Politecnico di Torino ICT School · – the log curve must be replicated in the III quadra nt » symmetric curve from I quadrant • Log 0 is undefined • near 0 the log curve can

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Politecnico di TorinoICT School

Telecommunication Electronics

C5 - Special A/D converters

» Logarithmic conversion» Approximation, A and µ laws» Differential converters» Oversampling, noise shaping

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Lesson C5: special A/D converters

• Logarithmic conversion– Piecewise approximation

– A and µ laws

• Differential converters– Sigma-delta converters– Oversampling

– Noise shaping

• Waveform encoding and model encoding– Voice LPC

• References sect. 4.5

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Radio systems: where are ADC/DAC ?

• Services– V battery, TX power, …..

• Baseband chain– A/D e D/A for voice signals

• Receiver chain:– A/D conversion of I/Q components in the IF channel

• Transmitter chain– D/A conversion of synthesized I/Q components

• Software Defined Radio architectures– Most functions by digital/programmable circuits

A/D or D/A conversion very close to antenna

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A/D and D/A conversion: where ?

A/D and D/A convertersfor voice signal.

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ADC and DAC system goals

• Improve cost/performance figure

– Cost factors» Complexity» Bit rate

– Performance parameters» Bandwidth » Precision

• Signals with known features– Amplitude distribution– Statistic parameters

– Model encoding

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Voice signal conversion

• Voice signal– exponential amplitude distribution

» more dense at lower levels

– wide dynamic range» SNRq low and variable with signal level

• Logarithmic analog to digital conversion – constant SNRq over a wide signal dynamic range– fewer bits for the same SNRq

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Linear and nonlinear A/D conversion

• Linear A/D conversion– all AD intervals have same amplitude

» quantization error does not depend on signal level

– poor results with signals at low levels for most time (voice)» high quantization noise power, low signal power

• Nonlinear A/D conversion– different AD intervals

» quantization error changes with signal level» the nonlinear relation can be chosen to optimize SNRq for a

specific signal type (PDF, amplitude distribution)

– for voice signals (exponential distribution)» logarithmic law

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Linear quantization

• AD intervals with constant width

• Constant quantization noise power

• SNRq varies withsignal level(worse for low-levelsignals)

A

D

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Nonlinear quantization

• AD intervals with variable width

• Quantization noise power related with signal level

• SNRq independentfrom signal level

A

D

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Standard conversion

• The A/D conversion adds εq noise to analog signal

– D = A + εq– AD is constant, therefore

» constant absolute error on D » % error (SNRq) is related with signal level A

A

εq

D

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Logarithmic conversion

• Conversion of signal logarithm:

– D = log A + εq

– sum of logs � log of product

» D = log A + εq = log K A (εq = log K)» multiplying error (1 - K)» constant % error, independent from signal level A

A

εq

Dlog

log A

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Nominal SNRq

• Log quantization causes a constant relative error– constant SNRq

SNRq

Level

Full scale

log

lin

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A and µµµµ approximation

– Audio signals are bipolar

– the log curve must be replicated in the III quadrant » symmetric curve from I quadrant

• Log 0 is undefined

• near 0 the log curve can be only approximated

– µ law» translate the positive and negative branches to get a

continuous curve in (0,0)

– A law» replace the curves near 0 with a straight line (crossing 0,0)

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A and µ approximation - graphs

• Translation (µ law) Replacement (A law)

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SNRq near 0

• A law: signals less than 1/A --> linear quantization– SNRq depends from signal level (6 dB/octave)

• µ law: almost linear quantization at low levels– similar effect: SNRq drop

SNRq

Level

Full scale (S)1/A

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SNRq with A and µ law

µ law

A law

linear

Linear behavior Log behavior Overload

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Log A/D approximation

• Obtaining calibrated continuous nonlinear behavior requires complex and expensive analog circuits

• Piecewise approximation

• The log curve is divided in linear segments– due to log scale, the same ratio of input signal corresponds

to the same shift in horizontal axis

– slope and starting point of each segment are sequenced as 2 powers (2, 4, 8, 16, ….)

– linear coding inside each segment

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Piecewise approximation

Compressed signal

Continuous log law

slope

slope

slope

slope

Input signal

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Log PCM format

• Each sample is coded on 8 bit

– MSB (bit 7): sign

– bit 6, 5, 4: segment

– bit 3, 2, 1, 0: level within the segment

7 6 5 4 3 2 1 0

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Piecewise approximation: SNRq

• Within each segment

– quantization error εq remains constant

– signal level changes � signal power changes

– SNRq changes with unity slope

• From each segment to the next one (from S to 0)

– quantization error εq is divided by 2

– signal level is divided by 2

– SNRq constant

• Near 0 same behavior as linear quantization

– constant εq

– signal level changes

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SNRq with A and µ µ µ µ approximation

µ 255 law

A (87.6) law

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Log conversion techniques

• Analog log circuit, followed by A/D– poor precision and stability in the analog circuit

– high cost

• High resolution A/D conversion, followed by digital log encoding

– makes available both the linear and log conversion

• Intrinsic log A/D converter– nonlinear law A/D or D/A conversion

– suitable for any type of nonlinear transfer function (DAC for DDS)

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A/D logarithmic converter

• A logarithmic A/D converter can use the D/A feedback technique: comparator-Approximation logic - D/A loop

– the D/A must have exponential transfer function

• How to build an exponential D/A (bipolar):

– sign bit: inverts the D/A reference voltage

– segment bits: provide a voltage with 2N steps» segment bits are decoded into linear code (3-8 decoder)» the 8 bit feed a linear 8-bit D/A » each segment generates outputs with a ratio 2 towards

adjacent ones

– level bits: fed directly to a linear D/A

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Logarithmic ADC

• Sign bit � inverts the reference voltage Vr• Segment bits � voltage Vs scaled with 2N steps (1, 2, 4, 8, …)

• Level bits � fed directly to a linear DAC using Vs as reference

Vs

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Nonlinear DAC

• Structure for nonlinear DAC and ADC with piecewise approximation

– Segment bit decoder

– Standard DAC + lookup table– Decoded DAC uniform elements

» To build starting point and slope of each segment

– Linear coding within each segment (level bits)– Output adder

» Shifts the segment starting point

• Technique used for DACs in DDS (sine generators)– Sine conversion law (piecewise approximation)

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Nonlinear DAC block diagram

• Piecewise nonlinear characteristic– Da: segment bits

– Db: level bits

+Vr

-Vr

Da

LEVELDAC

Db

SEGMENT SLOPE DAC

DECODER/LOOKUP

SEGMENTSTART DAC

+VO

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Lesson C5: special A/D converters

• Logarithmic conversion– Piecewise approximation

– A and µ laws– Logarithmic converters

• Differential converters– Sigma-delta converters

– Oversampling – Noise shaping

• Waveform encoding and model encoding– Voice LPC

– Comparison quality/bit rate

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Tracking converter

• The tracking ADC is a differential converter– The serial bit flow from the comparator output represents the

sign of A - A’ (current value – previous value)

Up/downcounter

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Differential converters

• Quantization of difference between previous and current values

– Dynamic reduction

– 1-bit A/D conversion (comparator)– Serial flow of uniform bits

CODER DECODER

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Delta converter

• Integrating differential converter– L is a sequence of positive or negative pulses, with rate

Fck = 1/Tck

– The recovered signal is S(L)

– On each pulse R changes of one step γ (posive or negative).

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Signal in the Delta converter

– L is a sequence of positive or negative pulses, with rate Fck = 1/Tck

– The recovered signal is S(L)

– On each pulse R changes of one step γ (posive or negative).

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Delta ADC dynamic

• Minimum signal (IDLE state)– Peak level γ/2; idle noise

• Maximum tracked signal– Slew rate γ/Tck overload

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Characteristic of Delta ( ∆) ADC

• A differential converter

– Does not require high precision devices

• but ….

– Provides limited dynamic range» From idle noise and overload

– For a specific SNRq, generates a bit flow with high rate

• Operates in oversampling mode

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Oversampling

• Sampling at a rate far higher than the Nyquist limit – Example: 3 kHz audio signal (Nyquist = 6 kS/s)

» 8 kS/s � Nyquist sampling» 1 MS/s � Oversampling

• Oversampling sends alias far away (1 MHz, 2MHz,…)– Relaxed specifications on the anti-alias input filter

• Reduced noise power density– Reduced inband noise power

– Requires good reconstruction filter

• Higher bit rate (more samples/s)– Can be reduced with digital filtering– Complexity: analog � digital domain

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Oversampling vs. Nyquist

X(ω)f

FS10

Main spectrum (baseband) First alias

2FS1

Second alias

X(ω)

f

FS20

First alias

Oversampling

Quantization noise (0-Fs1 band)

Nyquist

Quantization noise (0-Fs2 band)

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Oversampling vs. Nyquist filtering

X(ω)f

FS10 2FS1

X(ω)

f

FS20

Oversampling

Nyquist Steep filter

Smooth filter

Different filters:same quantization noise power (after reconstruction filter)

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Oversampling vs. Nyquist noise

X(ω)f

FS10 2FS1

Nyquist Steep filter

Same filter:reduced quantization noise power (after reconstruction filter)

X(ω)

f

FS20

Oversampling Steep filter

Removed quantization noise

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Which is the actual limit ?

• Actual Nyquist rule:– A signal must be sampled at least

twice the signal BANDWIDTH

– Example: a 1 GHz carrier, 100 kHz BW signal can be safely sampled at Fs > 200 ks/s

– Spectrum is folded around K Fs/2

• Less stringent specs for RF A/D converters– Sampling rate related with bandwidth, not carrier

• Tight specs for the S/H– sampling jitter related with carrier, not bandwidth

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Filter for Nyquist sampling

A/D

Complex analog LP filter

NYQUIST

X(ω)f

FS0 2FSFS/2

Spectrum segment folded to baseband(aliasing noise)

Steep antialias filter, to limit aliasing noise

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Reducing aliasing noise

• Lower outband signal level– More steep input filter

• Increase sampling rate Fs (oversampling)– Moves alias spectra far from baseband– Need for faster A/D converter– Higher bit rate

can be reduced by digital filtering

• Move complexity from the analog to the digital domain

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Oversampling: more simple filter

X(ω)

ω

2πFS20

Complex, steep digital filter:- reduce noise- reduce bit rate (decimation)

Alias is far away; antialias analogfilter can be simple

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Filters with oversampling

A/D

Simple analog filter

Complex digital filterCan reduce the bit rate (decimation)

NYQUIST �

OVERSAMPLING �

Complex analog LP filter

A/D

Move complexity from the analog to the digital domain

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∆ converter input dynamic range

• Range of input signals correctly handled– γ corresponds to

the quantization stepAD in a standard ADC

– Input dynamic range: Fck/πFs» Does not depend on γ

• To increase input dynamic range– γ constant

» possible only to change Fck

– γ variable (adaptive converters)» Minimum in idle condition (output sequence 0-1-0-1-0-…)» Maximum near overload (output sequences 000… or 111... )

– Remove dependency from signal frequency (ω)» Σ−∆ converters

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Adaptive converters

• Two integrators in the loop– Stability problems

– Integrator + predictor (pole/zero)

• Variable step γ , depending from – Signal level (power estimation)

» Syllabic adaptation

– Error sign sequences» Real-time adaptation

• Adaptation circuits must use the line signal– idle: alternated 0-1-0-1… sequence at output– overload: continuous streams 0000… or 1111...

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Adaptive converters

• DAC uses only line signal

Powerestimation

Powerestimation

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Differential converter architectures

• The differential converter can operate on many bits

– The comparator is replaced by an ADC

– A DAC drives the integrator

Integrator

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Digital differential converters

• Integration can occur in the digital domain

– Integrator becomes accumulator

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Σ∆Σ∆Σ∆Σ∆ converters

• The input dynamic range is limited by signal slew rate– For wider dynamic: limit slew rate

– Integrator on input signal» Decrease amplitude as frequency goes up (integrator) � constant slew rate

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Σ∆Σ∆Σ∆Σ∆ converters

• The input dynamic range is limited by signal slew rate– For wider dynamic: limit slew rate

– Integrator on input signal» Decrease amplitude as frequency goes up (integrator)

– To correctly rebuild the signal: derive the output

Standard differential chain

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Sigma-Delta ADC and DAC

• Move integrators on adder input � single integrator at the output

• Remove the integrator-derivator in DAC

• Keep antialias input and reconstruction output filters (not shown)

ADC DAC

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Quantization noise in ΣΣΣΣ−−−−∆∆∆∆

• In the ΣΣΣΣ−−−−∆∆∆∆ ADC quantization noise εq is generated after integraton

• Y/N transfer function is highpass

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Noise shaping

• Noise is shifted towards high frequencies

• Noise power spectrum density is higher at high frequencies:

– Noise shaping

• Noise power spectrum density in baseband is reduced

• Further reduction to output noise power– Or simpler reconstruction filter

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Oversampling vs. Nyquist noise

Noise power is moved to HF, lower power density in baseband

X(ω)

f

FS20

Noise shaping

Shaped quantization noise

X(ω)

f

FS20

Oversampling Reconstruction filter

Flat quantization noise

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Complete ΣΣΣΣ−−−−∆∆∆∆ conversion chain

• Anti aliasing filter» Oversampling allows simple filters

• ADC Σ−∆ order 1, 2, … N» Produces a high speed, non-weighted bit stream

• Decimator» Changes the high speed bit rate in low rate words

– ---- Channell -----

• Interpolator» Recreates the high speed serial flow

• Σ−∆ DAC» Rebuilds analog signal

• Reconstruction filter

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Bit rate reduction

ΣΣΣΣ−−−−∆∆∆∆ DECIMATOR

INTERPOLATOR ΣΣΣΣ−−−−∆∆∆∆

A

A filtered

D serial, High rate

D parallel, Low rate

A’

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Numeric example

• Audio signal– Fmax 3 kHz

– Sampled 8 ks/s, 8 bit quantization » Which SNRq ?» Which Fck to obtain the same SNRq with a differential

converter?

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Lesson C5: special A/D converters

• Logarithmic conversion– Piecewise approximation

– A and µ laws– Logarithmic converters

• Differential converters– Sigma-delta converters

– Oversampling – Noise shaping

• Waveform encoding and model encoding– Voice LPC

– Comparison quality/bit rate

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Model encoding vs waveform encoding

• Waveform encoding:– Sequence of number which represent the sequence of

values generated by sampling the time varying signal.

– Example: sine tone» Values of the sine signal at sampling times.

• Model encoding:– Define a “source model”

– Model parameters are derived from the signal– The signal is rebuilt from parameters using the model

– Example: sine tone» Model: sine generator» Parameters: amplitude, frequency, and phase» Rebuilt using a properly set signal generator.

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Waveform encoding

• Sequence of samples– example: sine tone

» Values of A sinωt for t = K Ts

• Values: 8, -1, -10, -7, +2, +10, +5, -6, -10, -4, ….

Ts = 0,2 ms

t(ms)

10 V

1

2

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Model and parameters

Model: v(t) = V sen (ω t + θ)

Parameters: V = 10 Vω = 2πf = 2π/T = 5.2 krad/sθ = 0,3π = 0,9 rad

6 decimal digits

Period TPhase θ

t[ms]1

2Peak value V

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SNR for model encoding

• Which factors influence SNR?

• Waveform encoding

– Sampling rate

– Resolution of samples (bit number)

• Model encoding

– Model accuracy– Correctness and resolution of parameters

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Example of model encoding

• LPC (Linear Predictive Coding) for voice signals– Based on a vocal segment model (larinx)

– Signal is divided in frames (10-30 ms)

– For each frame:» voiced/unvoiced decision» evaluation periodicity step (pitch)» Evaluation of adapted filter coefficients

– Voiced: complex waveforms repeated » Pulse generator at pitch rate» Filter to generate the waveform

– Unvoiced: filtered noise» Noise generator + filter

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Block diagram of LPC decoder

PULSE GENERATOR

NOISEGENERATOR

FILTER

PITCH

FILTER PARAMETERS

VOICED

UNVOICED

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Model encoding: performance

• Standard criteria:– speaker recognition (A)

– speech understanding (B)

• Waveform encoding speed (kbit/s)

– log PCM 64/32– Differential 32/16

– Adaptive Differential (ADPCM) 4 (only B)

• Model encoding– LPT (GSM phones) 9,6– Frequency slots vocoder 4,8

– LPC 2,4

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Lesson C5 – final test

• Which are the benefits of logarithmic conversion?

• What happens for signals close to 0 in a log ADC?

• Which are the differences between A and µ log approximations?

• Which parameter controls the dynamic range of a differential ADC?

• Explain structure of delta-sigma ADC.–

• Which are benefits and drawbacks of oversampling?

• Explain noise shaping.

• Which parameters influence S/N for model encoding?

• Describe features of waveform and model encoding techniques.