Thesis Organization

This thesis is organized into five chapters. In Chapter 1, this thesis and the hearing aids architecture are briefly introduced. In Chapter 2, the basic concepts of quantizer, comparison of oversampling and Nyquist converters, and sigma-delta modulator are introduced. Also, the stability of high-order sigma-delta modulator is considered. Finally, the performance of sigma-delta modulator is presented in detail.

In Chapter 3, the noise sources in sigma-delta modulator are introduced and the power efficiency of the integrators is estimated. The solutions for driving the switches

In Chapter 4, a 40µW sigma-delta modulator using inverter opamp is introduced.

The inverter opamp can reduce power consumption and chip area. The some techniques are presented to solve the problems using inverter opamp in this chapter.

In Chapter 5, the conclusions of this work are summarized. Moreover, the comparisons to the sigma-delta modulators are presented as well.

Chapter 2

Fundamentals of

Sigma-Delta Modulator

2.1 Introduction

Some of the fundamental issues in the design of sigma-delta modulators will be reviewed in this chapter. The discussion of oversampling converters is in Section 2.2.

It includes the basic concepts of quantization, the comparison of oversampling and Nyquist A/D converters, and oversampling techniques. In Section 2.3 and Section 2.4, noise-shaped sigma-delta modulator and high-order sigma-delta issues are introduced.

It discusses how the sigma-delta modulators work. The basic linear models are reviewed. The performance metrics of A/D converter is introduced in Section 2.5.

2.2 Overview of Oversampling Converters

2.2.1 Quantization Error

In A/D converters, quantizers are always needed. The analog signal is quantized by quantizer to a of digital code based on different signal levels. The performance of

x

Figure 2-1 Two types of quantization (a) midtread and (b) midrise.

the A/D converter depends on the precision of the quantization. According to the quantizer characteristic, it can be classified as either midtread or midrise. Figure 2-1 shows two types of quantizer. The output of midtread is a constant value and that for the midrise is a transition point at the middle of the input range. To understand the characteristics of the quantization error (also called quantization noise), the ramp waveform signal x is given, as the dashed line in Figure 2-1. The corresponding quantization error V_Q exhibits sawtooth-like variation and it is depicted at the bottom of Figure 2-1. The quantization error is limited to ±∆/2 where ∆ is the gap between output levels. It is also given by

N

where T is the period of the sawtooth wave. It is found that the power of the quantization error is in proportion to the square of ∆. From (2.1) and (2.2), it is obvious that quantization error can be reduced if the number of bits increases. If the

2

Figure 2-2 The probability density function for the quantization error.

quantization error V_Q is a uniformly distributed random variable, the interfering effect of V_Q is similar to thermal noise. The probability density function for such an error signal, ^f_Q

( )

^q , will be a constant value, as shown in Figure 2-2. The probability density function of the quantization error is given by

( )

q =

f_Q (2.3)

However, it must be ensured that the incoming signal does not overload the quantizer.

The power spectral density ^S_Q

( )

^f of the quantization noise is white and uniformly distributed between ± f_S /2, as shown in Figure 2-3. From (2.2), the noise power is

Thus, the height of the noise power spectral density equals to

( )

Figure 2-3 The power spectral density of quantization noise.

) 2

Quantizer

x(n) y(n)

y(n) x(n)

e(n)

Figure 2-4 Quantizer and its linear model.

From (2.5), the power spectral density is inversely proportional to the sampling frequency. If the input signal is a sinusoidal waveform between zero and maximum full-scale, which equals to 2^N

(

∆/ 2

)

, and the rms value is ^2N

(

^{∆/2 2}

)

. Thus, the

The above equation gives the best possible SNR for an N-bit A/D converter. However, the idealized SNR decrease from this best possible value for reduced input signal levels. The SNR values could be improved through the use of oversampling techniques which also means higher sampling frequency. In other words, the input signal’s bandwidth is much lower than the Nyquist rate.

The quantization noise can be regarded as an independent additive white-noise signal. Thus, a linear model is made about the statistical property of quantization error,

( )

n

e , as shown in Figure 2-4. This model is helpful to analysis of sigma-delta modulators.

2.2.2 Comparison of Oversampling and Nyquist Converters

The function block of the A/D converter is shown in Figure 2-5. The anti-aliasing filter limits the bandwidth of the input signal and keeps the out of band noise from entering the signal baseband in the sampling process. The analog signal is then sampled in discrete time by a sample-and-hold. Each sampled data is converted to the corresponding digital code by the amplitude quantizer. The design of the digital signal processor depends on the type of the converter.

Figure 2-6 shows the comparison of frequency spectrum between a conventional

Anti-aliasing

Filter

Sample

&

Hold

Amplitude Quantizer

Digital Signal Processor

Digital Output Analog

Input

Analog Digital

Figure 2-5 Function block of A/D converter.

Signal Bandwidth Transition band Anti-aliasing filter

Amplitude

fB 0.5f_S f_S=f_N f

Signal Bandwidth Transition band Anti-aliasing filter

Amplitude

fB f_N 0.5f_S f_S=Mf_N ^f

(a) (b)

Figure 2-6 (a) Conventional Nyquist ADC and (b) Oversampling ADC [5].

Nyquist A/D converter and an oversampling A/D converter. The bandwidth of the input signal is shown by the rectangle with diagonal lines. The upper limit of the frequency is indicated by f_B. In Figure 2-6 (a), the sampling rate is equal to the Nyquist rate ( f_N). It is made as close to 0.5f_N as possible to achieve maximum signal bandwidth ( f_B). In Figure 2-6 (b), the Nyquist rate is much less than the sampling rate. Anti-aliasing filter requirements of the oversampling A/D converters are much more relaxed than those of Nyquist rate converters. The reason for this is that the sampling frequency is much higher than the Nyquist rate in oversampling converters. Also, the transition band is quite wide and the transition is smooth so a simple first-order or second-order analog filters are usually sufficient to meet the relaxed anti-aliasing requirements [5]. Therefore, the less complex filter is adapted into the design of the anti-aliasing filter using an oversampling A/D converter.

2.2.3 Oversampling Technique

Oversampling means that the sampling frequency exceeds Nyquist frequency. In other words, the sampling frequency is greater than twice the signal bandwidth. And the oversampling ratio (OSR) is defined as

B S

f 2

OSR≡ f (2.7)

For a sinusoidal input signal, the maximum signal power is square of rms value (^2N

(

^{∆/2 2}

)

) and is rewritten as

For an oversampling system, the input signal with frequency below the bandwidth ( f_B) can pass through the lowpass filter without any decay. But, the out-of-band quantization noise is filtered out, as shown in Figure 2-7. Thus, the quantization noise power is given by

It can be found that the quantization noise power is reduced half or equivalently 3dB as OSR is doubled. The maximum SNR value can be calculated by 2.8) and (2.9) as

⎟⎟

The first two terms are the same as (2.6) from an N-bit quantization without oversampling and the final term is due to the use of oversampling techniques. The

2

Figure 2-7 Quantization noise power spectral density after low-pass filter.

oversampling techniques have an improvement of 3dB/octave or 0.5-bit/octave. In other words, if the oversampling ratio is increased by a factor of 4, the resolution can be improved by one bit. Thus, increasing OSR can improve SNR. However, oversampling is not an attractive way because higher sampling frequency and severe linearity are required for high resolution. Hence, the noise-shaped method will be introduced in the next section. It provides a reasonable oversampling frequency to achieve much higher dynamic range.

2.3 Noise-Shaped Sigma-Delta Modulators

2.3.1 Introduction to Noise-Shaped Sigma-Delta Modulators

Although oversampling technique can improve SNR value to achieve high resolution A/D converters, the extremely high sampling frequency is needed. To achieve reasonable sampling frequency and higher dynamic range, the feedback architecture is used to obtain the noise-shaping function. The oversampling converters with noise-shaping technique remove much noise away from the signal band. The oversampling converters with noise-shaping technique are implemented as sigma-delta modulator.

Figure 2-8 shows the system block diagram of sigma-delta modulator. The analog signal, ^x_in

( )

^t , is limited in the signal band by anti-aliasing filter to prevent the aliasing of high frequency noise into the signal band. After the anti-aliasing filter, the band limited signal, x

( )

t , is oversampled and processed by the sigma-delta modulator.

The sigma-delta modulators are usually implemented by a switched-capacitor circuits and then the analog signal is transformed into digital signal, x

( )

n . Finally, the digital signal is converted into digital code by decimation filter.

A general architecture of sigma-delta modulator with noise-shaping and its linear

Sigma-Delta Modulator Anti-aliasing

Filter

Low-Pass Filter

Down Sampling

Analog Digital

Digital Output Analog

Input

Decimation Filter ( )t

x_in x( )t x( )n x_n( )t

Figure 2-8 The system block diagram of sigma-delta modulator.

Y(n) H(Z) X(n)

+

-Quantizr

D/A Converter u(n)

(a)

Y(n)

u(n)

+

H(Z) X(n)

-+ e(n)

(b)

Figure 2-9 (a) The architecture and (b) linear model of the sigma-delta modulator.

model are shown in Figure 2-9. This feedback topology is analogous to the amplifier constructed by the feedback of the opamp. The feedback reduces the low frequency noise when the opamp gain is high enough. At high frequency, the noise is not reduced due to low opamp gain. In other word, a large gain in forward path can lessen the quantization error [6]. The feedback gain equal to unit is utilized to prevent the DAC errors from influencing the performance of sigma-delta modulator. Hence, a great effort should be devoted to the linearity of the DAC for high resolution converters, especially in multi-bit sigma-delta modulator. Single-bit sigma-delta modulator is a good method to solve the linearity requirement of DAC due to only two output levels.

Considering the linear model in Figure 2-9 (b), there are two independent inputs.

The signal transfer function S_TF and noise transfer function N_TF can be calculated as

( ) ( )

( ) ( )

( )

^z

H 1

z H z

U z z Y S_TF

= +

≡ (2.11)

( ) ( )

( )

^{z 1 H1}

( )

^z

E z z Y N_TF

= +

≡ (2.12)

From (2.12), the zeros of the noise transfer function, N_TF

( )

z , is equal to the poles of

( )

z

H . Since the signal transfer function, S_TF

( )

z , approaches unity and the noise

transfer function, N_TF

( )

z , approximates zero when H

( )

z goes to infinite. band. With such a choice, the signal transfer function, S_TF

( )

z , will approximate unity over the signal frequency band and the noise transfer function, N_TF

( )

z , will approximate zero over the same band. Hence, the quantization noise is reduced over the signal frequency band.

2.3.2 First-Order Sigma-Delta Modulator

Figure 2-10 shows the architecture of a first-order sigma-delta modulator. To realize first-order noise-shaping, the noise transfer function, N_TF

( )

z , should have a zero at dc (i.e., z=1) so it is high pass for the quantization noise [1]. Because the zeros of N_TF

( )

z are equal to the poles of H

( )

z , H

( )

z with noise-shaped function is

Figure 2-10 The Architecture of first-order sigma-delta modulator.

Thus, the output of sigma-delta modulator is

The magnitude of noise transfer function is

( )

⎟

From (2.5) and (2.19), the noise power is derived as

( ) ( )

Hence, the maximum SNR is derived as

⎟⎟

From (2.22), a first-order sigma-delta modulator has an improvement of 9dB, or equivalently, 1.5 bits in SNR while OSR is doubled.

2.3.3 Second-Order Sigma-Delta Modulator

Figure 2-11 shows a second-order sigma-delta modulator. Its fundamental theory is the same as a first-order sigma-delta modulator. However, to realize a second-order noise shaping, the noise transfer function, N_TF , must be a second-order high-pass function. Thus, the output can be derived as

( )

^{z U}

( )

^{z z 2 E}

( )

^z

(

^{1 z 1}

)

²

Y = ⋅ ⁻ + ⋅ − ⁻ (2.23)

The magnitude of noise transfer function is

( )

²

Hence, the quantization noise power can be calculated as

( ) ( )

Figure 2-11 The Architecture of second-order sigma-delta modulator.

Second-order First-order No noise shaping

( )

^f

Figure 2-12 Noise-shaping transfer function curves.

As the approximation in previous section, due to _⎟⎟

⎠

, the quantization noise power in the signal band can be derived as

4 5 Hence, the maximum SNR is derived as

⎟⎟

The above equation shows that an improvement of 15dB, or equivalently, 2.5 bits is achieved for the second-order noise-shaping while the OSR is doubled.

Figure 2-12 shows that the second-order modulator has steeper shaping slope for the noise in the signal band than the first-order modulator. In other words, the noise power decreases as the noise-shaping order increase. However, the out-of-band noise increases for the higher-order modulators.

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}

)

^L

Y = ⋅ ⁻ + ⋅ − ⁻ (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, N_TF

( )

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 ^N_TF

( )

^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 ^N_TF

( )

^z is all one needs to know to describe the stability properties of a single-bit modulator. However, the properties of

( )

z

N_TF 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, N_TF

( )

z , is likely to be stable if max N_TF(e^jω ) 1.5

ω < .

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

( )

₁

n 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 ^N_TF

( )

^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].

2.5 Performance Metrics

2.5.1 Resolution

The resolution of a converter is defined to be the number of distinct analog levels corresponding to the different digital expression. Resolution is usually expressed by the base 2 logarithm. That is, there are 2^N distinct analog levels it can resolve for a resolution of N-bit. Resolution is usually affected by the noise and the nonlinearity of the quantization or analog circuits. Hence, the real resolution of a converter will be a little degradation. Sometimes it is also called effective number of bits (ENOB).

2.5.2 Signal-to-Noise Ratio

The signal-to-noise ratio (SNR) is the ratio of signal power to noise power in the system. In a sigma-delta modulator, the SNR value is measured at the output of modulator. The noise sources include all noise in a converter except the noise of harmonic distortion. The peak value of the SNR is also the performance of the system.

2.5.3 Signal-to-Noise plus Distortion Ratio

The signal-to-noise plus distortion ratio (SNDR) is the ratio of the signal power to the sum of all noise power and the harmonic distortion. The peak SNDR is an important criterion to evaluate the capability and the acceptable linearity of a sigma-delta modulator in the signal band. Note that the peak SNDR is frequency dependent and can be used to measure the degradation of the modulator performance as the input signal increases in frequency.

2.5.4 Dynamic Range

Figure 2-16 shows that SNDR/SNDR versus the input signal power. Dynamic range (DR) denotes the range of the input signal amplitude from which useful output can be obtained from a sigma-delta modulator system. The definition of DR is the difference between the input signal level of peak SNR and the input level where x-axis intercepts to the SNDR curve. The minimum detectable input signal power is at

dB 0

SNDR= . If the noise power is independent of the signal amplitude, the dynamic range would be equal to the SNR in the full range.

0 -10 -20 -30 -40 -50 -60 -70 -80 -100 -90

10 20 30 40 50 60 70 80 90 100

SNR SNDR Dynamic Range

Normalized Input Power (dB)

SNDR/SNR (dB)

Peak SNR

Overload

Figure 2-16 SNDR/SNR versus input signal power.

2.6 Summary

The sampling rate of Nyquist A/D converters is only twice of the signal band.

However, it is difficult to achieve high resolution A/D conversion due to requirements

However, it is difficult to achieve high resolution A/D conversion due to requirements

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