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Data conversion for band-limited signals with sampling frequency higher than the Nyquist rate is termed as oversampling. The oversampling ratio is defined as OSR = _2f^fs

0, where

4.1 Principle of analog-to-digital conversion 41

Figure 4.3: Spectral distribution of noise

f0 is the input signal bandwidth. An ADC structure based on oversampling is shown in Fig. 4.4. It uses an additional filter at the ADC output which is essential to remove the out-off-band noise.

In oversampling ADCs also an uniform distribution of noise spectrum is assumed, hence it is distributed all over the new sampling range as shown Fig. 4.5. In the figure, it is clearly visible that the mean value of quantization noise over the new frequency range is reduced by the factor of OSR. With the assumption of an ideal filtering at the output of the ADC, the noise which is outside the signal band is attenuated completely, and the corresponding noise spectral power is evaluated as in (4.3).

Pe =

From (4.3) it is easily understandable that the OSR appears at the denominator to reduce the noise power. Similar to (4.2) the peak SNR of an oversampling ADC can be evaluated as in (4.4).

SNRpeak = 10loge²_S

P_e² = 6.02N + 1.76 dB + 10log(OSR) (4.4) From (4.4), every doubling of OSR will increase the SNR by 3 dB, which is approximately 0.5 bit in resolution. In order to extract best capabilities out of oversampling ADCs one needs to analyze the input and noise transfer behavior. The output can be expressed in terms of input and transfer functions as y(z) = x(z)Hx(z) + e(z)He(z), where Hx(z) is

: Over sampled output : Decimated output

Low-pass filter (Decimator)

y (n)d

Figure 4.4: Structure of oversampling ADC

the signal transfer function and He(z) is the noise transfer function. In conventional oversampling ADCs these transfer functions are Hx(z) = He(z) = 1. This need not be the case always, and in fact oversampling ADCs can be designed with different Hx(z) and He(z). This means Hx(z) leaves the input signal unchanged while He(z) reduces the in-band noise to allow a high resolution output [CT1992] [ST2004] [NS1996] [JC1985]. E.g.

after introducing a loop filter G(z) before the quantizer and taking a negative feedback at y(n) as shown in Fig. 4.6, the corresponding input and noise transfer functions are given in (4.5).

Hx(z) = G(z)

1 + G(z), He(z) = 1

1 + G(z) (4.5)

Choosing G(z) such that it has very large gain within the band of interest and small gain outside the band, then from (4.5) follows that Hx(z) ≈ 1 and He(z) ≈ 0. If G(z) is chosen as an integrator, then Hx(z) = z^-1and He(z) = 1−z^-1, which implies that the input signal is transferred to the output with a sample delay, while the quantization noise is applied to a first-order z domain differentiator or a high-pass filter. The corresponding output function in z and time domain can be given as in (4.6).

y(z) = x(z)z^-1+ e(z)(1 − z^-1)

y(n) = x(n − 1) + e(n) − e(n − 1) (4.6) Replacing G(z) in Fig. 4.6 by an integrator results in a first-order ∆Σ modulator and similarly by introducing another integrator and a feedback, a second-order modulator is realized as shown in Fig. 4.7. The DAC used in the feedback has a single-bit resolution, is perfectly linear and is realized by a simple comparator. Consequently, if the sampling frequency is high enough, the ∆Σ-ADC will allow to use a single-bit quantizer to achieve an overall high resolution. By inserting more loops with integrators, it is possible to achieve higher-order ∆Σ modulators. The signal transfer function of a n^th-order modulator can be given as Hx(z) = z^-n and He(z) = (1 − z^-1)ⁿ. The total quantization noise power within

Noise spectrum of Nyquist rate ADC Noise spectrum of oversampled ADC Noise spectrum of oversampled ADC with ideal filtering

Figure 4.5: Noise spectrum of oversampling ADC

4.1 Principle of analog-to-digital conversion 43

x(n) y(n)

e (n)q

N-bit

Quantizer Low pass filter (Decimator)

y (n)d

G(z)

y(n): Over sampled output DAC

-y (n)d : Decimated output

Figure 4.6: Noise-shaping oversampling ADC

x(n) y(n)

Figure 4.7: First- and second-order ∆Σ modulator

the signal band is given in (4.7). The evaluated noise power in the equation is actually an ideal value, because it is only possible with ideal filtering.

Pe =

Comparing (4.2) and (4.7), an oversampling ADC with noise shaping gives 2n + 1 times better performance in terms of noise reduction. The corresponding peak SNR for an n^th-order modulator can be evaluated as in (4.8).

SNRpeak = 10log

"

where N is the number of bits in the quantizer and n is the modulator order. Note that this is the performance of an ideal n^th-order modulator which can be improved by increasing the modulator order and the OSR.

Using (4.8), one can verify that the first-order modulator’s SNR increases by 9.03 dB for every doubling of sampling frequency and for the second-order modulator even further to 15.05 dB, which corresponds to an increase of 1.5 and 2.5 bits in resolution, respectively.

In general, noise shaping helps to enhance the resolution of oversampling ADCs by a factor of 2n + 1 compared to conventional oversampling ADCs without noise-shaping.

However, the integrator gains have to be adjusted if the modulator order is higher than two to ensure stability. Even though the term sigma-delta (Σ∆) was coined by some of the early researchers in the field [JC1985], nowadays the term delta-sigma (∆Σ) is mostly synonymous with noise-shaping ADCs. The noise spectrum of ∆Σ modulators with different orders is shown in Fig. 4.8 on linear and log scale. For the sake of comparison, the spectrum of the conventional oversampling ADC is also shown in the figure. From these plots it is clear that noise shaping suppresses the noise to a great extent in the low-frequency range.