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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

- 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

- 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.

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

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 Modeling

- Quantizer is non linear system.
- Linearized and modeled by noise source

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

SNR

- Every increment in there is improvement in SNR.
- Comparison between ADC\'s: each in SNR is referred as one bit higher resolution.

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

- Technique that improves resolution by oversampling.
- Total amount of noise is like in Nyquist rate conversion, but its frequency distribution is different.

Oversampled PCM conversion

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

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.

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

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 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 Modulation

- Modulator output (assuming ideal DAC):
- STF – simple delay
- NTF – containing zero at DC frequency

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 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

- Over sampling ratio:
- SNR:
- For every doubling of the oversampling ratio, the SNR improves by , the resolution improves by 1.5bit.

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 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 Behavior

- Sinusoidal input.
- Sampling frequency:

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

- 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.

High Order Sigma-Delta Modulation

- A straightforward extension to the first sigma-delta:
- STF:
- NTF:

High Order Sigma-Delta Modulation

- Example: Second 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 revisited

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

Other topologies

- Distributing zeros over the signal band
- Example: forth order topology

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

- 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

- Generalized block diagram:

Modulation based Parallel Sigma-Delta

- The modulation sequences are rows of a unitary matrix:
- Signal path (without noise):

Modulation based Parallel Sigma-Delta

- The filter must satisfy:
- The filter should be design to minimize the quantization noise

Time-interleaved ADC

- Using the unitary matrix:

Examples

- TI

Examples

- High bandwidth Sigma-Delta

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

PRFQ

- General FQ configuration:

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.

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

PRFQ - Optimization Constraints

- Transparent converter.
- The filters and are stable.
- The filter is strictly causal.

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?

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