1. DOLPHIN INTEGRATION TAMES-2 workshop 23/05/2004 Corsica1 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

2. DOLPHIN INTEGRATION © Lancaster University 2 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 3 TAMES-2 workshop 23/05/2004 Corsica The Sigma-Delta A/D Converter (I) A device comprised by three stages Anti-aliasing filter Sigma-Delta modulator Decimation phase

4. DOLPHIN INTEGRATION © Lancaster University 4 TAMES-2 workshop 23/05/2004 Corsica The Sigma-Delta A/D Converter (II) Sampling at a frequency much higher then the Nyquist, where f s is sampling frequency and f i is the input frequency

5. DOLPHIN INTEGRATION © Lancaster University 5 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 6 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 7 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 8 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 9 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 10 TAMES-2 workshop 23/05/2004 Corsica Signal Reconstruction using Walsh A simple case of a sine-wave signal approximated with 3 Walsh functions

11. DOLPHIN INTEGRATION © Lancaster University 11 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 12 TAMES-2 workshop 23/05/2004 Corsica 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 FFT Analysis Setup

13. DOLPHIN INTEGRATION © Lancaster University 13 TAMES-2 workshop 23/05/2004 Corsica Fault Set For FFT Input Offsets Integrator Gain Variations Corrupted feedback paths Presence of noise on the modulator input Integrating capacitor mismatch Faulty capacitor Gain mismatch

14. DOLPHIN INTEGRATION © Lancaster University 14 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 15 TAMES-2 workshop 23/05/2004 Corsica 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. DOLPHIN INTEGRATION © Lancaster University 16 TAMES-2 workshop 23/05/2004 Corsica 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 OvershootsDamping (ms)SNR (dB) Ideal-97.6 Max. amplitude 2.05 V0.0695,9 0.1494.7 Max. amplitude 2.15 V0.0695.5 0.1494.6

17. DOLPHIN INTEGRATION © Lancaster University 17 TAMES-2 workshop 23/05/2004 Corsica Test Stimulus Accuracy (II) SNR with respect to different input amplitudes Analysis for both 16384 and 65536 samples Square wave amplitude (V)# samplesSNR (dB) 1.91638493.9 2.097.6 2.194.8 2.293.2 2.39.7 1.965536104.4 2.0107.5 2.1107.6 2.2102.4 2.33.4

18. DOLPHIN INTEGRATION © Lancaster University 18 TAMES-2 workshop 23/05/2004 Corsica Ideal Walsh Sequency Power Spectrum BW = 24 kHz (0 kHz to 24 kHz) 0 dB -130 dB 16384 samples 65536 samples -120 dB -12 dB Walsh transform for 2Vpk square wave @ 1.5 kHz SNR = 95 dB SNR = 107 dB

19. DOLPHIN INTEGRATION © Lancaster University 19 TAMES-2 workshop 23/05/2004 Corsica Gain Deviation in 2 nd Integrator Gain deviation of 7.4% -72 dB -120 dB 0 dB Ideal Deviated output BW = 24 kHz (0 kHz to 24 kHz)

20. DOLPHIN INTEGRATION © Lancaster University 20 TAMES-2 workshop 23/05/2004 Corsica Walsh Noise Test (I) 0 dB -130 dB For both cases N = 65536 Noise modelled at input can cause catastrophic failure (SNR ~30dB) BW = 24 kHz (0 kHz to 24 kHz) Ideal Catastrophic failure

21. DOLPHIN INTEGRATION © Lancaster University 21 TAMES-2 workshop 23/05/2004 Corsica Walsh Noise Test (II) FFT – Smoother transition to performance failure Walsh – Sudden transition to catastrophic failure

22. DOLPHIN INTEGRATION © Lancaster University 22 TAMES-2 workshop 23/05/2004 Corsica 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