Enhanced Random Equivalent Sampling Based on Compressed ...

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 3, MARCH 2012

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Enhanced Random Equivalent Sampling Based on Compressed Sensing Yijiu Zhao, Yu Hen Hu, Fellow, IEEE, and Houjun Wang

Abstract—The feasibility of compressed-sensing-based (CS) waveform reconstruction for data sampled from the random equivalent sampling (RES) method is investigated. RES is a wellknown random sampling method that samples high-frequency periodic signals using low-frequency sampling circuits. It has been incorporated in modern digital oscilloscopes. However, the efficiency and accuracy of RES may be sensitive to timing uncertainty of analog RES circuits. For signals with sparsely populated harmonic components, the CS-based signal reconstruction method promises to mitigate the inherent timing error and to enhance the overall performance. A novel measurement matrix motivated by the Whittaker–Shannon interpolation formula is proposed for this purpose. Experiments indicate that, for spectrally sparse signal, the CS-reconstructed waveform exhibits a significantly higher signal-to-noise ratio than that using the traditional time-alignment method. A prototype realization of this proposed CS-RES method has been developed using off-the-shelf components. It is able to capture analog waveforms at an equivalent sampling rate of 25 GHz while sampled at 100 MHz physically. Index Terms—Compressed sensing (CS), nonuniform sampling, random equivalent sampling (RES), signal reconstruction, sparsity.

I. I NTRODUCTION

R

ANDOM EQUIVALENT SAMPLING (RES) [1]–[4] is a random sampling method that enables digital sampling of a periodic analog signal using an analog-to-digital converter (ADC) clocked at a frequency much lower than the Nyquist rate. This sub-Nyquist rate sampling property makes the family of random sampling methods such as RES, as well as the equivalent time sample (ETS) method [5], attractive options for the implementation of low-cost high-performance digital storage oscilloscopes. Random sampling has also found many important applications such as signal integrity measurement [6] and ultrasonic instrumentation [7], among others. RES acquires signal samples at random positions within a cycle of the sampling clock. The randomness is realized by dithering the phase of the sampling clock [8]. The relative

Manuscript received April 16, 2011; revised September 6, 2011; accepted September 7, 2011. Date of publication October 27, 2011; date of current version February 8, 2012. The Associate Editor coordinating the review process for this paper was Dr. Kurt Barbe. Y. Zhao and H. Wang are with the School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: [email protected]; [email protected]). Y. H. Hu is with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706 USA (e-mail: hu@engr. wisc.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIM.2011.2170729

position of a sample within a cycle of the sampling clock can be measured using time-to-digital converter (TDC) circuitry. The output of TDC gives the address in a waveform buffer. The sampled waveform value then will be stored in this address. Each address of the waveform buffer corresponds to a specific time interval (bin) within a cycle of the sampling clock. If there are N addresses in the waveform buffer, i.e., corresponding to N bins per cycle of the sampling clock, then the average equivalent sampling rate of RES would be N times of the sampling frequency. Hence, this method is called RES. Compared with uniformly spaced sampling technique, RES often takes considerably longer sampling time to produce a sufficiently accurate reconstructed waveform. The main cause for such a delay is the uneven random distribution of the relative positions of RES samples within a cycle of the sampling clock. Due to the randomness, some positions may be repeatedly sampled, whereas other positions are left blank. As such, to achieve desired accuracy, one often has to wait patiently until a sufficient number of samples populate the waveform buffer. This long sampling latency may limit potential applications of RES for real-time acquisition of high-frequency signals. In this paper, we propose to use compressed sensing (CS) to reconstruct a spectrally sparse periodic analog waveform to reduce this unwanted delay. CS [9]–[11] has been proposed as an efficient signal reconstruction method for random samples of spectrally sparse analog signals [12], [13]. A periodic analog signal is spectrally sparse if it consists of very few significant harmonics. For spectrally sparse analog signals, the CS theory ensures that very moderate number of random samples (usually four times the number of nonzero sparse spectral components [11]) will be sufficient to reconstruct the analog signal with desired accuracy. Previously, the feasibility of applying CS to sample spectrally sparse periodic signals has been demonstrated [12], [13]. However, both these earlier methods require preprocessing of the analog signal using additional special-purpose random sampling circuitry. In particular, a pseudorandom sequence clocked at the Nyquist rate is needed to modulate the analog signal. Implementation of such a pseudorandom sequence generator would be nontrivial and costly. A key contribution of this paper is the successful demonstration of the feasibility and potential advantage of incorporating CS with RES sampling. Specifically, RES samples will be regarded as random time samples of a spectrally sparse periodic analog signal. The CS procedure will be used to recover the underlying spectral lines, which then can be transformed into a corresponding time signal. During this process, a novel measurement matrix motivated by the Whittaker–Shannon

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Fig. 2.

RES block diagram.

The ADC is clocked at a sampling frequency fs f0 (subNyquist rate sampling). Let Q be a positive integer; the relation between f0 and fs may be explicitly expressed as T s = Q · T0 + δ

Fig. 1. Illustration of RES. A waveform with the high equivalent sampling frequency is composited after multiple acquisitions.

interpolation formula is proposed to facilitate CS reconstruction. The CS reconstruction algorithm has been incorporated into a prototype of a RES-enabled digital oscilloscope [14] developed recently by the first author. This oscilloscope is capable of capturing analog waveform at an equivalent sampling rate of 25 GHz while sampled at 100 MHz. By using the RES sampling circuitry of this prototype oscilloscope, waveforms reconstructed using traditional TDC time-alignment method and CS are compared. It is observed that the CS-reconstructed waveform exhibits a higher SNR while requiring fewer RES samples. In the remaining part of this paper, the RES sampling method is reviewed and analyzed in Section II. A recent effort of developing a RES oscilloscope prototype is also reviewed briefly. The CS algorithm is introduced in Section III. The CS-RES algorithm and several experiments using the hardware prototype oscilloscope are reported in Section IV. Further discussions are summarized in Section V.

II. RES The basic principle of RES sampling is illustrated in Fig. 1. Given a periodic analog signal, as shown in the solid line on the top row, level-trigger circuitry compares this analog waveform against a reference voltage shown as the horizontal dashed line. A trigger pulse will be generated whenever the voltage of the analog signal rises crossing the reference voltage. It is assumed that exactly one trigger pulse will be generated per cycle of the analog signal. If so, the trigger signal would be activated at a rate equal to the fundamental frequency f0 of the analog signal. The trigger pulse provides a fixed reference point to align samples. A block diagram of a RES device is shown in Fig. 2.

(1)

where Ts = 1/fs is the sampling interval, T0 = 1/f0 is the period of the periodic analog signal, and 0 ≤ δ = Ts mod T0 < T0 . Let us define ∆t (0 ≤ ∆t ≤ Ts ) to be the time difference between the trigger pulse and the immediate sampling clock edge. Therefore, ∆t gives the relative position of the sample within one cycle of the sampling clock. Denote q and r to be two positive integers, the trigger pulse occurs at r · T0 , and the sampling clock edge has a time index of q · Ts + β, where 0 < β < Ts is a randomly dithered phase shift of the sampling clock. Hence, ∆t can be expressed as ∆t = (q · Ts + β − r · T0 ) mod Ts = (β − (r · T0 mod Ts )) mod Ts β − (r · T0 mod Ts ) if β ≥ (r · T0 mod Ts ), = Ts − β + (r · T0 mod Ts ) otherwise. (2) The success of RES hinges upon the even distribution of ∆t of different samples over the sampling interval [0, Ts ]. This, in turn, depends on the uniformity and randomness of the distribution of the realizations of β in interval [0, Ts ]. The random phase dithering can be achieved by passing the sampling clock through variable delay circuitry before feeding into the ADC. The variable delay circuitry may be realized with a field-programmable gate array (FPGA) where a number of selectable delay paths are implemented. By randomly selecting one of these different delay paths each time a sample is taken, a different value of β may be used in (2). Moreover, the value of (r · T0 mod Ts ) will assume different values as long as δ = 0. In practical application, interval Ts is partitioned into equally spaced bins of duration Te . Value fe = 1/Te then becomes the equivalent sampling frequency of RES. If Te < T0 /2 (i.e., fe > 2f0 ), then the analog waveform may be sampled without aliasing. In other words, the RES should not be applied to sample analog signals whose bandwidth exceeds half of the equivalent sampling frequency. Fast-charging slow-discharging TDC circuitry [14] is required to measure the relative sampling position ∆t. When enabled for sampling, the TDC starts counting the elapse time

ZHAO et al.: ENHANCED RANDOM EQUIVALENT SAMPLING BASED ON COMPRESSED SENSING

Fig. 3. Distribution of time intervals. Multiple time intervals are measured and displayed in the same screen by an Agilent oscilloscope MSO6102A. Time intervals are measured after superimposing interval Tc /2, and Tc is the period of the counter in FPGA. After being magnified by a TDC with magnification of R, time intervals unevenly distribute in the interval [R · Tc /2, R · (Tc /2 + Ts )].

upon a trigger signal and stops when the next sampling clock pulse arrives, and a sample is taken. The TDC will quantize ∆t into an integral multiple of Te . This integer-valued bin count will be used as the address in the waveform buffer where the sample is to be stored. Once the TDC reports the estimated ∆t, it will be ready to be triggered again by a trigger pulse to acquire the next sample. However, if ∆t is too small, the TDC circuitry may not perform properly. To remedy this problem, the TDC is initialized with an offset voltage corresponding to an offset of time delay Tc /2 where Tc is the cycle time of a counter clock. This offset value is automatically compensated when the TDC output is converted into an address in the waveform buffer. The waveform buffer is controlled by a RES controller that may be implemented using the FPGA. Its functions include the following: 1) to store sampled data into waveform memory buffer to an address determined by the output of the TDC unit; and 2) to enable the TDC and ADC to capture the next sample when the previous sample is written into the buffer. The size of the waveform buffer should be sufficient to hold the waveform samples of a complete cycle of the sampling clock at increment of Te . Clearly, the minimum size of the waveform buffer should be N = Ts /Te .

(3)

Once a fixed number of samples are taken, the RES unit will reconstruct the analog waveform by reading samples from the waveform buffer and by interpolating missing values where samples are unavailable. Due to the uneven sampling interval, some bins may have more than one sample falling into them, whereas some bins may be left unfilled for a long duration. An illustrating example is shown in Fig. 3 where multiple time intervals are measured and displayed in the same screen by an Agilent MSO6102A oscilloscope [15]. As discussed earlier, the offset period of Tc /2 in this figure is not part of the sampling cycle [0, Ts ] and should be ignored. Due to this uneven distribution of sampling intervals, RES would require a rather large number of samples to fill the empty bins and thereby achieve the desired accuracy. This is illustrated in Fig. 4. With

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Fig. 4. Relationship between the numbers of the unfilled bins and the numbers of samples acquired using RES.

Ts = 10 ns and Te = 40 ps, there are N = 250 bins in the waveform buffer to be filled. After 1500 samples are acquired by the RES circuitry, there are still about 40 bins left empty. On average, one useful sample is obtained for every seven random samples. Clearly, the inefficiency of the RES sampling method is demonstrated. To make the matter even worse, note thatposition ∆t is estimated using analog TDC circuitry whose accuracy may be compromised due to variations of manufacturing and operating conditions. In fact, in view of the narrow bandwidth of the ADC, each sample assumes a low-pass filtered value of the analog signal around the sampling position. Thus, these errors manifest themselves as distortions after reconstruction of the analog waveform. Below, we propose to use a CS reconstruction method to improve the efficiency and accuracy of the RES sampling when the underlying analog signal has a spectrally sparse frequency spectrum. III. C OMPRESSED S ENSING S IGNAL R ECONSTRUCTION CS is an emerging data sampling paradigm that has received much attention [9], [11]. For a class of signals that exhibit a “spectral sparseness” property, CS method promises perfect reconstruction of the signal using very few measurements. Sparsity expresses the idea that the “information rate [16]” of a continuous-time signal may be much smaller than its bandwidth presented in the signal or that a discrete-time signal depends on a number of degrees of freedom that is comparably much smaller than its (finite) length. More precisely, CS exploits the fact that many signals are sparse or compressible in the sense that they have concise representations when expressed in an appropriate basis, such as Fourier basis, wavelet basis, etc. The basis can be selected according to the signal’s features. With instrumentation applications, an analog signal is sparse if it is periodic and contains few significant harmonic components. Since RES also requires the underlying analog signal to be periodic, it is natural to consider the feasibility of applying CS to improve the quality of reconstructed signals. In CS, the signal to be reconstructed is denoted by an N dimensional vector x. In the current application, x would be

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the unknown analog signal sampled at the equivalent sampling rate fe . Using RES or other methods, a set of measurements of the elements of x is obtained and denoted by vector y. In RES, each element of y is the output of a low-rate ADC. It can be represented as a weighted linear combination of elements in x that contains evenly spaced samples of the unknown waveform, with the sampling rate fe > fNyquist (fNyquist is the Nyquist rate of the signal to be measured). Let these weights be arranged in the measurement matrix Φ; one may express the relation between x and y as follows: y = Φx.

(4)

Since the signal is periodic, it has a discrete Fourier spectrum (line spectrum). Such a signal is spectrally sparse if only very few Fourier coefficients have significant magnitude values while other are nearly zero. In other words, the energy of the signal is concentrated on few spectral coefficients. In particular, a K-sparse signal x has K significant spectral coefficients where K is the sparsity level. If x is sampled from a K-sparse periodic analog signal x(t), it can be approximated by a linear combination of K (K N ) discrete Fourier basis functions, i.e., x=

K

αni ψni

(5)

i=1

where ψni ∈ Ψ, ni ∈ {1, 2, . . . , N }, and Ψ is the inverse discrete Fourier transform matrix [17]. Let α = [α1 , α2 , . . . , αN ]T be the coefficients vector of x in Ψ, and let α is a sparse vector consisting of K nonzero Fourier coefficients. Equation (4) can be rewritten as y = Φx = Dα

(6)

where D (= Φ · Ψ) is the equivalent measurement matrix. Given the observations y, the measurement matrix Φ, and the transform matrix Ψ, the purpose of CS reconstruction is to find α such that α0 is minimized subject to y − Dα2 ≤ η.

(7)

Here, the l0 norm counts the number of nonzero elements in a vector α, and η is a predefined acceptable recovery error. Minimizing α0 is equivalent to minimize the number of nonzero elements of the solution α# and therefore force the solution to be a sparse vector. Enforcing the constraint y − Dα2 ≤ η would ensure the reconstructed signal x# = Ψα# will yield (almost) the same measurements y if subjected to the same RES sampling process. A sufficient condition for a sparse solution of α to exist is that matrix D must satisfy a restricted isometry property (RIP) [10]. The RIP condition can also be evaluated in terms of incoherence between the measurement matrix Φ and the representation basis Ψ [11]. The coherence between Φ and Ψ is defined as √ (8) µ(Φ, Ψ) = N · max |ϕi , ψk | . 1≤i,k≤N

To ensure that the sparse solution α# exists, µ(Φ, Ψ) should be as small as possible. A. Measurement Matrix A key contribution of this paper is the development of a special kind of measurement matrix. Unlike general CS approaches where random measurement matrices are used, this special measurement matrix for RES sampled signals is motivated by the low-rate ADC sampling mechanism. More specifically, the measurement matrix is formed by the wellknown Whittaker–Shannon interpolation formula. According to the Shannon sampling theorem [18], the observation y and the original signal x satisfy the following formula: N ∆tm x(nTe )sinc −n (9) y(∆tm ) = Te n=1 where 1 ≤ m ≤ M ; 1 ≤ n ≤ N ; M is the number of random measurements (M < N ); ∆tm is time interval for the mth sample, as described in Fig. 1; and Te is the equivalent sampling period. We require no more than c · K · log(N ) (c is a constant, chosen as 5 in this paper) [9] random measurements to recover signal with overwhelming probability. From (9), the relation between the nonuniformly (randomly) sampled signal y of size M (i.e., composite acquisitions in Fig. 1) and the uniform equivalent sampling signal x of size N can be represented by the following matrix–vector representation:   x(T )   y(∆t )   φ φ ··· φ 1

1,1

 y(∆t2 )   φ2,1 = .  ..   .  . . y(∆tM ) φM,1

1,2

φ2,2 .. . φM,2

··· .. . ···

1,N

e

φ2,N   x(2Te )    .. ..    . . x(N · Te ) φM,N

⇔ y = Φx. (10) In (10), Φ is the measurement matrix with (m, n)th element as follows: ∆tm φm,n = sinc −n Te = sinc(Bm − n).

(11)

Due to the dependence on individually estimated ∆tm , each φm,n must be computed after the sample is acquired. However, since 0 ≤ Bm = ∆tm /Te ≤ Ts /Te = N , the values of sinc(Bm − n) may be stored in a table and be accessed using a simple lookup table operations. This will significantly reduce the computational complexity. B. OMP Recovery Algorithm As described in the introduction of Section III, the dimension of y is often much lower than that of x (M < N ). As such, recovering x from y is an ill-posed problem, and there may be many possible solutions of (7). Among them, the CS theory favors the mostly sparse solution that contains minimum number of nonzero elements in α# . However, the l0 -norm


Fig. 5.

Picture of a hardware implementation of the RES circuitry.

regularization problem is N P -hard and is computationally challenging when N increases. A variety of signal recovery algorithms have been proposed to alleviate this difficulty. To solve the optimization problem of (7), one may either apply convex programming [19] or use a family of greedy pursuit algorithm [20]. For RES signal reconstruction, the large computation cost of convex programming makes it a less appealing approach. In this paper, we adopt the orthogonal matching pursuit (OMP) algorithm [21], which is a variant of the greedy pursuit algorithm for solving the sparse vector α. Equation (7) implies that the sparse solution vector α should use the fewest possible columns of D matrix to approximate the observation y. This can be formulated as a subset selection problem where a minimum subset of columns of the D matrix is chosen to approximate the observation vector y in the least squares sense. The OMP algorithm successively chooses an additional column of the D matrix to reduce the approximation error. Equivalent, the OMP method begins with a tentative solution of α with a single nonzero entry and gradually adds nonzero entries one by one until the approximation error of y meets a predetermined criterion. A comprehensive reference of the OMP algorithm can be found in [21]. IV. E XPERIMENT Experiments are performed to compare the performance of the proposed CS-RES method against conventional interpolation-based RES method. Two kinds of synthetic analog waveforms are generated and applied to a prototype RES sampling board to yield a list of time-indexed samples. The first kind of an analog waveform is a spectrally sparse signal, and the second kind is a spectrally dense signal. These synthesized analog waveforms are fed into a RES prototype board [13]. A picture of this prototype is shown in Fig. 5. The block diagram of the major component of this prototype is shown in Fig. 2. In this system, fs = 100 MHz, and fe = 25 GHz. Thus, the number of temporal bins N = fe /fs = 250. The phase of the sampling clock is dithered by passing the clock signal through variable delay circuitry consisting of 20 different delay paths. For each new sample acquisition run, one of these 20 paths is

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randomly selected. Thus, β may assume one of 20 different values. After a fixed number of acquisition runs, a sequence of acquired RES samples and corresponding positions within the sampling period are obtained. Then, these RES samples will be used to reconstruct the original analog waveform using both the conventional interpolation method as well as the CS reconstruction method. With the conventional interpolation method, a linear interpolation is applied to the RES sample to fill in sample values of empty bins. Then, a low-pass digital filter is applied to smooth the resulting waveform. With the CS reconstruction method, the positions of the RES samples are first utilized to compute the measurement matrix using the lookup table method. Then, the OMP reconstruction is applied to obtain the reconstructed signal x. SNR is used as a metric to compare the quality of the reconstructed analog waveform, as defined in x (12) SN R = 20 · log10 x − x# where · is the Euclidean norm, x is the signal as measured by a sampling oscilloscope, and x# is the reconstructed signal vector. The first test signal is a spectrally sparse amplitudemodulated (AM) analog waveform shown in the following: fAM (t) = A cos(2πfc t) · (m(t) + C)

(13)

where m(t) is the modulation message signal, fc = 0.8 GHz is the carrier frequency, and A and C are the constants. The AM signal has 2K + 1 nonzero frequency components (K = 1 in this example). For the traditional time-alignment-based reconstruction method, M = 118 RES samples are acquired. For the CS-based reconstruction method, M = 64 RES samples are used. The reconstructed signal using the traditional timealignment method achieves an SNR = 14.56 dB, whereas the CS reconstruction yields an SNR = 31.51 dB. The original waveform, the time-alignment-based reconstructed waveform, and the CS-reconstructed waveform are depicted in Fig. 6(a). Hence, the CS-RES achieves a much higher SNR using fewer than half of the RES samples. Next, the accuracy of reconstructed analog waveforms at different lengths of the RES sample sequence is compared. For the range of 20 to 240 RES samples in increment of 20 samples, 100 random trials are performed for each specific length. The mean and variance of the SNRs of 100 trials at each length of the random samples are plotted in Fig. 6(b) and (c), respectively. It is clear that after the length of random samples increases beyond 40, the CS method yields much higher SNR with very small variance. Thus, for spectrally sparse signal (with very few harmonics), the advantage of the CS method is clearly demonstrated. In the second experiment, we consider a spectrally dense square wave. M = 118 RES samples are used for both time alignment and CS reconstruction. The original square waveform, the time-alignment-based waveform, and the CS-based reconstructed waveform are plotted in Fig. 7(a). The RES time-alignment method yields an SNR = 12.71 dB, whereas

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Fig. 6. AM signal reconstruction. (a) Performance of AM signal reconstruction via the time-alignment method (SNR = 14.56 dB) and the CS method (SNR = 31.51 dB). (b) Relationship between the average of SNR and the numbers of random samples. (c) Relationship between the variance of SNR and the numbers of random samples.

the CS method yields an SNR = 14.35 dB. These are errors calculated on the sampling point. Obviously, due to aggressive truncation of the Fourier series, the so-called Gibb’s phenomenon caused significant distortion of the CS-reconstructed signal near the transition edges of the square wave. The mean and variance of the SNR values averaged over 100 repetitive trials with different numbers of samples are shown in Fig. 7(b) and (c), respectively. Clearly, for a spectrally dense squarewave signal, the performance benefit of the CS method is less prominent.

Fig. 7. Square-wave reconstruction. (a) Performance of square-wave signal reconstruction via the time-alignment method (SNR = 12.71 dB) and the CS method (SNR = 14.35 dB). Both waveforms were reconstructed from 118 samples. (b) Relationship between the average of SNR and the numbers of random samples. (c) Relationship between the variance of SNR and the numbers of random samples.

Finally, the impacts of Gaussian distributed noise on the quality of reconstructed signal are evaluated. A spectrally sparse multisinusoidal signal with five tones is used in this experiment. We set Te = 40 ps, Ts = 10 ns, N = 250, and M = 128 (128 RES samples are used in both the RES reconstruction and the CS-RES reconstruction). The measurement matrix is constructed according to (11). Noise levels over the range of 5 to 50 dB in increment of 5 dB are tested. 100 trials are repeated for each specific noise level, and the averaged reconstruction


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R EFERENCES

Fig. 8.

Reconstruction performance for different noise levels.

SNR is plotted in Fig. 8. Note that the CS reconstruction method exhibits better robustness against additive Gaussian noise. V. C ONCLUSION A CS-based signal reconstruction method is proposed for reconstructing periodic waveforms sampled using the RES method. Compared with the traditional time-alignment RES reconstruction method, the CS approach yields better performance when the underlying waveform is spectrally sparse in the sense that it contains few significant Fourier coefficients. To this aim, a novel measurement matrix structure motivated by the Shannon’s sampling theory is proposed. A popular OMP CS reconstruction algorithm is applied to expedite computation. A hardware RES prototype system is used to capture RES samples from synthesized analog waveforms. We have tested this method using these RES samples with satisfactory results. Future work includes a comprehensive analysis of the impact of the finite bandwidth of the ADC on the reconstructed signal. It is hypothesized that the CS-RES reconstruction may overcome this difficulty and offers a super-resolution reconstruction result. It is planned to construct different representation bases to better handle spectrally nonsparse (in Fourier domain) signals such as the square waves. In addition, needed are practical guidelines (e.g., the number of sample points needed for reconstruction) to facilitate turn-key implementation of the CS-RES reconstruction. Along this direction, it is our hope to explore opportunities to implement CS-RES in an embedded hardware platform. Finally, as indicated in Section I, in addition to RES, there are other random sampling techniques for broadband signals such as ETS [5]. The applicability of CS to these random sampling methods certainly deserves further investigation. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive comments that are most helpful for the authors to significantly improve the presentation of this paper. They are also very grateful to Dr. Y. Eldar, Dr. M. F. Duarte, and W. E. I. Sha for informative discussions.

[1] P. D. Hale, C. M. Wang, D. F. Williams, K. A. Remley, and J. D. Wepman, “Compensation of random and systematic timing errors in sampling oscilloscopes,” IEEE Trans. Instrum. Meas., vol. 55, no. 6, pp. 2146–2154, Dec. 2006. [2] F. M. C. Clemencio, C. F. M. Loureiro, and C. M. B. Correia, “An easy procedure for calibrating data acquisition systems using interleaving,” IEEE Trans. Nucl. Sci., vol. 54, no. 4, pp. 1227–1231, Aug. 2007. [3] M. O. Sonnaillon, R. Urteaga, and F. J. Bonetto, “High-frequency digital lock-in amplifier using random sampling,” IEEE Trans. Instrum. Meas., vol. 57, no. 3, pp. 616–621, Mar. 2008. [4] P. J. Pupalaikis, Random Interleaved Sampling. [Online]. Available: http://www.lecroy.com/files/WhitePapers/WP_Ris_102203.pdf [5] Technique Primer, Sampling Oscilloscope Technique, Tektronix, Beaverton, OR, 1989. [Online]. Available: http://www.cbtricks.com/ miscellaneous/tech_publications/scope/sampling.pdf [6] R. A. Witte, “Sample rate and display rate in digitizing oscilloscopes,” Hewlett-Packward J., vol. 43, no. 1, pp. 18–19, Feb. 1992. [7] V. G. Ivchenko, A. N. Kalashnikov, R. E. Challis, and B. R. Hayes-Gill, “High-speed digitizing of repetitive waveforms using accurate interleaved sampling,” IEEE Trans. Instrum. Meas., vol. 56, no. 4, pp. 1322–1328, Aug. 2007. [8] D. E. Toeppen, “Acquisition clock dithering in a digital oscilloscope,” Hewlett-Packward J., vol. 48, no. 2, pp. 1–4, Apr. 1997. [9] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, Apr. 2006. [10] E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [11] E. Candès and M. Wakin, “An introduction to compressive sampling [a sensing/sampling paradigm that goes against the common knowledge in data acquisition],” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21–30, Mar. 2008. [12] M. Mishali, Y. C. Eldar, and J. A. Tropp, “Efficient sampling and stable reconstruction of wide band sparse analog signals,” in Proc. 25th IEEE Conv. Elect. Electron. Eng. Israel, Dec. 2008, pp. 290–294. [13] J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: Efficient sampling of sparse bandlimited siganls,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 520–544, Jan. 2010. [14] Y. Zhao, X. Zhuang, and L. Wang, “The research and application of random sampling in digital storage oscilloscope,” in Proc. IEEE Circuits Syst. Int. Conf., Chengdu, China, Apr. 28–29, 2009, pp. 1–3. [15] User Manual. Agilent 6000 Series Digital Oscilloscope, Agilent, Palo Alto, CA, Mar. 2011. [Online]. Available: http://cp.literature.agilent.com/ litweb/pdf/54695-97026.pdf [16] T. Blu, P. L. Dragotti, M. Vetterli, P. Marziliano, and L. Coulot, “Sparse sampling of signal innovations,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 31–40, Mar. 2008. [17] A. C. Gilbert, M. J. Strauss, and J. A. Tropp, “Improve time bounds for near-optimal sparse Fourier representation via sampling,” in Proc. Wavelets XI SPIE Opt. Photon., San Diego, CA, 2005. [18] M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE, vol. 88, no. 4, pp. 569–587, Apr. 2000. [19] J. A. Tropp, “Algorithms for simultaneous sparse approximation. Part II: Convex relaxation,” Signal Process.—Special Issue on Sparse Approximations in Signal and Image Processing, vol. 86, no. 3, pp. 589–602, Mar. 2006. [20] J. A. Tropp, “Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit,” Signal Process.—Special Issue on Sparse Approximations in Signal and Image Processing, vol. 86, no. 3, pp. 572–588, Mar. 2006. [21] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655–4666, Dec. 2007.

Yijiu Zhao received the B.S. and M.S. degrees in automation engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 2004 and 2007, respectively, where he is currently working toward the Ph.D. degree. His research interests include signal processing, compressed sensing, and analog signal sampling.

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Yu Hen Hu (M’83–SM’88–F’99) received the B.S.E.E. degree from National Taiwan University, Taipei, Taiwan, in 1976 and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern California, Los Angeles, in 1980 and 1982, respectively. He was with the Department of Electrical Engineering, Southern Methodist University, Dallas, TX. He is currently a Professor with the Department of Electrical and Computer Engineering, University of Wisconsin–Madison, Madison. His research interests include multimedia signal processing and communication, design methodology and implementation of embedded algorithms and systems, and nanoscale IC design methodologies. He is the author of more than 300 journal and conference papers and editor and coauthor of several books. Dr. Hu served as Secretary of the IEEE Signal Processing Society, the Board of Governors of the IEEE Neural Networks Council, Chair of IEEE Multimedia Signal Processing Technical Committee, and the IEEE Neural Network Signal Processing Technical Committee. He served as an Associate Editor for the IEEE T RANSACTIONS ON S IGNAL P ROCESSING, the IEEE Signal Processing Letters, the Journal of VLSI Signal Processing, IEEE Multimedia Magazine, and the European Journal of Applied Signal Processing.

Houjun Wang received the M.S. degree in measuring and testing technology and instruments and the Ph.D. degree in signal and information processing from the University of Electronic Science and Technology of China, Chengdu, China, in 1982 and 1991, respectively. He is currently a Professor of the School of Automation Engineering, University of Electronic Science and Technology of China. His research interests include signal processing, signal testing, and design for tests. Dr. Wang is a member of the National Electronic Measuring Instrument Standardization Technology Committee. He was selected in the “Cross-training plan” held by the Ministry of Education of China.