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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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Sigma-Delta Converters

Kfir Gedalyahu


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Outline

  • Quantization and performance modeling

  • Oversampled PCM conversion

  • Sigma Delta Modulators:

    • First order

    • High order

    • Parallel

  • Perfect Reconstruction Feedback Quantizers (PRFQ)


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References

  • 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


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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.


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Quantization

  • 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


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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)


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Performance Modeling

  • Quantizer is non linear system.

  • Linearized and modeled by noise source


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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


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SNR

  • Every increment in there is improvement in SNR.

  • Comparison between ADC's: each in SNR is referred as one bit higher resolution.


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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.


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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.


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Oversampled PCM conversion

  • Noise power outside the signal band can be attenuated by a digital low-pass filter.


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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.


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Example

  • Audio signal: audio range, needed resolution (CD quality audio) or

  • Using ADC, the needed sampling rate is

  • Using ADC, the needed sampling rate is


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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.


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First order Sigma-Delta Modulation


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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.


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First order Sigma-Delta Modulation

  • Modulator output (assuming ideal DAC):

  • STF – simple delay

  • NTF – containing zero at DC frequency


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NTF of first order sigma-delta modulator

  • Example:

  • Better noise reduction in the signal band.

  • Noise "pushed" out the signal band.

Signal Band


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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


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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.


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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….).


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Qualitative time domain Behavior


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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.


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Qualitative time domain Behavior

  • Sinusoidal input.

  • Sampling frequency:


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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


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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.


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Non linear behavior

Wanted signal

Wanted signal

  • Input signal frequency:


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High Order Sigma-Delta Modulation

  • A straightforward extension to the first sigma-delta:

    • STF:

    • NTF:


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High Order Sigma-Delta Modulation

  • Example: Second order sigma-delta modulator


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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.


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Example revisited

  • Audio signal: audio range, needed resolution (CD quality audio) or


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Other topologies

  • Distributing zeros over the signal band

  • Example: forth order topology


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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.


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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.


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Modulation based Parallel Sigma-Delta

  • Generalized block diagram:


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Modulation based Parallel Sigma-Delta

Signal path

Quantization noise path


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Modulation based Parallel Sigma-Delta

  • The modulation sequences are rows of a unitary matrix:

  • Signal path (without noise):


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Modulation based Parallel Sigma-Delta

  • The filter must satisfy:

  • The filter should be design to minimize the quantization noise


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Time-interleaved ADC

  • Using the unitary matrix:


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Hadamard-Modulated Sigma-Delta


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Examples

  • TI


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Examples

  • High bandwidth Sigma-Delta


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Examples


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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


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PRFQ

  • General FQ configuration:


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PRFQ

  • 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.


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PRFQ

  • 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


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PRFQ - Optimization Constraints

  • Transparent converter.

  • The filters and are stable.

  • The filter is strictly causal.


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Discussion

  • 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?