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Abstract – We introduce a novel randomization method for time- interleaved analog-to-digital converters (TIADCs). In contrast to conventional randomization ...
IMTC 2005 - Instrumentation and Measurement Technology Conference Ottawa,Canada, May 17–19, 2005

A Novel Channel Randomization Method for Time-Interleaved ADCs Christian Vogel, Viktoria Pammer, and Gernot Kubin Christian Doppler Laboratory for Nonlinear Signal Processing , Graz University of Technology, Austria, Phone: +43 316 873 4443, Fax: +43 316 873 4381, E–mail: [email protected], [email protected], [email protected] Abstract – We introduce a novel randomization method for timeinterleaved analog-to-digital converters (TIADCs). In contrast to conventional randomization methods our method improves the signalto-noise and distortion ratio (SINAD) and the spurious free dynamic range (SFDR) by combining randomization of channel ADCs and spectral shaping of timing mismatches. The new method can exploit information about timing mismatches of the channel ADCs in order to optimize the randomized sequence. We provide the theoretical background, discuss aspects of the advanced randomization, and present simulation results.

fs /M

2π 0 M

ADC0 fs /M analog input xa (t)

ϕ=

2π M1

ADC1 fs /M

ϕ=

2π Mm

MUX

digital output y[n] fs

ADCm fs /M

Keywords – time-interleaved, randomization, timing mismatches, spectral shaping, compensation

ϕ=

2π M (M

− 1)

ADCM −1

Fig. 1. The principle of a non-randomized time-interleaved ADC with M channel ADCs. All channel ADCs sample with the same sampling rate but with different phases [1].

I. INTRODUCTION A time-interleaved analog-to-digital converter (TIADC) consists of M parallel channel ADCs sampling with the same s sampling frequency Ω M but with different sampling phases [1], which is depicted in Fig. 1. Without any mismatches among the channel ADCs, a TIADC works like a single channel ADC. Unfortunately, there are channel mismatches that produce additional error power, which significantly decreases the signal-to-noise ratio (SINAD) [2, 3]. Moreover, this error power is not, like for example clock jitter, uniformly distributed over the frequency band. It results in recognizable spectral peaks in the output spectrum, which significantly reduce the spurious free dynamic range (SFDR) [3] as well. Randomization is one useful method for distributing the mismatch power over the frequency band and increasing the SFDR [4–6]. However, it does not improve the SINAD, since the mismatch error power is only distributed but not reduced in any way [7, 8]. To improve the SINAD we can oversample the randomly sampled input signal and filter out the remaining input signal free frequency band. With ideal filters and white noise we can gain 3dB for every doubling of the oversampling ratio. Nevertheless, high oversampling is not in line with the design concept of a TIADC, which is to increase the sampling rate by parallelization. Therefore, we look for methods to improve the SINAD by more than the 3dB when doubling the sampling frequency.

0-7803-8879-8/05/$20.00©2005 ©2005IEEE IEEE 0-7803-8879-8/05/$20.00

ϕ=

The most critical error source in TIADCs are timing mismatches. They are difficult to identify and even harder to compensate. In the paper we describe how we can combine randomization and spectral shaping of timing mismatches in order to improve the SINAD and the SFDR. II. COMBINING RANDOMIZATION AND SPECTRAL SHAPING First, we briefly introduce randomization of channel ADCs in TIADCs and establish a notation for the output spectrum. Then, we show how to select the channel ADCs to obtain both randomization and spectral shaping of timing mismatches. A. Randomized Time-Interleaved ADCs In Fig. 2(a) we see the principle of a randomized TIADC. For each sampling instant a new channel ADC is randomly selected, whereas we have some constraint on the selection process. If we have a redundant array of (M + R) channel ADCs, operating with a minimum channel sampling period of M Ts , and we would like to build a randomized TIADC with an overall sampling period of Ts , we can choose at each time

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clk

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ADC1

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ADC2

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ADC0

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M

xa(t)

ADC2 y[n]

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low-pass filter

randomized digital output

ADC1

y˜[n]

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(a) Architecture

(b) Principle

Fig. 2. The basic concept of channel randomization. (a) The channel ADC sequences are arbitrarily selected. No information about the TIADC is used to improve the selection process. (b) Conventional randomization of channel ADCs with M = 2 and R = 1. After the first two channel ADCs (ADC0 , ADC1 ) have taken a sample we can choose between ADC0 and ADC2 , without violating our sampling constraints. After we have, for example, chosen ADC2 we can decide between ADC0 and ADC1 and so forth (dashed line).

instant among R + 1 channel ADCs without violating the sampling constraint, M Ts , for one channel ADC. This is illustrated in Fig. 2(b) for M = 2 and R = 1. Furthermore, we see from Fig. 2(a) that the TIADC is followed by a low-pass filter, which filters out some portion of the uniformly shaped mismatch spectrum in order to increase the SINAD. To represent the output spectrum of a randomized TIADC we start with the well-known output spectrum of a nonrandomized TIADC with M channels and timing mismatches ∆tm , which is [2] ∞   1  Y ejΩTs = Ts p=−∞       Ωs Ωs αp j Ω − p Xa j Ω − p , M M (1)

where

M −1 1  −jΩ∆tm −jpm 2π M . e e αp (jΩ) = M m=0

(2)

In order to describe randomized TIADCs as well we use N −1 the sequence υ = υn n=0 , where each element υn ∈ 0, 1, . . . , M + R − 1 stands for an index number, which maps each sampling instant to a channel ADC. The sequence has N = (M + R)L elements and is repeated forever. Therefore, (1) and (2) can be rewritten as [9] ∞   1  Y ejΩTs = Ts p=−∞

      Ωs Ωs αp j Ω − p Xa j Ω − p , N N (3) where αp (jΩ) =

N −1 1  −jΩ∆tυn −jpn 2π N . e e N n=0

(4)

For a non-randomized TIADC, i.e., to obtain (1) and (2), the N −1 sequence would be υ = 0, 1, . . . , M − 1n=0 , where R = 0, L = 1 and N = M . When we use a random sequence υ to describe a randomized TIADC it has to fulfill the sampling constraints, i.e., a minimum sampling period of M Ts for all channel ADCs. Furthermore, the number of elements N of the sequence should be very large or in the ideal case infinite. For such a sequence we see from (4) that we increase the number of aliased spectra, whereby the power of each spectrum decreases with N and the mismatch power becomes more uniformly distributed over the frequency band [7, 9]. B. Spectrally Shaped and Randomized Time-Interleaved ADCs In Fig. 3 we see the principle of spectrally shaped randomization. Fig. 3(a) shows how we can use information from identified timing mismatches to improve the channel selection process in order to achieve a spectrally shaped and randomized output spectrum. The shaped timing mismatch error power can be filtered out by a simple low-pass filter.

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select next channel

min(∆t0, ∆t3)≥ max(∆t1, ∆t2) ADC0

y[n]

xa(t)

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analog input

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filtered digital output

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(a) Architecture

(b) Principle

Fig. 3. The basic concept of spectrally shaped channel randomization. (a) The selection process uses information about the identified timing mismatch to achieve a spectrally shaped randomization. (b) Advanced randomization of channel ADCs with M = 2 and R = 2. For the odd samples we can choose from channel ADCs belonging to group A and for the even samples we can choose from channel ADCs belonging to group B. Therefore, at each sampling instant we can choose between two channel ADCs. All channel ADCs of Group A have timing mismatches greater than the timing mismatch of each channel ADC in group B. The dashed line shows one possible way to choose a random sequence.

α0 (jΩ) Xa (jΩ)

  α2 j Ω −   α1 j Ω −   Xa j Ω −

0

Ωs 4

Ωs 2



  Xa j Ω −



However, we can go one step further and combine spectral shaping and randomization. To show this, we approximate the timing mismatch by the first two terms of a Taylor series expansion and can write (4) as

α0 (j (Ω − Ωs)) Xa (j (Ω − Ωs ))

input spectrum

Ωs 2



αp (jΩ) = δ[p] −

   α3 j Ω − 3 Ω4s    Ωs Xa j Ω − 3 4

Ωs 4  Ωs 4

Ωs 2

3 Ω4s

N −1 2π jΩ  ∆tυn e−jpn N , N n=0

(5)

where δ[p] is the discrete impulse function, and recognize that the central coefficient α N (jΩ) of (5) can be further reduced 2 to N 2 −1   jΩ  ∆tυ2n − ∆tυ2n+1 . (6) α N (jΩ) = − 2 N n=0

Ωs

Fig. 4. Output spectrum of a non-randomized time-interleaved ADC with four channels (M = 4). When the input signal is bandlimited to Ω4s the “ “ ”” does not overlap with the aliased spectrum at α M (jΩ) Xa j Ω − Ω2s 2

input spectrum and can be filtered out.

According to (1) and (3) timing mismatches cause weighted aliased    spectra of the input signal αp (jΩ) s Xa j Ω − p Ω [10–12], which is illustrated in Fig. 4. For M an even number of channel ADCs M of a non-randomized TIADC and a bandlimited input signal, i.e., Xa (jΩ) = 0 for |Ω| ≥ Ω4s , the central aliased spectrum (p = M ) becomes 2       Ωs Ωs does not α M (jΩ) Xa j Ω − 2 , where Xa j Ω − 2 2 contribute to the input signal frequency band. Thus, for a non-randomized TIADC we can shape timing mismatch power towards the central spectrum by maximizing α M (jΩ) [13].

Then we build two index groups A and B out of all possible channel ADCs, i.e., 0, 1, . . . , M + R − 1, with correspondB ing timing mismatches ∆tA m and ∆tm where the size of each 2 2 group is M 2+R , e.g., A = 1, 4, 5m=0 and B = 0, 2, 3m=0 for M = 4 and R = 2. We further assume that the frequency N −1 of occurrences of the elements in υn n=0 are uniformly distributed, which is feasible for a fair randomization and large N , and all even indices υ2n are elements from group A and all odd indices υ2n+1 are elements from group B. Therefore, we can simplify (6) to

2

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αN 2

jΩ (jΩ) = − M +R =−

jΩ M +R

M +R −1 2



m=0 M +R −1 2

L−1  1  A ∆tm − ∆tB m L

 

l=0

m=0

 B ∆tA m − ∆tm .

(7)

From (6) and (7) we see that α N (jΩ) is the difference of 2 the sum of all even indexed timing mismatches, i.e., all timing mismatches ∆tA m from group A, and the sum of all odd indexed timing mismatches, i.e., all timing mismatches ∆tB m from group B. Therefore, the magnitude of α N (jΩ) is maxi2 mized if B (8) ∆tA m ≥ ∆tn , or vice versa, for all possible m and n. Thus, although we can apply an arbitrarily large random N −1 sequence υn n=0 to obtain uniformly distributed timing mismatches we can maximize the central aliased spectra, when we ensure that every 2nth sample is taken from group A and every (2n + 1)th sample is taken from group B. Within the groups we can still randomize, which is depicted in Fig. 3(b). In order to provide randomization within the two groups we need at least two additional channel ADCs (R = 2) to fulfill the sampling constraint of M Ts for each channel ADC. For R additional channel ADCs we can choose between R2 + 1 channel ADCs for each sampling instant without violating the sampling constraint. Because the selection of the two ADC groups is only based on the rank order of the timing mismatches, the absolute accuracy of the mismatch identification method does not matter and, therefore, the requirements on the timing mismatch identification are very low. Moreover, for larger numbers of channel ADCs and a typical symmetrical probability density function of the timing mismatch distribution, the difference between the mean and the median value of the timing mismatches becomes very small and we only have to determine the sign of the timing mismatches in order to divide them into the groups A and B and obtain spectral shaping. III. SIMULATION RESULTS In Fig. 5 we illustrate the effect of combining randomization and spectral shaping. For all three plots the input signal consists of three sine waves with different frequencies. In Fig. 5(a) we see the output spectrum of a TIADC (M = 6, R = 2) with timing mismatches. Additional aliased spectra of the input signal noticeably distort the whole spectrum. In Fig. 5(b) we have used conventional randomization to distribute the mismatch power. A low-pass filter at Ω4s can reduce the SINAD at most by 3dB. Finally, in Fig. 5(c) we have combined randomization and spectral shaping of timing mismatches. Hence, we have maximized the central aliased spectrum by optimizing the channel selection process and still have a uniformly distributed mismatch noise spectrum, which is considerably reduced (dashed and dashed-dotted line). With a simple low-pass filter we can filter out the spectrally shaped mismatch power and increase the SINAD and the SFDR compared to Fig. 5(b). In order to compare the performance of spectrally shaped randomization and conventional randomization we have simulated TIADCs with different numbers of channel ADCs M .

For each number of channel ADCs M we have simulated 1000 outcomes and have averaged the measured SINAD and the measured SFDR. We have assumed normally distributed timing mismatches with a standard deviation of 0.01Ts and a si2039 nusoidal input with frequency Ω0 = 16384 Ωs , from which we have taken 16384 samples. To allow advanced randomization we have used two additional channel ADCs (R = 2) for each simulation. Furthermore, we have used an ideal low-pass filter with cut-off frequency Ωcut = Ω4s . For each of the following figures we have plotted three curves: the randomized unfiltered case, the randomized filtered case, and the randomized, spectrally shaped, and filtered case. Fig. 6(a) shows the SINAD in effective number of bits (ENOB) [3] against the number of channel ADCs. The unfiltered average SINAD is 0.5 ENOB below the filtered SINAD. The spectrally shaped curve is 1.2 ENOB better than the unfiltered one. Therefore, we gain about 0.7 ENOB compared to the non-optimized randomization. Fig. 6(b) shows the average SFDR against the number of channels ADCs. The spectrally shaped and randomized TIADC increases the SFDR by about 5dB compared to the non-optimized one. Furthermore, we recognize that the filtered and the unfiltered TIADC have approximately the same average SFDR. This is because, on average, there are as many peaks below as above Ω4s , whereby the filtering has minor influence on the SFDR. The SFDR decreases with the number of channel ADCs, because we have used a finite number of samR decreases (R has been ples for the simulation and the ratio M constantly set equal to 2), whereby the distribution statistics of the mismatch error power become worse [7, 8]. A further problem is the change of group membership within the randomization process. Each time we have new estimates for the timing mismatches (cf. Fig. 3(a)), which differ from the last ones, we have to change the group membership of the channel ADCs, i.e., ADCs from group A go to group B and vice versa. In order to change the membership we have to give up fair randomization because the change of membership is deterministic. Nevertheless, we still have to ensure a minimal sampling period of M Ts for each channel ADC. It is easy to show that such an algorithm, which fulfills the above requirements and halt in finite time, exists. In Fig. 7 we see the execution time of a group change algorithm in relation to the number of channels. The time is measured in cycles, where one cycle consists of M processed channel ADCs. For each number of ADCs we have simulated 10000 arbitrary group membership changes. Even for a large number of channel ADCs the algorithm performs very well. Hence, the change of group membership should not influence the randomization statistics significantly. IV. CONCLUSION We have introduced an advanced randomization method that combines randomization and spectral shaping of timing

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Fig. 5. Effect of spectrally shaped randomization for a TIADC (M=6,R=2) with dominating timing mismatches. In the band of interest [0, reduce the mismatch noise.

SINAD in ENOB

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(b) The average SFDR against the number of channel ADCs M .

(a) The average SINAD in ENOB against the number of channel ADCs M .

Fig. 6. Performance enhancement due to spectral shaping. For each number of channel ADCs M we have simulated 1000 outcomes and have averaged the measured SINAD and the measured SFDR. We have assumed normally distributed timing mismatches with a standard deviation of 0.01Ts and a sinusoidal 2039 input with frequency Ω0 = 16384 Ωs , from which we have taken 16384 samples. To allow randomization, we use two extra channel ADCs (R = 2) for all simulations. The spectrally shaped and randomized algorithm performs remarkably better than pure randomization. (a) As expected, the randomized and filtered SINAD curve is roughly 0.5 ENOB above the randomized unfiltered SINAD curve. The randomized and spectrally shaped curve is about 1.2 ENOB better than the unfiltered curve. (b) Also for the SFDR we notice a significant improvement of about 5dB.

mismatches. With a simple low-pass filter the method can, compared to conventional randomization algorithms, improve the SFDR as well as the SINAD. The method assumes a dominating mismatch source, which is given by the timing mismatch. To find optimal shaping sequences, we only have to sort the channel ADCs into two groups, where one group contains all channel ADCs with a greater timing mismatch than the channel ADCs in the other group. Because we only compare

timing mismatches the absolute accuracy of the identification method does not matter and the requirements on the timing mismatch identification are very low. ACKNOWLEDGMENT Support of our research by Infineon Technologies AG, especially the useful comments from Dieter Draxelmayr, is gratefully acknowledged.

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11 9 Number of cycles

of the 2005 IEEE International Symposium on Circuits and Systems, ISCAS 2005, May 2005, to be published.

Worst Case Average (median) Average (mean) Best Case

7 5 3 1 8

16 32 64 128 Number of channel ADCs

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Fig. 7. Relation between the number of ADCs M and the needed number of cycles to change the group membership of the channel ADCs. We see that the change of membership should not influence the randomization.

REFERENCES [1] W. C. Black Jr. and D. A. Hodges, “Time-interleaved converter arrays,” IEEE Journal of Solid State Circuits, vol. 15, no. 6, pp. 1024–1029, December 1980. [2] C. Vogel, “The impact of combined channel mismatch effects in timeinterleaved ADCs,” IEEE Transactions on Instrumentation and Measurement, vol. 54, no. 1, pp. 415–427, February 2005. [3] “IEEE standard for terminology and test methods for analog-to-digital converters,” IEEE Std 1241-2000, June 2001. [4] H. Jin, E. K. F. Lee, and M. Hassoun, “Time-interleaved A/D converter with channel randomization,” in Proceedings of 1997 IEEE International Symposium on Circuits and Systems, 1997, ISCAS ’97, vol. 1, June 1997, pp. 425–428. [5] M. Tamba, A. Shimizu, H. Munakata, and T. Komuro, “A method to improve SFDR with random interleaved sampling method,” in International Test Conference, 2001, Proceedings., October 2001, pp. 512–520. [6] K. El-Sankary, A. Assi, and M. Sawan, “New sampling method to improve the SFDR of time-interleaved ADCs,” in Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS ’03., vol. 1, May 2003, pp. 833– 836. [7] J. Elbornsson, F. Gustafsson, and J.-E. Eklund, “Analysis of mismatch noise in randomly interleaved ADC system,” in 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP’03)., vol. 6, April 2003, pp. 277–280. [8] J. Elbornsson, “Analysis, estimation and compensation of mismatch effects in A/D converters,” Ph.D. dissertation, Link¨opings universitet, 2003. [9] C. Vogel and G. Kubin, “Analysis and compensation of nonlinearity mismatches in time-interleaved ADC arrays,” in Proceedings of the 2004 IEEE International Symposium on Circuits and Systems, ISCAS 2004, vol. 1, May 2004, pp. 593–596. [10] C. Vogel, D. Draxelmayr, and F. Kuttner, “Compensation of timing mismatches in time-interleaved analog-to-digital converters through transfer characteristics tuning,” in Proceedings of the 47th IEEE International Midwest Symposium On Circuits and Systems, MWSCAS, vol. 1, July 2004, pp. 341–344. [11] C. Vogel and G. Kubin, “Time-interleaved ADCs in the context of hybrid filter banks,” in Proceedings International Symposium on Signals, Systems, and Electronics (ISSSE), August 2004, pp. 214–217. [12] ——, “Modeling of time-interleaved ADCs with nonlinear hybrid filter banks,” International Journal of Electronics and Communications, vol. 05, 2005, to be published. [13] C. Vogel, D. Draxelmayr, and G. Kubin, “Spectral shaping of timing mismatches in time-interleaved analog-to-digital converters,” in Proceedings

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