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Sigma-Delta Converters

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

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  1. Sigma-Delta Converters Kfir Gedalyahu

  2. Outline • Quantization and performance modeling • Oversampled PCM conversion • Sigma Delta Modulators: • First order • High order • Parallel • Perfect Reconstruction Feedback Quantizers (PRFQ)

  3. 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

  4. 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.

  5. 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

  6. 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)

  7. Performance Modeling • Quantizer is non linear system. • Linearized and modeled by noise source

  8. 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

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

  10. 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.

  11. 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.

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

  13. 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.

  14. 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

  15. 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.

  16. First order Sigma-Delta Modulation

  17. 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.

  18. First order Sigma-Delta Modulation • Modulator output (assuming ideal DAC): • STF – simple delay • NTF – containing zero at DC frequency

  19. NTF of first order sigma-delta modulator • Example: • Better noise reduction in the signal band. • Noise "pushed" out the signal band. Signal Band

  20. 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

  21. 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.

  22. 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….).

  23. Qualitative time domain Behavior

  24. 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.

  25. Qualitative time domain Behavior • Sinusoidal input. • Sampling frequency:

  26. 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

  27. 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.

  28. Non linear behavior Wanted signal Wanted signal • Input signal frequency:

  29. High Order Sigma-Delta Modulation • A straightforward extension to the first sigma-delta: • STF: • NTF:

  30. High Order Sigma-Delta Modulation • Example: Second order sigma-delta modulator

  31. 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.

  32. Example revisited • Audio signal: audio range, needed resolution (CD quality audio) or

  33. Other topologies • Distributing zeros over the signal band • Example: forth order topology

  34. 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.

  35. 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.

  36. Modulation based Parallel Sigma-Delta • Generalized block diagram:

  37. Modulation based Parallel Sigma-Delta Signal path Quantization noise path

  38. Modulation based Parallel Sigma-Delta • The modulation sequences are rows of a unitary matrix: • Signal path (without noise):

  39. Modulation based Parallel Sigma-Delta • The filter must satisfy: • The filter should be design to minimize the quantization noise

  40. Time-interleaved ADC • Using the unitary matrix:

  41. Hadamard-Modulated Sigma-Delta

  42. Examples • TI

  43. Examples • High bandwidth Sigma-Delta

  44. Examples

  45. 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

  46. PRFQ • General FQ configuration:

  47. 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.

  48. 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

  49. PRFQ - Optimization Constraints • Transparent converter. • The filters and are stable. • The filter is strictly causal.

  50. 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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