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Lecture 3 - Data Converters

James Barnes (James.Barnes@colostate.edu)

Spring 2014

Lab2 Info

Lab2 Info

❖ Lissajou Figures

❖ Support Files Update

Introduction to ADCs

Quantization Noise andEffective Number of Bits

Oversampling ADC

Noise Filtering andDecimation

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Lissajou Figures

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Lissajou figures: a way of combining two phasors. Plot x-component of onephasor on x axis and x-component of other phasor on y-axis.

● Ref: http://www.vias.org/basicradio/basic_radio_23_09.html

Support Files Update

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Minor tweaks to three support files to

● Fix compile problem when using input_sample()● Increase heap size to 0x800.

Please copy Support.zip and replace the eight support files used in Lab1.

Introduction to ADCs

Lab2 Info

Introduction to ADCs

❖ References

❖ Classes of ADCs

❖ ADC Families

❖ Example: Flash ADC

Quantization Noise andEffective Number of Bits

Oversampling ADC

Noise Filtering andDecimation

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References

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[1] David Jarman, “A Brief Introduction to Sigma Delta Conversion”, IntersilApplication Note AN9504, May 1995.

http://www.engr.colostate.edu/ECE423/docs/intersil_delta_sigma_tutorial.pdf

[2] Sangil Park, “Principles of Sigma-Delta Modulation for Analog-to-DigitalConverters”, Motorola APR8.

http://www.engr.colostate.edu/ECE423/docs/APR8-sigma-delta.pdf

[3] “Demystifying Delta-Sigma ADCs”, Maxim Application Note 1870, 2003

http://www.engr.colostate.edu/ECE423/docs/SD01AN1870.pdf

Classes of ADCs

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We can divide ADCs into classes as follows:

1. Full Resolution Quantization

(a) Nyquist rate sampling. The first designs. Still used in applicationswhich can afford the cost and power.

(b) Oversampling - can increase effective resolution but at the cost ofpower and expense. For applications where ultimate performance isneeded.

2. Coarse Quantization - uses oversampling to increase the effective number ofbits. Newer technology, enabled by VLSI development.

ADC Families

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● Parallel processing => higher conversion rate● Because hardware is expensive (chip cost, power), there is a speed-accuracy tradeoff.

Example: Flash ADC

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● Conversion consists of (1) sampling, (2) slicing, and (3) encoding.● Questions:

✦ Where is the sampler in this diagram?✦ Why the 3-input OR gates?

Quantization Noise and Effective Numberof Bits

Lab2 Info

Introduction to ADCs

Quantization Noise andEffective Number of Bits❖ Quantization as aWhite Noise Source❖ Quantization NoisePower❖ Signal-to-Noise Ratioand Effective Number ofBits of an ADC

Oversampling ADC

Noise Filtering andDecimation

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Quantization as a White Noise Source

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Consider quantization by a Nyquist-rate ADC with a sampling frequency Fs.Quantization can be modelled as introducing additive white noise distributed overa range −

∆

2to ∆

2, where ∆ is the quantization level spacing and B is the number

of bits.

Quantization Noise Power

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The total quantization noise power (variance) is

< e2 >=1

∆

∫ ∆/2

−∆/2

e2qdeq =e3q3∆

∣

∣

∣

∣

∣

∆/2

eq=−∆/2

=∆2

12.

With xr = xmax − xmin , the total noise power N and noise power density N(f)over the frequency range [−Fs/2, Fs/2] are given by

N =x2r · 2

−2B

12

and

N(f) =x2r · 2

−2B

12Fs.

Signal-to-Noise Ratio and Effective Numberof Bits of an ADC

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The signal-to-noise ratio for a known input, commonly a sine wave, can be usedto determine the effective number of bits of a quantizer. The measured ENOBmay be less than the design specification because of other noise contributors.The SNR is given by

SNR(db) = 10log10(< signal power >

< noise power >)

For a full range sine wave x(t) = xr/2 sin(2πft), the signal power is< x2 >= x2

r/8. Therefore

SNR(db) = 10log10(3

2· 22B) = 6.02B + 1.76 (1)

and

ENOB =SNR(db)− 1.76

6.02(2)

Studies have cast doubt on the accuracy of this formula because of unmodelledeffects, but it does show that the SNR doubles for every bit added to thequantizer, as expected.

Oversampling ADC

Lab2 Info

Introduction to ADCs

Quantization Noise andEffective Number of Bits

Oversampling ADC

❖ Oversampling FullResolution ADC❖ Reduced ResolutionOversamplingConverters❖ Delta-SigmaPredecessor - DeltaModulator❖ Frequency DomainAnalysis of DM

❖ Possible CircuitImplementations ofIntegrator

❖ Delta-Sigma Converter

❖ Matlab Example - x̂ vsxs

❖ Second OrderDelta-Sigma Modulator

❖ Frequency Responsein Z Domain

Noise Filtering andDecimation

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Oversampling Full Resolution ADC

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A design challenge for Full Resolution Nyquist Rate sampling is the anti-aliasingfilter (AAF). Oversampling can simplify the design of the AAF and also improvenoise performance because the total noise power remains the same but isspread over a wider frequency range..

Because the AAF can now have a very small flat region, it can be implementedwith a simple RC low pass filter.The total noise is still ∆

2

12but the inband noise (dark region) is reduced to

∆2

12·OSR , where OSR is the oversampling ratio.A simple comb filter decimator, which performs a moving average, can recoverlog2(N) additional bits which can be appended to the “physical” wordlength of theconverter.

Reduced Resolution OversamplingConverters

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Sampling at a rate higher than the input signal bandwidth:

● the signal change per sample time will small compared to the total range ofthe signal

● ⇒ a lower resolution (# of bits) converter can be used.● But the quantizer should in some sense operate on the change in the signal

since the last sample. The quantizer should utilize past values of the signal to“predict” the current value and the difference between the actual andpredicted value to “correct” the prediction.

Delta-Sigma Predecessor - Delta Modulator

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● AKA ”predictive signal quantizer” or ”delta modulation” (DM). First used in telephony.● Integrating path subtracts off the average value computed from past samples, adjusting

threshold of slicer

● Problems: “Slope Overload”, “Granularity”. Increasing integrating path gain A helps the first

problem but hurts the second.

Frequency Domain Analysis of DM

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Possible Circuit Implementations ofIntegrator

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These conceptual: there are a lot of practical details to be considered.

Delta-Sigma Converter

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Move Integrator in front of Quantizer.

Noise shaping:

● Signal transfer function is a lowpass filter● Noise transfer function is highpass filter

⇒ Noise is pushed out of signal band! Attenuated by following LPF.

Matlab Example - x̂ vs xs

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● OSR=256, A=0.2

Second Order Delta-Sigma Modulator

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Higher-order loops can be created by cascading integrators. The order =number of integrators in series.

Ref[1]

Frequency Response in Z Domain

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There are two ways the first order loop is drawn. They give different results forthe signal transfer function but same result and implication for the noisetransfer function . The analysis is as follows (whiteboard):

Noise Filtering and Decimation

Lab2 Info

Introduction to ADCs

Quantization Noise andEffective Number of Bits

Oversampling ADC

Noise Filtering andDecimation❖ Delta-Sigma Signaland Noise Spectra

❖ SNR (and ENOB) vsOSR for Delta-SigmaModulators❖ Decimation andFiltering

❖ Comb Filter❖ Comb Filter asDecimator❖ Final Operation - FIRFilter

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Delta-Sigma Signal and Noise Spectra

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Digital low-pass filter removes most of quantization noise while preserving signalspectrum

Ref[3]

SNR (and ENOB) vs OSR for Delta-SigmaModulators

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Ref[3]ENOB formula given earlier ⇒ 16 bit resolution requires SNR of 98dB

Decimation and Filtering

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This is one way of doing decimation and filtering (the Motorola way).

The comb filter sums the number of “1s” in consecutive samples, irrespective oftheir locations in the sequence, essentially doing a moving average over thelength of the filter.Ref[2], Section 7 has a very nice description of the whole process.

Comb Filter

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The comb filter looks like this (example for N=4):

Ref[2]

Comb Filter as Decimator

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The filter operation can be considered as anintegration-decimation-differentiation process

Ref[2]

Final Operation - FIR Filter

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The comb filter

● does not have enough suppression of the out-of-band quantization noise● has less than flat response in the baseband

Therefore, the final FIR filter must boost the low-frequency response whileproviding additional suppression of out-of-band quantization noise.Ref[2]