principles of sigma delta conversion for analog to digital converters

Table of Contents IntroductionConventional Analog-to-Digital ConvertersQuantization Error in A/D ConversionOversampling and Decimation BasicsDelta Modulation Sigma-Delta modulation for A

bySangil Park, Ph D.

Strategic ApplicationsDigital Signal Processor Operation

Motorola Digital Signal Processors

Principles of Sigma-Delta Modulation for Analog-to- Digital Converters

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Table

of Contents

IntroductionConventional Analog-to-Digital ConvertersQuantization Error in A/D ConversionOversampling and Decimation BasicsDelta Modulation

Sigma-Delta modulation for A/D Converters (Noise Shaping)

6.1 Analysis of Sigma-Delta Modulation inZ-Transform Domain

Digital Decimation Filtering7.1 Comb-Filter Design as a Decimator7.2 Second Section Decimation FIR FilterMode Resolution by Filtering the Comb-Filter Out put with Half-Band Filters

Summary

SECTION 1 SECTION 2 SECTION 3 SECTION 4 SECTION 5 SECTION 6

6-6

7-17-57-108-1

9-1

References-1

MOTOROLA v

Illustrations

Generalized Analog-to-Digital Conversion ProcessConventional Analog-to-Digital Conversion ProcessSpectra of Analog and Sampled Signals

Quantization ErrorNoise Spectrum of Nyquist Samplers

Comparison Between Nyquist Samplers and 2X Oversamplers

Anti-Aliasing Filter Response and Noise Spectrum of Oversampling A/D Converters

Frequency Response of Analog Anti-Aliasing FiltersSimple Example of Decimation Process

Delta Modulation and Demodulation

Derivation of Sigma-Delta Modulation from Delta Modulation

Block Diagram of Sigma-Delta Modulation S-Domain Analysis of Sigma-Delta Modulator Block Diagram of First-Order Sigma-Delta A/D Converter

Input and Output of a First-Order Sigma-Delta Modulator

Z-Domain Analysis of First-Order Noise ShaperSpectrum of a First-Order Sigma-Delta

Noise Shaper

2-22-32-5

3-33-3

4-34-54-64-7

5-2

6-16-26-36-56-66-76-9

Figure 2-1 Figure 2-2 Figure 2-3

Figure 3-1 Figure 3-2

Figure 4-1 Figure 4-2 Figure 4-3 Figure 4-4

Figure 5-1

Figure 6-1 Figure 6-2 Figure 6-3 Figure 6-4 Figure 6-5 Figure 6-6 Figure 6-7

Transfer Function of a Comb-FilterCascaded Structure of a Comb-FilterAliased Noise in Comb-Filter Output(a) Comb-Filter Magnitude Response inBaseband

(b) Compensation FIR Filter Magnitude Response

FIR Filter Magnitude ResponseAliased Noise Bands of FIR Filter Output

Spectrum of a Third-Order Noise Shaper (16384 FFT bins)

Spectrum of Typical Comb-Filter Output (4096 FFT bins)

Decimation Process using a Series of Half Band Filters

Figure 6-8 Figure 6-9 Figure 6-10

Figure 7-1 Figure 7-2 Figure 7-3 Figure 7-4 Figure 7-5 Figure 7-6

Figure 7-7 Figure 7-8

Figure 8-1 Figure 8-2 Figure 8-3

6-106-136-13

7-47-67-77-87-97-117-117-127-13

8-28-38-5

MOTOROLA vii

Tables

Table 8-1 Parameters for Designing Half-Band Filters 8-4

“Since the Σ−∆

A/D converters are based on digital filtering techniques, almost 90% of the die is implemented in digital circuitry which enhances the prospect of compatibility.”

SECTION 1

The performance of digital signal processing andcommunication systems is generally limited by theprecision of the digital input signal which is achieved

at the interface between analog and digital tion Sigma-Delta (Σ−∆) modulation based analog-to-digital (A/D) conversion technology is a cost effectivealternative for high resolution (greater than 12 bits)converters which can be ultimately integrated on dig-ital signal processor ICs

informa-Although the sigma-delta modulator was first duced in 1962 [1], it did not gain importance untilrecent developments in digital VLSI technologieswhich provide the practical means to implement thelarge digital signal processing circuitry The increas-ing use of digital techniques in communication andaudio application has also contributed to the recent in-terest in cost effective high precision A/D converters

intro-A requirement of analog-to-digital (intro-A/D) interfaces iscompatibility with VLSI technology, in order to providefor monolithic integration of both the analog and digi-tal sections on a single die Since the Σ−∆ A/Dconverters are based on digital filtering techniques,almost 90% of the die is implemented in digital circuit-

ry which enhances the prospect of compatibility Introduction

1-2 MOTOROLA

Additional advantages of such an approach clude higher reliability, increased functionality, andreduced chip cost Those characteristics are com-monly required in the digital signal processingenvironment of today Consequently, the develop-ment of digital signal processing technology ingeneral has been an important force in the devel-opment of high precision A/D converters which can

in-be integrated on the same die as the digital signalprocessor itself The objective of this applicationreport is to explain the Σ−∆ technology which is im-plemented in the DSP56ADC16, and show thesuperior performance of the converter compared

to the performance of more conventional mentations Particularly, this application notediscusses a third-order, noise-shaping oversam-pling structure

imple-Conventional high-resolution A/D converters, such

as successive approximation and flash type verters, operating at the Nyquist rate (samplingfrequency approximately equal to twice the maxi-mum frequency in the input signal), often do notmake use of exceptionally high speeds achievedwith a scaled VLSI technology These Nyquist sam-plers require a complicated analog lowpass filter(often called an anti-aliasing filter) to limit the maxi-mum frequency input to the A/D, and sample-and-hold circuitry On the other hand, Σ−∆ A/D convert-ers use a low resolution A/D converter (1-bitquantizer), noise shaping, and a very high oversam-pling rate (64 times for the DSP56ADC16) The highresolution can be achieved by the decimation (sam-ple-rate reduction) process Moreover, since

con-precise component matching or laser trimming is

not needed for the high-resolution Σ−∆ A/D

convert-ers, they are very attractive for the implementation

of complex monolithic systems that must

incorpo-rate both digital and analog functions These

features are somewhat opposite from the

require-ments of conventional converter architectures,

which generally require a number of high precision

devices This application note describes the

con-cepts of noise shaping, oversampling, and

“Most A/D converters can

be classified into

two groups according to the

sampling rate

criteria: Nyquist

rate converters

and oversampling

converters ”

Conventional Analog-to-Digital Converters

SECTION 2

Signals, in general, can be divided into two ries; an analog signal, x(t), which can be defined in acontinuous-time domain and a digital signal, x(n),which can be represented as a sequence of numbers

catego-in a discrete-time domacatego-in as shown catego-in Figure 2-1 Thetime index n of a discrete-time signal x(n) is an integernumber defined by sampling interval T Thus, a dis-crete-time signal, x*(t), can be represented by asampled continuous-time signal x(t) as:

Eqn 2-1

where:

A practical A/D converter transforms x(t) into a crete-time digital signal, x*(t), where each sample isexpressed with finite precision Each sample is ap-proximated by a digital code, i.e., x(t) is transformed

fs is the sampling frequency [2] Meanwhile, sampling converters perform the samplingprocess at a much higher rate, fN << Fs (e.g., 64times for the DSP56ADC16), where Fs denotesthe input sampling rate

Figure 2-1 Generalized Analog-to-Digital Conversion Process

Figure 2-2 illustrates the conventional A/D

con-version process that transforms an analog input

signal x(t) into a sequence of digital codes x(n) at

a sampling rate of fs = 1/T, where T denotes the

sampling interval Since in Eqn 2-1 is

a periodic function with period T, it can be

repre-sented by a Fourier series given by [5]:

Eqn 2-2

Combining Eqn 2-1 and Eqn 2-2, we get:

Eqn 2-3

Multi-level Quantizer

Analog

Signal

Digital Signal

001 010 001 000 111 110 101

x (n)

x (t)

Band-limiting

Successive Approximation Flash Conversion

Dual Slope Method e.g.:

Figure 2-2 Conventional Analog-to-Digital Conversion Process

Sample and Hold Circuit

2-4 MOTOROLA

Eqn 2-2 states that the act of sampling (i.e., thesampling function):

is equivalent to modulating the input signal by

carri-er signals having frequencies at 0, fs, 2fs, Inother words, the sampled signal can be expressed

in the frequency domain as the summation of theoriginal signal component and signals frequencymodulated by integer multiples of the sampling fre-quency as shown in Figure 2-3 Thus, input signalsabove the Nyquist frequency, fn, cannot be properlyconverted and they also create new signals in thebase-band, which were not present in the originalsignal This non-linear phenomenon is a signal dis-tortion frequently referred to as aliasing [2] Thedistortion can only be prevented by properly low-pass filtering the input signal up to the Nyquistfrequency This lowpass filter, sometimes called theanti-aliasing filter, must have flat response over thefrequency band of interest (baseband) and attenu-ate the frequencies above the Nyquist frequencyenough to put them under the noise floor Also, thenon-linear phase distortion caused by the anti-alias-ing filter may create harmonic distortion and audibledegradation Since the analog anti-aliasing filter isthe limiting factor in controlling the bandwidth andphase distortion of the input signal, a high perfor-mance anti-aliasing filter is required to obtain highresolution and minimum distortion

x t( )δ(t–nT)

n = – ∞

∞

∑

| X (f) |

fN

f (a) Frequency response of unlimited signal

(c) Frequency response of band-limited analog signal

(d) Frequency response of sampled digital signal

Anti-Aliasing Filter (Band-Limiting)

sample-Each of these reference levels is assigned a digitalcode Based on the results of the comparison, a dig-ital encoder generates the code of the level theinput signal is closest to The resolution of such aconverter is determined by the number and spacing

of the reference levels that are predefined Forhigh-resolution Nyquist samplers, establishing thereference voltages is a serious challenge

For example, a 16-bit A/D converter, which is thestandard for high accuracy A/D converters, requires

216 - 1 = 65535 different reference levels If theconverter has a 2V input dynamic range, the spac-ing of these levels is only 30 mV apart This isbeyond the limit of component matching tolerances

of VLSI technologies [4] New techniques, such aslaser trimming or self-calibration can be employed

to boost the resolution of a Nyquist rate converterbeyond normal component tolerances However,these approaches result in additional fabricationcomplexity, increased circuit area, and higher cost

■

The process of converting an analog signal (whichhas infinite resolution by definition) into a finite rangenumber system (quantization) introduces an error sig-nal that depends on how the signal is beingapproximated This quantization error is on the order

of one least-significant-bit (LSB) in amplitude, and it isquite small compared to full-amplitude signals How-ever, as the input signal gets smaller, the quantizationerror becomes a larger portion of the total signal

When the input signal is sampled to obtain the quence x(n), each value is encoded using finite word-lengths of B-bits including the sign bit Assuming thesequence is scaled such that for fraction-

se-al number representation, the pertinent dynamicrange is 2 Since the encoder employs B-bits, thenumber of levels available for quantizing x(n) is The interval between successive levels, q, is there-fore given by:

Eqn 3-1

which is called the quantization step size The pled input value is then rounded to the nearestlevel, as illustrated in Figure 3-1

sam-x n( ) ≤1

2B

2B 1– -

=

x∗( )t

Quantization Error in A/D Conversion

3-2 MOTOROLA

From Eqn 3-2, it follows that the A/D converter put is the sum of the actual sampled signal and an error (quantization noise) component e(n);that is:

Eqn 3-3

where: E denotes statistical expectation

We shall refer to in Eqn 3-3 as being the state input quantization noise power Figure 3-2shows the spectrum of the quantization noise Sincethe noise power is spread over the entire frequencyrange equally, the level of the noise power spectraldensity can be expressed as:

steady-Eqn 3-4

The concepts discussed in this section apply in

x∗( )t

x n( ) = x∗( )t +e n( )

σe 2

σe2 E e[ ]2 1

q

q –

σe2

N f( ) q2

12fs - 2

2B

–3fs -

Figure 3-1 Quantization Error

Figure 3-2 Noise Spectrum of Nyquist Samplers

SECTION 4

The quantization process in a Nyquist-rate A/D verter is generally different from that in anoversampling converter While a Nyquist-rate A/Dconverter performs the quantization in a single sam-pling interval to the full precision of the converter, anoversampling converter generally uses a sequence ofcoarsely quantized data at the input oversampling rate

con-of followed by a digital-domain decimationprocess to compute a more precise estimate for theanalog input at the lower output sampling rate, fs,which is the same as used by the Nyquist samplers Regardless of the quantization process oversamplinghas immediate benefits for the anti-aliasing filter To il-lustrate this point, consider a typical digital audioapplication using a Nyquist sampler and then using atwo times oversampling approach Note that in the fol-lowing discussion the full precision of a Nyquistsampler is assumed Coarse quantizers will be con-sidered separately

The data samples from Nyquist-rate converters aretaken at a rate of at least twice the highest signal fre-quency of interest For example, a 48 kHz samplingrate allows signals up to 24 kHz to pass without

Oversampling and Decimation Basics

20 kHz in digital audio applications) To prevent nal distortion due to aliasing, all signals above 24kHz for a 48 kHz sampling rate must be attenuated

sig-by at least 96 dB for 16 bits of dynamic resolution These requirements are tough to meet with an an-alog low-pass filter Figure 4-1(a) shows therequired analog anti-aliasing filter response, whileFigure 4-1(b) shows the digital domain frequencyspectrum of the signal being sampled at 48 kHz.Now consider the same audio signal sampled at2fs, 96 kHz The anti-aliasing filter only needs toeliminate signals above 74 kHz, while the filter hasflat response up to 22 kHz This is a much easier fil-ter to build because the transition band can be 52 kHz(22k to 74 kHz) to reach the -96 dB point However,since the final sampling rate is 48 kHz, a samplerate reduction filter, commonly called a decimationfilter, is required but it is implemented in the digitaldomain, as opposed to anti-aliasing filters whichare implemented with analog circuitry Figure 4-1(d)and Figure 4-1(e) illustrate the analog anti-aliasingfilter requirement and the digital-domain frequencyresponse, respectively The spectrum of a requireddigital decimation filter is shown in Figure 4-1(f) De-tails of the decimation process are discussed in

SECTION 7 Digital Decimation Filtering.

(a) Anti-aliasing filter response for Nyquist samplers

f

(d) Anti-aliasing filter response for 2x over-samplers

4-4 MOTOROLA

This two-times oversampling structure can be

extend-ed to N times oversampling converters Figure 4-2(a)shows the frequency response of a general anti-alias-ing filter for N times oversamplers, while the spectra

of overall quantization noise level and basebandnoise level after the digital decimation filter is illustrat-

ed in Figure 4-2(b) Since a full precision quantizerwas assumed, the total noise power for oversamplingconverters and one times Nyquist samplers are thesame However, the percentage of this noise that is inthe bandwidth of interest, the baseband noise power

NB is:

Eqn 4-1

which is much smaller (especially when Fs is muchlarger than fB) than the noise power of Nyquist sam-plers described in Eqn 3-4

Figure 4-3 compares the requirements of the aliasing filter of one times and N times oversampledNyquist rate A/D converters Sampling at theNyquist rate mandates the use of an anti-aliasing fil-ter with very sharp transition in order to provideadequate aliasing protection without compromisingthe signal bandwidth

anti-N

f B

–

fB

F s

Fs/2 Fs

fB

Overall Noise Level:

In-Band Noise Level:

Note: One R-C lowpass filter is sufficient for the anti-aliasing filter

fB

∫

qfB 2 12

-2

F s

4-6 MOTOROLA

The transition band of the anti-aliasing filter of anoversampled A/D converter, on the other hand, ismuch wider than its passband, because anti-aliasingprotection is required only for frequency bands be-tween and , when N = 1, 2, ,

as shown in Figure 4-2(b) Since the complexity ofthe filter is a strong function of the ratio of the width

of the transition band to the width of the passband,oversampled converters require considerably sim-pler anti-aliasing filters than Nyquist rate converterswith similar performance For example, with N = 64,

a simple RC lowpass filter at the converter analog put is often sufficient as illustrated in Figure 4-2(a)

in-(a) Nyquist rate A/D converters

(b) Oversampling A/D converters

The benefit of oversampling is more than an

eco-nomical anti-aliasing filter The decimation process

can be used to provide increased resolution To see

how this is possible conceptually, refer to Figure 4-4,

which shows an example of 16:1 decimation

pro-cess with 1-bit input samples Although the input

data resolution is only 1-bit (0 or 1), the averaging

method (decimation) yields more resolution (4 bits

[24 = 16]) through reducing the sampling rate by

16:1 Of course, the price to be paid is high speed

sampling at the input — speed is exchanged for

7

=================>>

1 multi - bit output

Figure 4-4 Simple Example of Decimation Process

The work on sigma-delta modulation was developed

as an extension to the well established delta tion [6] Let’s consider the delta modulation/demodulation structure for the A/D conversion pro-cess Figure 5-1 shows the block diagram of the deltamodulator and demodulator Delta modulation isbased on quantizing the change in the signal fromsample to sample rather than the absolute value ofthe signal at each sample

modula-Since the output of the integrator in the feedback loop

of Figure 5-1(a) tries to predict the input x(t), the grator works as a predictor The prediction error term,

inte-, in the current prediction is quantizedand used to make the next prediction The quantizedprediction error (delta modulation output) is integrated

in the receiver just as it is in the feed back loop [7].That is, the receiver predicts the input signals asshown in Figure 5-1

The predicted signal is smoothed with a lowpass filter.Delta modulators, furthermore, exhibit slope overloadfor rapidly rising input signals, and their performance

is thus dependent on the frequency of the input signal

In theory, the spectrum of quantization noise of theprediction error is flat and the noise level is set by the1-bit comparator Note that the signal-to-noise ratiocan be enhanced by decimation processes as shown

x t( ) –x t( )

SECTION 5

Delta Modulation

input

y(t)

Analog Signal

Figure 5-1 Delta Modulation and Demodulation

x(t)

x t ( )

Sigma-Delta Modulation and Noise Shaping

Delta modulation requires two integrators for lation and demodulation processes as shown inFigure 6-1(a) Since integration is a linear operation,the second integrator can be moved before the mod-ulator without altering the overall input/outputcharacteristics Furthermore, the two integrators inFigure 6-1 can be combined into a single integrator bythe linear operation property

modu-1-bit quantizer

Channel

Σ

+

Demodulation

(a)

Σ

+ Analog

(b)

Note: Two Integrators (matched components)

Figure 6-1 Derivation of Sigma-Delta Modulation from Delta Modulation

6-2 MOTOROLA

The arrangement shown in Figure 6-2 is called aSigma-Delta (Σ−∆) Modulator [1] This structure,besides being simpler, can be considered as being

a “smoothed version” of a 1-bit delta modulator

The name Sigma-Delta modulator comes from puttingthe integrator (sigma) in front of the delta modulator.Sometimes, the Σ−∆ modulator is referred to as an in-terpolative coder [14] The quantization noisecharacteristic (noise performance) of such a coder isfrequency dependent in contrast to delta modulation

As will be discussed further, this noise-shaping erty is well suited to signal processing applicationssuch as digital audio and communication Like delta

Note: Only one integrator

modulators, the Σ−∆ modulators use a simple coarse

quantizer (comparator) However, unlike delta

modu-lators, these systems encode the integral of the signal

itself and thus their performance is insensitive to the

rate of change of the signal

The noise-shaping principle is illustrated by a

simpli-fied “s-domain” model of a first-order Σ−∆ modulator

shown in Figure 6-3 The summing node to the right

of the integrator represents a comparator It’s here

that sampling occurs and quantization noise is

add-ed into the model The signal-to-noise (S/N)

1 1s

+

+

6-4 MOTOROLA

transfer function shown in Figure 6-3 illustrates themodulator’s main action As the loop integrates theerror between the sampled signal and the input sig-nal, it lowpass-filters the signal and highpass filtersthe noise In other words, the signal is left unchanged

as long as its frequency content doesn’t exceed thefilter’s cutoff frequency, but the Σ−∆ loop pushes thenoise into a higher frequency band Grossly over-sampling the input causes the quantization noise tospread over a wide bandwidth and the noise density

in the bandwidth of interest (baseband) to

significant-ly decrease

Figure 6-4 shows the block diagram of a first-orderoversampled Σ−∆ A/D converter The 1-bit digitaloutput from the modulator is supplied to a digital dec-imation filter which yields a more accuraterepresentation of the input signal at the output sam-pling rate of fs The shaded portion of Figure 6-4 is afirst-order Σ−∆ modulator It consists of an analog dif-ference node, an integrator, a 1-bit quantizer (A/Dconverter), and a 1-bit D/A converter in a feed backstructure The modulator output has only 1-bit (two-levels) of information, i.e., 1 or -1 The modulator out-put y(n) is converted to by a 1-bit D/Aconverter (see Figure 6-4)

The input to the integrator in the modulator is thedifference between the input signal x(t) and thequantized output value y(n) converted back to thepredicted analog signal, x(t) Provided that the D/Aconverter is perfect, and neglecting signal delays,

x t( )