Multirate DSP, part 3: ADC oversampling

Figure 12-25. Block diagram of SDM ADC.

First, the analog signal is sampled to obtain the discrete-time signal x(n). This discrete-time signal is subtracted by the analog output from the m-bit DAC, converting the m bit oversampled digital signal y(n). Then the difference is sent to the discrete-time analog integrator, which is implemented by the switched-capacitor technique, for example. The output from the discrete-time analog integrator is converted using an m-bit ADC to produce the oversampled digital signal. Finally, the decimation filter removes outband quantization noise. Further decimation process can change the oversampling rate back to the desired sampling rate for the output digital signal w(m). To examine the SDM, we need to develop a DSP model for the discrete-time analog filter described in Figure 12-26.

Figure 12-26. Illustration of discrete-time analog integrator.

As shown in Figure 12-26, the input signal c(n) designates the amplitude at time instant n, while the output d(n) is the area under the curve at time instant n, which can be expressed as a sum of the area under the curve at time instant n – 1 and area increment:

d(n) = d(n – 1) + area incremental. (12.22)

Using the extrapolation method, we have

d(n) = d(n – 1) + 1 × c(n). (12.23)

Applying the z-transform to Equation (12.23) leads to a transfer function of the discrete-time analog filter as

H(z) = D(z)/C(z) = 1/(1 – z^–1). (12.24)

Again, considering that the m-bit quantization requires one sample delay, we get the DSP model for the first-order SDM depicted in Figure 12-27, where y(n) is the oversampling data encoded by m bits each, and e(n) represents quantization error.

Figure 12-27. DSP model for first-order SDM ADC.

The SDM DSP model represents a feedback control system. Applying the ztransform leads to

Y(z) = ((1/(1 – z^–1))(X(z)) – z^–1Y(z)) + E(z). (12.25)

After simple algebra, we have

In Equation (12.26), the indicated highpass filter pushes quantization noise to the high-frequency range, where later the quantization noise can be removed by the decimation filter. Thus we call this highpass filter (1 – z₁) the noise shaping filter, illustrated in Figure 12-28.

Figure 12-28. Noise shaping of quantization noise for SDM ADC.

Shaped-in-band noise power after use of the decimation filter can be estimated by the solid area under the curve. We have

Using the Maclaurin series expansion and neglecting the higher-order term due to the small value of Ω_max, we yield

1 – e^–jΩ = 1 – (1 + (–jΩ)/1! +(–jΩ)²/2! + …) ≈ jΩ.

Applying this approximation to Equation (12.27) leads to

After simple algebra, we have

Shaped-in-band noise power ≈ ((π²σ_q²)/3) × ((2f_max)/f_s)³
= (π²/3) × ((A²2^–2m)/12) × ((2f_max)/f_s)³. (12.29)

If we let the shaped-in-band noise power equal the quantization noise power from the regular ADC using a minimum sampling rate, we have

(π²/3) × ((A²2^–2m)/12) × ((2f_max)/f_s)³ = (A²/12) × 2^–2n. (12.30)

We modify Equation (12.30) into the following useful formats for applications:

n = m + 1.5 × log₂ (f_s/(2f_max)) – 0.86 (12.31)

(f_s/(2f_max))³ = (π²/3) × 2^{2(n – m)}. (12.32)