1. Y. C. Jenq1 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

2. Y. C. Jenq2 Outlines n Non-uniform Sampling Signals n Digital Spectrum of Non-uniformly Sampled Signal n Timing Error Estimation n Reconstruction of Digital Spectrum

3. Y. C. Jenq3 Non-uniform Sampling time, t Waveform amplitude, x(t) with FT = X c (  ) t0t0 t1t1 t5t5 t4t4 t3t3 t2t2 t7t7 t6t6 t8t8 T = nominal sampling period  n = t n - nT, r n =  n / T T 11 22 M=4

4. Y. C. Jenq4 Non-uniform Sampling Clock t0t0 t1t1 t5t5 t4t4 t3t3 t2t2 t7t7 t6t6 t8t8 T

5. Y. C. Jenq5 Non-uniform Sampling Examples n Random Equivalent–time Sampling n Interleaved ADC Array n Direct Digital Synthesizer

6. Y. C. Jenq6 Random Equivalent-Time Sampling Triggering Level Triggering Time Instances Sampling Time Instances

7. Y. C. Jenq7 Random Equivalent-Time Sampling

8. Y. C. Jenq8 Interleaved ADC Arrays ADC Signal in Delay elements Memory OR with a 4-phase clock Sampling Clock

9. Y. C. Jenq9 Interleaved ADC Arrays ADC Signal in Memory 4-phase clock

10. Y. C. Jenq10 Direct Digital Synthesizer (DDS) Waveform Memory Waveform Memory Phase Accumulator D/A Converter D/A Converter Low-Pass Filter Low-Pass Filter

11. Y. C. Jenq11 Direct Digital Synthesizer (DDS) Waveform Memory Waveform Memory D/A Converter D/A Converter Low-Pass Filter Low-Pass Filter Integer PartFraction Integer PartFraction + Address Accumulator Address Increment Register

12. Y. C. Jenq12 Direct Digital Synthesizer (DDS) Waveform Memory Fs: Master Clock Frequency f:Sine Wave Frequency TL:Table Length

13. Y. C. Jenq13 Direct Digital Synthesizer (DDS) Frequency Resolution Integer PartFraction W + L/M B bits Frequency Resolution = Fs/2 B-1 Sine wave Frequency f = (W+L/M)Fs/TL

14. Y. C. Jenq14 Non-uniform Sampling Model T = nominal sampling period t n = nT +  n, and  n is periodic with period M. Let n = k M + m where k ranges from –∞ to +∞ and m ranges from 0 to (M-1), Then t n = ( k M + m )T +  (kM+m) = k M T + m T +  m = k M T + m T + r m T where r m =  m /T

15. Y. C. Jenq15 n 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. n 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 Digital Spectrum of Non-uniformly Sampled Signals

16. Y. C. Jenq16 Digital Spectrum of Non-uniformly Sampled Signals If we use x(t n ) to compute the digital spectrum, X d (  ), as if the data points were sampled uniformly, i.e., X d (  ) =  n x(t n ) e -j  n Then, it can be shown that X d (  T) = (1/T)  k A(k,  ) X c [  -k(2  /MT)] Where A(k,  ) = (1/M)  m=0,(M-1) e -j[  -k(2  /MT]r m T e -jkm(2  /M) Notice that A(k,  ) is the m-point DFT of e -j[  -k(2  /MT]r m T

17. Y. C. Jenq17 Digital Spectrum of Non-uniformly Sampled Sinusoid Input Signal x(t) = exp(j  o t), And X c (  )=2  (   ) ThenX d (  ) = (2  /T)  k A(k)  [  -  o -k(2  /MT)] where A(k) =  m=0,(M-1) (1/M)e jr m  o T e -jkm(2  /M) Notice that A(k) is no longer a function of  and A(k) is a M-point DFT of e jr m  o T, m=0, 1,…,M-1

18. Y. C. Jenq18 Digital Spectrum of Non-uniformly Sampled Sinusoid A(0) A(1) A(2) A(3) M=4

19. Y. C. Jenq19 Digital Spectrum of Non-uniformly Sampled Sinusoid M=8

20. Y. C. Jenq20 Estimation of Timing Errors - r m A(k) =  m=0,(M-1) [(1/M)exp(jr m  o T)]e -jkm(2  /M) A(0) A(1) A(2) A(3)

21. Y. C. Jenq21 Reconstruction of Digital Spectrum Once the timing errors are known, can we reconstruct the correct digital spectrum?

22. Y. C. Jenq22 Selecting Test Frequencies A(0) A(1) A(2) A(3) Higher frequency  more sensitive to timing error Using FFT  spurious harmonics should be on the bins Windowing function selection

23. Y. C. Jenq23 Estimation of r m - Synchronous Case Residual Timing Error timing offset error RMS value before Adjust- ment 30% 20% 10% 5% 4x10 -11 3x10 -11 2x10 -11 0.9x10 -11 RMS value after (4 bits) RMS value after (6 bits) RMS value after (8 bits) RMS value after (10 bits) RMS value after (∞ bits) 2.4x10 -12 3.1x10 -12 2.3x10 -12 2.6x10 -12 4.4x10 -13 5.6x10 -13 6.1x10 -13 5.4x10 -13 1.1x10 -13 1.6x10 -13 1.3x10 -13 1.4x10 -13 2.9x10 -14 3.0x10 -14 2.7x10 -14 3.6x10 -14 2.6x10 -24 2.2x10 -24 1.8x10 -24 2.0x10 -24 Residual timing errors are independent of initial timing errors!

24. Y. C. Jenq24 Estimation of r m - 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

25. Y. C. Jenq25 1 10 -1 10 -2 10 -3 10 -4 10 -5 46810bits Residual Timing Error: RMS r m Residual Timing Error One order of magnitude improvement per 3 effective bits increase Residual RMS r m ~ 10 -3 at 7 Bits

26. Y. C. Jenq26 Perfect Reconstruction of Digital Spectrum n Yih-Chyun Jenq, “Perfect Reconstruction of Digital Spectrum from Non-uniformly Sampled Signals,” IEEE Transactions on Instrumentation and Measurement, vol. 46, no. 3, 1997.

27. Y. C. Jenq27 Reconstruction of Digital Spectrum with Residual Timing Error S/N ~ 20*log(1/  ) -16 dB SNR = 6.02* (number of bits) + 1.76 dB (Residual  ~ (Initial  /1000 at 7 Bits and improve one order of magnitude per 4 bits increase  = standard deviation of r m Reconstruction noise due to quantization error: Reconstruction noise due to residual timing error:

28. Y. C. Jenq28 Reconstruction of Digital Spectrum with Residual Timing Error n Yih-Chyun Jenq, “Improveing Timing Offset Estimation by Aliasing Sampling,” IMTC’05, May 2005, Ottawa, Canada.