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

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

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

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

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

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

Y. C. Jenq7 Random Equivalent-Time Sampling

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

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

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

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

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

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

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

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 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

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

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

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

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

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)

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

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

Y. C. Jenq23 Estimation of r m - Synchronous Case Residual Timing Error timing offset error RMS value before Adjust- ment 30% 20% 10% 5% 4x x x x RMS value after (4 bits) RMS value after (6 bits) RMS value after (8 bits) RMS value after (10 bits) RMS value after (∞ bits) 2.4x x x x x x x x x x x x x x x x x x x x Residual timing errors are independent of initial timing errors!

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

Y. C. Jenq bits Residual Timing Error: RMS r m Residual Timing Error One order of magnitude improvement per 3 effective bits increase Residual RMS r m ~ at 7 Bits

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.

Y. C. Jenq27 Reconstruction of Digital Spectrum with Residual Timing Error S/N ~ 20*log(1/  ) -16 dB SNR = 6.02* (number of bits) 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:

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.