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Sigma-Delta Converters. Kfir Gedalyahu. Outline . Quantization and performance modeling Oversampled PCM conversion Sigma Delta Modulators: First order High order Parallel Perfect Reconstruction Feedback Quantizers (PRFQ). References.

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  • Quantization and performance modeling
  • Oversampled PCM conversion
  • Sigma Delta Modulators:
    • First order
    • High order
    • Parallel
  • Perfect Reconstruction Feedback Quantizers (PRFQ)
  • Aziz, P.; Sorensen, H. & vn der Spiegel, J. An overview of sigma-delta converters Signal Processing Magazine, IEEE,1996, 13, 61-84.
  • Derpich, M.; Silva, E.; Quevedo, D. & Goodwin, G. On Optimal Perfect Reconstruction Feedback Quantizers Signal Processing, IEEE Transactions on,2008, 56, 3871-3890
  • Eshraghi, A. & Fiez, T. A comparative analysis of parallel delta-sigma ADC architectures Circuits and Systems I: Regular Papers, IEEE Transactions on,2004, 51, 450-458
  • Galton, I. & Jensen, H. Delta-Sigma modulator based A/D conversion without oversampling Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on,1995, 42, 773-784
a d architectures trading resolution for bandwidth
A/D architectures – trading resolution for bandwidth
  • Trade off between signal bandwidth, output resolution and complexity of the analog and digital hardware.
  • Sigma-delta attains highest resolution for relatively low signal bandwidths.
  • Non-invertible process.
  • The quantized output amplitudes are represented by a digital code word – pulse code modulation (PCM).
  • Quantizer with output levels is said to have bits of resolution
performance modeling
Performance Modeling
  • Quantized output level bounded between and
  • LSB is equivalent to
  • ADC not overloaded when
  • Quantization error doesn\'t not exceed half LSB (when not overloaded)
performance modeling7
Performance Modeling
  • Quantizer is non linear system.
  • Linearized and modeled by noise source
assumptions about the noise processes
Assumptions about the noise processes
  • is a stationary random process
  • is uncorrelated with
  • The error is uniformly distributedon
  • The noise process is white
  • These assumptions are reasonable:
    • Quantizer not overloaded
    • is large
    • Successive signal values are not excessively correlated
  • Every increment in there is improvement in SNR.
  • Comparison between ADC\'s: each in SNR is referred as one bit higher resolution.
dynamic range
Dynamic Range
  • A measure of the range of amplitudes for which the ADC produces a positive SNR
  • For Nyquist rate ADC the dynamic range is the same as its peak SNR.
  • Sigma-delta converters do not necessarily have their peak SNR equal to their dynamic range.
oversampled pcm conversion
Oversampled PCM conversion
  • Technique that improves resolution by oversampling.
  • Total amount of noise is like in Nyquist rate conversion, but its frequency distribution is different.
oversampled pcm conversion12
Oversampled PCM conversion
  • Noise power outside the signal band can be attenuated by a digital low-pass filter.
performance modeling for oversampled pcm converters
Performance modeling for oversampled PCM converters
  • Oversampling ratio
  • SNR:
  • Every doubling of the oversampling ratio, improvement in SNR or half bit in resolution.
  • Trading speed for resolution.
  • Trading analog circuit complexity for digital circuit complexity.
  • Audio signal: audio range, needed resolution (CD quality audio) or
  • Using ADC, the needed sampling rate is
  • Using ADC, the needed sampling rate is
sigma delta modulation a d conversion
Sigma-Delta Modulation A/D Conversion
  • Output of ADC in the Z-domain:
  • Signal transfer function (STF) –
  • Noise transfer function (NTF) –
  • For oversampled PCM converter
  • can be designed to be different from allowing high resolution output.
  • Noise shaping – attenuating noise in the signal band, amplifies it outside the signal band.
first order sigma delta modulation17
First order Sigma-Delta Modulation
  • The quantized signal is a filtered version of the difference between the input and an analog representation of the quantized output.
  • The filter – feedforward loop, a discrete time integrator.
  • The integrator and the rest of the analog circuit are implemented in a sampled data switched capacitor technology.
  • The sampling operation is not shown explicitly.
first order sigma delta modulation18
First order Sigma-Delta Modulation
  • Modulator output (assuming ideal DAC):
  • STF – simple delay
  • NTF – containing zero at DC frequency
ntf of first order sigma delta modulator
NTF of first order sigma-delta modulator
  • Example:
  • Better noise reduction in the signal band.
  • Noise "pushed" out the signal band.

Signal Band

dac non linearities
DAC non-linearities
  • DAC non-linearities can be modeled as an error source added to the input:
    • Benefits from oversampling
    • Not subject to noise shaping
  • 1 bit DAC is perfectly linear.
  • It\'s common to use 1 bit DAC and a corresponding 1 bit quantizer – a comparator
performance of first order sigma delta modulator
Performance of first order sigma-delta modulator
  • Over sampling ratio:
  • SNR:
  • For every doubling of the oversampling ratio, the SNR improves by , the resolution improves by 1.5bit.
example revisited
Example revisited
  • Audio signal: audio range, needed resolution (CD quality audio) or
  • First order sigma-delta modulator with 1 bit comparator require
  • 1 bit comparator can operate at this speed in 1996 CMOS technology.
  • The sampled data analog switched capacitor can\'t operate at this speed (1996….).
qualitative time domain behavior24
Qualitative time domain Behavior
  • Over period of time the proportion (or density) of 1\'s and -1\'s will be relate to the DC input value.
  • The output of a sigma-delta modulator using 1 bit quantizer is said to be in pulse density modulated (PDM) format.
  • By averaging the modulator output over a period of time, we can approximate the input.
  • Averaging operation is done using the digital LPF block.
qualitative time domain behavior25
Qualitative time domain Behavior
  • Sinusoidal input.
  • Sampling frequency:
implementation imperfection
Implementation Imperfection
  • Integrator with gain and leakage:
  • DAC with gain
  • STF:
    • Pole is stable for
  • NTF:
    • Zero inside the unit circle – degradation in noise attenuation
non linear behavior
Non linear behavior
  • Sigma-delta modulator is a non-linear system with feedback.
  • May display limit cycle oscillations that results in the presence of periodic components in the output.
  • The quantizer output is not white. Successive quantizer input samples may be correlated:
    • Only two output levels
    • Oversampling
  • Because of the significant tone structure at the output, first order sigma-delta ADC is rarely used in audio and speech applications.
non linear behavior28
Non linear behavior

Wanted signal

Wanted signal

  • Input signal frequency:
high order sigma delta modulation
High Order Sigma-Delta Modulation
  • A straightforward extension to the first sigma-delta:
    • STF:
    • NTF:
high order sigma delta modulation30
High Order Sigma-Delta Modulation
  • Example: Second order sigma-delta modulator
performance of high order sigma delta modulator
Performance of high order sigma-delta modulator
  • Over sampling ratio:
  • SNR:
  • For every doubling of the oversampling ratio, the SNR improves by , the resolution improves by bits.
example revisited32
Example revisited
  • Audio signal: audio range, needed resolution (CD quality audio) or
other topologies
Other topologies
  • Distributing zeros over the signal band
  • Example: forth order topology
parallel sigma delta system
Parallel Sigma-Delta System
  • One of the drawbacks of the sigma-delta modulators we have seen so far is the need for oversampling.
  • Parallelism can be used to improve the performance of sigma-delta modulators.
  • For a given signal bandwidth, modulator order and sampling frequency, higher resolution can be attained.
  • The cost is extra hardware needed for each parallel channel.
multi band sigma delta modulation
Multi-band Sigma Delta Modulation
  • Each channel NTF reject different portion of the signal band.
  • A bank of FIR filters attenuates the out of band noise for each band.
  • SNR improves at rate of per octave increment in the number of channels.
modulation based parallel sigma delta
Modulation based Parallel Sigma-Delta
  • Generalized block diagram:
modulation based parallel sigma delta37
Modulation based Parallel Sigma-Delta

Signal path

Quantization noise path

modulation based parallel sigma delta38
Modulation based Parallel Sigma-Delta
  • The modulation sequences are rows of a unitary matrix:
  • Signal path (without noise):
modulation based parallel sigma delta39
Modulation based Parallel Sigma-Delta
  • The filter must satisfy:
  • The filter should be design to minimize the quantization noise
time interleaved adc
Time-interleaved ADC
  • Using the unitary matrix:
  • High bandwidth Sigma-Delta
perfect reconstruction feedback quantizes prfq
Perfect Reconstruction Feedback Quantizes (PRFQ)
  • Feedback quantizer (FQ):
    • ADC architecture wherein a quantizer is placed within a linear feedback loop.
  • Examples:
    • DPCM converters
    • Sigma delta modulators
    • D-modulators
  • General FQ configuration:
  • The filters and allows exploiting the predictability of the input signal, in order to reduce the variance of .
  • The error-feedback filter is used for shaping the quantization noise spectrum.
  • is a weighting filter, used to define a frequency weighted error criterion.
  • Assumption:
    • The spectral properties of the signal are known –
  • The goal is to minimize the variance of weighted error, by choosing the filters , and .
  • A transparent converter:
  • Only two degrees of freedom:
    • andor
    • and
prfq optimization constraints
PRFQ - Optimization Constraints
  • Transparent converter.
  • The filters and are stable.
  • The filter is strictly causal.
  • Sigma Delta:
    • Continuous-Time Sigma-Delta Converters ?
    • Generalized sampling – reducing the noise energy in the signal space (not necessarily low frequencies).
  • PRFQ:
    • The IIR filters and "live" is a sampled but not quantized domain - is it realizable?