Non-uniform Sampling Signals and Systems (A/D & D/A Converters)

1. Non-uniform Sampling Signals and Systems(A/D & D/A Converters) Y. C. Jenq Department of Electrical & Computer Engineering Portland State University P. O. Box 751 Portland, OR 97207 jenq@ece.pdx.edu Y. C. Jenq

2. Outlines • Non-uniform Sampling Signals • Digital Spectrum of Non-uniformly Sampled Signal • Timing Error Estimation • Reconstruction of Digital Spectrum Y. C. Jenq

3. Non-uniform Sampling Waveform amplitude, x(t) with FT = Xc(W) M=4 T = nominal sampling period Dn = tn- nT, rn = Dn / T T D1 D2 time, t t0 t1 t2 t3 t4 t5 t6 t7 t8 Y. C. Jenq

4. Non-uniform Sampling Clock T t0 t1 t2 t3 t4 t5 t6 t7 t8 Y. C. Jenq

5. Non-uniform Sampling Examples • Random Equivalent–time Sampling • Interleaved ADC Array • Direct Digital Synthesizer Y. C. Jenq

6. Random Equivalent-Time Sampling Triggering Level Triggering Time Instances Sampling Time Instances Y. C. Jenq

7. Random Equivalent-Time Sampling Y. C. Jenq

8. Interleaved ADC Arrays Sampling Clock Signal in Memory ADC Memory ADC Delay elements Memory ADC Memory ADC OR with a 4-phase clock Y. C. Jenq

9. Interleaved ADC Arrays Memory ADC Memory ADC Signal in Memory ADC Memory ADC 4-phase clock Y. C. Jenq

10. Direct Digital Synthesizer (DDS) Waveform Memory Low-Pass Filter D/A Converter Phase Accumulator Y. C. Jenq

11. Direct Digital Synthesizer (DDS) Waveform Memory Low-Pass Filter D/A Converter Address Accumulator Integer Part Fraction + Address Increment Register Integer Part Fraction Y. C. Jenq

12. Direct Digital Synthesizer (DDS)Waveform Memory Fs: Master Clock Frequency f: Sine Wave Frequency TL: Table Length Y. C. Jenq

13. Direct Digital Synthesizer (DDS)Frequency Resolution W + L/M Integer Part Fraction B bits Frequency Resolution = Fs/2B-1 Sine wave Frequency f = (W+L/M)Fs/TL Y. C. Jenq

14. Non-uniform Sampling Model T = nominal sampling period tn = nT + Dn , and Dn is periodic with period M. Let n = k M + m where k ranges from –∞ to +∞ and m ranges from 0 to (M-1), Then tn = ( k M + m )T + D(kM+m) = k M T + m T + Dm = k M T + m T + rm T where rm = Dm/T Y. C. Jenq

15. Digital Spectrum of Non-uniformly Sampled Signals • Yih-Chyun Jenq, “Digital Spectra of Non-uniformly Sampled Signals - Fundamentals and High-Speed Waveform Digitizers,” IEEE Transactions on Instrumentation and Measurement, vol. 37, no. 2, June 1988. • Yih-Chyun Jenq, “Digital Spectra of Non-uniformly Sampled Signals: A Robust Time Offset Estimation Algorithm for Ultra High-Speed Waveform Digitizers Using Interleaving,” IEEE Transactions on Instrumentation and Measurement, vol. 39, no. 1, February 1990 Y. C. Jenq

16. Digital Spectrum of Non-uniformly Sampled Signals If we use x(tn) to compute the digital spectrum, Xd(w), as if the data points were sampled uniformly, i.e., Xd(w) = Sn x(tn) e-jwn Then, it can be shown that Xd(WT) = (1/T)Sk A(k,W) Xc[W-k(2p/MT)] Where A(k,W) = (1/M)Sm=0,(M-1) e-j[W-k(2p/MT]rmTe-jkm(2p/M) Notice that A(k,W) is the m-point DFT of e-j[W-k(2p/MT]rmT Y. C. Jenq

17. Digital Spectrum of Non-uniformly Sampled Sinusoid Input Signal x(t) = exp(jWot), And Xc(W)=2pd(W-Wo ) Then Xd(WT) = (2p/T) Sk A(k) d[W-Wo-k(2p/MT)] where A(k) =Sm=0,(M-1)(1/M)ejrmWoTe-jkm(2p/M) Notice that A(k) is no longer a function of W, and A(k) is a M-point DFT of ejrmWoT, m=0, 1,…,M-1 Y. C. Jenq

18. Digital Spectrum of Non-uniformly Sampled Sinusoid M=4 A(0) A(2) A(1) A(3) Y. C. Jenq

19. Digital Spectrum of Non-uniformly Sampled Sinusoid M=8 Y. C. Jenq

20. Estimation of Timing Errors - rm A(k) =Sm=0,(M-1)[(1/M)exp(jrmWoT)]e-jkm(2p/M) A(0) A(2) A(1) A(3) Y. C. Jenq

21. Reconstruction of Digital Spectrum Once the timing errors are known, can we reconstruct the correct digital spectrum? Y. C. Jenq

22. Selecting Test Frequencies Higher frequency  more sensitive to timing error Using FFT spurious harmonics should be on the bins Windowing function selection A(0) A(2) A(1) A(3) Y. C. Jenq

23. Estimation of rm- Synchronous Case Residual Timing Error RMS value before Adjust-ment RMS value after (4 bits) RMS value after (6 bits) RMS value after (8 bits) RMS value after (10 bits) RMS value after (∞ bits) timing offset error 30% 4x10-11 2.4x10-12 4.4x10-13 1.1x10-13 2.9x10-14 2.6x10-24 20% 3x10-11 3.1x10-12 5.6x10-13 1.6x10-13 3.0x10-14 2.2x10-24 10% 2x10-11 2.3x10-12 6.1x10-13 1.3x10-13 2.7x10-14 1.8x10-24 5% 0.9x10-11 2.6x10-12 5.4x10-13 1.4x10-13 3.6x10-14 2.0x10-24 Residual timing errors are independent of initial timing errors! Y. C. Jenq

24. Estimation of rm- Synchronous Case Sensitivity to Quantization Noise in A/D Converter Residual Timing Error is relatively independent of initial timing error, but it is quite sensitive to the effective-bit of ADC Y. C. Jenq

25. Residual Timing Error Residual Timing Error: RMS rm 1 Residual RMS rm ~ 10-3 at 7 Bits 10-1 10-2 One order of magnitude improvement per 3 effective bits increase 10-3 10-4 10-5 4 6 8 10 bits Y. C. Jenq

26. Perfect Reconstruction of Digital Spectrum • Yih-Chyun Jenq, “Perfect Reconstruction of Digital Spectrum from Non-uniformly Sampled Signals,” IEEE Transactions on Instrumentation and Measurement, vol. 46, no. 3, 1997. Y. C. Jenq

27. Reconstruction of Digital Spectrumwith Residual Timing Error Reconstruction noise due to residual timing error: S/N ~ 20*log(1/s) -16 dB s = standard deviation of rm (Residual s) ~ (Initial s)/1000 at 7 Bits and improve one order of magnitude per 4 bits increase Reconstruction noise due to quantization error: SNR = 6.02* (number of bits) + 1.76 dB Y. C. Jenq

28. Reconstruction of Digital Spectrumwith Residual Timing Error • Yih-Chyun Jenq, “Improveing Timing Offset Estimation by Aliasing Sampling,” IMTC’05, May 2005, Ottawa, Canada. Y. C. Jenq