Institut fur Integrierte Systeme

Integrated Systems Laboratory

Fachpraktikum 5. / 6. Semester

IS 6:

Sigma-Delta Analog-to-DigitalConverters

Spring Semester 2020

Advisors: Pascale Meier ETZ J61

Jonathan Bosser ETZ J61

1 Introduction

Analog-to-digital (AD) and digital-to-analog (DA) converters provide the link between the ana-log world of transducers and the digital world of signal processing, computing, and other digitaldata collection or data processing systems. AD and DA-converters are among the most criticalcomponents in a data processing system as they determine to a large extent the overall qualityof such a system.Among the usual converters types, converter based on Sigma-Delta (Σ∆) modulation are themost viable for high resolution, “low-frequency” applications. Σ∆ converters achieve high-resolution without using precision resistive or capacitive ladder networks, and require no ad-justments to achieve best performance. They are therefore especially well suited for monolithicintegration in low cost CMOS technologies. These kind of peculiarities makes Σ∆ modula-tion the technique of choice for audio applications. In fact, almost all consumer digital audiosystems like CD, DAT, DVD, MD and computer audio cards employ Σ∆ AD and DA-converters.

This laboratory starts with the presentation of the two basic operations of AD conversion,namely the sampling and the quantization. We will see that oversampling can increase reso-lution without increasing the number of bits in the quantizer. Σ∆ modulation is then finallyintroduced as a necessary improvement of the oversampling technique. Advantages and limi-tations of the Σ∆ modulation are presented. A small set of MATLAB functions will then beused to simulate a 2nd order modulator. Finally, in a small experiment, you will listen to thedefects of a commercial product, the DA converter of our workstation audio-card.The laboratory is organized in such a way that a new concept is always introduced by a ques-tion. Answering the question will help to understand the subsequent mathematical explanationwhich is followed by a small experiment that demonstrates its practical implementation.

2 AD Conversion

The aim of an AD conversion system is to convert a real world physical quantity into a digitalnumber. As one can expect, this operation cannot be performed in a single step and impliesprogressive loss of information.The first step is represented by the sensor. The sensor transforms a real world physical quantityinto an electrical signal. Let’s take sound as an example. Sound is physically generated by acontinuous local change of air pressure. Basically, sound has a large dynamic range (can be veryloud and very quiet) and a large frequency range. The human ear can distinguish tones if theyare stronger than 0 dBSPL (sound pressure level) and it does it up to 120 dBSPL without beingdamaged and this in a frequency range that starts at 20 Hz to around 20 kHz. It is sensible totake the dynamic range and bandwidth of the human ear as the dynamic range and bandwidthof sound.The microphone is a sensor that converts air pressure variations into an electrical signal. Atthis level a loss of information occurs. Normally the noise (thermal noise) generated in the mi-crophone and in the preamplifier is around 15 dBSPL. Also the microphone’s transfer functionis not constant over the entire frequency band, and finally for strong sounds (> 100 dBSPL)distortion is introduced.

2

y[k]

1 0 1 10 1 1 01 1 1 0

Anti-AliasingFilter

SamplingCircuit

Quantizer Decoder

fs

DigitalFilter

x(t) x’(t) u[k] v[k] v’[k]

Figure 1: Conceptual blocks of an AD converter.

1

C

S

S/H

0 t [s]1

S openS closed

x’(t)

t [s]0

1

0 k10

u[k]

fs

Figure 2: Sample and Hold operation.

The electrical signal generated by this system is a continuous-time, continuous-amplitude, fi-nite dynamic range and finite bandwidth signal. This is the kind of analog signal we usuallyencounter as an input x(t) of an AD-converter. Figure 1 shows the conceptual building blocksof a generic converter.

2.1 Sampling

The first operation in a AD-converter is the sampling operation. It consists of taking samplesof the continuous time input signal x(t) at fixed time intervals Ts. The obtained signal u[k] iscalled discrete-time signal and represents x(t) at the time instants t = k · Ts.The sampling operation is done by the sample-and-hold (S/H) circuit depicted in figure 2. The

switch S is closed and opened rapidly with a frequency fs = 1Ts

. During the time the switch isclosed, C is charged to the momentary value of x(t). When the switch is open the value is heldon the capacitor C and buffered so that it can be processed by the next stage.

Before continuing with the theory let’s answer the following questions:

Q1.1 The output values u[k] of a sampling circuit that samples a sinusoidal input signal x(t)at a frequency fs = 4 Hz are shown in figure 3. What are the amplitude, the frequencyand the phase of the sampled signal?

Q1.2 Do the displayed samples represent a non ambiguous sinusoidal input curve?

Q1.3 Which condition must be imposed on the input signal to make the plot non ambiguous?

3

0 k

3

u[k]

1 2 4

f = 4Hzs

1f =......A=......

ϕ =......

Figure 3: Sampled signal.

u[k]

k0 0 f

s2f

s

U(f)

a)

0 fs

2fs

0

t [s]

u(t) U(f)sin(x)

x

b)

Figure 4: a) Discrete-time signal. b) S/H-signal.

Answering these questions reveals that there is no unequivocal relationship between the setof samples and the sinusoidal input signal, unless you specify that you are looking for thesinusoidal signal with the lowest possible frequency. If we carefully analyze the set of all possiblefrequencies represented by the samples, we find that they are a replica of the fundamentalfrequency (lowest frequency) shifted by multiples of fs

1.

This gives rise to the rule that an analog signal with maximum frequency fb (bandwidth) has tobe sampled at a frequency fs ≥ fN = 2fb, where fN is called Nyquist frequency. If this conditiondoes not hold, all the frequency components of the input signal that are higher than fN

2 are

mirrored down to a frequency lower than fN2 generating a distortion that is called aliasing.

1This is true as long as only the discrete-time signal u[k] is considered. If we take into account the completewaveform generated by the S/H circuit, a sin(x)/x-like function with zeros at all multiples of fs has to besuperimposed to the periodic spectrum (see figure 4). But this is not a matter for us, because we are onlyinterested in time-discrete values. Therefore, we can assume that the spectrum is perfectly periodic.

4

E1.1 Read appendix A and prepare the described setup. Set the generator frequency to 3 kHz.Change the sampling frequency to 8 kHz and record the WFG-output. Sweep the fre-quency up to 5 kHz. Click the stop button to finish the recording. Play the recordedsound file. You will hear that after reaching the maximum frequency of 4 kHz (fs/2)the tone starts to decrease. ’Unfortunately’ (for our experiment) its amplitude decreasesrapidly as the frequency increases. Moreover, the first image tone, that is not completelycanceled by the image filter of the D/A converter, is also present.

The effect you have heard is aliasing. As one can easily see, an analog signal is never perfectlyband limited (it can contain higher harmonics, high frequency noise, and other types of out-of-band distortions). As a consequence it must always be low-pass filtered before being sampled.This kind of filter is called anti-aliasing filter (see first block of diagram 1).

Q2.1 The audio bandwidth ranges from 20 Hz to 20 kHz. Assume you sample a signal at44.1 kHz (CD quality) and that the aliased components have to be suppressed by at least70 dB. The filter ripple in the passband has to be less than 0.5 dB. With the aid of thenomograph in figure 5, find the minimum order needed by a Chebycheff-Cauer filter tosatisfy these requirements.

Q2.2 Assume now that only the components aliased frequencies below 20 kHz have to be sup-pressed by 70 dB. How many filter orders can you save?

Q2.3 You increase the sampling frequency to 96 kHz. Which filter order do you need in thiscase, if only the aliased frequencies below 20 kHz have to be attenuated by 70 dB?

As it can be seen from the last exercise, the anti-aliasing filter is not at all a negligible part ofthe system. If one chooses a sampling frequency that is too tight for the required bandwidth,one is obliged to design a high order, high quality filter that can be very expensive and takemuch area on the PCB. On the other hand, choosing a too high sampling frequency will puthigher requirements (higher speed) on the rest of the AD-converter circuit and on the digitallogic that processes the signal.

2.2 Quantization

Each sample taken by the S/H circuit has to be quantized i.e., approximated with a level froma finite set of references that can be encoded into a digital number.The basic element of the quantization circuit is the comparator (see figure 6). The comparator

output is “1” if the input signal IN is larger than the reference level REF, and it is “0” otherwise.The name of the AD converter is sometimes given by the way the comparator is used. Forexample flash converters employ comparators in parallel. Successive approximation convertersemploy only one comparator and change its reference voltage using a binary algorithm.Let’s take the quantizer of a flash converter of figure 7a as an example. Its ideal characteristic

can be expressed mathematically as a rounding of a continuous amplitude signal u to the setof odd integers in the range ±5 (see figure 7b). In this case the level spacing ∆ is 2 and thenumber of steps is 6, which corresponds to a quantizer of nearly n = 3 bits (number of bitsrequired to encode the output). The quantization is not a linear operation but we will find ituseful to represent the quantized signal v by a linear function Gu with an error e: that is,

v = Gu+ e (1)

5

fstop

fpass

Ω=

A(f)

[dB]

ffstopfpass0

Amin

Amax

Example ( )

A = 1dBmax A = 30dBmin

f = 4kHzpass f = 5kHzstop

Needed filter order:

n = 4

Figure 5: Nomograph for Chebycheff-Cauer filters.

The gain G is the slope of the straight line that passes through the center of the quantizationcharacteristic. Assuming the quantizer does not saturate (i.e., when −6 ≤ u ≤ +6), the erroris bounded by ±∆

2 (±1 here). Notice that the above consideration remains applicable to atwo-level (1-bit) quantizer, but in this case the choice of the gain G is arbitrary.Strictly speaking, the error is completely defined by the input. But if the input changes ran-domly between samples by amounts comparable with or greater than the threshold spacing,without causing saturation, then the error is largely uncorrelated from sample to sample andhas equal probability of lying anywhere in the range ±∆

2 . Therefore, if we assume that the error

6

+

−

"1"

"0"

+

− REF

IN

OUT

OUT

IN

RE

F

"1"

"0"

Figure 6: The comparator

has statistical properties that are independent of the signal, we can represent it by a noise: thequantization noise. Since we have assumed that the quantization error e has equal probabilityof lying anywhere in the range ±∆

2 , its rms value is given by

e2rms =

∫ ∆2

−∆2

e2 · p(e) de =1

∆

∫ ∆2

−∆2

e2 de =∆2

12. (2)

(For the ensuing discussion of spectral densities of the quantization noise, we shall employ aone-sided representation of frequencies: that is, we assume that all the power is in the positiverange of frequencies.) When a quantized signal is sampled at a frequency fs, all of its powerfolds into the frequency band 0 ≤ f ≤ fs

2 . Then, if we assume that the quantization noise iswhite, the spectral density of the quantization noise is constant and is given by

E(f) =e2rmsfs2

. (3)

Let’s define the signal-to-noise ratio (SNR) of a device as the ratio

SNR =PsinPnoise

[dB], (4)

where Psin is the power of the input sinus signal and Pnoise is the quantization noise at theoutput integrated over the band of interest.

Q3.1 Calculate the maximum SNR, achievable by an ideal quantizer with a step size ∆ = 1µV,as a function of n, where n = log2N with N being the number of steps. Naturally, themaximum SNR is achieved by the sinus signal with the maximum amplitude that doesnot saturate the quantizer.

Q3.2 Does the maximum SNR ideally depend on the step size?

Q3.3 What limits the minimum step size and therefore the maximum achievable number ofbits in a practical implementation?

If you have done correct calculations you should have found that the SNR and the quantizernumber of bits n are two interchangeable variables that are related by

SNR = 1.76 + n · 6.02 [dB], (5)

7

+

−

"1"

"0"

+

−

"1"

"0"

+

−

"1"

"0"

+

−

"1"

"0"

+−

+−

+−

+−

x

0

−4

De

co

de

r

y

+

−

"1"

"0"

+−

−2

2

4

−6

"1"

"3"

y

x

"5"

"−1"

"−3"

"−5"

−4 −2 2 4 6

y=Gx+e

−6

1

e

x−1

−3

−4 −2 2 4 6

e=y-Gx

Input Range = ±6 b)

a)

Figure 7: a) Comparator of a flash A/D converter. b) Output characteristic and error.

independently of the size of the minimum step ∆ (at least as long as no thermal noise isconsidered).By modeling the quantization error we assumed that, for a fast enough changing input signal,the error is uncorrelated to the input signal and that it shows a white spectrum in the frequencyband 0 . . . fs2 . Now it is time to listen to this noise! The first thing you have to do is to readappendix B and prepare the described setup.

E2.1 First we listen to white noise. Set NAME=0, load the sample (rs) and play it (po). Whatyou have heard is white noise, generated in MATLAB by the function randn().

Load the sound sample number 1 (NAME=1; rs) and quantize it with 10 bits (Qbit=10;

8

qt). Listen to the quantization noise (pn). If the volume is too low you can increase it bysetting a gain of e.g. 20 dB (NVol=20). You will hear that this quantization noise is verysimilar to white noise.

Now decrease the number of bits in the quantization. Set Qbit=5, quantize the originalsignal (qt), and listen to the quantization noise (pn). Is it still white noise?

If you want you can try the same experiment with other types of sound samples (NAME=2,3, 4, 5, 6, 7, 8).

As you heard, the quantization noise is not really white (especially if it is generated by aquantizer with a low number of bits). Nevertheless, this assumption will be very useful forthe following discussion. It is important that you always keep in mind that this is only amathematical assumption.An important (and necessary) technique to whiten the quantization noise is to add a dithersignal (usually white noise) to the quantizer input.

E2.2 First we listen to a coarsely quantized signal. Reset all variables with startup. Loadthe sound sample number 1 (NAME=1; rs), set the number of quantization bits to 4,and quantize the signal (Qbit=4; qt). Listen to the quantized sample (pq) and to thequantization noise (pn).

Now we introduce a dither signal. Set DTH=-25 (4 bit) and repeat the quantizationoperation (qt). If you listen to the quantized sample (pq) and to the quantization noise(pn), you will notice that they sound much better than before, despite the fact that othernoise has been introduced.

If you want you can try the same experiment with other types of sound samples (NAME=2,3, 4, 5, 6, 7, 8).

Another way to highlight the beneficial effect of dithering can be done using a single tone signal.

E2.3 Reset all parameters with startup. First generate a tone with frequency 200 Hz andamplitude −24.1 dB (F=200; A=-24.1; cs). Set Qbit=4 and quantize the signal (qt).Listen to the original signal (po) and to the quantized signal (pq). What has happened?

Now quantize the same signal with the same number of bits as before but with a dithersignal of −24 dB (DTH=-24); the tone is “magically” back. We didn’t increase the signalenergy, we only added noise(!).

You can try to reduce the tone amplitude until you are not more able to hear it any more.Determine at how many dBs under the quantizer step are you still able to hear the tone?

As you have heard in the last experiment, a dithered signal fades smoothly also if its amplitudeis smaller than the step size of the quantizer.

After the quantizer the signal has a digital format. It is discrete-time, discrete-amplitude andit is already coded as a digital number. Sometimes this stream has to be encoded in a differentway to make it compatible with further digital signal processors. This is done by the encoder(see figure 1).At this point of the description, one could think that everything on AD-converters has beensaid (at least from the theoretical point of view). To tell the truth, it is only now that thediscussion becomes really interesting.

9

2.3 Oversampling

We have seen that the sampling process does not introduce distortion into the signal, if a goodanti-aliasing filter is employed, while the quantization process, as it is non linear, introducesdistortion in the form of a quantization error that can also be modeled as white noise with aconstant power density. The frequency range is 0 . . . fs2 and total power is proportional to thequantization step (see equations (2), (3)).

Q4.1 Assume an A/D converter with a sampling frequency fs = 8 kHz and a bandwidth ofapprox. fb = 4 kHz. Assume that nothing changes except for the sampling frequency,which is now four times higher (f ′s = 4 · fs). What happens to the peak SNR with respectto the first system? (Notice that the band of interest is still fb = 4 kHz.)

Let’s define the oversampling ratio

OSR =fs2

fb, (6)

as the maximum frequency that does not cause aliasing (system bandwidth) divided by themaximum effective signal frequency (signal bandwidth).

Q4.2 Extend the equation (5) so that the OSR is taken into account.

The idea behind question Q4.1 and Q4.2 is that the total quantization noise power does notchange when fs is increased. But as the quantization noise density is constant in the frequencyrange 0 . . . fs2 the total noise power in the band of interest decreases (see figure 8). Thereforewe have:

SNR = 10 log12(22(n−1)∆2)

∆2

12 OSR

= 1.76 + n · 6.02 + 10 log OSR [dB]. (7)

Each time the OSR is multiplied by 4, the SNR improves by 6 dB. That means that the systemlooks like a non-oversampled AD-converter with a quantizer with 1 bit more than it really has.This is the first evidence that it is possible to increase the system resolution without increasingthe number of steps in the quantizer.Now there is a problem: if we maintain the oversampled digital stream we need an OSR-timesfaster digital signal processor to operate on the digitized data. On the other hand, we do notneed so much bandwidth because most of it is “filled” with noise only. But if we re-samplethe oversampled signal at a lower frequency, all out-of-band noise will fold back into the signalband, making the oversampling useless. As for the sampling operation in the analog domain, thesolution is to low-pass filter the signal before re-sampling it. The structure of an oversampledAD-converter is depicted in figure 9.There is another great advantage of oversampled compared to non-oversampled (also calledNyquist-rate) AD-converters. Oversampled ADCs require only a very simple, low-order anti-aliasing filter that prevents the noise above the frequency OSR · fs − fb to alias down into thesignal band (see figure 10). In fact, all the noise between fb and OSR · fs2 is suppressed by thedigital decimation filter.Now we want to make use of this mechanism to solve a practical problem.

Q5.1 A signal has to be digitized with CD audio quality (fs = 44.1 kHz, n = 16 bits). We wantto use a 1 bit quantizer. Which sampling frequency do we need?

10

∆

12

2

e =2rms

fsOSR⋅f2

fsig

fb

2

fs ffsig

fb

Figure 8: Oversampling principle.

y[k]

Anti-AliasingFilter

x(t)

x’(t) u[k] v[k]

OSR⋅fb

S/H QuantizerDecimation

Filter

fb

Figure 9: Oversampling AD converter.

If you have calculated exactly you have found a pretty difficult to realize sampling frequency(more than 47 THz). Maybe in the next decades, frequencies of this order of magnitude willbe achieved. Nevertheless with Σ∆ AD-converters we are able (today) to achieve even betterresolutions using the same 1 bit quantizer (and some other simple circuitry), but at a samplingfrequency of only a few MHz.

3 Σ∆ AD Converter

If you look at figure 8 you can notice that in an oversampled converter there is a wide frequencyband which does not carry any information and which we are not interested in. Therefore wecan use a low-pass filter to suppress all frequencies higher than fb, and whereby also a bigportion of the quantization noise is removed.Equation (5) relates the ADC resolution in bits to the ratio signal power to noise power inthe band of interest but no conditions are set for the shape of the noise spectrum. By intro-ducing the quantizer into a regulation loop, a correlation from sample to sample is achieved,which is a necessary condition for changing the spectral shape of the quantization noise (noise

11

f2fsfb

2OSR⋅fs

−70dB

Low Order Anti-aliasing Filter

Oversampled ConverterA)

f2fsfb

−70dB

High Order Anti-aliasing Filter

Nyquist ConverterB)

Figure 10: Oversampled vs. Nyquist AD converter.

u[k] v[k]

OSR⋅fb

S/HLoop Filter

DecimationFilter

fb

Σ H(z) G Σ

e[k

]

Quantizer

Σ∆ Modulator

U(z) V(z)

− E(z

)

DA

Figure 11: Generic Σ∆ modulator with quantizer linearized model (see equation 1).

shaping). To analyze the sampled data regulation loop, we make use of the Z-transform. TheZ-Transform extremely simplifies the calculation of a cascade of discrete-time filters very easy(simple multiplication). Moreover, if we substitute

z = e2π f

fs (8)

we obtain the frequency response of the transformed signal.Consider figure 11, where the quantizer is placed in a regulation loop. In fact, its output

V (z) connects not only to the decimation filter but it is also sent back to the system input. Aclever choice of the loop filter H(z) leads to a system that can modulate low-frequency noise sothat it is shaped out of the signal band. As a result the SNR (and with it the ADC resolution)improves, although the total noise power is still the same or even higher (due to the out-of-bandgain of the noise transfer function). Let’s see how to choose H(z).

12

Q6.1 Consider the dashed part of the diagram 11. Find the signal transfer function

STF(z) =V (z)

U(z)

∣∣∣∣∣E(z)=0

(9)

and the noise transfer function

NTF(z) =V (z)

E(z)

∣∣∣∣∣U(z)=0

(10)

as a function of the loop filter H(z) (set G=1).

By answering question Q6.1, you have certainly noticed that noise and signal have differenttransfer functions. Since we want to shape the quantization noise out of the band of interestwhile maintaining the input signal unchanged, the requirements for the frequency responses ofNTF and STF are:

in-band out-of-band

‖STF(f)‖ ∼ 1 don’t care

‖NTF(f)‖ 1 > 1

Q6.2 Find the transfer function requirements of the loop filter H that fulfill the above conditionsfor the NTF and STF.

in-band out-of-band

‖H(f)‖

Which function does it represent?

3.1 First Order Σ∆

As you have probably discovered from question Q6.2, the ‖H(f)‖ has to be very high in thesignal band and very low outside. The integrator is the mathematical operator with the neededfrequency response (see figure 12). Let’s consider now the simplest Σ∆ modulator (see figure13) that is of order one and therefore it contains one integrator in the forward path. STF andNTF of this modulator are

STF(z) =H(z)

1 + H(z)= z−1 (11)

NTF(z) =1

1 + H(z)= 1− z−1, (12)

and their magnitudes are

‖STF(f)‖ = 1 (13)

‖NTF(f)‖ = 2 sinπf

fs. (14)

13

DΣ

x[k]

X(z)

H(z)

y[k]=y[k-1]+x[k]

Y(z)=z-1

1−z-1

X(z)

H(z)

-6

20

40

60

fs

2

fs

20

fs

200

fs

2000

0

[dB

]

[Hz]

H(f)

Figure 12: Time-discrete integrator.

u[k] v[k]Σ Σ

U(z) V(z)

−

Σ D

Integrator Quantizer

w[k]

W(z)

e[k

]

E(z

)

Figure 13: First order Σ∆.

If we calculate the SNR as a function of the number of bits n of the quantizer and of theoversampling ratio OSR, taking into account the effect of the first order noise shaping, we find[1]

SNR = −3.41 + n · 6.02 + 3 · 10 log OSR (15)

which means that each time the OSR is doubled, the resolution increases by 1.5 bits. This isan enormous improvement compared to equation (7).

Q7.1 A signal has to be digitized with CD audio quality (fs = 44.1 kHz, n = 16 bits). We wantto use a 1 bit quantizer, first order Σ∆ AD converter. Which sampling frequency do weneed?

If you have calculated exactly you have found that you need a sampling frequency of 67MHz.A quite big improvement compared to what we found for question Q5.1 (!).

14

At this point a skeptic reader can argue that the idea of noise shaping is theoretically very at-tractive but in practice, what about the increased complexity in the realization of the convertercompared to the simple oversampled converter? In most of the existing Σ∆ AD convertersthe shaping filter is implemented with switched capacitor (SC) integrators. The complexity ofan SC-integrator is comparable to that of a S/H circuit, but in an SC-integrator the sample-and-hold operation is intrinsic (for this reason the S/H block in figure 11 is crossed).Hence nocomplexity is introduced in the overall circuit. The only true difference is given by the feedbackpath and the input summing node. But the advantages brought by the Σ∆ modulation aremuch more significant than the required slight increase in circuit complexity.

Up to now, we always assumed a linear quantizer and a quantization noise that is white anduncorrelated with the input signal. However, from section 2.2 we know that this is only amathematical assumption. In the next sections we will look at some “non-idealities” of Σ∆converters that are caused by the deterministic nature of the quantization noise.

3.1.1 Limit Cycles

Let’s begin with a small example.

Q8.1 Consider the first order Σ∆ modulator of figure 13. Assume you have a real single-bitquantizer, this means that the quantizer output is “-1” if its input is less than 0 and “+1”otherwise. Assume a constant input signal u[k] = 3

7 and calculate the output sequence.(w[−1] = 1

7 .)

k 0 1 2 3 4 5 6 7 8 9 . . .

u[k] 3/7 3/7 3/7 3/7 3/7 3/7 3/7 3/7 3/7 3/7 . . .

w[k]

v[k]

Is there anything strange in the output sequence?

Q8.2 What happens if the repeating sequence is longer than 2·OSR? Give a value of u[.] thatcan generate such a sequence (OSR= 32).

If you calculated correctly, you found that the input signal u[.] = 37 gives rise to a repeating

sequence of 7 elements, namely “−1”, “+1”, “+1”, “−1”, “+1”, “+1”, “+1”. If you lookcarefully at that sequence, you can notice that the mean value of these 7 numbers is exactly 3

7 .This reasoning can help you to find new input signals that generate other repeating sequencesof arbitrary lengths and forms.A problem arises when the frequency of repetition of the generated sequence is in the signalband. In this case a very annoying disturbance at well defined frequencies is generated. Thisrepeating sequence is called limit cycle.A useful method to look for limit cycles in a Σ∆ modulator is to put a high amplitude, slowlychanging sinus at the input of the modulator and then to listen to the output signal. Becausethe input is very slow it cannot be heard. On the other hand, he generated limit cycles are at

15

u[k] v[k]Σ

U(z) V(z)−

Σ

D

Integrator 1

Σ

Quantizer

Σ DΣ

−

Integrator 2

e[k

]

E(z

)

Figure 14: Second order Σ∆ modulator.

much higher frequencies and can be heard very well. We will make use of this method later onin this laboratory.

3.1.2 Dead Zones

Let’s begin with another small example.

Q9.1 Assume you have the same system and the same input signal as in Q8.1, but now at timek = 2 a small change of the input occurs. Calculate the output sequence like? (fill thetable.)

k 0 1 2 3 4 5 6 7 8 9 . . .

u[k] 3/7 3/7 3/7 + 1/8 3/7 3/7 3/7 3/7 3/7 3/7 3/7 . . .

w[k]

v[k]

If you have calculated exactly, you have found that the output sequence remains identical ascompared to that of question Q8.1, although the input sequence was not the same. In this caseyou say that the modulator has a dead zone. In a dead zone, a small and fast change of theinput does not change the output sequence. Dead zones become wider if the integrator hassome leakage.We can evoke this phenomenon by applying a slowly changing sinus superimposed with a small,high frequency tone to the modulator input. At the output you will not hear the low-frequencysignal, but when a dead zone is reached the high-frequency tone disappears completely.

3.2 Second Order Σ∆

In section 3.1 we have seen that a simple first order noise shaping filter enormously improvesthe performance of an oversampled converter. But noise could be better shaped with higherorder filters. Let’s consider the second order Σ∆ modulator of figure 14. STF and NTF of thismodulator are

STF(z) = z−1 (16)

NTF(z) = (1− z−1)2, (17)

16

and their magnitude is

‖STF(f)‖ = 1 (18)

‖NTF(f)‖ = 4 sin2 πf

fs. (19)

If we calculate the SNR as a function of the number of bits n in the quantizer and of theoversampling ratio OSR taking into account the effect of the second order noise shaping, wefind

SNR = −11.13 + n · 6.02 + 5 · 10 log OSR (20)

which means that each time the OSR is doubled, the resolution increases by 2.5 bits. Now asampling frequency of less than 5.7 MHz is enough to digitize at CD quality (16 bits, 44.1 kHz),using an AD converter with internal single-bit quantizer.

3.2.1 The Simulation

In section 3.1.1 and 3.1.2 we have seen that the first order Σ∆ modulator is very likely togenerate limit cycles and to fall into a dead zone. For this reason, you cannot find any imple-mentation of such modulator in industrial products. With 2nd order modulators the situationimproves, nevertheless it is still possible to observe these kind of anomalies.For this reason we have implemented a small program to simulate the modulator of figure 14.The simulation environment is MATLAB, which has the advantage that the results can beeasily displayed in graphical form. The core routine (modulator and decimator) is written in Cso that speed optimization can be achieved.For the simulation we assume that the sine-wave generated by the MATLAB sin -function is our“real world”, continuous-time continuous-amplitude signal. The coefficients of the decimatorFIR filter are generated automatically, starting from maximum ripple and cut off frequencyspecifications, and they are passed to the simulation routine. An FFT of the output sequenceis calculated, and the amplitude spectrum is displayed. A small introduction in the simulationenvironment is presented in the appendix C. You need to read it before continuing with thelaboratory.

Let’s begin with the first experiment, in which we will look for the limit cycles of the modulator.

E3.1 Let all the parameters but the amplitude and the frequency of the primary sine-wave asthey are. Use a relatively big input amplitude (e.g., A1=−10 dB) and a low frequency(e.g., F1=0.2 Hz). If you never generated the filter coefficients execute cf; then generatethe input signal (cs). Now you can simulate the modulator (cad) and listen to its outputsignal (pd) (if the quantization noise is too low you can amplify it by setting NVol=20).You will certainly notice that around the zero-crossing of the input signal, a mixture oftones are generated. To increase the tones duration, reduce the amplitude of the inputsignal (e.g., A1=−40 dB) and repeat the same procedure done before. These tones are ourwell known limit cycles.Now we know that limit cycles are generated around the zero amplitude. Try very smallDC values (F1=0); good amplitudes are in the range A1=−70 . . .− 50 dBFS. You will beable to generate any kind of limit cycle. Imagine what happens if, by chance, the inputDC offset of a commercial Σ∆ modulator is exactly one of these values!

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E3.2 Try to play with the program parameters, e.g., you can change the decimation filtercharacteristic (BW, Rpass, Rstop), you can change the oversampling ratio (OSR), you canvary the sampling frequency (Fs), you can add another signal (A2, F2), etc.

In the second experiment we look for dead zones. To do this, we need to introduce some leakagein the integrators. The routine cadl (instead of cad) provides this feature.

E3.3 Reset the parameters of the simulation (startup), and type hh to display the defaultparameters. Change A1=−55 dB, F1=0.2 Hz, A2=−80 dB, F2=300 Hz, Leakage=0.98 andNVol=20 dB. Execute cf, then generate the input signal (cs), simulate the modulatorwith leakage (cadl) and listen to its output signal (pd). You will notice that around thezero-crossing of the input signal every sound disappears. This is a dead zone.

3.3 Higher Order Σ∆

We have seen in section 3.1 that with a first order Σ∆ one can gain 1.5 bits of resolution for eachdoubling of the oversampling ratio. Then, in section 3.2 we decided to increase the order by 1,and this time the improvements was 2.5 bits for each doubling of the oversampling ratio. Nowyou could think that a third order modulator can achieve 3.5 or more bits per oversamplingratio doubling, but this time things are more complicated. In fact, a third order system withnegative feedback is only conditionally stable. This means that if we choose the loop filter forthe best noise shaping curve the obtained modulator will be certainly unstable. As a result, theimprovement in signal-to-noise ratio brought by higher order modulators are less important, butnot at all negligible. Moreover, higher order modulators are less likely to generate limit cyclesand it is nearly impossible that they get stuck in a dead zone. For these reasons, nowadaysmost of the existing Σ∆ modulators are of order three or higher.The topologies chosen for the realization of the filter can be of several types. There are thesingle-loop modulators which are subdivided in feed forward (see figure 15(A)) and feedback(see figure 15(B)) structures, and the cascaded modulators (also called MASH modulators) thattry to overcome stability problems cascading two or more stable modulators with order less thanthree (see figure 15(C)).

4 Conclusions

We conclude the laboratory with a small history of Σ∆ modulation. This is an excerpt of [1].The basic concept underlying ∆Σ converters is the use of feedback for improving the effectiveresolution of a coarse quantizer. An early description of this concept was given in a patent byCutler [2], which was filed in 1954 and granted in 1960. The Σ∆ idea was first proposed in1962 by Inose, Yasuda and Murakami [3]. It was a first order modulator with a 1 bit quantizer.Since this system contained a delta modulator and an integrator, they named it a delta-sigmamodulator, where the “sigma” denoted the summation performed by the integration. In the 34years since its first description, the basic delta-sigma converter has been modified many timesand in many ways. The first important change was suggested in 1977 by Richie [4]. He proposedusing several integrators in cascade in the forward path to create a higher order loop filter witheach integrator receiving an additional input from the DAC (our feedback structure of figure15). The latter was needed to prevent instability. In an influential paper published in 1985 [5],Candy gave extensive design information on the double-integrator loop. Even so, for more than

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u[k] v[k]

U(z) V(z)Σ D Σ DΣ Da1 a2 a3 b1

b1

b1

QΣΣ

u[k]

v[k]

U(z)

V(z)Σ D Σ DΣ Da1 a2 a3 QΣ ΣΣ

b1 b2 b3

b1 b2 b3

Q

Q

Σ

H (z)1

ord<3

H (z)2

ord<3

DigitalErrorCancel-lationLogic

u[k]

U(z)v[k]

V(z)

(A)

(B)

(C)

Figure 15: (A) Single loop feed-forward. (B) Single loop feedback. (C) MASH.

two integrators in the loop, the stability was conditional and had to be verified by numericalsimulation. In 1987, Lee and Sodini gave design techniques for stable high-order loops [6], [7].Based on these, delta-sigma ADCs with fourth- and fifth-order loop filters, containing severalcascaded switched capacitor integrators and resonators, have been successfully designed andproduced by several integrated circuit (IC) companies.

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References

[1] S. R. Norsworthy, R. Schreier, and G. Temes, “Delta-Sigma Data Converters (Theory, De-sign, and Simulation),” IEEE Press.

[2] C. C. Cutler Transmission System Employing Quantization, U. S. Patent No. 2,927,962,March 8, 1960 (filed 1954).

[3] H. Inose, Y. Yasuda, and J. Murakami, A Telemetering System by Code Modulation – ∆-Σmodulation, IRE Trans. Space Electron. Telemetry, vol. SET-8, pp. 204-209, Sept. 1962.

[4] G. R. Richie, “Higher Order Interpolation Analog to Digital Converters,” Ph.D. Disserta-tion, University of Pennsylvania, 1977.

[5] J. C. Candy, A Use of Double Integration in sigma-delta Modulation, IEEE Trans. Commun.,Vol. 33, no. 3, pp. 249-258, March 1985.

[6] W. L. Lee, “A Novel Higher Order Interpolative Modulator Topology for High ResolutionOversampling A/D Converters,” Master’s Thesis, Massachusetts Institute of Technology,Cambridge, MA, June 1987.

[7] K. C. H. Chao, S. Nadeem, W. L. Lee, and C. G. Sodini, A Higher Order Topology forInterpolative Modulators for Oversampling A/D Conversion, IEEE Trans. Circuits Syst.,vol. 37, pp. 309-318, March 1990.

[8] Anatol I. Zverev, “Handbook of Filters”, John Wiley and Sons, Inc., New York, London,Sydney.

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Appendix

A Setup for Experiment E1

Needed material:

• Windows workstation,

• Wavetek 20MHz Pulse/Function Generator (WFG),

• 3.5 mm Jack to BNC male, 2 m cable,

• Headphone.

• Audio Adaptor

Hardware setup:

• Connect the Audio Adaptor (similar to figure 17) to NUC audio jack (see figure 16) andplug headphones and cable to the adaptor. Do not connect the cable to the WFG yet!

• Set all buttons and wheels of the WFG as shown in figure 18. This will generate a sinesignal with frequency 200 Hz and amplitude attenuation of 20 dB.

• Connect the cable with the WFG FUNCTION OUT. Be careful, too high amplitudescan damage your workstation so remember not to change the attenuation setting!

• To sweep frequencies, turn wheel A in the wanted direction.

Figure 16: NUC audio connector

Software setup:

• If you are not yet logged into the workstation, read appendix D.

• Open the Sound Setting by a right click on the speaker in the right lower corner ofyour screen (figure 19) to have access to the System Sound settings of Windows10.Be sure that a window like figure 20 has opened. Ensure that the Headphones and theMicrophone are active. Choose an appropriate volume level so as not to harm yourhearing.

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Figure 17: Audio Adapter

.02

20

ATTENUATIONFUNCTIONDC OFFSETTRIGGER

CAL

CONT

10K

FREQ. MULT.

POWER

WAVETEK

OFF

A

FUNCTION

OUT

Figure 18: Wavetek Function Generator.

Figure 19: Volume Control.

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Figure 20: Sound Setting

23

• Open Audacity by opening the Audacity Folder and double clicking on audacity.exe.A window like figure 21 should open. Open the Audacity Preferences, Edit - Preferencesor Bearbeiten - Einstellungen, to control the basic setting as shown in figures 22 and23. You can change the sample rate for every new project by choosing the desired rate inthe bottom left corner of the Audacity window.

Figure 21: Audio Sound Editor.

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Figure 22: Audacity Audio Preferences.

Figure 23: Audacity Quality Preferences.

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B The Audiolab

• If you are not yet logged into the workstation read appendix D.

• Before starting the audiolab you have to set the audio output of your workstation asdescribed in appendix A (except the Audacity part).

• To start audiolab, you have to go to the audiolab directory (cd ∼/sigmadelta/audiolab/),then you can start MATLAB directly with startup.m. Now your screen should look likein figure 24. You will see a MATLAB shell and a graphical window containing threeempty plots. In the MATLAB shell you can see all existing simulation commands andtheir meaning as well as an editor window which you please close by pressing X in theright upper corner of the editor window. The simulation commands do not need explicitarguments as they use global variables as input. The global variables can be changed inthe MATLAB shell like any other workspace variables (e.g., pippo = 10;).The following table presents the list of commands, including a small explanation for each.

Figure 24: AudioLab screen snapshot.

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Command Meaning Explanation

rs read sig-nal

Reads a previously saved audio signal. The input argumentNAME is the label of the sample that has to be loaded.rs also updates the top plot of the graphical window with thetime behavior of the loaded signal.

cs Calculatesignal

calculates a sinusoidal signal with the specified amplitude A

[dBFS], frequency F [Hz] and duration D [s]. The samplingfrequency Fs [Hz] is also needed.cs also updates the top plot of the graphical window with thetime behavior of the calculated signal.

qt quantize Quantizes the signal generated by cs or rs. Arguments of thiscommand are: the resolution of the quantizer Qbit [bit] andthe power of the dithering noise that it is added to the originalsignal before it is sampled [DTH dBFS].qt also updates the center plot with the quantized signal andthe bottom plot with the quantization noise, i.e. the differenceof the original and the quantized signal.

q2o quantizedto origi-nal

Copies the current quantized signal to the original signal.q2o also updates the top plot with the new original signal.

po playsoriginalsample

Plays the signal generated by cs or rs. The input argument isthe sampling frequency Fs [Hz] (to be played, a signal has to besampled at standard rates, namely 8 kHz, 16 kHz, 22.05 kHz,32 kHz, 44.1 kHz or 48 kHz).

pq playquan-tizedsample

Plays the signal generated by qt. Input argument is the sam-pling frequency Fs [Hz] (to be played, a signal has to besampled at standard rates, namely 8 kHz, 16 kHz, 22.05 kHz,32 kHz, 44.1 kHz or 48 kHz).

pn playquanti-zationnoise

plays the quantization noise generated by command qt. Inputarguments are: Fs the sampling frequency [Hz] (to be played,a signal has to be sampled at standard rates, namely 8 kHz,16 kHz, 22.05 kHz, 32 kHz, 44.1 kHz or 48 kHz) and the noiseamplification NLev [dB].

hh help Displays the command list and a list with the current valuesof the global simulation parameters.

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C The Simulator

• If you are not yet logged into the workstation read appendix D.

• Before starting the audiolab you have to set the audio output of your workstation asdescribed in appendix A (except the Audacity part).

• To start the simulator, you have to go to the simulator directory (cd ∼/sigmadelta/simulator/), then you can start MATLAB directly with startup.m. Now your screenshould look like in figure 25, where you have a MATLAB shell and a graphical windowcontaining three empty plots as well as an editor window which you please close by pressingX in the right upper corner of the editor window. In the MATLAB shell you can see allexisting simulation commands and their meaning. The simulation commands do not needexplicit arguments as they use global variables as input. The global variables can bechanged in the MATLAB shell like any other workspace variables (e.g., pippo = 10;).The following table presents the list of commands, including a small explanation for each.

Figure 25: Screen snapshot of the simulator.

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Command Meaning Explanation

cf calculatefilter

Calculates the coefficients of the decimation FIR filter. Inputarguments are: the oversampling ratio OSR (an integer), theused bandwidth BW (this is a positive real < 1; it gives thepercent of the signal band that is really used for the signal,the rest is used for the transition band of the decimator filter),the maximum pass-band ripple Rpass [dB],and the stop-bandripple Rstop [dB].cf also updates the upper plot of the graphical window withthe frequency response of the filter.

cs calculatesignal

Prepares the oversampled signal that can be passed to themodulator. Input arguments are: the oversampling ratio OSR

(an integer), the sampling frequency Fs [Hz], the signal dura-tion D [s], the amplitude of the first input sin wave A1 [dBFS],the frequency of the first input sin wave F1 [dBFS], the ampli-tude of the second input sin wave A2 [dBFS], and the frequencyof the second input sin wave F2 [dBFS].cs also updates the center plot of the graphical window withthe input signal.

cad calculateADsignal

Simulates the modulator using the filter coefficients generatedby cf and the input signal generated by cs. No other argu-ment is needed.cad also updates the bottom plot of the graphical window withthe spectrum of the modulator output signal.

cadl calculateAD withleakage

Simulates the modulator with leakage using the filter coeffi-cients generated by cf and the input signal generated by cs.Leakage is an additional argument that sets the leakage of theintegrators. Leakage is the part of input signal that is reallyadded to the integrator state (always < 1).cadl also updates the bottom plot of the graphical windowwith the spectrum of the modulator output signal.

po play orig-inal sam-ple

Plays the signal generated by cs. Input argument is the sam-pling frequency Fs [Hz] (to be played, a signal has to besampled at standard rates, namely 8 kHz, 16 kHz, 22.05 kHz,32 kHz, 44.1 kHz or 48 kHz).

pd playdigitizedsample

Plays the digitized signal generated by cad or cadl. Inputarguments are the sampling frequency Fs [Hz] (to be played,a signal has to be sampled at standard rates, namely 8 kHz,16 kHz, 22.05 kHz, 32 kHz, 44.1 kHz or 48 kHz) and the gainfactor for the quantization noise NVol [dB].

hh help Displays the command list and a list with the current valuesof the global simulation parameters.

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D Account Instructions

Login & Password:

Please login into D Domain with your nethz account and password.

Account setup:

Open following folder in a browser,

https://iis-people.ee.ethz.ch/˜kleiert/IS6

download the file IS6.zip to the Apps Folder and extract the files here.

Please check your file permissions before you continue and contact an assistent if a permis-sion is missing.

30