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Kostas Georgopoulos, Martin Burbidge, Andreas Lechner and Andrew Richardson

Behavioural Error Injection, Spectral Analysis and Error Detection for a 4 th order Single-loop Sigma-delta Converter Using Walsh transforms. Kostas Georgopoulos, Martin Burbidge, Andreas Lechner and Andrew Richardson. Presentation Overview. The Sigma-Delta A/D Converter

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Kostas Georgopoulos, Martin Burbidge, Andreas Lechner and Andrew Richardson

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  1. Behavioural Error Injection, Spectral Analysis and Error Detection for a 4th order Single-loop Sigma-delta Converter Using Walsh transforms Kostas Georgopoulos, Martin Burbidge, Andreas Lechner and Andrew Richardson

  2. Presentation Overview • The Sigma-Delta A/D Converter • The Walsh functions and Walsh series • Motivation for work • The FFT error simulation analysis • The Walsh error simulation analysis • Conclusions • Future Work

  3. The Sigma-Delta A/D Converter (I) • A device comprised by three stages • Anti-aliasing filter • Sigma-Delta modulator • Decimation phase

  4. The Sigma-Delta A/D Converter (II) • Sampling at a frequency much higher then the Nyquist , where fs is sampling frequency and fiis the input frequency

  5. The Sigma-Delta A/D Converter (III) • Transfer function: , where L is the order of modulator • Noise floor is moved out of bandwidth of interest by noise shaping Bandwidth of interest, 0-24 kHz

  6. The Walsh Functions (Theory) • Walsh functions form an ordered set of rectangular orthogonal waveforms • Only two amplitude values, +1 and –1 • Fast Walsh transforms exist • Any given signal can be represented through the combination of two or more Walsh functions

  7. The Walsh Series (I) • The Walsh series is similar to the Fourier Series expansion where parameter α determines the amplitude or weighting of each Walsh function and

  8. The Walsh Series (II) • Walsh functions can also be expressed in terms of even and odd waveform symmetry , where Walsh functions SAL and CAL can be visualised as the respective sine and cosine basis functions in Fourier Series

  9. The Walsh Series (III) • Employing SAL and CAL functions a Walsh Series similar to the sine-cosine series is given where f(t) is the sum of a series of square-wave shaped functions

  10. Signal Reconstruction using Walsh • A simple case of a sine-wave signal approximated with 3 Walsh functions

  11. Motivation & Methodology • FFT converges rapidly to sine wave hence use for classic dynamic performance testing • Walsh converges rapidly to square wave: • Idea to use square wave for input to modulator • Walsh transform of bit-stream should give single spectral peak • All other peaks in spectrum are due to noise and non-idealities • Higher potential for on-chip transform of fewer samples Methodology: • Determine modulator behavior and model parameters that lead to performance failure in FFT domain • Analyse effect of these failure modes on Walsh results

  12. FFT Analysis Setup • Use of initial C-based model provided by Dolphin • Ideal model FFT S/(N + THD) results: • Input 2.5 Vpk sine @ 1 kHz • BW approx 24 kHz (150 Hz to 24 kHz) • S/(N+THD) approx 100dB • Next step: Analyse how Walsh transforms could compare to FFT

  13. Fault Set For FFT Faulty capacitor • Input Offsets • Integrator Gain Variations • Corrupted feedback paths • Presence of noise on the modulator input • Integrating capacitor mismatch Gain mismatch

  14. Effect of Integrator Offset • 40 mV offset on the modulator input, S(N/THD) 99,6 dB • 50 mV offset on the modulator input, 50 dB drop in S(N/THD) Bandwidth of interest, 46 – 24 kHz

  15. Analyses of Walsh Testing • Setup: • Square wave test stimulus, @ 1.5kHz, 1.9 to 2.3 V amplitude • Bit-stream frequency of 3.072 MHz, analysis on the bit stream with 16384 and 65536 (1-bit) samples. • Analyses: • Investigation into test stimulus accuracy requirements • Assessment of Walsh-based modulator performance tests • Analysis of Walsh test coverage against modulator failure modes

  16. Test Stimulus Accuracy (I) • Finite rise/fall time: No significant effect • Overshoots: 4% of maximum amplitude, i.e. 0.08 V for 2 V input 16384 samples

  17. Test Stimulus Accuracy (II) • SNR with respect to different input amplitudes Analysis for both 16384 and 65536 samples

  18. Ideal Walsh Sequency Power Spectrum Walsh transform for 2Vpk square wave @ 1.5 kHz 0 dB -12 dB SNR = 95 dB SNR = 107 dB 16384 samples -120 dB -130 dB 65536 samples BW = 24 kHz (0 kHz to 24 kHz)

  19. Gain Deviation in 2nd Integrator • Gain deviation of 7.4% 0 dB Deviated output -72 dB -120 dB Ideal BW = 24 kHz (0 kHz to 24 kHz)

  20. Walsh Noise Test (I) Noise modelled at input can cause catastrophic failure (SNR ~30dB) 0 dB Catastrophic failure -130 dB For both cases N = 65536 Ideal BW = 24 kHz (0 kHz to 24 kHz)

  21. Walsh Noise Test (II) • FFT – Smoother transition to performance failure • Walsh – Sudden transition to catastrophic failure

  22. Summary and Future Work Summary • Failure Insertion in C-based models => FFT results • Usage of Walsh Transforms with square wave inputs for spectral analysis • Initial Potential for Walsh SNR test assessed • Test Stimulus Requirements • Challenges and Limitations Identified Future Work • Expansion of existing fault simulation data for Walsh applicability • Investigation into hardware implementation and test stimulus generation • Investigation into hybrid test solution

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