IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER A Flexible and Scalable Structure to Compensate Frequency Response Mismatches in Time-Interleaved ADCs Christian Vogel, Member, IEEE, and Stefan Mendel, Member, IEEE Abstract In this paper, we present a flexible and scalable structure to compensate frequency response mismatches in time-interleaved analog-to-digital converters (TI-ADCs). The flexibility of the structure allows for designing compensation filters independent of the number of channels that can achieve any desired signal-to-noise ratio due to the scalability of the structure. Therefore, the compensation structure may be used to compensate time-varying frequency response mismatches in TI-ADCs, as well as to reconstruct uniform samples from nonuniformly sampled signals. We analyze the compensation structure, investigate its performance, and demonstrate application areas of the structure through numerous examples. Index Terms Analog-to-digital converter (ADC), compensation, frequency response mismatches, reconstruction, time-interleaved, time-varying filter. I. INTRODUCTION T IME-INTERLEAVED analog-to-digital converters (TI-ADCs) could be a feasible solution for emerging communication standards requiring high-performance ADCs. They can achieve high-speed sampling rates [1] and low power consumption [2]. Moreover, advanced analog design techniques [3], as well as integrated digital calibration [4], allow for medium-to-high-resolution TI-ADCs. As illustrated in the model shown in Fig. 1, a TI-ADC is an array of -parallel-channel ADCs operating in a TI mode [6]. Each sample of the analog input signal is taken by a different channel ADC in a periodic manner. At the output of the TI-ADC, the samples among all channels are combined and result in the digital output signal. Hence, compared to a single-channel system, time interleaving relaxes the conversion requirements. To be specific, the time for the quantization process is relaxed by the factor, where is the number of channels. The sampling process itself, however, has to resolve the full input-signal bandwidth. Therefore, samples are taken at an -times lower rate, but the switching requirements are the Manuscript received July 18, 2008; revised November 06, First published February 18, 2009; current version published November 04, The work of C. Vogel was supported by the Austrian Science Fund (FWF) s Erwin Schrödinger Fellowship J2709. This paper was recommended by Associate Editor H. Johansson. C. Vogel is the Signal and Information Processing Laboratory, Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology Zurich (ETH Zurich), 8092 Zurich, Switzerland ( c.vogel@ieee.org). S. Mendel is the Signal Processing and Speech Communication Laboratory, Graz University of Technology, 8010 Graz, Austria ( stefan. mendel@tugraz.at). Digital Object Identifier /TCSI Fig. 1. Model of a TI-ADC M linear channels. The box named IT/SC is an impulse-train-to-discrete-time-sequence converter (IT/SC) [5]. same as for a single-channel system. The drawback of the TI-ADC structure is additional errors caused by mismatches among the channel ADCs [7] [10]. Since each sample is taken by a different channel ADC in a periodic manner and all channels are slightly different due to mismatches, a TI-ADC can be considered as a periodically time-varying system. In consequence, besides the expected spectrum of the sampled signal, we also observe modulated spectral components in the output spectrum. These modulated components, i.e., the mismatch errors, mainly degrade the performance of TI-ADCs and limit their applicability. Fortunately, the aggressive scaling of integrated-circuit technology makes digital correction of analog circuits increasingly attractive [11] [13] and therefore the digital correction of TI-ADCs as well [14]. Previously, the compensation of gain, offset, and timing mismatches of TI-ADCs has been considered [15] [21]. Particularly, the digital compensation of timing mismatches (linear-phase mismatches [22]) has been extensively investigated [23] [32]. To further push the limits of TI-ADCs, however, we have to consider the compensation of the complete frequency response mismatches among all channels [14]. In [33], the authors have corrected frequency response mismatches in the frequency domain but have not shown how to extend the method to continuous data processing. A comprehensive /$ IEEE

2 2464 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER 2009 method to compensate frequency response mismatches by using least squares filter design methods has been introduced in [34]. The filter design method depends on the number of channels and has polynomial complexity. It is therefore useful for offline design for a relatively small number of channels. A different least squares filter design method reduced complexity, which is a straightforward extension of [25], is investigated in [35]. Another interesting approach has been demonstrated in [36]. The authors have analyzed bandwidth mismatches and introduced an equivalent discrete-time filter model. By solving polynomial equations, they have derived bandwidth-mismatch compensation filters. As in a typical single-channel ADC, we do not explicitly equalize the frequency response; it is not necessary in a TI-ADC either. If such an equalization is needed, as, for example, in communication systems, a channel equalizer can be used [37]. In [38], this fact and a first-order approximation of the design equations simplify the filter design and the final correction structure in the two-channel case. Unfortunately, the authors in [38] did not show how to extend the method to channels. Finally, it should be noted that, during the revision of this paper, we became aware of a related frequency response-mismatch compensation method [39] that introduces an efficient realization of the general compensation idea developed in this paper. We introduce a flexible and scalable structure, which has its roots in the th-order inverse introduced by Schetzen [40], to compensate frequency response mismatches. In Section II, we present a system model of a TI-ADC frequency response mismatches. By explicitly defining the mismatches in the frequency responses, the model allows us to formally decompose the output signal of a TI-ADC into a signal that is caused by mismatches, i.e., the error signal, and a signal that is not affected by them, i.e., the desired signal. Furthermore, the model explains the relation to periodically time-varying systems and the thereby arising modulation effects. In Section III, we introduce the compensation structure. The compensation structure basically reconstructs the error signal and subtracts it from the TI-ADC output signal to obtain the desired signal. The achievable performance in terms of a signal-to-noise ratio (SNR) for arbitrary finite-energy signals is analyzed in Section IV. We consider the flexibility of the structure in Section V. In particular, we show that, in general, the filter design complexity linearly scales the number of channels and that, for certain types of frequency response mismatches, the filter design complexity becomes independent of. Therefore, the structure can compensate an arbitrary number of time-varying channels since the compensation filters can be designed in real time on chip. The compensation of time-varying channel-frequency response mismatches in TI-ADCs and the reconstruction of uniformly sampled signals from nonuniformly sampled signals are examples. In Section VI, the scalability of the structure is highlighted, which allows one to achieve arbitrary SNRs by cascading compensation filters. We exemplify the flexibility and scalability of the structure by compensating frequency response mismatches and bandwidth mismatches and by reconstructing uniformly sampled signals from nonuniformly sampled signals in Section VII. Finally, we give some concluding remarks in Section VIII. II. SYSTEM MODEL In this section, we derive a system model of a TI-ADC that explicitly introduces an error signal accounting for frequency response mismatches and a signal representing the desired signal. Furthermore, the TI-ADC model shows the relation to linear periodically time-varying filters. Assuming linear time-invariant channels, we can model an -channel TI-ADC through the impulse responses,,of linear filters. Therefore, the output of the th channel can be written as (cf. Fig. 1) where is the overall sampling period of the TI-ADC and is the Dirac-delta function. The time shift accounts for the time interleaving, and denotes the sampling period of each channel. Hence, each channel runs on an -times lower rate compared to the overall sampling rate. Adding the outputs of the channels gives the sampled output of the TI-ADC in (2) (1) and by intro-, we obtain Alternatively, by replacing ducing the index (1) (2) where for. Hence, the timevarying impulse response is the -periodic extended version of, i.e.,, which is given by the discrete-time Fourier series (DTFS) pair (4) and (5) [41, Ch. 4.2]. By taking the continuous-time Fourier transform (CTFT) of (3), we can express the output in the frequency domain as [42] The frequency responses are the DTFS of the channel frequency responses. If we assume a band-limited input signal, i.e., the CTFT of fulfills (3) (4) (5) (6) (7) (8)

3 VOGEL AND MENDEL: FLEXIBLE AND SCALABLE STRUCTURE TO COMPENSATE MISMATCHES IN TI-ADCS 2465 we can write (6) in discrete time as [5] (9) (10) (11) The discrete-time channel frequency response is the -periodic extended version of the band-limited channel frequency response. In a TI-ADC out mismatches, all channel frequency responses are identical, i.e., for, where is the desired frequency response. Any deviation from the desired frequency response results in mismatches causing an error signal at the output of the TI-ADC. To introduce the error signal, we define the difference between the th channel frequency response and the desired frequency response as the channel-frequency response mismatch (12) Applying the DTFS to (12) results in the transformed channelfrequency response mismatch (13) being the discrete-time impulse. Using (13), we can rewrite (9) as where (14) (15) is the discrete-time Fourier transform (DTFT) of the desired signal and (16) is the DTFT of the error signal. For a system out mismatches, for, and the error signal vanishes. The system model reduces to the one of a singlechannel ADC frequency response. Applying the inverse DTFT to (14) results in where is the inverse DTFT of (15) and is given by (17) (18) as shown in the Appendix A. In (18), the error signal is expressed in terms of the -periodic time-varying frequency Fig. 2. Discrete-time system model of a TI-ADC M channels for bandlimited input signals x(t). response, i.e., the channel-frequency response mismatch. This representation reflects the periodically time-varying nature of a TI-ADC mismatches. For each time index, the signal excites a different frequency response, where the channel-frequency response mismatch accounts for the mismatches between the channel frequency response and the desired frequency response. The final discrete-time system model of the TI-ADC is shown in Fig. 2. The band-limited continuous-time signal is uniformly sampled the sampling period and converted by the IT/SC into the discrete-time signal [5] (19) The signal is fed into the time-invariant filter and results in the desired signal out mismatches. Furthermore, the signal is filtered by complex discrete-time filters frequency responses and is modulated afterward by. These signals are added up and result in the error signal. The structure of time-invariant filters and modulators can also be represented by the single time-varying filter, which is indicated by the dashed box in Fig. 2. Finally, the desired signals and the error signal add up to the output signal. III. COMPENSATION STRUCTURE The compensation structure basically tries to reconstruct the error signal from the TI-ADC output signal and then subtracts the reconstructed error signal from. Fig. 3 shows the principle of the compensation structure. The discrete-time system model of a TI-ADC introduced in Section II is shown on the left, and the compensation structure the time-varying compensation filter frequency response (20) is shown on the right. The frequency response of the compensation filter is the channel-frequency response mismatch given by (12) that is normalized by the desired frequency

4 2466 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER 2009 Fig. 5. Model for mismatch error propagation. Fig. 3. Discrete-time system model of an M-channel TI-ADC and the proposed compensation structure. Fig. 4. Equivalent signal-flow graph of Fig. 3. response, i.e., the normalized channel-frequency response mismatch. For the following discussion, we assume that the compensation filter has been sufficiently approximated so that the error due to the filter design can be neglected. According to Fig. 3, the signal after compensation is (21) The output of the TI-ADC is used to obtain the reconstructed error signal, which is ideally identical to the error signal. In this case, the subtraction after the compensation filter given by (21) would lead (17) to and would cancel the error signal. However, as the derivations will show, the signal is slightly different from, and we will not achieve perfect mismatch compensation. For the analysis of the difference, we use the signal-flow graph shown in Fig. 4. It explicitly indicates all involved signal components of the linear system model shown in Fig. 3. As shown in Fig. 4, we can write in (21) as (22) where is the output due to and is the output due to. The output in (22) is given (20) and (18) as (23) and since we exactly obtain, the subtraction in (21) cancels the mismatch error in. Unfortunately, the term in (22) results in (24) and the final output after compensation becomes (21), (17), (22) (24) (25) The remaining error after compensation is the output of the compensation filter driven by the error signal, which is the output of the filter. As for a reasonably designed TI-ADC, the energy of the error signal is much smaller than the energy of the desired signal, and because, for a reasonably chosen desired frequency response, the error signals and are basically determined by the same time-varying filter, the energy of the error signal will be considerably smaller than the energy of the error signal. A detailed performance analysis of the compensation structure will be given in the next section. IV. PERFORMANCE ANALYSIS To evaluate the performance of the compensation structure, we relate the desired signal to the error signal by defining the SNR before compensation as where the signal energies are given by [41] db (26) (27) (28) where we have used (17) for the simplification. Accordingly, we define the SNR after compensation as db (29) (30) For sufficiently approximated compensation filters, the remaining error after compensation is, and (30) simplifies (25) to (31) The error signal is fed into the compensation filter and gives the new error signal, which is explicitly shown in Fig. 5 and can be written as (32)

5 VOGEL AND MENDEL: FLEXIBLE AND SCALABLE STRUCTURE TO COMPENSATE MISMATCHES IN TI-ADCS 2467 After replacing in (31) by (32), we obtain (33) The frequency response of the compensation filter can be split into a magnitude response and a phase response. By taking the maximum magnitude response over all frequencies s and channels s, i.e.,, we can bound (33) by Fig. 6. Equivalent representation of the time-varying compensation filter Q (e ) M time-invariant filters Q (e ) and M modulators e. (34) Assuming that the normalized phase mismatches are small so that holds, we can bound the energy of the error signal from above (28) as where, according to (20) and (12) (35) (36) is the maximum absolute value of the normalized channel-frequency response mismatch between the channel frequency response and the desired frequency response. Therefore, the SNR after compensation becomes (26), (29), (35), and (36) (37) Hence, for all finite-energy input signals, starting and small normalized phase mismatches, we obtain an improvement in the SNR. The better the initial mismatch, the better the absolute SNR improvement after compensation. For example, a maximum mismatch of 3% will result in an improvement of at least 30 db, while a maximum mismatch of 1% will result in an improvement of at least 40 db. As exemplified in Section VII-A, however, the relative SNR improvements regarding the SNRs before compensation are similar. From (37), we can conclude that a bad choice of the desired frequency response can influence the performance of the structure. In general, one should choose that is close to all to obtain good results. V. FLEXIBILITY OF THE COMPENSATION STRUCTURE The frequency response of the compensation filter only depends on the channel-frequency response mismatch and the chosen desired frequency response. db Hence, we have to design mutually independent filters. Therefore, out any modification, the design complexity linearly scales the number of channels. In order to implement the compensation filter, we can use a multirate filter bank, a system of filters and modulators, or a time-varying filter coefficient update [43]. For a finite number of channels, we can go from one representation to another by applying the DTFS and multirate theory [43], [44]. As shown in Section V-C, for certain channel-frequency response mismatches, the design complexity becomes independent of the number of channels and feasible for real-time on-chip filter design. A. Implementation as Multirate Filter Bank For a finite number of channels and by using multirate theory, the time-varying filter may be represented as an -channel maximally decimated multirate filter bank analysis filters and synthesis filters for [44]. An implementation on the low rate using polyphase filters may reduce the requirements on the digital circuit design. B. Implementation as System of Filters and Modulators For a finite number of channels and by using the DTFS as in (10), the compensation filter can be represented time-invariant filters and modulators, as shown in Fig. 6. Unfortunately, in this form, the compensation filter is not of practical use. Even though the input signal and the output signal are real signals, the compensation filter contains complex intermediate signals generated by the complex filters and the complex modulators. We can, however, transform the complex filters to obtain real filters and modulators and, consequently, real intermediate signals. To this end, we express the output of the compensation filter for an odd number of channels in the time domain as where is the inverse DTFT of. Since (38) (39)

6 2468 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER 2009 as shown in Appendix B, we obtain by applying the inverse DTFT to (39) (40) where refers to the complex conjugation of. According to [5], the complex impulse response can be split into an even and an odd part Fig. 7. Cascaded compensation filters. (41) where and are the inverse DTFTs of the frequency responses, respectively (42) (43) By using (40) and (41) and the fact that, where denotes the real part of, we can rewrite (38) as (44) Following the same argumentation for an even number of channels, we obtain (45) This representation of the filters can be exploited for the adaptive identification and compensation of gain and timing mismatches in TI-ADCs [31], [32]. C. Implementation as Time-Varying Filter We can implement the time-varying compensation filter frequency response as a filter where, for each time instant, an appropriate coefficient set is used. Consequently, we are not limited by the number of frequency responses that we are able to compensate for, but the design complexity linearly scales the number of channels. Although the design complexity is an improvement to many methods in the literature, e.g., [33], [34], [45], and [46], the online design of the filters may still be too complex because of speed and power constraints. The online design of the compensation filters, however, is necessary if the environment changes and, therefore, the frequency response mismatches change accordingly. Fortunately, if the frequency response mismatches only depend on a small set of parameters, we can use a prototype filter, e.g., a Farrow filter [47], to efficiently minimize the complexity of the filter redesign process of.for example, first-order frequency response mismatches, i.e., bandwidth mismatches [36], [38], can be compensated by a Farrow filter, as demonstrated in Section VII-C. The Farrow filter is parameterized by the normalized cutoff frequency, reducing the complexity of the filter redesign process to alter the values of some multipliers in the Farrow filter. Hence, we use the same prototype filter for all channels but parametrize it the th normalized cutoff frequency to compensate the th channel-frequency response mismatch. Thanks to the compensation structure, the design complexity for such filters becomes independent of the number of channels and other environmental changes. Indeed, it does not make a difference if we change from one channel to another or out environmental changes. This flexibility of the structure and the scalability of the structure introduced in Section VI have been exploited for efficiently reconstructing nonuniformly sampled signals [28], [29]. The relation of this work to [28] and [29] is shown in Section VII-D. VI. SCALABILITY OF THE COMPENSATION STRUCTURE A major advantage of the proposed compensation structure is its scalability. Any desired SNR can be achieved by cascading the compensation filters, as shown in Fig. 7. The output signal of the first stage is used as input to the second stage and so forth. The output is subtracted from the output of the TI-ADC. According to (25) and (37), the signal is a much better approximation of the desired signal, and repeated filtering will lead to progressively better approximations,, and so on. By defining the SNR after stages as db (46) (47) and following the performance analysis in Section IV, we can bound the SNR after compensation filters from below as (48) In practice, however, the approximation error in the filter design of limits the achievable, so that an increasing number of filter coefficients is needed each additional stage. db

7 VOGEL AND MENDEL: FLEXIBLE AND SCALABLE STRUCTURE TO COMPENSATE MISMATCHES IN TI-ADCS 2469 Fig. 8. SNR before and after compensation out a compensation filter (wo comp.), for one compensation filter (1 CF), for two cascaded compensation filters (2 CF), and for three cascaded compensation filters (3 CF). For evaluating each SNR, we have used N =1024samples. VII. APPLICATION EXAMPLES A. Compensation of Frequency Response Mismatches: Simulations For the numerical simulations MATLAB, we have assumed channel frequency responses given by (49) where is the nominal 3-dB cutoff frequency of the first-order filter, denotes the normalized relative frequency offsets from the cutoff frequency, and represents the relative time offsets [9]. The relative time offsets s constitute the deviation from the nominal sampling instant and can be modeled as linear-phase shifts of the input signal [42]. We did not include gain mismatches, as there is no advantage in using the compensation structure for them. Indeed, gain mismatches can be compensated by a single time-varying multiplier. The desired frequency response has been chosen to be the average frequency response among all channels (50) where is given by (7). We have investigated an eightchannel TI-ADC and have used a sinusoidal input signal to determine the SNR, as given in (26). For a sinusoidal input signal and the chosen desired frequency response, the SNR defined in (26) agrees the definition of the signal-to-noiseand-distortion ratio in [48]. The frequency of the sinusoidal input signal was very close to the Nyquist frequency. The nominal cutoff frequency in (48) was set to three times the sampling frequency, i.e.,. For each simulation, two sets of individual relative normalized frequency offsets and relative time offsets have been generated as random numbers from a Gaussian distribution standard deviations ranging from 0.1 to For the simulations, we have not considered quantization effects. Fig. 8 shows the SNR improvement of the presented compensation structure. We see the SNR out compensation, the after applying one compensation filter (output stage Fig. 9. SNR before and after compensation, as in Fig. 8. The additional crosses indicate the lower SNR bound given by (48) for K =1; 2; 3. 1), the after applying two cascaded compensation filters (output stage 2), and the after applying three cascaded compensation filters (output stage 3) according to Fig. 7. We have used ideal digital filters for this plot to avoid any influences related to the filter design. In each stage, we obtain a significant improvement of the SNR, which depends on the initial matching of the filters. The absolute SNR improvements, i.e., the differences between the SNRs after and the SNRs before compensation, increase the SNRs before compensation. However, the relative SNR improvements, i.e., the ratios between the absolute SNR improvements and the SNR before compensation, are similar and therefore depend only weakly on the SNRs before compensation. For clarity, we use a separate plot shown in Fig. 9 to verify the lower SNR bound given in (48) by comparing it to the simulated SNR shown in Fig. 8. The lower bounds hold for all three plots. s In Fig. 10, we show the SNR for one particular realization of the simulated eight-channel TI-ADC over the frequency. In contrast to the former plot, we have approximated the frequency responses of the compensation filter by a finite-impulse response (FIR) approximation. To be specific, we have approximated the frequency responses in the Chebyshev sense by using MATLAB and the CVX tool [49], i.e., (51) where is the definition domain of the angular frequencies s. From Fig. 10, we notice that one time-varying compensation filter 15 coefficients is sufficient to roughly double the SNR for the given simulation parameters. Hence, we have increased the SNR above 70 db over the definition domain. When we use two cascaded compensation filters 23 coefficients each, we obtain an SNR of about 100 db, and for three cascaded compensation filters 31 coefficients each, it is about 140 db. The dashed lines crosses represent the lower bound on the SNR, as given in (48). The gap between the simulated results and the lower SNR bounds indicates that, on average, a much better performance can be achieved, as predicted by the lower bound. Magnitude response-mismatch compensation, as a special case of frequency response-mismatch compensation, has been investigated in [50] and [51].

8 2470 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER 2009 Fig. 10. SNR at the output of the eight-channel TI-ADC and the compensation structure for different numbers of filter coefficients and compensation filters. The normalized relative frequency offsets have been set to = [0:006; 00:01; 0:009; 00:012; 0:018; 0:007; 0:02; 00:01], and the relative time offsets have been adjusted to r = [0:016; 00:013; 0:008; 0:009; 0:012; 00:007; 0:009; 00:013]. For each evaluated SNR, we have processed 4096 samples. The dashed lines crosses are the corresponding lower SNR bounds given by (48). B. Compensation of Frequency Response Mismatches: Measurements The subject of this section is to demonstrate the introduced compensation structure for the measured data rather than to develop a full system off-the-shelf ADCs and digital correction that are comparable state-of-the-art TI-ADC designs [52], [53]. We have used two 10-bit 105-MS/s AD converters from Analog Devices (AD9218) operated in a TI manner so that the overall sampling rate was 210 MS/s. For the signal generation, we have used an Agilent E8267C PSG vector signal generator. In order to design our compensation filters from measurements, we have to identify the frequency responses of the channels sufficient frequency resolution in the band of interest. In fact, to obtain a specification for our compensation filter, we are not interested in the frequency responses but in the normalized channel-frequency response mismatches or their DTFS. As the derivations will show, a good choice for the desired frequency response is. According to (13), is therefore identically zero, and the entire energy of the error signal is modulated, as we recognize from Fig. 2. Applying a sinusoidal input signal to the TI-ADC, we obtain (14) (16) Fig. 11. Output energy density spectrum of the two-channel 10-bit TI-ADC. The upper plot shows the uncompensated output, and the lower plot shows the output compensated a time-varying filter 51 coefficients. The spurs due to frequency response mismatches, marked circles, have been reduced below the noise floor. mutually spectrally separated. Therefore, we can identify the frequency responses at frequency by relating the output spectrum as (cf. Fig. 2) (53) and by definition of the desired frequency response. By repeating this identification for a sufficient number of frequencies s, we can characterize the frequency responses of our compensation filters. As we see from (53), the sinusoidal input signal itself cancels out and does not influence the identification process. Therefore, we can relax the requirements on the input-signal source and still obtain accurate identification points. Nevertheless, the noise floor limits the achievable identification accuracy. For the system performance evaluation, we have measured another series of frequencies uniformly distributed over the frequency band. In Fig. 11, we see one measured output spectrum before and after compensation. Other than the spurs caused by frequency response mismatches, we see strong harmonics and offset mismatches. The frequency response-mismatch compensation will only slightly improve the SNR, as it is then limited by harmonics, offset mismatches, and noise. However, by defining a different measure, namely, the signal-to-mismatch-error ratio (SMER) for sinusoidal input signals db (54) (52) The output spectrum consists of the spectrum of the sinusoidal input signal filtered by and of modulated by filtered spectral copies of it. As long as the input frequency differs from, the spectral components are we can explicitly evaluate the performance of the frequency response-mismatch compensation method. The SMER relates the energy of the desired signal to the energy of all spurs caused by frequency response mismatches, i.e., the energy of the error signal. For the compensation, we have designed a compensation filter in the Chebyshev sense, as in (51) but

9 VOGEL AND MENDEL: FLEXIBLE AND SCALABLE STRUCTURE TO COMPENSATE MISMATCHES IN TI-ADCS 2471 Fig. 12. Normalized channel magnitude response jh (e )j=j H (e )j and normalized channel-phase response arg(h (e ))0arg( H (e )) of the measured two-channel TI-ADC. The solid lines represent the normalized channelmagnitude response and channel-phase response mismatches before compensation, and the lines hexagrams represent them after compensation by a compensation filter 51 coefficients. for different numbers of coefficients. In Fig. 12, we can see the magnitude and the phase of the normalized channel frequency response before and after compensation by a compensation filter 51 coefficients. The magnitude and the phase of the normalized frequency response mismatches have been considerably reduced over the frequency band. In Fig. 13, we see the SMER plot of the TI-ADC for one compensation filter 11, 31, and 51 coefficients. As expected, the SMER increases the number of filter coefficients. Further increasing the filter order above 51 does not significantly improve the SMER performance since the compensation quality is limited by the noise floor shown in Fig. 11, which is about 80 db. For the same reason, it does not make sense to employ cascaded compensation filters in this case. Furthermore, as we can observe from Fig. 13, the noise floor causes fluctuations of the SMER for higher filter orders because the attenuated mismatch spurs are in the range of the noise floor. We have also plotted the SMER for a TI-ADC ideal gain and timing-mismatch compensation. Ideal means that we have not used a filter design but compensated each frequency point by the identified average gain and average timing mismatch (linear-phase mismatch). As we can see from the plot, the SMER improvements through pure-gain and timing-mismatch compensation are limited. The measurements show that, for a reasonably designed medium-resolution TI-ADC, one stage of the compensation structure is sufficient. For high-resolution TI-ADCs [16], more than one compensation stage can be necessary. Furthermore, the technology scaling gap between analog and digital circuits can be exploited by intentionally designing simple and cheap but impaired analog circuits and using more digital signal processing to compensate their impairments [11], [12]. Therefore, in the near future, more compensation stages could be useful for medium-resolution TI-ADCs as well. Fig. 13. Compensation of frequency response mismatches for a two-channel 10-bit TI-ADC. We see the SMER before compensation and the SMER after compensation compensation filters consisting of 11, 31, and 51 coefficients. C. Compensation of Bandwidth Mismatches: Simulations Bandwidth mismatches are a special case of frequency response mismatches and are often modeled by a first-oder system [8], [42]. In [36], it has been shown, however, that, for bandwidth mismatches whose nominal cutoff frequencies are close to the sampling frequency, the transient response in the system cannot be neglected. We will apply the introduced compensation filter to the model developed in [36] and compare the simulation results from [36] to our results. It should be noted that, for the two-channel case, the authors in [38] have simplified the filter design compared to [36]; we are, however, interested in the general -channel case. According to [36], the channel model for a four-channel TI-ADC is given by (55) For an increasing ratio between the cutoff frequency and the sampling frequency, the second fraction in (55) tends to one. Hence, in many practical cases, for cutoff frequencies that are significantly larger than the sampling frequency, the second fraction and, therefore, also the dependence on the number of channels can be neglected. In [36], the individual cutoff frequencies have been set to, which means that the cutoff frequencies are about half the sampling frequency, and the second term in (55) cannot be neglected. The input signal in [36] was a multitone signal frequencies, which results the given TI-ADC model and TI-ADC parameters in the output spectrum shown in Fig. 14 in the top subplot. The largest undesired spur is about 38 db. We also see the influence of the channel on the input signal itself as the signal is slightly attenuated at higher frequencies. As for the frequency response mismatches in Section VII-A, we can design selected compensation filters to compensate for the given bandwidth mismatches. To stress the flexibility of the compensation structure, however, we will use a Farrow filter as compensation filter. First, we have to define

10 2472 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER 2009 where denotes the definition domain of the spectral parameter and is the frequency response of the Farrow filter. The frequency response is given by (62) where is the number of subfilters and (63) Fig. 14. Compensation of bandwidth mismatches by using a Farrow filter as compensation filter. The top subplot shows the uncompensated output spectrum, the middle plot shows the output spectrum after compensation one compensation filter (output stage 1), and the bottom plot shows the output spectrum after compensation two cascaded compensation filters (output stage 2). our desired frequency response to obtain the frequency response of the compensation filters, which is chosen to be (56) As suggested by the performance analyses in Section IV, the desired frequency response is chosen to be close to the expected frequency responses of the channels. A closer examination of (55) reveals that the frequency response depends on the frequency and one additional spectral parameter [54]. Therefore, we can express (55) in discrete time according to (11) as (57) (58) and circumvent (58) the dependence on. Accordingly, the frequency responses of the th compensation filter results (20) and (56) in (59) (60) For the practical realization of the Farrow filter, we have to approximate the frequency response given in (60). For our purpose, the ideal frequency response is approximated in the least squares sense [54], i.e., (61) is the DTFT of the th FIR subfilters impulse response and length. For the design example, the definition domain of the angular frequencies was, and the range of possible cutoff frequencies was. We have designed compensation structures consisting of one stage and two stages Farrow filters according to Table I. After designing the Farrow filters, we have parametrized them according to (59) and compensated the mismatches. The results are shown in Fig. 14, where, in the middle subplot, we see can the compensated output (output stage 1) for one compensation filter 4 15 coefficients and, in the bottom subplot, we can see the compensated output (output stage 2) for two cascaded compensation filters 4 15 (compensation filter stage 1) and 4 25 (compensation filter stage 2) coefficients. For one compensation filter, the largest undesired spur is about 78.3 db, and for two cascaded compensation filters, the largest undesired spur is about db, which means improvements of 40.3 and 68.9 db, respectively. In Table I, we compare our results that in [36]. Although we are able to compensate for any bandwidth mismatch in the range of, and the shown comparison is only a particular configuration of the Farrow filter, our method achieves similar or even better results. On balance, the filter design in [36] was based on frequency sampling, and a better suited filter design method would improve the results. It should be noted that optimal filter designs [34], [35] can achieve a higher SNR the same number of coefficients; however, contrary to optimal designs, our design does not, in the design margins of the Farrow filter, require a full redesign of the filters when the cutoff frequency drifts because of environmental changes. Moreover, as already discussed, for reasonably large nominal cutoff frequencies, the second fraction in (55) becomes one and can be neglected. Therefore, we can design Farrow filters independent of the number of channels and can compensate any number of time-varying channels by applying the same filter. D. Reconstruction of Nonuniformly Sampled Signals The differentiator multiplier cascade described in [28] and [29] can be seen as a special case of the discussed compensation structure. It can be used to reconstruct uniformly sampled signals from nonuniformly sampled band-limited signals and can be easily interpreted by using the introduced general framework. The time offset from the ideal sampling instant at time can be modeled in the frequency domain by the time-varying filter (64)

11 VOGEL AND MENDEL: FLEXIBLE AND SCALABLE STRUCTURE TO COMPENSATE MISMATCHES IN TI-ADCS 2473 TABLE I COMPARISON BETWEEN [36] AND THIS PAPER FOR A FOUR-CHANNEL SYSTEM We recognize from (64) that the frequency response depends on the frequency and one additional spectral parameter. Therefore, we may express (64) (11) in discrete time as (65) (66) and being an ideal discrete-time differentiator [41, Ch. 8.2]. According to the discussed bandwidth mismatches, we see that the th channel frequency response becomes a particular result of. By defining the desired frequency response as (67) we obtain (20) the frequency response of the compensation filter as (68) (69) Finally, we can approximate the frequency response in (69) by a Farrow filter given (62). By using the so-designed compensation filter, the flexibility of the structure allows for compensating the time offsets of an arbitrary number of nonuniform samples and, therefore, reconstructing the uniformly sampled signal. To obtain a better SNR, we can use the scalability of the structure and cascade the compensation filters, as shown in Section VI. As the authors in [29] have shown, it is not necessary to approximate each compensation filter in the structure the same accuracy. On that account, they have used a Taylor series approximation of the filter to obtain the frequency response of the compensation filter for the first stage as and that for the second stage as (70) (71) and so on. This has further simplified the filter design process to the design of a discrete-time differentiator since the number and the frequency response of the subfilters in the Farrow structure in (62) are thereby fixed, i.e., for the first stage, and for the second stage, and so forth. In contrast, by using optimization techniques to find the optimal number of subfilters s and their corresponding frequency responses, the SNR may improve by some decibels compared to that in [28] and [29]. As the reader can find plenty of simulation results in [28] and [29], we have omitted simulation results here. VIII. CONCLUDING REMARKS We want to conclude by summarizing the results and by pointing out some further aspects. We have presented a flexible and scalable structure to compensate frequency response mismatches. We have started by introducing a system model that establishes the relation between TI-ADCs and time-varying systems. Afterward, we have presented the compensation structure, the mismatch-compensation principle, and the achievable performance. We have further shown the flexibility of the compensation structure to compensate time-varying frequency responses and the scalability of the compensation structure to achieve arbitrary SNRs. On balance, the structure is a suboptimal solution and needs more coefficients than optimal designs [34], [35] for the given frequency response mismatches. As indicated, the structure can also be used for identifying mismatches, but this was beyond the scope of this paper, and examples for gain and timing-mismatch identification can be found in [31], [32]. As demonstrated in the example section, the flexibility allows for utilizing the structure to compensate more general time-varying systems like nonuniform sampling systems; however, a thorough characterization of the class of systems has to be further investigated. APPENDIX A RELATION BETWEEN MODULATED AND TIME-VARYING FILTERS By applying the inverse DTFT to (16), we obtain where (72) (73) From (73), we see that and are DTFS pairs [41, Ch. 4.2]. By changing the order of summation in (72), we can write (74) In (74), we take the inverse DTFS of the coefficients given by (73) for, which results in the coefficients. Since the given inverse DTFS is periodic, the resulting coefficients are periodic as well, i.e., for all. Finally, we can rewrite (74) as (75)

12 2474 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 11, NOVEMBER 2009 APPENDIX B EQUIVALENCE BETWEEN AND IN (39) The frequency responses are the DTFS of the normalized channel-frequency response mismatches. Therefore, is given by (76) After replacing and using complex conjugation, we obtain (77) Because the frequency responses of real filters fulfill the relation we can insert (78) into (77) and immediately obtain as required. (78) (79) ACKNOWLEDGMENT The authors would like to thank S. Tertinek for the fruitful discussions and suggestions for this paper. Furthermore, the authors would like to thank C. Krall for helping them the measurements. Finally, the authors would like to thank the anonymous reviewers for their useful and detailed comments. REFERENCES [1] K. Poulton, R. Neff, B. Setterberg, B. Wuppermann, T. Kopley, R. Jewett, J. Pernillo, C. Tan, and A. Montijo, A 20 GS/s 8 b ADC a 1 MB memory in 0.18 m CMOS, in Proc. IEEE Int. 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Circuits Syst. II, Analog Digit. Signal Process., vol. 49, no. 11, pp , Nov Christian Vogel (S 02 M 06) received the Dipl.Ing. degree in telematics and the Dr.Techn. degree in electrical and information engineering from Graz University of Technology, Graz, Austria, in 2001 and 2005, respectively. Since 2008, he has been a Postdoctoral Researcher the Signal and Information Processing Laboratory, Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology Zurich (ETH Zurich), Zurich, Switzerland. His research interests include the design and theory of digital-, analog-, and mixed-signal processing systems, special emphasis on communication systems and digital enhancement techniques for analog-signal processing systems. Stefan Mendel (S 07 M 07) was born in Graz, Austria, in He received the Bakk.Techn. and Dipl.Ing. degrees in telematics from Graz University of Technology, Graz, in 2004 and 2006, respectively. During his master thesis, he worked on postcompensation of channel mismatches in time-interleaved analog-to-digital converters. Since 2006, he has been working toward the Ph.D. degree in the Signal Processing and Speech Communication Laboratory, Graz University of Technology, working on digital synthesizers for gigahertz-range fast-frequencyhopping systems. His research interests are all-digital phase-locked loops, analog-to-digital converters, digital enhancement of analog circuits, mixed-signal systems, and systems for communications.