{V(z)}{U(z)}=\frac{z^{-1}}{1-z^{-1}}\]

The transfer function from input, x(n), to output, y(n), is equal to
\[\frac{Y(z)}{X(z)}=\frac{H(z)}{1+H(z)}=z^{-1}\]
which tell us there is signal delay but no signal distortion.

e(n) added to 1-bit ADC in the diagram represents quantization noise. The transfer function from e(n) to y(n) is
\[\frac{Y(z)}{E(z)}=\frac{1}{1+H(z)}=1-z^{-1}\]
Frequency response of Y(z)/E(z) is shown below. Its amplitude increases with frequency which means sigma delta modulator pushes out noise from in-band to out-of-band. Reducing in-band noise level is the major benefit of sigma delta. What we discussed here is first-order sigma delta. If we goes to higher order sigma delta, the in-band noise level will be further reduced but the cost paid is more sophisticated hardware and higher out-of-band noise level.