Time Domain Characterization and Test of High Speed Signals Using ...

2011 Asian Test Symposium

Time Domain Characterization and Test of High Speed Signals Using Incoherent Sub-sampling Debesh Bhatta, Joshua W. Wells, Abhijit Chatterjee School of Electrical and Computer Engineering, Georgia Institute of Technology Atlanta, Georgia, USA

optimization algorithm implementation. Previous papers, have used both time domain as well as frequency domain cost functions to achieve waveform reconstruction under reasonable assumptions. To reduce the computation cost and to improve the accuracy of waveform period estimation with a given number of samples we use a time domain cost metric. The time domain cost metric is computed with the same effort as computing only the dc coefficient of the spectrum. This significantly reduces the computation cost per iteration as well as the number of data points required to achieve a given accuracy in waveform reconstruction. A brief comparison of the three techniques is presented in Table I. Prior work in incoherent sub-sampling is reviewed in Sec II followed by a detailed discussion of the proposed signal reconstruction algorithm and associated cost function in Sec III. Hardware validation measurements are discussed in Sec IV, followed by conclusions.

Abstract—High speed signal acquisition and characterization contributes a significant amount to the total test cost of the finished product in modern high speed systems. Incoherent under-sampling allows robust and low cost signal acquisition without requiring a prior accurate knowledge of signal period. In this paper we propose a frequency estimation and signal reconstruction technique for incoherently sub-sampled periodic waveforms that is based on a time domain cost function. The method reduces the per iteration cost by a factor of log N compared to frequency domain cost functions. The proposed method estimates the period of a test signal with much fewer samples without degradation of accuracy. Index Terms—Incoherent sub-sampling, High speed analog test, Signal acquisition, Frequency estimation

I. I NTRODUCTION Different under-sampling techniques are used to reduce the sample rate in high speed signal waveform acquisition. This reduces the bandwidth requirement and hence equipment cost without sacrificing much of the signal fidelity. At high speeds Nyquist Rate Sampling (NRS), which demands a sampling rate twice the signal frequency, is not practical. Time interleaved sampling (TIS) [1] can be used to reduce the sampling rate. In time interleaved sampling, N parallel channels sampling at N distinct phases are combined to achieve an effective sampling rate, higher than the Nyquist rate. However, time interleaved sampling suffers from sensitivity to phase error, gain mismatch and clock jitter [2], [3]. In random interleaved sampling (RIS), sampling phases are randomly permuted to distribute the distortion power from spurs at harmonic frequencies to a flat noise floor [4], [5]. To achieve this, extra sampling channels are required. The sampling phases also needs to be temporally coherent with the waveform phase. In absence of accurate prior knowledge about the signal period, incoherent sub-sampling (ISS) [6]–[10] provides a robust and cost effective method of waveform acquisition. The sampling clock need not be temporally coherent with the waveform time base and can be, in theory, arbitrarily slow compared to the signal period as long as enough samples are acquired and the frequency remains reasonably stable. The period of the signal waveform is extracted by back end signal processing algorithms. Most of the acquisition cost is shared by the front end track and hold amplifier and the back end

II. I NCOHERENT SUB - SAMPLING AND WAVEFORM RECONSTRUCTION

In this section the specific case of sampling a periodic waveform of minimum period Tp is discussed. The sampling period, Ts , is chosen in such a way that Tp /Ts is irrational, ( TABLE I A COMPARISON OF DIFFERENT SAMPLING TECHNIQUES FOR PERIODIC WAVEFORMS

Minimum sample clock rate

NRS

2 times bandwidth

TIS

2 times bandwidth /N 2 times bandwidth /N limited by accuracy of back end algorithm

RIS ISS

This work is supported by SRC project task id 1836.072 and NSF Grant EHCS 0834484

1081-7735 2011 U.S. Government Work Not Protected by U.S. Copyright DOI 10.1109/ATS.2011.77

Sampling method

21

Hardware Prior requirements information of signal time base required(no aliasing)? High speed Yes single channel Low speed Yes N channels Low speed > N channels Low speed single channel

Yes No

the fact that to obtain the histogram, a large number of samples are required. In other previous work [10] and [11], this was done in two steps. A coarse estimation of Tp was obtained first by taking the fast Fourier transform (FFT) of the signal followed by identifying the peak of the power spectrum in the aliased data through interpolation. It is assumed that the peak corresponds to the fundamental component of the signal. T˜p , a finer estimate of Tp was obtained by minimizing the sum of absolute values of the spectrum (L1 norm) of the reconstructed waveform as a function of the estimated period T˜p . The coarse estimation process can be ambiguous for waveforms which have relatively high harmonic content. For the finer estimation we need to repeatedly estimate the spectra of the reconstructed signal. The reconstructed waveform is not, in general, sampled on a uniform grid. This makes the FFT calculation in each iteration extremely costly or requires a large number of points to be used.

Fig. 1. The concept of phase remapping as used to reconstruct the waveform from incoherently sampled data

Ts and Tp are linearly independent on the field of rationals .) If Ts is chosen at random, the probability that it is co-linear with Tp is negligible and Tp /Ts is irrational with overwhelmingly large probability. For a finite length representation of numbers a more correct statement is, Tp /Ts cannot be represented by using less than the full word length within the limit of quantization error. In general, periodic signal waveforms are used to characterize system performance. To fully capture the effects such as rise time, fall time, jitter and distortion we need to look at features with spectral content much higher than the period or the data rate of the signal. To reduce the cost of acquisition it is desirable to be able to use Ts