Thesis Organization

In Chapter 1, this thesis is briefly introduced.

Chapter 2 is the overview of sigma-delta data converters and other analog-to-digital data converters. It consists of classification and characteristics of performance and speed. Then, the oversampling technique and noise-shaping strategy are introduced and described mathematically. Finally, the various architectures of sigma-delta modulators will be introduced and compared.

Chapter 3 discusses the basic concept of continuous-time SDM which consisting the power consumption, clock jitter effect, and excess loop delay issues. Then, the advantages and disadvantages of continuous-time and discrete-time SDMs will be discussed. We will use the continuous-time technique to apply in audio-frequency domain in order to obtain the advantage of very low power consumption.

Chapter 4 presents the system level design consideration. After building the behavior model, we continue the circuit level design, including the operation amplifier, comparator, and feedback digital-to-analog converter (DAC). The circuits and simulation results will be shown in this chapter.

Chapter 5 presents the testing environment, including the instruments and external circuits on the printed circuit board (PCB). Measured results for the continuous-time SDM, which is fabricated in a standard TSMC 0.18μm CMOS mixed-signal process, will be plotted and summarized

Finally, the conclusions of this thesis are summarized in Chapter 6.

Chapter 2

An Overview of Sigma-Delta Data Converters

2.1 Introduction

This chapter reviews the basic concept of design the sigma-delta data converter.

The discussion begins with a brief overview of data converter in the aspects of speed, resolution, and architecture. After this issue, the theories of how sigma-delta modulators work including sampling, quantization, oversampling, and noise shaping will be discussed. Following the introduction, tradeoffs of various sigma-delta modulator architectures will be discussed.

2.2 Overview of Analog-to-Digital Data Converters

The operations of analog-to-digital data converters can be roughly separated into two steps: sampling and quantization. The process of sampling transforms continuous time analog signals into discrete time step-like signals. The process of quantization converts the step-like signals to a set of discrete levels. Then, these discrete levels signals can be coded and be transmitted into DSP units or digital systems.

2.2.1 Categories of Analog-to-Digital Data Converters

According to operation, speed, and accuracy, there are three categories of analog-

to-digital converter shown in Table 2.1. Each category is applied in different field. But the demarcation for some structures nowadays is a little blurred.

TABLE 2.1 Various Kinds of Analog-to-Digital Data Converters

Speed Resolution Structure

Slow High Integrating

Oversampling

2.2.2 Oversampling Ratio (OSR)

The oversampling ratio (OSR) of a data converter is defined as

2 however, when the OSR is great than 1, it means the data converter is the oversampling data converter. The OSR is the important parameter for oversampling data converters. The OSR increases the SNR by (2n+ ⋅1) 3dB or by 2n+1 per octave, where n is the order of loop-filter.

The larger the OSR, the larger the sampling frequency when the signal bandwidth is fixed. Thus, it will need faster circuit and consume more power consumption. But the OSR need to keep as low as possible for high signal bandwidth consideration. In order to obtain the advantages of using noise- shaping strategy, the OSR should be at least 4 [01].

2.2.3 Signal to Noise Ratio (SNR)

The signal-to-noise ratio (SNR) of a data converter is the ratio of the signal power to the noise power, which measured at the output of the data converter. The maximum SNR that a converter can achieve is called the peak signal-to-noise ratio. Generally, the theoretical value of SNR for an N-bit Nyquist-rate ADC is given by

6.02 1.76

SNR= ⋅ +N dB (2.2)

But for oversampling ADC, the theoretical value of SNR is

6.02 1.76 10log( )

SNR= ⋅ +N + OSR dB (2.3)

2.2.4 Signal to Noise and Distortion Ratio (SNDR)

The signal to noise and distortion ratio (SNDR) of a data converter is the ratio of the signal power to the power of the noise plus the harmonic distortion components, which measured at the output of the data converter. The maximum SNDR that a converter can achieve is called the peak signal to noise and distortion ratio. Generally, SNDR is lower than SNR.

2.2.5 Spurious Free Dynamic Range (SFDR)

The spurious free dynamic range (SFDR) is defined as the ratio of rms value of amplitude of the fundamental signal to the rms value of the largest harmonic distortion component in a specified frequency range. SFDR may be much larger than SNDR of a data converter.

2.2.6 Dynamic Range at the input (DR)

The dynamic range is defined as the ratio between the power of the largest input signal which didn’t significantly degrade the performance and the power of the smallest detectable input signal which is determined by the noise floor of converters.

2.2.7 Effective Number of Bits (ENOB)

For data converter, a specification often used in place of the SNR or SNDR is ENOB, which is a global indication of how many bits would be required to get the same performance as the converter. ENOB can be defined as follows:

1.76 6.02 ENOB SNDR−

= bits. (2.4)

2.2.8 Overload Level (OL)

OL is defined as the relative input amplitude where the SNR is decreased by 6dB compared to peak SNR value.

2.3 Sampling Theorem

Naturally, signals transmitted in the air are analog whether they originate from.

The analog signals need to be sampled to become the digital signals for suitability in processing in the digital system. Thus, sampling is a very important procedure in the front end of the overall system. How much information can be preserve from the original signals depend on how fine to sample the signals and deal. It is crucial to choose the sampling frequency with a fixed signal bandwidth. And the relationship between the sampling frequency, f , and the signal bandwidth,_s f , is shown as _b follows :

s 2 b

f ≥ f (2.5)

At least the sampling frequency must be greater than twice the input signal bandwidth to avoid aliasing. If f is smaller than twice the signal bandwidth, aliasing will occur _s at the output signal spectrum as shown in Figure 2.1.

fb

Figure 2.1 Illustration of the aliasing of the sampling process ( f_s <2f_b)

There are two ways to deal with these problems. One is to cut the input signal bandwidth to f to make _nb f greater than twice the input signal bandwidth. But the _s disadvantage is that some high frequency information from the original signal will be loss (Figure 2.2). Another is to increase f to new sampling frequency _s f in order _ns to match the equation. This is more popular to deal with aliasing problems because there is no information of original input signal loss as shown in Figure 2.3. And just a low pass filter at the output is needed to recover the original signal.

fb

Figure 2.3 Illustration of the aliasing of the sampling process ( 2f_b ≤ f_ns)

2.7 Quantization Noise

The quantizer is the interface between analog and digital domain. Once the analog signals pass through the quantizer, the signals will be digitized and separated into several different levels. The space between two adjacent levels is called a step size, Δ.

There are two types of quantizer. One is uniform, and another is nonuniform. In a uniform quantizer, the distance between two adjacent levels is uniform; otherwise it is a nonuniform quantizer.

The process of quantization introduces an error, q(n). The error is defined as the difference between the input signal, x(n), and the output signal, y(n). And it is called the quantization error. Figure 2.4 and Figure 2.5 show the quantization process and assume the quantizer is uniform.

Figure 2.4 Quantized signal

Figure 2.5 Quantization error

Many of the original results and insights into the behavior of quantization error are due to Bennett [02]. Bennett first developed conditions under which quantization noise could be reasonably modeled as additive white noise. A common statement of the approximation is that the quantization error has the following properties, which we call it the “input-independent additive white-noise approximation” [03]:

Property 1. q[n] is statistically independent of the input signal Property 2. q[n] is uniformly distributed in [-Δ/2, Δ/2]

Property 3. q[n] is an independent identically distributed sequence or q[n] has a flat power spectral density (white).

Since the quantization noise, q(n), is equal to y(n)-x(n), a quantizer can be modeled as shown in Figure 2.6 [04] . For a uniform quantizer, if the input signal does not overload, the quantization error will be bounded by ±Δ/2. If the Δ is very small, it is convenient and reasonable to assume the quantization noise is zero mean and uniform distribution (Figure 2.7). The probability density function (pdf) of the quantization noise can be express as

1 , - 2 ( ) 2 ( ) 0, otherwise

Q

f q ⎧ Δ Δ ≤q n ≤ Δ

= ⎨⎩ (2.6)

Figure 2.6 Quantizer and its linear model

Q( ) f q

− Δ 2

1 Δ

Δ 2 Figure 2.7 The pdf of quantization noise

From Figure 2.7, the power of quantization noise can be shown as follows: the equation (2.8) is

2 2

From equation (2.9) we show that power spectrum density is inversely proportional to sampling frequency shown in Figure 2.8. The larger the sampling frequency is, the less the noise amplitude is.

Figure 2.8 Power spectrum density of q(n)

Assume the quantization signal is uniformly distributed over the range± , and N is V_A the bits per sample. The step size can be write as

2 2

A N

Δ = V (2.10)

According to the equations above, the SNR can be shown as

2

Equation (2.11) shows that increasing the number of bits per sample in the qunatizer increases the accuracy of the converter by 6dB for each extra bit.

2.7 Oversampling Technique

Oversampling is an important technique for sigma-delta ADCs. It can release the requirement of anti-aliasing filter. And it also can improve the resolution of a sigma-delta ADC. This improvement is achieved by oversampling the signal. In other words, the sampling rate is much greater than Nyquist-rate. The definition of oversampling ratio (OSR) is

2 _s

b

OSR f

= f (2.12)

where f is the sampling frequency and _s f is the input signal bandwidth. _b Assuming the quantization noise is white noise. It means that noise power is uniformly distributed between −f_s 2 and f_s 2. It had shown that total amount of noise power injected into the quantized signals are the same whether they are oversampling or Nyquist-rate conversions. But the distributions are different due to different sampling frequencies. Figure 2.9 shows the power spectrum density of

quantization noise S_Q( )f for conversion of Nyquist-rate (dotted line) sampling with sampling frequency, f_{s NR,} , and oversampling (solid line) sampling with sampling

frequency, f_{s OS,} , which is much greater than input signal bandwidth, f . The power _b spectrum density of input signal bandwidth for Nyquist rate is much greater than oversampling.

Figure 2.9 Quantization noise power spectrum density for Nyquist-rate (dotted line) and oversampling (solid line) conversion

The area of the both two rectangles meaning the total amount of noise power are the same and equal to Δ² 12. From Figure 2.9, it shows that the quantization noise power has spread to f_{s OS,} 2 and only a small fraction of quantization noise fall into the range of − and f_b f . And the quantization noise outside the signal band will be _b attenuated by a digital low-pass filter as shown in Figure 2.10. Recollecting the quantization noise power spectrum density in equation (2.9) then we can show that the quantization noise is becoming

LP( )

Figure 2.10 (a) Oversampling conversion with digital low-pass filter (b) magnitude of frequency response of digital low-pass filter

2

According equation (2.7), (2.10), (2.12) and (2.14), the SNR of oversampling conversion is

2

2 ,

10log 10log 2 6.02 1.76 10log( )

1

The first term of equation (2.15) denotes the contribution of N-bit quantizer and the

last term is the enhancement of oversampling technique. For every doubling the OSR, the SNR improve by 3dB corresponding to improve the resolution by 0.5 bit. Besides, since the resolution of N-bit quantizer is lower than overall resolution of system, it could reduce the complexity of analog circuit and power of overall system.

2.7 Noise Shaping Strategy

A general noise-shaped sigma-delta modulator and its linear model have been shown in Figure 2.11.

(a)

(b)

Figure 2.11 Block diagram of a noise-shaped SDM and its linear model

We can show that

( )= ( − +1) ( )− ( − (2.16) 1)

After transforming equation (2.16) by Z-transform, we obtain

The equation (2.17) will become

( ) _TF( ) ( ) _TF( ) ( )

Y z =S z ⋅X z +N z Y z⋅ (2.20)

The STF generally have all-pass or low-pass frequency response and the NTF have high-pass frequency response. In other words, the STF will be approximately unity over the signal band and the NTF will be approximately zero over the same frequency band. The quantization noise will be removed to high frequency band when using noise-shaping strategy [05]. The quantization noise over the frequency band of interest will be reduced and do not affect the input signal. This would improve the SNR significantly for overall system.

2.6.1 First-Order Sigma-Delta Modulator

z

⁻1

Figure 2.12 The First-Order SDM

In Figure 2.12, it is a simple block diagram of the first-order SDM [06]. It includes an integrator and 1-bit quantizer. The noise-transfer function,N_TF( )z , should have a zero at DC. And zeros of N_TF( )z are equal to poles of the H z (ie., ( ) has a pole at z=1). Therefore, the quantization noise will be high-pass filtered. In other words, the H z will be small and the ( ) N_TF( )z will large over the frequency band of interest. Thus, the discrete time integrator with a pole at DC can be expressed as

According to equation (2.18) and (2.19) we obtain

1 1

The total transfer function of system is

1 1

( ) ( ) ( ) (1 )

Y z =X z ⋅z⁻ +Q z ⋅ −z⁻ (2.24)

From equation (2.24) we know the STF is just a delay and NTF is a high-pass filter.

In another word, the output signal comprises the delayed input signal and high-pass filtered quantization noise. Now, we may consider the amplitude of the noise transfer function, N_TF( )z . Let z=e^{j Tω} =e^{j2πf fs}, equation (2.23) will becomes

The quantization noise power over the signal band is shown as follows:

2 2

Because OSR 1for oversampling conversion, f would be much larger than_s f . _b

Thus, ^sin

(

π f fs

)

can be approximated to π f f_s . Equation (2.26) will become

Using the equation (2.10) and (2.27), we obtain the SNR of first-order SDM

( )

For every doubling the OSR, the SNR will improve by 9dB (ie., resolution will increase 1.5 bits). This result can be compared with equation (2.15), the SNR only can improve by 3dB when oversampling conversion do not use noise-shaping strategy.

It will be much efficiency when using noise-shaping technique.

2.6.2 Second-Order Sigma-Delta Modulator

z

⁻1

z

⁻¹

Figure 2.13 The Second-Order SDM

In Figure 2.13, it is a block diagram of a second-order SDM. It is popular and widely used in SDM designing. It includes two integrators and a 1-bit quantizer. Its fundamental theorem is the same as the first-order SDM. Thus, the transfer function can be expressed as

Y z( )= X z( )⋅z⁻²+Q z( ) (1⋅ −z^{−1 2}) (2.29)

And we can show the S_TF( )z and N_TF( )z

S_TF( )z =z⁻² (2.30) ( ) (1 1 2)

NTF z = −z⁻ (2.31)

Thus we obtain the magnitude of N_TF

2

The quantization noise power over the signal band is shown as following

2 2

With the same method, we can obtain the SNR of the second-order SDM as

For every doubling the OSR, the SNR will improve by 15dB (ie., resolution will increase 2.5 bits). This result can be compared with equation (2.15) and (2.28), the second-order SDM can provide more suppression over the same band, and thus more noise power outside the signal band. Figure 2.14 shows the phenomenon of using noise-shaping technique or not. For a fixed signal band, the case of no noise-shaping has the largest quantization power over the signal band. The second and the third are

fb

Figure 2.14 Power spectrum density of 1^st order, 2^nd order noise-shaping and non noise-shaping strategy

the first-order SDM and the second-order SDM respectively. As the number of order increasing, the quantization noise power will decrease over the same signal band. The simulation result can show as Figure 2.15.

Figure 2.15 Magnitude of NTF

2.6.3 Higher-Order Sigma-Delta Modulator

z⁻1 z⁻¹

Figure 2.16 The Higher-Order SDM

Higher-order SDMs are divided into single-stage and multi-stage structures [07].

Figure 2.16 is the system block diagram of single-stage Lth-order SDM. Here, we will discuss the change of quantization noise and SNR when the number of order increases. With the same approach, we obtain the noise-transfer function,N_TF,of the Lth-order SDM as follows:

(

¹

)

^L

NTF = −z (2.35)

In a similar manner, the quantization-noise power over the signal band of the single-stage Lth-order SDM with N-bits quantizer is

2 2

Then the SNR of the single-stage Lth-order SDM is

2

Finally, we get the result

( ) ( )

2

6.02 1.76 10log 20 10 log dB

2 1

L

SNR b L OSR

L π

⎛ ⎞

= + + ⎜⎝ + ⎟⎠+ + (2.37)

From equation (2.37), we know that for every doubling the OSR, the SNR will improve by (6L+3) dB (ie., resolution will increase L+0.5 bits). There are three ways to increase the SNR of a SDM. First, we can increase the bits of quantizer. The disadvantage is that multi-bit quantizer would induce harmonic distortion because of mismatch. Second, we can increase the number of order of a SDM. But it may have stability problem when the order is greater than 2. Third, increasing the OSR is the most popular way to improve the performance But for low power design, increasing

Figure 2.17 Plot of SNR versus SDM

the OSR is not suitable because the requirement of the integrators, such as settling time, bandwidth, and slew rate will be increased. Beside, the power of decimation filter will also be increased because of high sampling frequency.

Figure 2.17 is the SNR of SDM. This plot provides a tradeoff between order, OSR and the power dissipation.

2.6.4 Multi-Stage Sigma-Delta Modulator (MASH) (Cascaded)

According the discussion above, we know that increasing the order of the SDM results in stability problems. There are many solutions to resolve the stability problem but may degrade the SNR of the system. This would significantly limit the benefits of increasing the order of the SDM. This problem can be overcome by employing cascaded-type structure. The cascaded SDM consists of several stages of first-order or second-order SDM and each stage converts the quantization error of the previous stage. Since the lower order SDMs are more stable, the system should remain stable [08] [09].

It is attenuated by the noise-shaping function of order equal to overall of cascaded [10]-[12]. Figure 2.18 Block diagram of Multi-Stage SDM

Figure 2.18 is the block diagram of a third-order cascaded 2-1 SDM [01] [13].

The first stage is a second-order SDM and the second stage is a first-order SDM. The input of the second stage is the quantization noise from the first stage. The digital output of the second stage may contains both shaped and unshaped quantization noise of the first stage and also first-order shaped quantization noise form the second stage. The outputs of both the first stage and second stage will be combined and processed by the noise-cancellation logic which is shown in the right of Figure 2.18 (dashed line block) to cancel the shaped and unshaped quantization noise of the first stage. The operation will be discussed. For the first stage, the output of second-order SDM is

( )

²

2 1

1( ) ( ) 1 1( )

Y z =z⁻ ⋅X z + −z⁻ ⋅E z (2.38)

And for the second stage, the output of first-order SDM is

( )

1 1

2( ) ( ) 1 2( )

Y z =z⁻ ⋅X z + −z⁻ ⋅E z (2.39)

Here, the E z and ₁( ) E z represent the quantization error of the first stage and ₂( ) second stage. And the output of the cascaded 2-1 structure can be written as

1 1 2 2

And the resulting output of the cascaded 2-1 topology is calculated by combining

So the output of the overall system will only contain a delayed version of the input signal and the quantization noise of the second stage to achieve third-order noise-shaping. Unfortunately, the cascaded topology is not so perfect. If non-idealities appear in the SDM, the quantization error of the first stage will not be perfectly removed by the digital noise-cancellation logic and can become visible at the output of the converter. The degree of the cancellation depends on how well analog implementation of the ( )H z match the inverse of the digital implementation of

So the output of the overall system will only contain a delayed version of the input signal and the quantization noise of the second stage to achieve third-order noise-shaping. Unfortunately, the cascaded topology is not so perfect. If non-idealities appear in the SDM, the quantization error of the first stage will not be perfectly removed by the digital noise-cancellation logic and can become visible at the output of the converter. The degree of the cancellation depends on how well analog implementation of the ( )H z match the inverse of the digital implementation of

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