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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TLCE - C5 14/10/2009
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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.
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