ΣΔ A/D converters are based on oversampling and noise shaping to reach high resolution.
Oversampling means the sampling rate is much higher than Nyquist rate, about 8~512 times in general applications. The goal of oversampling is to expand quantization noise to wider range. It can reduce the quantization noise in signal bandwidth and increase the DR (Dynamic range) of input signal. Noise shaping is a technique that moves noise to high frequency, which is done by using discrete time filter and feedback technique. After noise shaping, the noise in high frequency can be filtered out by a digital filter [19].
2.3.1 Oversampling Technique
First, we made the assumption that quantization noise is a uniform distribution in sampling spectrum so its mean is zero and is a white noise [20]. The system in Fig. 2.6 just has oversampling function and does not have noise shaping effect. If a A/D converter is sampled in Nyquist rate, then the quantization noise is uniform distributed between ±fB ; if it is sampled by oversampling technique, then quantization noise is uniform distributed between± fS2/2s, which is much larger than fB. As shown in Fig. 2.7, if the signal bandwidth is between ±fB, then quantization noise in this bandwidth will be reduced by using oversampling technique, which will raise PSNR significantly.
Fig. 2.6 Sampling system
Fig. 2.7 Noise distribution after sampling
In the condition of oversampling, the PSD (Power Spectrum Density) of quantization noise is as Se2(f) in Fig. 2.7 and can be represented as:
From (2.7) we can estimate the quantization noise in 2fB after oversampling
PQ =
∫
−B ⋅ =In (2.8), we define a parameter OSR (Oversampling Ratio) as OSR =
Finally, we can get PSNR from (2.5) and (2.8)
PSNR = 10 log（
Q signal
P
P ）= 6.02B + 1.76 + 10 log（OSR） （2.10）
From (2.10), we can find that doubling OSR will increase 3dB in PSNR, which is about 0.5 bit increase in resolution. Although oversampling can reduce quantization noise, it is
difficult to reach high SNR when using a low bit quantizer. For example, if we need a 16bit A/D converter, then SNR must be equal to 98dB, if the signal bandwidth is 20KHz, then the sampling rate must equal to 2 × 109 × 20KHz, it is impossible to implement. Because at such high frequency, quantization noise is no longer a white noise, it is correlated with input signal. So there is not only oversampling technique, we must add noise shaping technique also, if we want to achieve high resolution.
2.3.2 Noise Shaping
From Fig. 2.8(a), we can derive output Y(z) as (2.11) Y(z) =
and define Signal Transfer Function STF and Noise transfer function NTF as STF (z)=
where H(z) is the transfer function of a discrete time filter. There have two important meanings in (2.12), (2.13). If we want to obtain highest SNR, STF must be equal to 1, that means the input signal can transfer to output without attenuating; and NTF (z) must be equal to 0, because the quantization noise will not affect output SNR.
In order to make NTF (z) be a high pass filter, so at DC(z = 1), NTF must be 0, and z = 1 is a pole of H(z), so the transfer function H(z) of the discrete filter is as
H(z) = 1 Z
1
− = 11 Z 1
Z
−
−
− （2.14）
Substitute (2.14) into (2.12) and (2.13), we can get STF (z) =
z
1 （2.15）
NTF (z) = z
1− （2.16） 1
And we substitute z with fs
f j2
e
π
, then we can plot STF(f)2 and NTF(f) 2 in frequency domain, as Fig. 2.9. We can find NTF(f)2 also increases with frequency, and STF(f)2 is always equal to 1, if we choose signal bandwidth in low frequency, then we can get highest signal power and lowest noise power, from this figure we see that quantization noise is moved to higher frequency significantly, this is the noise shaping effect.
2 TF(f) N
2 TF(f) S
Fig. 2.9 Noise shaping
After noise shaping, we can filter out the noise in high frequency by using digital filter, and we will illustrate its architecture more detail in the next chapter.
3
Architectures of Sigma-Delta Modulator
Before we introduce various architectures of ΣΔ modulators, we must to realize the basic architecture of a general A/D converter. Fig. 3.1 is a complete block diagram of a A/D converter [18], and we can divide it into two different parts. First part is the modulator. The main function of this part is doing oversampling and noise shaping to the input analog signal. Second part is the decimation filter. The main function of this part is to remove noise in high frequency and down sampling the sampling frequency to base band frequency.
ΣΔ ΣΔ
ΣΔ
Fig. 3.1 Block diagram of ΣΔ A/D converter
First, the input signal Xin(t) pass an Anti-aliasing filter, the 3dB frequency of this filter is about few times of Nyquist frequency, so signal and noise out of Nyquist frequency is filtered roughly, and this signal goes into the ΣΔ modulator after goes through a S/H circuit. However, in the circuits implement situation, the sample and hold function is included in the circuits of modulator, so the signal Xc(t) will pass this modulator and produces a high speed data code Xdsm(n), because of noise shaping, the quantization noise will appear in high frequency. Finally, we must filter the noise in high frequency and reduce the sampling frequency to Nyquist frequency by a decimator, and passes the digital signal to
ΣΔ
the output [18].
In this chapter, we will focus on the architectures of ΣΔ modulator, because that the noise model and optimal method is focus on this part, we must understand the theorem, benefits and drawbacks of each kinds of ΣΔ modulators. In addition, the implement of decimator is very typical [21][22]. In today’s technology, DSP processors are also used to replace decimators, so we will introduce this part roughly.