# Blind Timing Skew Estimation Using Source Spectrum Sparsity in

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Blind Timing Skew Estimation Using Source Spectrum Sparsity in

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 2401 Blind Timing Skew Estimation Using Source Spectrum Sparsity in Time-Interleaved ADCs Yue Xian Zou, Senior Member, IEEE, and Xiang Jun Xu Abstract—Timing skews in time-interleaved analog-to-digital converter (ADC) (TIADC) greatly degrade the spurious-free dynamic range of the TIADC, which can be improved effectively by digital compensation algorithms. Literature review shows that many nonblind compensation algorithms required the accurate timing skew parameters measured before the compensation process. In this paper, the blind timing skew estimation (TSE) for the TIADC system is investigated. With the evaluation of the working rationale of the TIADC system and the undersampling nature in each sub-ADC, the relation between the output digital spectra of the sub-ADCs and the source analog spectrum is formulated. A novel subband partition approach has been proposed, and the corresponding expression of the output digital subband spectra of the sub-ADCs in terms of the output digital subband spectra of the TIADC is obtained. With the assumption of the source spectra sparsity, the nonoverlapping frequency points can be determined from the output digital subband spectra of the sub-ADCs. Making use of the determined nonoverlapping frequency points, the phase information of the output digital subband spectra of the sub-ADCs can be employed to provide a good metric to estimate the timing skews, and a closed-form expression of the TSE has been derived. As the result, the source spectrum sparsity-based blind TSE (SS-BLTSE) algorithm has been developed and evaluated. Computer simulation results show that the SS-BLTSE algorithm provides good TSE performance for arbitrarily narrow or wideband source signals. Moreover, it has many merits over the existing blind TSE algorithms, such as there is no prior requirement of the source signal except that it has certain spectrum sparsity, no limit to the number of the sub-ADCs, and the frequency of the input signal can be approaching the Nyquist frequency. Index Terms—Blind estimation, spectrum sparsity, timeinterleaved analog-to-digital converter (ADC) (TIADC), timing skews, undersampling. I. I NTRODUCTION A Fig. 1. Conceptual illustration of TIADC and timing mismatches with M = 4. (a) Conceptual illustration of M -channel TIADCs. (b) Illustration of the timing skew effect in TIADC output spectra. Manuscript received September 13, 2011; revised January 29, 2012; accepted February 24, 2012. Date of publication May 3, 2012; date of current version August 10, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 60775003 and in part by the Shenzhen Science and Technology Program. The Associate Editor coordinating the review process for this paper was Dr. Rik Pintelon. The authors are with the Advanced Digital Signal Processing Laboratory, Shenzhen Graduate School, Peking University, Shenzhen 518055, China (e-mail: [email protected]; [email protected]). 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.2012.2192337 interleaved technique proposed by Black and Hodges [1] that results in a time-interleaved ADC (TIADC) may be considered. In principle, the TIADC is formed by using M identical parallel ADC (sub-ADC or channel) each with a sampling rate of fs /M as shown in Fig. 1(a). As the result, an overall sampling rate of fs can be achieved. Each channel samples the input signal xa (t) in turn, and the outputs are combined into one discrete output signal y(n). Ideally, all the M parallel channels are assumed to be linear and identical. They should have the same gain (gi = constant, i = 0, . . . , M − 1) and should be operating at the precise equally displaced sampling time instants. However, due to the practical implementation constraints in the fabrication process, the TIADC system exhibits HIGH-speed and high-resolution analog-to-digital converter (ADC) is a key component for many modern electronic systems such as radars, communication systems, and medical instruments. For a given fabrication technology and ADC word length requirement, there is a limit to the maximum achievable sampling speed. To achieve a conversion speed higher than that achievable by a single ADC, the time- 0018-9456/$31.00 © 2012 IEEE 2402 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 the “channel gain mismatch,” “channel bandwidth mismatch,” “channel dc offset mismatch,” and the “channel timing skew mismatch” [2]. For illustration purpose, the effect of timing skew mismatch is shown in Fig. 1(b) with a sinusoidal input signal. It is noted that, in Fig. 1(b), three distortion components caused by the timing skew mismatch severely reduce the spurious-free dynamic range of the TIADC [3]. More detailed discussions on mismatches can be found in [4]–[6]. The digital compensation of TIADC channel mismatches has attracted great attention recently. It is noted that several authors have proposed blind equalization techniques [5], [7]. Blind techniques do not require using a known signal to estimate the channel mismatch parameters and so do not suffer from “downtime problem,” but blind techniques use the excess bandwidth as a result of oversampling to estimate the mismatch parameter online. Experimental results showed that the blind equalization techniques, in general, asked for extra hardware (such as more ADCs and low-pass filter [9]). However, their performance of the timing mismatch compensation is inferior to that of the nonblind mismatch compensation techniques. In contrast to the blind techniques, many nonblind timing mismatch compensation algorithms have been proposed [10]–[12], where the channel timing mismatch parameters are needed. In [10], Johansson and Löwenborg cascaded fractional delay filters to compensate the timing mismatch. In [11], a multichannel-filtering approach is used for the TIADC mismatch compensation, which is computationally less complex and more robust than the filter-bank approach. Recently, Zou et al. introduced a novel multichannel Lagrange polynomial interpolation-timing mismatch compensation algorithm [12], which leads to a real-time implementation filter structure with low computational complexity. Careful evaluation reveals that these nonblind compensation methods all ask for the good measurement of the channel mismatch parameters. However, the measurement of the channel mismatch parameters using either online or offline approaches is a very challenging task [14]–[16]. In principle, the offline approaches need to know the input calibration signal. On the contrary, the blind parameter estimation methods only request statistical information on the input signal in an online manner, which has the advantages of being able to track the channel mismatch varying without interruption of the operation of the TIADC system. The blind timing skew estimation (TSE) has attracted many researchers working on it from different perspectives [13], [17]–[19]. Jamal et al. [17] and Huang and Levy [18] employed an oversampling technique and used the alias-free bandwidth to estimate the channel mismatch parameters in a blind estimation manner. Unfortunately, these methods presented poor scalability. Zou et al. [13] proposed an efficient blind TSE method for the TIADC systems by evaluating the autocorrelation of the analog input signal and the mean squared difference between the output samples of the two adjacent subADCs. However, this approach only works well when the frequency bandwidth of the analog input signal is lower than half of the Nyquist frequency. In [8], Elbornsson et al. integrated the channel mismatch parameter estimation and compensation into a uniform framework known as blind adaptive equalization. The mismatches are compensated using their proposed iterative stochastic gradient minimization algorithm, which made use of the signal continuity, and it works well for signals whose energy concentrates in low-frequency bands. In this paper, we focus on the blind TSE problem of the TIADC systems, where the channel gain mismatch and dc offset mismatch are not taken into account, although it has been shown that either gain mismatch or dc offset mismatch will have no impact on the performance of our proposed algorithm. In one word, our task is to estimate the channel timing skews Δtm (m = 0, 1, . . . , M − 1) in a blind manner. With careful evaluation of the working rationale of the TIADC system and the undersampling nature in each sub-ADC, we are able to determine some nonoverlapping frequency points from the output spectra of the TIADC system when the input analog signal has a certain frequency spectrum sparsity attribute. With the determined nonoverlapping frequency points, it is found that the phase information of the spectra can be employed to provide a good metric to estimate the timing skews. For getting the tractable solution, we have made the following assumptions. 1) Input signal xa (t) is bandlimited with the highest frequency smaller than fs /2. 2) The system parameters of the TIADC system are known, such as the number of sub-ADCs (M ) and the overall sampling frequency (fs ). 3) The discrete output signals (ym (n), y(n)) are available. 4) No channel dc offset and gain mismatches are considered since they can be compensated using the simple subtraction and scalar gain approaches [23]. 5) No additive noise is considered. 6) As the timing skew is usually much smaller than the sampling period (smaller than 10%), the input analog signal is much stronger than the disturbance introduced by TIADC channel timing skews [7]. Literature review shows that the aforementioned assumptions are commonly satisfied in the research of the TIADC technology [11], [19]. This paper is organized as follows. The formulation of the digital spectra of the ideal TIADC system is presented in Section II. The spectrum sparsity-based blind TSE (SS-BLTSE) algorithm is derived in detail and summarized in Section III. Computer simulation results are presented in Section IV. Section V gives the conclusions of our work. II. D IGITAL S PECTRA M ODELING FOR I DEAL TIADC In this section, we will present the formulation of the output digital spectra of the overall and sub-ADCs of the ideal TIADC system, in which no channel mismatches, additive thermal noise, and quantization error are considered. A. Digital Spectra Representation for the Ideal TIADC and Sub-ADC In this paper, the architecture of the TIADC considered is illustrated in Fig. 1(a), where the analog input xa (t) is a bandlimited signal with the highest frequency of fmax (Hz). The analog spectrum (Fourier transform) of xa (t) is denoted as Xa (jΩ) [21], where Ω = 2πf is the analog radian frequency in ZOU AND XU: BLIND TSE USING SOURCE SPECTRUM SPARSITY IN TIME-INTERLEAVED ADCs radians per second and f is the analog frequency in cycles per second (in hertz). The output discrete-time signal of the ideal ADC with the sampling period of Ts can be denoted as xd (n) = xa (t)|t=nTs , n = 0, 1, . . . , ∞. (1) The digital spectrum [the discrete-time Fourier transform (DTFT)] of xd (n) is denoted as Xd (ejω ) [21], where ω is the digital frequency in radians per sample associated with the sampling frequency of fs , which has the following relation: ω = ΩTs = 2πf /fs (2) where fs is the sampling frequency and fs = 1/Ts . Therefore, the relation between the analog and the digital spectrum can be denoted as [21] +∞ ω 1 2πk Xd (e ) = Xa j − . Ts Ts Ts jω (3) 2403 where Tss is the sampling period of the sub-ADCs, N is the total number of samples for TIADC, and m is the channel index. Let us define the digital frequency associated with the sampling period of Tss as , which has the following relation with the analog frequency Ω associated with the sampling frequency fss : = ΩTss = 2πf /fss . (7) From (6) and (7), the DTFT of ym (n) can be expressed as ∞ 2πk 2πk 1 j Ym (e ) = Xa j − ej ( Tss − Tss ) mTs . Tss Tss Tss k=−∞ (8) It can be seen that Ym (ej ) is also a summation of the frequency-scaled and the shifted version of Xa (jΩ), but the digital frequency is specified in (7). Obviously, Ym (ej ) is also a periodic function of with a period of 2π. k=−∞ From (3), it is noted that the digital spectrum Xd (ejω ) is a summation of the frequency-scaled and the shifted version of Xa (jΩ), where the frequency scaling is specified in (2). It is also a continuous periodic function of ω with a period of 2π. Since the frequency range of xa (t) is bandlimited to (−fmax , fmax ), the digital spectrum of xd (n) in its basic band can be denoted as ⎧ ⎨ 1 Xa j ω , ω ∈ (0, π) T T s s (4) Xdp (ejω ) = 1 ω−2π ⎩ Xa j , ω ∈ (π, 2π) Ts Ts where Xdp (ejω ) specifically represents the principal component of Xd (ejω ). According to the Nyquist sampling theorem, the analog signal xa (t) can be perfectly reconstructed from its ideal sampling sequence xd (n). From the TIADC system shown in Fig. 1(a), in the ideal case, the output of the TIADC will be identical to its counterpart of an ideal ADC, i.e., y(n) = xd (n). (5) It is noted that the overall sampling frequency of an ideal M -channel TIADC system is the same as that of an ideal ADC system. Hence, it is also denoted as fs , while the sampling frequency for each subchannel (sub-ADC) in the TIADC system denoted as fss turns to be fss = fs /M . In the following context, we will derive the relation between the output digital spectra of the sub-ADCs and the input analog spectrum using the TIADC system architecture shown in Fig. 1(a). From Fig. 1(a), it is straightforward to get the time domain expression of the output of the mth sub-ADC ym (n) = xa (t)|t=nTss +mTs , n = 0, 1, . . . , N/M − 1, Tss = 1/fss , m = 0, 1, . . . , M − 1 (6) B. Digital Frequency Subband Modeling in Ideal TIADC System From the working principle of the TIADC system, we have the following observations: 1) If xa (t) is bandlimited with the upper bound of fs /2, the xa (t) can be reconstructed perfectly from the output xd (n) of the ideal TIADC, and 2) for each sub-ADC in TIADC system, it operates in undersampling situation with an undersampling factor of M . In what follows, the relation between xa (t) and ym (n) for an ideal TIADC system will be explored in the time domain and spectrum domain. A new spectrum partition approach is proposed, and the relation between the partitioned spectra of the sub-ADCs and the partitioned spectra of the TIADC has been derived. As discussed earlier, the analog frequency Ω and the digital frequency ω is interconnected by the sampling frequency, such as specified in (2) or (7). Let us partition the analog-frequency range (−Ωs /2, Ωs /2) into 2M equal bandwidth bands, where Ωs = 2πfs is the sampling angular frequency of the TIADC system. We name each band as one analog-frequency subband (AFSB). The bandwidth of each AFSB is given by Δfa = Ωs /2/M = πfs /M. (9) Therefore, the principle digital frequency ranging from 0 ∼ 2π correspondingly can be partitioned into 2M digital bands with equal bandwidth, which is defined as one digital-frequency subband (DFSB) correspondingly. It is obvious that the bandwidth of a DFSB associated with the sampling period of Ts can be determined as Δfd = Δfa × Ts = π . M (10) Similarly, the bandwidth of a DFSB associated with the sampling period of Tss can be determined as Δfm = Δfa × Tss = π. (11) 2404 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 As the result, the principle digital spectrum of xd (n) can be partitioned into 2M DFSBs with a bandwidth of Δfd in (10); meanwhile, the principle digital spectrum of ym (n) only can be partitioned into two DFSBs with a bandwidth of Δfm in (11). Specifically, let us define the kth component of Xdp (ejω ) in (4) at the kth DFSB as follows: (k−1)π kπ jω (k) jω X (e ) ω ∈ , dp M M , k = 1, 2, . . . , M Xdp (e ) = 0 others. (12) Similarly, the first and second Ym (ej ) in (8) are given by j (1) j Ym (e ) = Ym (e ) 0 j Ym(2) (ej ) = Ym (e ) 0 DFSB components of ∈ (0, π) others ∈ (π, 2π) others. (13) Next, we will investigate the relations between Xdp (ejω ) and (2) and Ym (ej ). Here, we define ω2π = ((ω + π) mod 2π) − π. According to assumption 1), Xa (jΩ) = 0 for Ω > |2πfs |, and (8) can be rewritten as follows: (1) Ym (ej ) M j Ym (e 1 )= Tss 2 k=−M/2=1 2πk Xa j − Tss Tss ×ej ( Tss − Tss )mT , 2πk ∈ (0, 2π). (14) Replacing Xa (jΩ) by Xdp (ejω ) using the relation expressed in (4), together with Tss = M ∗ Ts , (14) can be derived as follows: 1 Ym (ej ) = M 0 Xdp ej ( M − 2πk M ) ej ( in (13) can be rewritten as follows: M − 2πk M )m . k=−M +1 (15) With variable exchange, (15) can be equivalently rewritten as j Ym (e M −1 2πk 2πk 1 )= Xdp ej ( M + M ) ej ( M + M )m M k=0 2π(k−1) M 2π(k−1) 1 j M + j + m M M = Xdp e . e M M k=1 (16) It is noted that, if ∈ (0, π), we have ((/M ) + (2(k−1)π/M )) ∈ ((2(k−1)π/M ), ((2k−1)π/M )), k = 1, (1) 2, . . . , M . According to (12) and (16), Ym (ej ) in (13) can be rewritten as follows: Ym(1) (ej ) = M π 1 (2k−1) j ( M +2(k−1) M ) Xdp e M k=1 π jm( M +2(k−1) M ) ×e 2π , Fig. 2. Conceptual illustration of AFSBs and DFSBs and the corresponding spectra (M = 4, Ωmax < Ωs /2). (a) |Xa (jΩ)|. (b) |Xd (ejω )| in (0 ∼ 4π). (c) |Ym (ej )| in (0 ∼ 4π). ∈ (0, π). (17) In the same way, if ∈ (π, 2π), then ((/M ) + (2(k − 1)π/ (2) M )) ∈ ((2k−1)π/M ), (2kπ/M )), k = 1, 2, . . . , M , Ym (ej ) Ym(2) (ej ) = M π 1 (2k) j ( M +2(k−1) M ) e Xdp M k=1 π jm( M +2(k−1) M )2π , ∈ (π, 2π). ×e (18) To get more understanding of the aforementioned derivation, an illustration is shown in Fig. 2, where we set M = 4. The partition of the analog amplitude spectrum (|Xa (jΩ)|) and the eight equal bandwidth AFSBs ranging from the −3th AFSB to the 4th AFSB are plotted in Fig. 2(a). The corresponding partition of |Xd (ejω )| in (0, 2π) and the eight equal bandwidth DFSBs are illustrated in Fig. 2(b). Correspondingly, the partition of |Ym (ej )| in (0, 2π) and two equal bandwidth DFSBs are presented in Fig. 2(c). Obviously, from Fig. 2(a), the highest frequency component is smaller than Ωs /2, so there is no spectrum overlapping in Xd (ejω ) since its sampling frequency is Ωs . However, from Fig. 2(c), since the subchannel sampling frequency is reduced to Ωs /4, we can see that the spectrum overlapping occurs in Ym (ej ). It is clear to see that there are still some frequency components that are not overlapped such as the spectrum line labeled as Ym (ej1 ) in Fig. 2(c). We can infer that, if the analog spectrum Xa (jΩ) has certain spectrum sparsity property, which we will discuss in the next section, most probably, there are certain nonoverlapping frequency components ωp existing in Ym (ej ) where ωp ∈ (0, 2π). ZOU AND XU: BLIND TSE USING SOURCE SPECTRUM SPARSITY IN TIME-INTERLEAVED ADCs III. S PARSE S PECTRUM -BASED B LIND TSE A LGORITHM In this section, we will derive a blind TSE algorithm jointly making use of the TIADC system structure and the relation between the partitioned spectra derived in the previous section. A. SS-BLTSE Algorithm Take sub-ADC-0 as the reference channel without lost of generality. Define the channel relative timing skew Δm as Δm = Δtm (tm −tm ) = , m = 0, 1, . . . , M −1, Δ0 = 0 Ts Ts (19) where Δtm is the timing skew for the mth sub-ADC and tm and tm are the real and the ideal sampling instant for the mth sub-ADC, respectively. From (6), ym (n) with timing skews can be reformulated as ym (n) = xa (nM Ts +(m+Δm)Ts), n = 0, . . . , N/M −1. (20) (1) Correspondingly, from (17) and (18), Ym (ej ) and (2) j Ym (e ) with timing skews will have the following form: M π 1 (2k−1) j ( M +2(k−1) M ) e Xdp M k=1 π j(m+Δm )( M +2(k−1) M )2π , ∈ (0, π) (21) ×e M π 1 (2k) ej ( M +2(k−1) M ) Ym(2) (ej ) = Xdp M k=1 π j(m+Δm )( M +2(k−1) M )2π , ∈ (π, 2π). (22) ×e Ym(1) (ej ) = Comparing (17) and (21), it is noted that the timing skews (1) (2) only introduce phase change in Ym (ej ) and Ym (ej ). (1) Carefully evaluating (21), we observe that Ym (ej ) is the summation of the frequency-expanded and the shifted version of Xdp (ejω ) at the (2k − 1)th DFSB in (0, 2π), where k is (2) an integer. Meanwhile, Ym (ej ) is the summation of the frequency-expanded and the shifted version of Xdp (ejω ) at the 2kth DFSB in (0, 2π), where k is an integer. Therefore, if the signal spectrum Xdp (ejω ) has certain sparse property, we can infer that there exist some nonoverlapping frequencies either in (0, π) or (π, 2π). Without loss of generality, here, we consider the nonoverlapping frequency denoted as ωp , which exists in (0, π). From (21), for arbitrary ωp , we have π 1 (2kp −1) j ( ωMp +2(kp −1) M ) Xdp Ym(1) (ejωp ) = e M ω π j(m+Δm )( Mp +2(kp −1) M )2π , ωp ∈ (0, π) (23) ×e where 2kp − 1 is the DFSB index associated with ωp . (1) YM (ejωp ) (1) Y0 (ejωp ) 2405 (1) (2) Similarly, the derivation of the Δm using Ym (ej ) gives (2) jωp ln Ym(2) (ejωp ) Y0 (e ) Δm = ωp −m, ωp ∈ (π, 2π). (26) π j M +2(kp −1) M 2π In the following section, we will introduce a systematic method to determine the nonoverlapping frequencies ωp and the related parameter kp . Moreover, it is noted that ωp is computed by (23) where Xd (ejω ) is assumed to be known. Carefully evaluating Xd (ejω ) when Δm is much smaller than Ts and with no gain mismatch and dc offset mismatch considered, according to assumption 6) in the Introduction, Xd (ejω ) can be approximately estimated by Y (ejω ), which is the DTFT of the output of the TIADC. According to this derivation, the estimate of Δm from (25) and (26) can be categorized as a blind TSE algorithm since it only requires the output spectra of the TIADC system and has no requirement on the input spectrum. Moreover, we name this proposed algorithm as an SS-BLTSE algorithm since it essentially was derived in the frequency domain and with the spectra sparsity assumption of the analog signal. B. Estimation of Nonoverlapping Frequencies in Frequency Domain In this section, we will introduce a method to determine the nonoverlapping frequency ωp using the relation between the spectra. Here, we still consider ωp locals in (0, π). The (1) first DFSB spectrum component Ym (ej ) is given in (17) and illustrated in Fig. 2(c). According to this formulation, we have the following facts: 1) If ωp ∈ (0, π) is a nonoverlap(1) (1) ping frequency, then Ym (ejωp ) = 0, and 2) since Ym (ejωp ) is the summation of the frequency-expanded and the shifted version of Xdp (ejω )|ω=ωp /M +2π(k−1)/M at the (2k − 1)th DFSB in (0, 2π), where k is an integer, therefore, for spectra components in the odd DFSBs, there should be only the corresponding (2kp − 1)th DFSB spectrum component (2k −1) Xdp p (ej((ωp /M )+2(kp −1)(π/M )) ) of Xdp (ejω ) that has a ω π ωp π j(m+Δm )( Mp +2(kp −1) M )2π ej ( M +2(kp −1) M ) e = ωp ω π π jΔ0 ( Mp +2(kp −1) M (2k −1) )2π ej ( M +2(kp −1) M ) e Xdp p ω π j(m+Δm )( Mp +2(kp −1) M )2π ω π e j(m+Δm )( Mp +2(kp −1) M )2π = = e ωp π jΔ0 ( M +2(kp −1) M )2π e (2kp −1) (1) From (23), dividing Ym (ejωp ) by Y0 (ejωp ), (m = 0, 1, . . . , M − 1), we have (24) shown at the bottom of the page. The last equation in (24) is obtained by using Δ0 = 0. After simple manipulation, the closed-form expression of Δm can be derived as (1) jωp ln Ym(1) (ejωp ) Y0 (e ) Δm = ωp − m, ωp ∈ (0, π). (25) π j M + 2(kp − 1) M 2π Xdp (24) 2406 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 value, and the other components are zero in the ideal case. The nonoverlapping frequencies can be determined by Algorithm I presented hereinafter. Algorithm I: Determine the nonoverlapping frequency points Step 1) Scanning the digital frequency ranging from 0 to π, find the nonoverlapping frequency candidate set by (1) (27) i : f Y0 (eji ) = 0, i = 1, . . . , L1 . Step 2) For each i obtained in (27), the amplitude frequency spectrum (AFS) is calculated as follows: π (2k−1) j ( Mi +2(k−1) M ) , k = 1, . . . , M (28) e Nx (k, i) = Xdp (2k−1) where 2k − 1 is the DFSB index and Xdp (ejω ) is approximately estimated by the mismatched TIADC output Y (ejω ). From (28), for each i , it is noted that Nx (k, i) is an M × 1 vector. If and only if there is only one DFSB (the (2kp − 1)th) that gives Nx (kp , i) = 0 (the additive thermal noise and quantization error will be considered in the next section), then i is viewed as a nonoverlapping frequency candidate, and we denote the AFS of the jth nonoverlapping frequency as Np (kp , j) = Nx (kp , i), ωp (j) = i (29) where kp is determined from the DFSB index where i is taken as a nonoverlapping frequency and ωp (j) means the jth nonoverlapping frequency candidate recorded. Repeat the same procedure for all i ’s, and as the result, we may get the nonoverlapping frequency set {ωp (j), j = 1, . . . , HL , HL ≤ L1 } and its corresponding AFS set {Np (kp , j), j = 1, . . . , HL }. Step 3) Find the maximum of the AFS set {Np (kp , j), j = 1, . . . , HL }, and correspondingly, the nonoverlapping frequency ωp1 ∈ (0, π) and its DFSB index can be determined. Step 4) For the digital frequency ranging from π to 2π, we follow the same procedure described in Steps 1)–3) (2) (2k) using |Ym (ej )| and |Xdp (ej((/M )+2(k−1)(π/M )))|, k = 1, . . . , M , to determine the nonoverlapping frequency ωp2 ∈ (π, 2π). C. Estimation of Nonoverlapping Frequencies in Discrete Frequency Domain In Section III-B, we have presented the estimation of nonoverlapping frequencies in the frequency domain systematically, where the nonoverlapping frequency ωp and its related DFSB index 2kp − 1 is determined from Xdp (ejω ). However, practically, we use N -point discrete Fourier transform (DFT) of y(n) to approximately estimate Xdp (ejω ). Essentially, N -point DFT is the sample version of DTFT, and DFT is defined in the discrete frequency domain. Regarding the estimation accuracy of ωp , we need to consider two issues: 1) the spectrum leakage due to N -point DFT and 2) the discrete nonoverlapping frequency estimation bias due to the DFT. First, considering the spectrum leakage of the N -point DFT (although the Hann window can be employed to ease this effect) and other noises (such as quantization noise and thermal noise), in general, the estimated Xdp (ejω ) in its DFSBs may not be zeros for some frequencies, while they are zeros in the ideal case. Therefore, when the amplitude spectrum at a frequency is small to some extent, we consider that it is caused by spectrum leakage and noises, and its effect should be neglected. Hence, it is reasonable to employ a preprocessing before the selection of the nonoverlapping frequency ωp , where a soft thresholding is applied on Xdp (ejω ) Xdp (ejω )= 0 Xdp (ejω ) if |Xdp (ejω )|≤εx max |Xdp (ejω )| others (30) where the predefined threshold parameter εx is related to the window function used for DFT, signal-to-noise ratio (SNR), and quantization bits. Second, since the spectrum of y(n) only can be obtained by an N -point DFT, which is a discrete frequency spectrum, the nonoverlapping frequency estimation bias is inevitable. The larger N will lead to small nonoverlapping frequency estimation bias at the higher computational complexity. A tradeoff between estimation bias and the length of DFT will be taken into account in practical applications. Simulation results showed that the estimation accuracy of Algorithm I is acceptable. D. Analysis of Signal Spectrum Sparsity In this section, the definition of the spectrum sparsity is given, and the requirement of the spectrum sparsity for our proposed SS-BLTSE algorithm will be discussed. With the definition of the spectrum of Xdp (ejω ), which is approximately estimated by the output spectrum of TIADC denoted as Y (ejω ), the spectrum sparsity of the input signal is defined as SSL = Nk /NT , where Nk denotes the number of the nonzero spectrum components of Xdp (ejω ) and NT denotes the total number of the spectrum components of Xdp (ejω ). As discussed earlier, for our proposed SS-BLTSE algorithm, it asks for certain signal spectrum sparsity. For making the problem manageable, we define two types of amplitude spectrum of the analog input signal: the range continuous spectrum and the discrete spectrum. If the amplitude spectrum is the continuous function over certain positive frequency range Rf (in hertz), such spectrum is termed as a continuous spectrum defined over Rf˙ ; if the spectrum does not satisfy the definition of the range continuous spectrum, it is termed as the discrete spectrum. It is easy to understand that, if the continuous spectrum defined on (fx , fx + Rf ) and Rf < fs /2, fs is the sampling rate, then it is expected that its digital spectrum will exist in (ωx , (ωx + DRf )) in (0, π), where ωx = 2πfx /fs and DRf = 2πRf /fs . As the result, the spectrum sparsity SSL will be smaller than 1. From (16), it is clear to see that Ym (ej ) is the summation of Xdp (ejω ) and its shifted version at the frequencies of (/M + 2π(k − 1)/M ), where k = 1, 2, . . . , M . Without loss of generality, let us assume that the spectrum Xdp (ejω ) exists from ωx to (ωx + DRf ) in (0, π) (corresponds to the ZOU AND XU: BLIND TSE USING SOURCE SPECTRUM SPARSITY IN TIME-INTERLEAVED ADCs continuous amplitude spectrum case). If the nonoverlapping frequency exists in Ym (ej ) (ωp ∈ (0, π)) and its corresponding DFSB index is kp , then the frequency of (ωp /M + 2(kp − 1)π/M ) will locate in the range of (ωx , (ωx + DRf )), and we have kp < M/2 derived from (ωp /M + 2(kp − 1)π/M ) < π. According to the symmetry of Ym (ej ), ωn = 2π − ωp is also a nonoverlapping frequency, and its corresponding DFSB related index is kn . Since Xdp (ejω ) is also of symmetry, with the definition of DFSB, it has kn = M + 1 − kp . By the careful evaluation of the frequencies of (ωn /M + 2(kp − 2)π/M ) and (ωn /M + 2(kp − 1)π/M ), we have the following relation: (ωn /M + 2(kp − 2)π/M ) < (ωp /M + 2(kp − 1)π/M ) < (ωn /M + 2(kp − 1)π/M ). Moreover, from kp < M/2 and kn = M + 1 − kp , we can drive kn − 1 > kp − 1 > kp − 2. Therefore, we can directly get the frequency relations as follows: (ωn /M + 2(kn − 1)π/M ) > (ωn /M + 2(kp − 1)π/M ) > (ωn /M + 2(kp − 2)π/M ). As the result, according to Step 2) of Algorithm I, the amplitude of Xdp (ejω ) at the frequencies of (ωn /M + 2(kp − 2)π/M ) and (ωn /M + 2(kp − 1)π/M ) is zero. It comes to the conclusion that the frequencies of (ωn /M + 2(kp − 2)π/M ) and (ωn /M + 2(kp − 1)π/M ) do not locate in the range of (ωx , (ωx + DRf )). With these results, we get the following relations: a) ωx > ωn /M + 2(kp − 2)π/M , and b) ωx + DRf < ωn /M + 2(kp − 1)π/M . To get the expression of DRf , let DRf = DRf + ωx − ωx . Making use of the relations a) and b), with some derivations, we reach the following result: DRf < 2π/M . Equivalently, we have SSL = DRf /π < 2/M . It is clear that the aforementioned derivations are reversible, and when SSL < 2/M , we can determine the nonoverlapping frequency ωp ∈ (0, π) and its related DFSB index kp through Algorithm I and the relations (ωn /M + 2(kp − 2)π/M ) > (ωp /M + 2(kp − 2)π/M ) and (ωn /M + 2(kp − 1)π/M ) < (ωp /M + 2kp π/M ). As shown earlier, our SS-BLTSE algorithm works for the range continuous spectrum of the input signal when SSL is smaller than 2/M . In addition, if the spectrum of the signal is discrete (discrete spectrum), for the proposed SS-BLTSE algorithm, the nonoverlapping frequencies in Ym (ej ) are determined by SSL and the amplitude spectrum distribution of the analog input signal. So far, there is no close form of the SSL derived for the discrete spectrum, which needs further research. Evaluating (16) shows that, for the discrete spectrum case, the nonoverlapping frequencies normally exist since the summation of Xdp (ejω ) and its shifted version at the frequencies of (/M + 2π(k − 1)/M ) (k = 1, 2, . . . , M ) will not mix up all discrete amplitude spectrum components. Moreover, it is easy to understand that the requirement of DRf (or SSL ) will be looser than those of the continuous spectrum case (DRf < 2π/M or SSL < 2/M ). For example, the discrete spectrum can be viewed as the summation of multiple continuous spectra defined over bandwidth DRf (j), where j = 1, 2, . . . , K (K is an integer, which is related to the distribution of the input spectrum). As discussed earlier, if DRf (j) < 2π/M , then we can get the candidate nonoverlapping frequency denoted as ωp (j). From ωp (j)(j = 1, 2, . . . , K), we are able to determine the final nonoverlapping frequency ωp . Therefore, it is straightforward to reach the conclusion that, for the discrete spectrum, 2407 when all DRf (j) < 2π/M , the nonoverlapping frequency may exit. Since SSL = K j=1 DRf (j)/π, then SSL may be larger than 2/M (looser requirement). In general, the smaller SSL is, the better the performance that our SS-BLTSE algorithm gives. Experiments show that, if the signal is with the discrete spectrum, our proposed SS-BLTSE algorithm is able to achieve good TSE performance. E. Summary of the Proposed SS-BLTSE Algorithm In this section, we summarize the proposed SS-BLTSE algorithm. It is noted that the digital spectrum Xdp (ejω ) will be approximately estimated by the N -point DFT of the y(n). The SS-BLTSE includes the following steps: Step 1) to calculate the N -point DFT of the output of TIADC y(n) and employ a preprocessing on the resulting spectrum using (30); Step 2) to calculate the N/M -point DFT (Ym (ej )) of the output of sub-ADCs ym (n), m = 0, 1, . . . , M − 1; Step 3) to determined the nonoverlapping frequency points ωp1 and ωp2 and their DFSB-related indices kp1 and kp2 by Algorithm I; Step 4) to get Ym (ejωp1 ) and Ym (ejωp2 ); Step 5) to calculate Δm1 using (25) and Δm2 using (26); Step 6) To compute the mean of Δm1 and Δm2 as the estimated Δm . Finally, we simply discuss the computational complexity of the proposed SS-BLTSE algorithm. As described in the previous section, an M -channel TIADC system generates a total of N discrete data samples. The computational complexity of the SS-BLTSE algorithm mainly comes from three parts: 1) An N -point Fast Fourier Transform (FFT) to get Y (n) gives a computational complexity of N log N ; 2) for each sub-ADC output, the calculation of Ym (i) needs M times (N/M )-points FFT, which causes the computational complexity of N log(N/M ); and 3) the preprocessing and determination of the best ωp ask for the computational complexity of 2N . In conclusion, the computational complexity of this proposed SS-BLTSE algorithm is about N log N + N log(N/M ) + 2N . Hence, the SS-BLTSE algorithm can be considered as a low computational complexity estimation method compared with other blind TSE methods [15]. F. Effective Coarse-To-Fine SS-BLTSE Algorithm To further reduce the computational complexity of our SS-BLTSE algorithm, an effective coarse-to-fine SS-BLTSE algorithm can be developed for short-time stationary signals as follows: 1) Setting N = H ∗ NH = H ∗ M ∗ NM H as the total data length of the TIADC output, where H, NH , and NM H are integers. 2) Computing one full NH -point DFT to get the spectrum of the TIADC output and one full NM H -point DFT to get the spectrum of the ADC-0 output, denoted as Y (i)(i = 0, . . . , NH − 1) and Y0 (i) (i = 0, . . . , NM H − 1), respectively. 2408 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 TABLE I TSE R ESULTS OF THE SS-BLTSE A LGORITHM 3) Determining the nonoverlapping frequency ωp in (0, π) in Ym (ej ) and its DFSB-related index kp according to Algorithm I. It is noted that the values Y (i)(i = 0, . . . , NH − 1) and Y0 (i)(i = 0, . . . , NM H − 1) are available. If the pth frequency point in Y0 (i) is found as the nonoverlapping frequency point, then the nonoverlapping frequency ωp should locate in the range of (2π(p − 1)/NM H , 2πp/NM H ). 4) For the mth TIADC sub-ADC, it can be seen that only H frequency points (H = (N/M )/NM H ) will be determined accordingly in the range of (2π(p − 1)/NM H , 2πp/NM H ), denoted as i1 , i2 , . . . , iH . Therefore, we only need to compute the N/M -point DFT for these H frequencies. 5) Determining the largest spectrum value among the resultant spectra from step 4), which corresponds to the nonoverlapping frequency point that we are searching for. 6) Determining the nonoverlapping frequency ωp in (π, 2π) and its DFSB-related index kp similarly. It can be seen that the computational complexities of steps 2), 4), and 5) are NH log NH + NM H log NM H , H ∗ M ∗ N/M = H ∗ N , and H, respectively. The computational complexity of step 3) is less than 2NH . Therefore, the computational complexity of the coarse-to-fine SS-BLTSE algorithm is about NH log NH + NM H log NM H + HN + 2NH + H. It is clear that the coarse-to-fine SS-BLTSE algorithm reduces the computational complexity by scarifying some estimation accuracy considering the practical hardware constraint. Essentially, the choice of the value of H will affect the estimation of the nonoverlapping frequency (ωp ). If H is too large, it may lead to the wrong selection of ωp . If H is too small, then the reduction of the computational complexity is limited. Therefore, the choice of H should be a tradeoff between the estimation accuracy and the computational complexity. To evaluate the impact of H on the estimation accuracy, several experiments have been conducted, and one result is listed in Experiment 6. It recommended that, if the spectrum sparsity of the input signal satisfies the requirements and the TSE accuracy is predefined, H can be chosen as an integer equal or smaller than 16, which is able to give a good approximation of the estimation at the lower complexity of about 1/H of that of the SS-BLTSE algorithm. IV. S IMULATION AND E XPERIMENTAL R ESULTS This section presents several computer simulation results for evaluating the performance of the proposed SS-BLTSE algorithm for a four-channel TIADC system. In part A, we simulated an infinite precision TIADC system which is assumed to have only timing mismatches. In part B, an experiment is conducted to measure the timing skews of a real TIADC system. A. Performances of the Proposed DFT-BLTSE Algorithm for Simulated TIADC System In this part, an infinite precision TIADC system with only timing mismatches was simulated, where the timing skews were assumed to be Δt0 = 0, Δt1 = 0.03/fs , Δt2 = 0.05/fs , and Δt3 = −0.02/fs , respectively, where fs is the overall sampling frequency of the TIADC system and was set to be 4 GHz. Other channel mismatches in the TIADC system have not been addressed in this paper (readers can refer to [8] for more discussions). Experiment 1: This experiment is carried out to evaluate the TSE capability of the proposed SS-BLTSE algorithm under noiseless condition. The input signal is set as three cases. 1) A sinusoidal signal with the frequency of 1.3 GHz and phase of 0.3π. 2) The multifrequency input signals given by xa (t) = 32 ak sin(2πfk t + θk ) (31) k=1 where fk , ak , and θk are uniform distributed random sequences. The signal frequency fk ranges from 1 Hz to 1.95 GHz, ak ranges from 5 to 10, and θk ranges from −π to π, respectively. 3) A bandlimited chirp signal with frequency ranges from 1.7 to 1.9 GHz. Other simulation parameters are set as follows: fs = 4 GHz, N = 65536, M = 4, the sample length of each sub-ADC is N/M = 16384, εx = 0.06, and the Hann window is used. The simulation results are listed in Table I. From Table I, it is clear to see that the SS-BLTSE algorithm is able to estimate the channel timing skews accurately nearly without estimation error when the input signal is a singlefrequency sine wave (the spectrum sparsity of a sine wave is high). For a multifrequency sine wave (considered as a wideband input signal), the TSE accuracy degrades. However, it is encouraged seeing that the proposed SS-BLTSE algorithm performs quite well to give the high estimation accuracy with precision up to 2%. From Table I, it is also noted that, when the input is the wideband chirp signal, our proposed SS-BLTSE algorithm still works well. However, its estimation accuracy decreases compared with those of the other two cases, where the spectra of the signals are discrete. Experiment 2: This experiment aims at evaluating the impact of fi (input frequency of a sine wave) and N (DFT length) on the TSE accuracy of the SS-BLTSE algorithm when SNR is ZOU AND XU: BLIND TSE USING SOURCE SPECTRUM SPARSITY IN TIME-INTERLEAVED ADCs Fig. 3. RMSE versus input frequency. Fig. 4. RMSE versus εx . set to be 30 dB. The root-mean-square error (RMSE) is taken as the measure, which is defined as −1 1 M RM SE = 20 log10 (Δm − Δ m )2 (dB) (32) M − 1 m=1 where Δm and Δ m denote the true and the estimated relative timing skew for the mth sub-ADC, respectively. The other simulation parameters are set as the same as those in Experiment 1 with a single sine wave. The simulation result is shown in Fig. 3. From Fig. 3, it is noted that the RMSE over all frequencies is bounded by −40 dB and the TSE accuracy of the SS-BLTSE algorithm increases with the increase of the input frequency. This is because, from (25) and (26), the estimated Δm is inversely proportional to the input frequency. Moreover, it can be seen that the RMSE decreases with the increase of N . This result suggests to choose N as 65 536, which is the tradeoff between the TSE accuracy and the computational complexity. Experiment 3: This experiment evaluates the impact of the parameter εx on the TSE accuracy using the RMSE measurement defined in (32). The simulation parameters are set as the same as those in Experiment 1 except that the parameter εx varies from 0.02 to 0.13 with a step size of 0.002. The RMSE performance is illustrated in Fig. 4. It is clear to see that the TSE performance with the multifrequency sine wave input is inferior to that of the single-frequency sine wave input. Moreover, it can be seen that the selection of parameter εx has big impact on 2409 Fig. 5. RMSE versus Δt2 . TABLE II TSE R ESULTS (N OISELESS, B = 10 bit, fin = 1.3 GHz, AND N = 65536) the TSE performance of the SS-BLTSE algorithm, particularly when the multifrequency sine wave is used (the line with small ring). In this case, when εx is smaller than 0.04 or larger than 0.08, the TSE performance degrades, which indicates that the nonoverlapping frequency ωp. may be estimated wrongly. More experimental results show that, when the SSL of the input signal increases, we need to select εx more carefully. From our experimental results, we suggest to select εx in the range of 0.045–0.075. Experiment 4: This experiment aims at evaluating the impact of the variation of the timing skews on the TSE accuracy of the SS-BLTSE algorithm, where Δt1 = 0.03Ts , Δt3 = −0.02Ts , and Δt2 is set from −0.15Ts to 0.15Ts with a step size of 0.005Ts . Other simulation parameters are set as the same as those in Experiment 1 with the input signal generated by (31). The simulation result is shown in Fig. 5. It is interesting to see that the performance of the SS-BLTSE algorithm is robust 2410 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 Fig. 6. RMSE versus SNR for the SS-BLTSE algorithm. TABLE III TSE R ESULTS FOR SS-BLTSE A LGORITHM AND C OARSE -T O -F INE SS-BLTSE A LGORITHM to the timing skew variation for this specific experimental setup; the largest RMSE is about −63 dB. Experiment 5: This experiment is carried out to evaluate the influence of the quantization effect and the additive noise on the TSE accuracy of the SS-BLTSE algorithm. The simulation parameters are set as the same as those in Experiment 1 with a 1.3-GHz sine wave as the input signal. The input signal is full scale quantized with B bits (B can be selected to be a different integer and is specified in Table II). The simulation results are listed in Table II and Fig. 6. In Table II, the TSE performance of the SS-BLTSE algorithm is compared that of Sin-fit algorithm [22] (it is noted that the Sin-fit algorithm is a nonblind TSE algorithm). From Table II, we can see that the proposed SS-BLTSE outperforms the Sin-fit TSE algorithm, which means that the quantization has more impacts on the performance of the Sin-fit TSE algorithm than that of the SS-BLTSE. The worst estimation error of SS-BLTSE is about 0.5% for this experiment setup. Fig. 6 is obtained by 100 independent runs for each SNR condition. From Fig. 6, it is clear to see that the tendency of the impact of additive noise on the TSE algorithm accuracy is similar. For a given quantization bit, the TSE algorithm accuracy increases with the increase of SNR. For a given SNR, the quantization effect with different quantization level is almost the same. This experimental result reveals that the additive noise has stronger impact on the TSE algorithm accuracy than that of the quantization for the SS-BLTSE algorithm. It is clear to see that the SS-BLTSE algorithm is able to provide high TSE accuracy over a wide range of SNR from 20 to 50 dB in this experiment setup. Experiment 6: This experiment is carried out to evaluate the performance of the coarse-to-fine SS-BLTSE algorithm compared with that of the SS-BLTSE algorithm. The parameter settings are as follows: H = 16, NH = 4096, N = H ∗ NH = 65536, and M = 4. The testing input signal is a 32 multitone Fig. 7. Developed four-channel 12-b 320-MHz TIADC system. sinusoidal signal, which is randomly generated by using the signal defined in (31). The simulation results are listed in Table III. From Table III, it is clear to see that the performance of the coarse-to-fine SS-BLTSE algorithm matches very well with that of the SS-BLTSE algorithm. It is desired to note that the computation cost of the coarse-to-fine SS-BLTSE algorithm is about one-thirtieth of that of the SS-BLTSE algorithm. B. Performance of the SS-BLTSE Algorithm for a Real TIADC System Our research team has worked on the TIADC technology since 2007, including the TIADC data capture system design and implementation, TIADC system mismatch measurement, and compensation techniques. In this section, to further validate the performance of the proposed SS-BLTSE algorithm, we carried out one experiment using the captured output data from a 4 × 80 Million samples per second 12-b TIADC system developed by our team, which is shown in Fig. 7. ZOU AND XU: BLIND TSE USING SOURCE SPECTRUM SPARSITY IN TIME-INTERLEAVED ADCs 2411 TABLE IV E STIMATED Δm R ESULTS FOR R EAL TIADC S YSTEM Experiment 7: In this experiment, we just take the data captured from the output of the real TIADC system. The data used were captured when the input is a sine wave with the frequency of 48.65 MHz. The timing skews of this TIADC system have been estimated by three different TSE algorithms: the SS-BLTSE algorithm, the Sin-fit TSE algorithm [22], and the blind TSE algorithm proposed in [13]. The estimated timing skews are listed in Table IV. Since we do not know the true value of the timing skews of this real TIADC system, to validate the estimation effectiveness of the TSE algorithms, we take the estimated timing skew parameters listed in Table IV to compensate the timing mismatches by using the same compensation algorithm proposed in [20] where a multirate filter bank-based timing mismatch compensation method is used. For visualization purpose, the normalized output spectra of the TIADC system with and without timing skew compensation are plotted in Fig. 8. The output spectrum spurs without compensation are shown in Fig. 8(a). From Fig. 8(b)–(d), we can clearly see that the spectrum spurs due to the timing skews of the real TIADC system have been effectively compensated using the estimated timing skews by our proposed SS-BLTSE algorithm, the blind TSE method in [13], and Sin-fit TSE algorithm in [22]. Accordingly, the spur spectrum compensations are about 57.8, 20.8, and 62.8 dB, respectively. These experimental results indirectly validate the TSE accuracy of the proposed SS-BLTSE method. V. C ONCLUSION A novel real-time blind TSE algorithm, called the SS-BLTSE algorithm, has been proposed in this paper. The channel timing skew parameters can be determined effectively through the output digital spectra calculated by the DFT of the TIADC output data and its corresponding sub-ADC output data, which has the computational complexity of N log(N ). For the narrow or wideband input signals, the SS-BLTSE algorithm gives high TSE accuracy in the range of 0.01%–5% for the simulated TIADC systems under different noise levels and different input signals. Intensive simulation results validate the accurate TSE capability of the SS-BLTSE algorithm, and it is encouraging to see that the SS-BLTSE algorithm is robust to the variation of the channel timing skews, input SNR, and input quantization effect. Moreover, the captured data from a real four-channel 320-MHz 12-b TIADC system have been used to evaluate the TSE accuracy of the SS-BLTSE algorithm through the compensated output spectrum of the TIADC system, which indirectly shows that the TSE performance of the SS-BLTSE algorithm is comparable to that of the nonblind Sin-fit TSE algorithm [22]. We are quite confident to conclude that the proposed SS-BLTSE algorithm can be taken as a good candidate when Fig. 8. Output spectra of the real TIADC system. (a) Uncompensated. (b) Compensated (timing skew estimated by our proposed SS-BLTSE). (c) Compensated (timing skew estimated by blind TSE in [13]). (d) Compensated (timing skew estimated by Sin-fit in [22]) (compensation algorithm by Prendergast et al. [20]). 2412 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 61, NO. 9, SEPTEMBER 2012 the online high accurate TSE is required. It is also suitable for the hardware implementation since only DFT is involved. ACKNOWLEDGMENT The authors would like to thank their former master graduate students, Z. D. Zhu, S. L. Zhang, and X. Chen, for their contributions in the TIADC system development and the implementation of the timing mismatch compensation algorithms. The authors would also like to thank Prof. Y. C. Lim from Nanyang Technological University, Singapore, for his kind help in capturing the output data from the developed TIADC system. R EFERENCES [1] W. C. Black and D. A. Hodges, “Time interleaved converter array,” IEEE J. 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Yue Xian Zou (S’96–M’00–SM’08) received the B.Sc. degree in electrical engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1985, the M.Sc degree in signal and information processing from UESTC in 1989, and the Ph.D. degree in communication and digital signal processing from The University of Hong Kong, Pokfulam, Hong Kong, in 2000. From 1989 to 1996, she was a Lecturer with the Department of Electronic Engineering, UESTC. Since 2005, she has been with the Shenzhen Graduate School, Peking University, Shenzhen, China, where she is an Associate Professor, the Director of the Advanced Digital Signal Processing Laboratory, and the Associated Dean of the School of Computer and Information Engineering from 2007 to 2010. She serves as the evaluation/peer review expert for the National Natural Science Foundation of China program, Shenzhen Bureau of Science Technology and Information. She was the Founding Chair of Women in Engineering Singapore (2005). From 2006, she was actively involved in national and international academic activities. She serves as the paper reviewer for several IEEE journals and international conferences, the Organizing Cochairman for IEEE International Conference on Nano/Micro Engineered and Molecular System 2009, the Track Chair of IEEE International Conference on Information and Communications Security 2011, and a Technical Program Committee member of IEEE International Conference on Green Circuits and Systems. She is currently working on a high-speed and high-resolution analog-to-digital converter research project. Her research work includes digital signal processing, adaptive signal processing, array signal processing, and video signal processing. Dr. Zou was the recipient of the award named as “Leading Figure for Science and Technology” by Shenzhen Municipal Government (2009). Xiang Jun Xu received the B.Eng. degree in electronic and information science from Beijing Normal University, Beijing, China, in 2006. He is currently working toward the M.Sc. degree in the School of Electronics Engineering and Computer Science, Peking University, Shenzhen, China. His research interests include time-interleaved analog-to-digital conversion and digital signal processing.