Spectral Impact of Mismatches in Interleaved ADCs

Spectral Impact of Mismatches in Interlea v ed ADCs Jérémy Guichemerre ∗ , Robert Reutemann ♢ , Thomas Burger ∗ , and Christoph Studer ∗ ∗ ETH Zurich, Switzerland; ♢ Mir omico IC A G, Zurich, Switzerland emails: jer emyg@iis.ee.ethz.ch, rr e@mir omico-ic.ch, b ur gert@ethz.ch, studer@ethz.ch Abstract —Interleav ed ADCs are critical f or applications re- quiring multi-gigasample per second (GS/s) rates, but their performance is often limited by offset, gain, and timing skew mismatches across the sub-ADCs. W e propose exact but compact expressions that describe the impact of each of those non-idealities on the output spectrum. W e derive the distrib ution of the power of the induced spurs and replicas, critical f or yield-oriented derivation of sub-ADC specifications. Finally , we provide a practical example in which calibration step sizes ar e deri ved under the constraint of a target production yield. I . I N T RO D U C T I O N T ime interleaving enables analog-to-digital con verters (ADCs) to reach multi-gigasample per second (GS/s) rates [1], [2]. Such high sampling rates are essential in applications such as high-speed wireline transcei vers, direct RF sampling in wireless receivers, and radar systems. The drawback of interleaving is that mismatches among the sub-ADCs introduce distortion: affine errors at the sub-ADC le vel produce spurious tones and replicas at the system lev el. The most critical mismatches in real-world sub-ADC designs are offset, gain, and timing skew . Offset corresponds to a constant shift at the sub-ADC’ s output, creating spurious tones at multiples of the sub-ADC rate. Gain mismatches are scaling errors and yield replicas of the input spectrum. Ske ws are sampling-time errors and replicate a differentiated version of the input spectrum. As those mismatches are fixed after fabrication (or vary slowly), performance metrics and their design targets must be linked to production yield—knowledge of the distribution of the impact mismatches hav e on the output spectrum is therefore necessary . A. Contrib utions In this paper , we express the impact of mismatches through the discrete Fourier transform (DFT) of the mismatch sequences to deriv e compact but rigorous expressions that capture the ef fects of offset, gain, and ske w mismatches without relying on tone-based approximations. This provides simple and intuitiv e spectral expressions, which naturally extend to a statistical frame work. W e deri ve exact spur and replica po wer distrib utions under Gaussian mismatch assumptions, and further quantify the accuracy of Gaussian approximations in the case of uniform distributions for practical interleaving factors. B. Limitations of Prior Art V ogel [3] deriv ed expressions for the average SNDR, but expectations alone prevent a yield analysis. Ghosh [4] provided an exact deriv ation of SNDR degradation, but the method is intricate and restricted to single-tone inputs. Neither approach provides the statistical framew ork needed for yield-driven design. Monte–Carlo analysis alone is also insuf ficient, as system-le vel exploration requires fast and general predictions. C. Notation W e write matrices in bold uppercase and column vectors in bold lowercase. W e index the entries u k of a vector u ∈ C N with k ranging from 0 to N − 1 , and we write k % N for k mo d N . The N × N identity matrix is I N . W e define the discrete Fourier transform (DFT) and the DFT matrix F as ˜ u k = { F u } k = 1 N P N − 1 n =0 e − 2 π j kn N u n , (1) where ˜ u ∈ C N is the DFT of u ∈ C N and j is the imaginary unit. The 1 / N normalization in (1) ensures that the squared magnitudes | ˜ u k | 2 represent av erage powers, independent of the length N of the sequence. W e denote the Dirac distribution as δ ( t ) for t ∈ R ; the conv olution of g ( t ) with h ( t ) is g ( t ) ∗ h ( t ) . W e normalize the full-scale range of ADCs to [ − 1 , 1] ; 0 dBFS is the power of a full-scale sine wav e. Zero-mean real and complex circularly-symmetric Gaussian distributions of variance σ 2 are N (0 , σ 2 ) and C N (0 , σ 2 ) , respecti vely . The Gaussian error function is denoted by erf ( x ) . I I . I D E A L I N T E R L E A V I N G A N D M I S M A T C H E S W e first analyze the Fourier-domain representation of an ideal interleav ed ADC, showing how the outputs of the sub-ADCs perfectly recombine (see Fig. 1a). W e then extend our analysis to consider of fset, gain, and ske w mismatches, and deri ve expressions that characterize their output spectrum impact. A. Ideal Interleaving W e consider an interleav ed ADC consisting of N sub-ADCs, each operating at a sampling rate of f s / N . When sampling an input x ( t ) , the sampled output of the ideal n th sub-ADC is y id n ( t ) = x ( t ) P k δ  t − k N f s − n f s  . (2) Let X ( f ) be the Fourier transform (FT) of x ( t ) . Then, the FT of the n th sub-ADC output y id n is given by Y id n ( f ) = X ( f ) ∗ f s N P k e − j 2 π kn N δ  f − k f s N  , (3) where the con volution with the Dirac comb leads to the expected aliasing corresponding to the f s / N rate. Howe ver , when summing the output of all N sub-ADCs, the resulting FT should be identical to that of an f s -rate ADC. Indeed, from (3), we obtain Y id ( f ) = P N − 1 n =0 Y id n ( f ) = X ( f ) ∗ f s N P k h δ  f − k f s N  P N − 1 n =0 e − j 2 π kn N i . 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(a) Conceptual block-diagram of an interleaved ADC, (b) example of an ideally sampled single-sided output spectrum and DFT of a mismatch sequence for a 4 × interleav ed ADC. (c), (d), and (e) show the impact of the mismatch sequence of (b) in the case of offset, gain mismatch, and timing ske w , respectively . The sum in n in the right-hand side corresponds to the sum of the N th roots of unity . Therefore, when k is a multiple of N , the sum equals N ; in any other case, the sum equals zero. W e re-index the sum to ignore the zero entries and obtain Y id ( f ) = X ( f ) ∗ f s P k δ ( f − k f s ) , (5) which matches the output of an ideal ADC of rate f s ; see Fig. 1b for an illustration. B. Offset Offsets in interleaved ADCs originate both from devic e mismatches in the sub-ADCs themselves (e.g., comparator offsets), and from offsets among the parallel track-and-hold stages used at large interleaving factors. As offsets are additi ve, we can model the output of the interleaved ADC as y o ( t ) = y id ( t ) + P k o k % N δ  t − k f s  , (6) where y id ( t ) is the ideally-sampled output and o n the offset of the n th sub-ADC. Applying the FT , we obtain Y o ( f ) = Y id ( f ) + f s P k ˜ o k % N δ  f − k f s N  , (7) with ˜ o the DFT of the offset sequence o . W e conclude that offsets among the sub-ADCs lead to spurs at multiples of f s / N whose amplitudes correspond to the DFT of the of fset sequence, as illustrated in Fig. 1c. Due to their additiv e nature, offset-induced spurs are independent of the input signal and constrain the dynamic range of the system. C. Gain Mismatch Another common non-ideality in interleav ed ADCs are gain mismatches, which arise from sub-ADC gain errors (e.g., capacitor ratio mismatches in SAR ADCs) and from mismatches in device or routing losses preceding the sub- ADCs. W e model the output of the interleaved ADC as y g ( t ) = x ( t ) P k (1 + g k % N ) δ  t − k f s  = y id ( t ) + x ( t ) P k g k % N δ  t − k f s  , (8) where g n is the relative gain mismatch of the n th sub-ADC. The FT of the output is therefore Y g ( f ) = Y id ( f ) + X ( f ) ∗ f s P k ˜ g k % N δ  f − k f s N  , (9) with ˜ g the DFT of the mismatch sequence g . Similar to (7), the result in (9) features a Dirac comb with amplitudes corresponding to the DFT of the mismatch sequence. Howe ver , the comb is now con voluted to the FT of the input signal, leading to residual aliases from the f s / N rate, as illustrated in Fig. 1d. For a single-tone input at f in , those replicas are spurs located at nf s / N ± f in with n an integer . In practice, the DC component of ˜ g , i.e., the average gain error, can be neglected, as it has little impact on the main signal component and does not significantly affect the spectrum. D. T iming Skew T iming ske w causes each sub-ADC to sample at a small time offset from its nominal sampling instant, so the output depends on the continuous-time behavior of the input. In time domain, the output of the n th sub-ADC with a skew s n is y s n ( t ) = x ( t − s n ) P k δ  t − k N f s − n f s  . (10) Assuming that the input x ( t ) is band-limited to f max , and that the skew of each sub-ADC is small compared to the period 1 /f max , the FT of x ( t − s n ) can be approximated as X ( f ) e − 2 π jf s n ≈ X ( f )(1 − 2 π j f s n ) . (11) The FT of (10) can then be approximated as Y s ( f ) ≈ Y id ( f ) − f s (2 π j f X ( f )) ∗ P k ˜ s k % N δ  f − k f s N  , (12) where ˜ s the DFT of the timing ske w sequence s . Analogous to the case of gain mismatches, aliases appear with amplitudes determined by the DFT of the ske w sequence, but in this case, a differentiated v ersion of the input is replicated (cf. Fig. 1e). The effect of timing ske w thus grows linearly with frequency: doubling the input frequency for a giv en ske w sequence increases the alias power by 6 dB. By Bernstein’ s inequality [5], [6], the band-limited signal with the largest deri vati ve under an amplitude constraint is a single tone at f max , therefore representing the worst-case input for skew analysis. I I I . S P U R - L E V E L S TA T I S T I C S In Sec. II, we hav e shown that the impacts of offset, gain, or ske w mismatches are proportional to the DFT of the mismatch sequences. Since mismatches in an interleav ed ADC are fixed after production (or drift slowly), they act as device-specific − 4 − 2 0 2 4 6 8 10 12 14 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 Power of DFT entry [dB] CCDF Gaussian Uniform N = 8 Uniform N = 16 Uniform N = 32 Real entries Circ. sym. entries Fig. 2. Complementary CDF (CCDF) of the squared magnitude of DFT entries for i.i.d. uniform sequences (various N ) normalized to unit variance of the magnitude, compared with the Gaussian case. Already at N = 16 , the Gaussian approximation is accurate within 1 dB at 10 − 4 probability . constants rather than random noise. As a result, average metrics across realizations are not informativ e for design. T o link a production yield target to a mismatch variance, one instead needs the cumulativ e density function (CDF) of the power of each spur or replica as a function of the mismatch variance. W e will assume that the mismatch sequences are i.i.d. zero- mean real Gaussian random variables and deri ve the distribution of the DFT of such a vector . Lemma 1. Let x be a vector of even length N containing r ealizations of i.i.d. zer o-mean r eal-valued Gaussian random variables, each of variance σ 2 . Then, the entries of its DFT ˜ x = Fx are distributed as follows: ˜ x k ∼    N  0 , σ 2 N  k = 0 , N / 2 C N  0 , σ 2 N  k = 1 , . . . , N / 2 − 1 , (13) wher e the entries in (13) ar e mutually independent, and ˜ x k = ˜ x ∗ N − k for k = N / 2 + 1 , . . . , N − 1 . 1 Lemma 1 can be proven by noticing that, by linearity of the DFT , ˜ x is a Gaussian random vector , hence fully characterized by its cov ariance matrix K and pseudo-covariance matrix J [7] K = E  ˜ x ˜ x H  = F E  xx H  F H = σ 2 N I N , (14) J = E  ˜ x ˜ x T  = σ 2 F 2 = σ 2 N P . (15) Here, P is the permutation matrix with ones at entries ( k , ℓ ) whenev er ( k + ℓ ) % N , including the DC bin and, if N is even, also the Nyquist bin on the diagonal. The structure of K and J then allows us to conclude on the distribution of ˜ x . A. CDF of Offset Spurs As shown in Sec. II-B , offsets among sub-ADCs lead to input-independent spurs at multiples of f s / N . W e assume the offset sequence o to contain i.i.d. zero-mean Gaussians with variance σ 2 o per sub-ADC, and derive the CDF of the squared magnitude of the spurs to link a target yield to mismatch variance. In order to ensure that our results can be applied 1 If N is odd, then the result is similar but without a Nyquist bin: the DC bin is a real-valued Gaussian and the rest of the DFT entries are complex-v alued circularly-symmetric Gaussians arranged in complex conjugate pairs. directly by designers, we consider a single-sided spectrum: the power of the spurs originating from circularly-symmetric Gaussian distributions therefore include a factor of two. W e express the spur po wer relati ve to the full-scale po wer as defined in Sec. I-C ; another factor of two is accordingly applied to spur powers. For real zero-mean Gaussians–corresponding to the spur at DC and, when N is e ven, also the spur at the Nyquist frequency–the squared magnitude p of the spur follo ws a chi- squared distribution with CDF F o real ( p ) = erf  q N p 4 σ 2 o  p ≥ 0 , (16) where 10 log ( p ) is the spur power in dBFS . For the rest of the spurs, i.e., the circularly-symmetric Gaussian-distributed bins, their power p follo ws an exponential distribution of CDF F o circ ( p ) = 1 − e − N p 4 σ 2 o p ≥ 0 . (17) As can already be observed in (13), doubling the interleaving factor N statistically reduces the po wer of each spur by a factor of two. Howe ver , as the number of spurs also doubles, the total spur po wer does not change. B. CDF of Gain-Mismatch Replicas As shown in Sec. II-C , gain mismatch among sub-ADCs leads to aliasing replicas. W e are therefore interested in the power of those replicas relative to the power of the input, i.e., in dBc . W e assume the gain-mismatch sequence to contain i.i.d. zero-mean Gaussians with variance σ 2 g . Unlike the case of offset, no power scaling is required: the squared magnitudes of the DFT entries of the mismatch sequence directly yield the power ratios in dBc . The impact of the DC bin of the DFT of g can be ignored, but, in case N is even, the Nyquist bin creates a replica. Analog to (16), we obtain F g real ( p ) = erf  q N p 2 σ 2 g  p ≥ 0 , (18) where 10 log ( p ) is the replica power in dBc . For replicas originating from circularly-symmetric Gaussians, we obtain F g circ ( p ) = 1 − e − N p σ 2 g p ≥ 0 . (19) C. CDF of Skew Replicas W e hav e shown in Sec. II-D that skews have a very similar ef fect compared to gain mismatches. If we consider the power of the already-dif ferentiated replicas, (18) and (19) can directly be used replacing σ 2 g by σ 2 s , the skew variance of a sub-ADC. Alternativ ely , we consider the scenario of a single-tone at f sig . In this case, we simply re write (18) and (19) as F s real ( p ) = erf  q N p 8 π 2 f 2 sig σ 2 s  p ≥ 0 , (20) F s circ ( p ) = 1 − e − N p 4 π 2 f 2 sig σ 2 s p ≥ 0 , (21) where 10 log ( p ) is the replica power in dBc . − 85 − 80 − 75 0 . 2 0 . 3 0 . 5 0 . 8 1 99% -quantile of strongest spur power [dBFS] Offset cal. step size [LSB] Fig. 3. Offset calibration step size that ensures the strongest offset spur remains below a specified power limit with 99% probability in a 12 -bit 16 × interleav ed ADC. Note that LSB = 2 1 − B for a B -bit ADC. D. Combined CDF for Maximum Spur Constraints For many practical cases, specifications are giv en as total spurious-free dynamic range (SFDR), i.e., we are interested in the probability of the strongest spur being below a target lev el in terms of dBFS or dBc . Since all contrib utions are assumed to be independent (Lemma 1), the combined CDF is the product of all contributions. T aking offset spurs as an e xample, we have two real Gaussian contributions (DC and f s / 2 spurs) and N / 2 − 1 circularly- symmetric Gaussian contributions (for ev en N ). The combined CDF , i.e., the likelihood that the power of all those spurs remain under 10 log( p ) dBFS is then F o tot ( p ) = F o real ( p ) 2 · F o circ ( p ) N/ 2 − 1 . (22) In many practical cases, spurs at DC and f s / 2 are not relev ant, in which case the component F o real ( p ) 2 in (22) can be omitted. The same reasoning can be applied for gain and timing skew spurs, but with only one real Gaussian contribution, since the DC (average) term is not relev ant, leading to 2 F g tot ( p ) = F g real ( p ) · F g circ ( p ) N/ 2 − 1 , (23) F s tot ( p ) = F s real ( p ) · F s circ ( p ) N/ 2 − 1 . (24) W e note that we can freely consider specific combinations of spurs or replicas by including the rele vant terms. E. Case of a Uniform Distribution T o mitigate the impact of spurs and replicas, interleav ed ADCs are typically calibrated for sub-ADC mismatches. Resid- ual mismatches would then ideally follo w i.i.d. uniform distri- butions of support [ − ∆ cal / 2 , ∆ cal / 2] , with ∆ cal the calibration step size. By the central limit theorem, the DFT of a mismatch sequence with i.i.d. entries approaches the Gaussian distribution of Lemma 1 as N grows. Fig. 2 provides Monte–Carlo results showing the in verse CDF of the squared magnitude of the DFT entries of i.i.d. uniform sequences, normalized to unit bin variance, compared with the Gaussian analytical results from Lemma 1. W e observe 2 As illustrated in Fig. 1d, we obtain N − 2 replicas from the circularly- symmetric Gaussian entries of the DFT , but, due to their conjugate symmetric pairing, only half of them are independent contributors. − 75 − 70 − 65 − 60 − 55 0.1 0.2 0.3 0.5 0.7 1 99% -quantile of strongest replica level [dBc] Gain cal. step size [%] 10 20 30 50 70 100 T iming cal. step size [fs] Fig. 4. Gain and sampling instant calibration step size that ensures the strongest replica remains below a specified power limit relative to the input signal with 99% probability in a 12 -bit 16 × interleaved ADC. errors below 2 dB , 1 dB , and 0 . 5 dB at 10 − 4 probability for N = 8 , 16 , and 32 , respectively . W e also note that the Gaussian approximation serves as a worst-case analysis: intuitiv ely , this is because realizations of a Gaussian random variable are unbounded. In most practical cases, the mismatch distribution is not exactly uniform due to noise in the estimates, resulting in an effecti ve CDF in-between uniform and Gaussian. I V . A P R A C T I C A L E X A M P L E As an example, we consider a 12 -bit interleaved ADC for RF-sampling applications with N = 16 sub-ADCs; our design goal is to determine the calibration step size ∆ cal required for each mismatch type. T argets are set to − 80 dBFS for of fset spurs and − 65 dBc for both gain and timing-skew spurs, for input frequencies up to 12 GHz . Offset spurs at DC and f s / 2 are excluded, while gain and ske w replicas include the f s / 2 − f sig component. As discussed in Sec. III-E , we assume each mismatch to be Gaussian with v ariance σ 2 mis = ∆ 2 cal / 12 . Using (22), (23), and (24), the 99 % -quantile of the strongest spur or replica is obtained numerically as a function of ∆ cal . Results in Fig. 3 indicate that a calibration step size of about half an LSB is required to meet the − 80 dBFS offset-spur limit. As shown in Fig. 4, meeting the − 65 dBc target for gain and ske w requires step sizes of approximately 0 . 27 % and 35 fs , respectiv ely , highlighting the stringent calibration accuracy needed for high-speed interleav ed ADCs. V . C O N C L U S I O N S W e have derived compact expressions describing the impact of offset, gain, and timing skew mismatch in interleaved ADCs. W e have further characterized the distribution of the power of spurs and replicas in the ADC output spectrum originating from those non-idealities under a Gaussian assumption; we hav e also shown that this Gaussian assumption is rele vant in scenarios where mismatch calibration is employed, i.e., when mismatches follow a uniform distribution. T o illustrate the usefulness and fle xibility of our results, we ha ve derived a calibration step size requirement for a practical example. 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