Chapter 2 Fundamentals of Sigma-Delta Modulator
2.4 High-Order Sigma-Delta Modulators
2.4.1 Introduction to High-Order Sigma-Delta Modulators
In a sigma-delta modulator, the order of noise transfer function determines how much noise is placed outside the signal frequency band. The high-order signal and noise transfer function can be derived by the similar method as the former modulators.
Thus, the general form for the output of the Lth-order noise-shaping modulator can be given by
( )
z U( )
z z L E( )
z(
1 z 1)
LY = ⋅ − + ⋅ − − (2.28)
The quantization noise power in the signal frequency band can be derived as
( ) ( ) ∫
Thus, the maximum SNR for the Lth-order modulator is
⎟⎟
The above equation shows that an Lth-order noise-shaping modulator improves the SNR by 6L+3 dB as doubling the OSR, or equivalently L+0.5 bits/octave. However, the high-order sigma-delta modulator has a system stability issue. The following sections will describe stability considerations and two approaches (single-loop and multi-stage noise noise-shaping) for high-order sigma-delta modulator.
2.4.2 Stability Considerations in High-Order Modulator
Figure 2-13 shows a general structure of the single quantizer sigma-delta modulator. The linear model of the single quantizer modulator predicts that the stability of the modulator is determined by the loop gain, which is determined by the noise transfer function, NTF
( )
z . However, this argument ignores the effect of nonlinear quantizers. For example, if the input signal is so large such that the input to the first integrator is positive at every time step. Finally, the output of the integrator will monotonically increase without bound and the loop filter will be unstable [7].Thus, the range of input magnitude is also an important factor in system stability for a high-order sigma-delta modulator. The proper and stable modulator operation is assured if the loop filter remains linear and the internal quantizer is not severely overloaded. Since the stable input range of a sigma-delta modulator is primarily determined by the NTF
( )
z and the number of bits in single quantizer, the stability of single-bit and multi-bit quantizers will be discussed.According to the above argument, the NTF
( )
z is all one needs to know to describe the stability properties of a single-bit modulator. However, the properties of( )
zNTF are not necessary and sufficient requirements for the stable operation. Thus, the most widely-used approximate criterion is the Lee criterion [8] [9]:
A single-bit sigma-delta modulator with a noise transfer function, NTF
( )
z , is likely to be stable if max NTF(ejω ) 1.5ω < .
Note that this single-bit criterion is neither necessary, nor sufficient. Nevertheless, due to its simplicity, it is of some use [7].
Loop
Filter Quantizer
U X Y
Figure 2-13 General structure of a single quantizer sigma-delta modulator.
For the multi-bit sigma-delta modulators, the following theoretical result can be useful:
Considering a modulator with an M-step (i.e. (M+1)-level) quantizer. Let the initial input y(0) to the quantizer be within its linear (no-overload) range. Then, the modulator is guaranteed not to experience overload for any input u(n) such
that
( )
1n u n M 2 n
max ≤ + − , where n 1 =
∑
∞n=0 n( )
n . Here, n(n) is the inverse z-transform of the noise transfer function NTF( )
z .It is easy to use the above rule to establish a modulator which is implemented with stable noise transfer function.
2.4.3 Single-Loop High-Order Sigma-Delta Modulator
A high-order sigma-delta modulator can be constructed by connecting a series of the integrators in a single-loop. There are many different single-loop topologies to overcome the stability problem of the modulators. Figure 2-14 shows a high-order interpolative sigma-delta modulator. This architecture reduces the component sensitivity depending on inserting resonators to adjust the zeros of the noise transfer function in the signal band. However, the unavoidable spurious tones appear in the signal band and the dynamic range is decreased due to multi-path of feedback and feedforward. Hence, to increase dynamic range, the architecture is modified with
0
Figure 2-14 The high-order interpolative sigma-delta modulator.
2.4.4 Multi-Stage Noise-Shaping Sigma-Delta Modulator
Another approach for realizing high-order modulators is to use a cascade-type structure where overall high-order modulator is constructed using first-order or second-order modulator [1]. Since the lower-order modulators are more stable, the overall system should remain stable. This architecture has been called MASH (i.e.
Multi-stAge noise SHaping).
Figure 2-15 shows a sigma-delta modulator with MASH structure. The quantization noise of the first stage can be processed by the following stage. The output of the second modulator is combined with the first modulator output to cancel the first modulator error. Hence, the only quantization noise appears in the output of the last stage in an ideal modulator with MASH structure. The advantage of a MASH approach is that high-order noise-shaping can be achieved using low-order modulators.
The low-order modulators are more stable as compared to a high-order single-loop structure.
However, MASH approach is sensitive to the finite gain of OP amplifier and mismatches between every stages. Such mismatches cause noise to leak through from each stage and hence reduce dynamic range performance. To reduce this mismatch
H(z) A/D
D/A
H(z) A/D
D/A
DSP Analog
Input
Digital Output 1
Digital Output 2
...
...
...
Final Digital Output
Figure 2-15 A sigma-delta modulator with MASH structure.
problem, the first stage is often chosen to be a higher-order modulator such that any noise leak-through does not have a serious effect[1].