Analysis and compensation of nonlinearity mismatches ... - IEEE Xplore
ANALYSIS AND COMPENSATION OF NONLINEARITY MISMATCHES IN TIME-INTERLEAVED ADC ARRAYS Christian Vogel and Gernot Kubin Christian Doppler Laboratory for Nonlinear Signal Processing Institute of Communications and Wave Propagation Graz University of Technology, Austria [email protected] — [email protected] fs /M
ABSTRACT
ADC0
Time-interleaved ADCs (TIADCs) are used to achieve high sampling rates. The drawback of such an architecture are mismatch effects, which decrease the signal-to-noise and distortion ratio (SINAD) and the spurious-free dynamic range (SFDR). Many papers have investigated the problem of mismatches but very few have considered the problem of nonlinearity mismatches. We present a mathematical framework for nonlinearity mismatches, describe their main characteristics, and show an effective compensation method to increase the SFDR.
fs /M ADC1 analog input xa(t)
digital output y[n] fs
MUX
fs /M ADCl fs /M ADCM −1
Fig. 1. Time-interleaved ADC with M channels.
1. INTRODUCTION The analog-to-digital converter (ADC) is still the bottleneck of modern telecommunication systems. A well-known technology to increase its sampling rate is the employment of a time-interleaved architecture. The concept of the time-interleaved architecture is illustrated in Fig. 1. The architecture consists of M channel ADCs working in a time-interleaved manner. Each channel ADC has a sampling period of MTs whereas the system has a sampling period of Ts . Thus, the conversion time requirements on each converter are relaxed by a factor M [1]. In an earlier paper we have presented a comprehensive analysis of combined offset, gain and timing mismatches [2]. Nevertheless, only few papers have investigated the influence of static nonlinearities, which are often measured and characterized as integral nonlinearity (INL) and differential nonlinearity (DNL) [3]. In [4] the authors show that the nonlinearity of the TIADC measured in terms of INL is better than the INL of its worst channel ADC. Our analysis reveals that this is only partially true, since nonlinearities behave dynamically in a time-interleaved architecture and cannot be characterized by conventional INL measurements. In [5] some characteristics of nonlinearity mismatches are illustrated, but no mathematical framework is provided. This paper gives a thorough analysis of nonlinearity mismatches, explains their main characteristics, and proposes a compensation method for them.
2. ANALYSIS OF NONLINEARITY MISMATCHES IN TIME-INTERLEAVED ADCS To calculate the output of a TIADC with nonlinearity mismatches we extend the theory of linear hybrid filter banks (HFB) [6, 7, 8] by putting a nonlinearity Fm (.) in the signal path. The nonlinear HFB is illustrated for the mth channel in Fig. 2. With the analysis filters Hm ( jΩ) = e jΩTs m e− jΩ∆tm ,
(1)
where the expression e− jΩ∆tm takes timing deviations of the sample and hold circuits among the channels into account, and the synthesis filters Gm e jΩTs = e− jωm |ω=ΩTs = e− jΩTs m
(2)
the output is given by Y e jΩTs
=
M−1
∑ Ym
e jΩTs
m=0
Support of our research by Infineon Technologies AG is gratefully acknowledged.
;,(((
,
=
M−1
∑
e− jΩTs m X˜m ( jΩM) ,
(3)
m=0
,6&$6
xa(t)
M · Ts Hm(jΩ)
Fm(.)
xm[n] = xm(nM Ts)
M
Gm(ejΩTs )
ym[n]
(1)
and f m (0) = gm and can rewrite Eq. (9) as Y e jΩTs
x˜m[n]
Fig. 2. The mth channel of a time-interleaved ADC modeled with a nonlinear hybrid filter bank.
where X˜m is the output of the nonlinearity Fm (.) for the sampled input signal ∞ Ωs 1 Xm ( jΩM) = ∑ Xa j Ω − p M · MTs p=−∞ Ωs , (4) · Hm j Ω − p M where the overall sampling frequency is Ωs = 2π Ts . The sampling process introduces aliased spectral components shifted by integer s multiples of the channel sampling frequency Ω M. One possibility to characterize the input-output relation of a static nonlinearity is to use a Maclaurin series, which is a Taylor series around x0 = 0, ∞
∑
f (x) =
k=0
f (k) (0) k x . k!
(5)
The argument x is determined by the input signal xm (t) sampled at t = nMTs . Thus, we can rewrite Eq. (5) for the output of the mth channel nonlinearity Fm (.) as x˜m (t) =
∞
∑
k=0
(k)
fm (0) k xm (t) k!
∞
∑
k=0
+
1 Ts
+
1 Ts
v p,k ( jΩ) =
(7)
(∗k)
where Xm ( jΩ) is defined as the (k-1)-fold convolution of Xm ( jΩ) with itself: ⎧ ⎪ 2πδ (Ω) for k = 0 ⎪ ⎪ ⎪ ⎨Xm ( jΩ) for k = 1 (∗k) Xm ( jΩ) = 1 k−1 ⎪ (X ∗ X ∗ ... ∗ X ) ( jΩ) for k ≥ 2 m m m ⎪ ⎪ 2π ⎪ ⎩
α p ( jΩ) =
βp =
A 2j
k
Y e
jΩTs
To show the connection to existing analyses of gain (gm ), off(0) set (om ) and timing mismatches (∆tm ) [9, 2], we set f m (0) = om
(13)
k ∑ n (−1)k−n e jΩ0 (2n−k)t n=0 k
(14)
= ·
1 A k k k · ∑ ∑ ∑ p=−∞ k=0 k! 2 j n=0 n Ωs k−n · δ Ω − Ω0 (2n − k) − p (−1) M 2π Ts
∞
∞
2π 1 M−1 (k) ∑ fm (0) e− jpm M . M m=0
(k)
(9)
2π 1 M−1 ∑ om e− jpm M . M m=0
and obtain for the Fourier transform k k A k (∗k) Xa ( jΩ) = ∑ n (−1)k−n 2πδ (Ω − Ω0 (2n − k)) . 2 j n=0 (15) Substituting Eq. (15) in Eq. (9) leads to
·
2π 1 M−1 − j(Ω−p Ωs )∆tm (k) M fm (0) e− jpm M . ∑e M m=0
(12)
From Eq. (9,10) we recognize that additional spectral compo(k) nents occur, when the derivatives of the nonlinearity f m (0) or the timing deviations ∆tm do not match. In order to find out where these spectral components appear, we study a sinusoidal input signal xa (t) = A sin (Ω0 t) and assume vanishing timing deviations ∆tm . We can write the kth power of the input signal xa (t) as
(8)
·
Ωs 2π 1 M−1 ∑ gm e− j(Ω−p M )∆tm e− jpm M , M m=0
(11)
and
(k−1)−convolutions
After inserting Eq. (7) in Eq. (3) we obtain (∗k) ∞ Xa ∞ j Ω − p ΩMs 1 = Y e jΩTs · ∑ ∑ Ts p=−∞ k! k=0
k! Ωs j Ω − p α ( jΩ) X a ∑ p M p=−∞ ∞ Ωs ∑ β p 2πδ Ω − p M p=−∞ p=−∞ k=2 ∞
Ωs 2π 1 M−1 (k) fm (0) e− j(Ω−p M )∆tm e− jpm M , ∑ M m=0
(6)
(k)
j Ω − p ΩMs
where
xka (t) =
fm (0) (∗k) Xm ( jΩ) , k!
∑ ∑ v p,k ( jΩ)
Xa
(10)
and its Fourier transform as X˜m ( jΩ) =
1 Ts
=
(∗k)
∞
∞
(16)
When all derivatives f m (0) are identical the last row in Eq. (16) (k) evaluates to f m (0) and we obtain the case of a single ADC, where the harmonics appear at integer multiples of Ω0 k. As (k) soon as the derivatives f m (0) differ, we get additional spectral Ωs lines at ±Ω0 k + p M , where the values are weighted by the discrete Fourier transform (DFT) of the corresponding kth deriva(k) tives f m (0). Hence, the harmonic tones produced for p = 0 are (k) weighted by the average of the M derivatives f m (0).
,
(k)
ADC0
tion over all f sn (0) has to be close to zero in order to remove the harmonics related to the input frequency. A solution exploiting this property is to design channel ADCs, which have pairwise complementary nonlinearities, so that in the ideal case (k) (k) fm (0) = − f m+1 (0) holds for k = 1. Channel ADCs designed in this way and the method of randomization finally distribute the error energy of the nonlinearities over the entire frequency band, even for the harmonics of the p = 0 baseband.
ADC1
4. SIMULATION RESULTS
ADC1 ADC0 ADC0
ADC1 ADC2
1
2
3
ADC2
4
Sampling instant (nTs) Fig. 3. Randomization of the channel ADCs with M = 2 and R = 1. The dashed line shows one possible way to choose the random sequence.
3. COMPENSATION OF NONLINEARITY MISMATCHES It has been shown that channel randomization is one method to decrease the SFDR in a TIADC [10, 11]. For that purpose, we either need additional channel ADCs or we have to decrease the sampling rate. If we have a redundant array of (M + R) channel ADCs, operating with a sampling interval of MTs , and we would like to build a TIADC with a sampling interval of Ts , we can choose at each time instant among R + 1 channel ADCs without violating the sampling constraints. This is illustrated in Fig. 3, where 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. If we use a periodic pseudo random sequence with the period N, fulfilling the above requirements, where one period of the sequence is defined as {sn }0N−1 with sn ∈ {0, 1, ..., M + R − 1} to determine the sequence of channel ADCs, we can rewrite Eq. (9) as (∗k) Ωs ∞ ∞ j Ω − p X a N 1 · = Y N e jΩTs ∑ ∑ Ts p=−∞ k! k=0 ·
2π 1 N−1 − j(Ω−p Ωs )∆tsn (k) N fsn (0) e− jpn N . ∑e N n=0
In order to show the effects of nonlinearity mismatches we use a behavioral MATLAB simulation environment [12]. We have simulated a TIADC with 10bit resolution in three different channel configurations. For each configuration we have taken 1024 samples of a sinusoidal input signal and plot the output spectrum. In Fig. 4 we see a TIADC with four channels and nonlinearity mismatches. Beside the harmonics which are related to the input signal, we s notice additional aliased harmonics centered around p Ω M ± k f0 = [0 ± k f 0 , 0.25 ± k f 0 , 0.5 ± k f 0 , 0.75 ± k f 0 ]. In Fig. 5 we see a TIADC with the same nonlinearities as in Fig. 4 but with two additional channels (and nonlinearities) to allow randomization of the channels without decreasing the sampling rate. The non-aliased harmonics of the input signal are still there, as we would expect it from our analysis. In Fig. 6 we combine randomization of a six channel TIADC with pairwise complementary nonlinearities and get a nearly flat noise floor. There are still small harmonics in the baseband, since we did not simulate perfectly complementary pairs of nonlinearities, as this would not have reflected real ADC channels. Nevertheless, for the three configurations the SFDR has increased from 59.8dB to 72dB and the total harmonic distortion (THD) has decreased from -59.7dB to -69.7dB. 5. CONCLUSION We have analyzed nonlinearity mismatches in TIADCs. Therefore, we have extended the theory of linear hybrid filter banks by static nonlinearities and have applied it to TIADCs. We have developed a mathematical framework and have investigated the consequences of nonlinearity mismatches. Similar to the well-known linear mismatch effects, baseband frequencies are aliased in upper frequency bands. Based on our analysis, we have developed a randomization strategy for redundant time-interleaved ADC arrays, which significantly increases the SFDR and also decreases the THD.
(17) Through randomization we virtually increase the number of channels to N and, consequently, the number of aliased spectra. Nevertheless, the amplitude of the DFT coefficients, except for p = 0, decreases due to the factor N1 in Eq. (17) and, therefore, the distortion energy is distributed more evenly over the frequency band. For p = 0, the DFT returns the average value, which is advantageous if we only consider gain and timing mismatches, since we amplify the input spectra by the average gain and the average timing mismatch tends to zero. However, the other harmonics caused by the nonlinearities behave like in a single ADC. Thus, the randomization process distributes all mismatch depending harmonics over the frequency band, but harmonics related to the input frequency are still there and limit the SFDR. From Eq. (17) we see that for p = 0 the averaged summa-
,
6. REFERENCES [1] W. C. Black Jr. and D. A. Hodges, “Time-interleaved converter arrays,” IEEE J. Solid-State Circuits, vol. 15, no. 6, pp. 1024–1029, December 1980. [2] C. Vogel, “Comprehensive error analysis of combined channel mismatch effects in time-interleaved ADCs,” in Proc. IMTC 2003., May 2003, vol. 1, pp. 733–738. [3] “IEEE standard for terminology and test methods for analogto-digital converters,” IEEE Std 1241-2000, June 2001. [4] J. B. Sim˜oes, J. Landeck, and C. M. B. A. Correia, “Nonlinearity of a data-acquisition system with interleaving/multiplexing,” IEEE Trans. Instrum. Meas., vol. 46, no. 6, pp. 1274–1279, December 1997.
1 INL in LSB
INL in LSB
1 0.5 0 Ŧ0.5 Ŧ1 0
256
512
768
0.5 0 Ŧ0.5 Ŧ1 0
1023
256
(a) Nonlinearities
0
Ŧf
f0
0
Ŧ20
0
Ŧ20 Signal Power in dBc
Signal Power in dBc
1023
0 Ŧf
f
Ŧ40 harmonics related to Ŧ k f
harmonics related to k f0
0
harmonics related to ± k f +0.5 0
harmonics related to ± k f0+0.25
harmonics related to ± k f +0.75 0
Ŧ80
Ŧ100 0
768
(a) Nonlinearities
0
Ŧ60
512
Ŧ40 harmonics related to k f
harmonics related to Ŧ k f0
0
Ŧ60
aliased harmonics are randomly spread
Ŧ80
0.2
0.4 0.6 Normalized frequency
0.8
Ŧ100 0
1
0.2
0.4 0.6 Normalized frequency
0.8
1
(b) Output spectrum
(b) Output spectrum
Fig. 4. TIADC with four channels. The nonlinearities, characterized as integral nonlinearity (INL), of all channels are different but have similar general characteristics.
Fig. 5. TIADC with six channels to allow randomization without decreasing the sampling rate. The aliased frequencies are randomly spread over the frequency band but the harmonics of the input signal are still there.
[5] N. Kurosawa, H. Kobayashi, and K. Kobayashi, “Channel linearity mismatch effects in time-interleaved ADC systems,” in Proc. ISCAS 2001, May 2001, vol. 1, pp. 420–423.
INL in LSB
1
[6] P. L¨owenborg, H. Johansson, and L. Wanhammar, “On the frequency-response of M-channel mixed analog and digital maximally decimated filter banks,” in Proc. European Conf. Circuit Theory Design, Stresa, Italy, August 1999.
0 Ŧ0.5 Ŧ1 0
256
512
768
1023
(a) Nonlinearities
[7] S. R. Velazquez, T. Q. Nguyen, and S. R. Broadstone, “Design of hybrid filter banks for analog/digital conversion,” IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 956–967, April 1998.
0 Ŧf0
f0
[8] A. Petraglia and S.K. Mitra, “Analysis of mismatch effects among A/D converters in a time-interleaved waveform digitizer,” IEEE Trans. Instrum. Meas., vol. 40, no. 5, pp. 831– 835, October 1991.
Signal Power in dBc
Ŧ20
[9] M. Gustavsson, J. J. Wikner, and N.N. Tan, CMOS Data Converters for Communications, Kluwer Academic Publishers, 2000. [10] H. Jin, E. K. F. Lee, and M. Hassoun, “Time-interleaved A/D converter with channel randomization,” in Proc. ISCAS ’97, 1997, vol. 1, pp. 425–428.
Ŧ40
Ŧ60
harmonics related harmonics related to Ŧ k f0 to k f0 aliased harmonics are randomly spread
Ŧ80
Ŧ100 0
[11] M. Tamba, A. Shimizu, H. Munakata, and T. Komuro, “A method to improve SFDR with random interleaved sampling method,” in Proc. International Test Conference, 2001, October 2001, pp. 512–520. [12] C. Vogel and H. Koeppl, “Behavioral modeling of timeinterleaved ADCs using MATLAB,” in Proc. Austrochip 2003, October 2003, pp. 45–48.
0.5
0.2
0.4 0.6 Normalized frequency
0.8
1
(b) Output spectrum
Fig. 6. TIADC with six channels, pairwise complementary nonlinearities, and randomization. Most of the error energy is distributed over the frequency band.
,
ABSTRACT
ADC0
Time-interleaved ADCs (TIADCs) are used to achieve high sampling rates. The drawback of such an architecture are mismatch effects, which decrease the signal-to-noise and distortion ratio (SINAD) and the spurious-free dynamic range (SFDR). Many papers have investigated the problem of mismatches but very few have considered the problem of nonlinearity mismatches. We present a mathematical framework for nonlinearity mismatches, describe their main characteristics, and show an effective compensation method to increase the SFDR.
fs /M ADC1 analog input xa(t)
digital output y[n] fs
MUX
fs /M ADCl fs /M ADCM −1
Fig. 1. Time-interleaved ADC with M channels.
1. INTRODUCTION The analog-to-digital converter (ADC) is still the bottleneck of modern telecommunication systems. A well-known technology to increase its sampling rate is the employment of a time-interleaved architecture. The concept of the time-interleaved architecture is illustrated in Fig. 1. The architecture consists of M channel ADCs working in a time-interleaved manner. Each channel ADC has a sampling period of MTs whereas the system has a sampling period of Ts . Thus, the conversion time requirements on each converter are relaxed by a factor M [1]. In an earlier paper we have presented a comprehensive analysis of combined offset, gain and timing mismatches [2]. Nevertheless, only few papers have investigated the influence of static nonlinearities, which are often measured and characterized as integral nonlinearity (INL) and differential nonlinearity (DNL) [3]. In [4] the authors show that the nonlinearity of the TIADC measured in terms of INL is better than the INL of its worst channel ADC. Our analysis reveals that this is only partially true, since nonlinearities behave dynamically in a time-interleaved architecture and cannot be characterized by conventional INL measurements. In [5] some characteristics of nonlinearity mismatches are illustrated, but no mathematical framework is provided. This paper gives a thorough analysis of nonlinearity mismatches, explains their main characteristics, and proposes a compensation method for them.
2. ANALYSIS OF NONLINEARITY MISMATCHES IN TIME-INTERLEAVED ADCS To calculate the output of a TIADC with nonlinearity mismatches we extend the theory of linear hybrid filter banks (HFB) [6, 7, 8] by putting a nonlinearity Fm (.) in the signal path. The nonlinear HFB is illustrated for the mth channel in Fig. 2. With the analysis filters Hm ( jΩ) = e jΩTs m e− jΩ∆tm ,
(1)
where the expression e− jΩ∆tm takes timing deviations of the sample and hold circuits among the channels into account, and the synthesis filters Gm e jΩTs = e− jωm |ω=ΩTs = e− jΩTs m
(2)
the output is given by Y e jΩTs
=
M−1
∑ Ym
e jΩTs
m=0
Support of our research by Infineon Technologies AG is gratefully acknowledged.
;,(((
,
=
M−1
∑
e− jΩTs m X˜m ( jΩM) ,
(3)
m=0
,6&$6
xa(t)
M · Ts Hm(jΩ)
Fm(.)
xm[n] = xm(nM Ts)
M
Gm(ejΩTs )
ym[n]
(1)
and f m (0) = gm and can rewrite Eq. (9) as Y e jΩTs
x˜m[n]
Fig. 2. The mth channel of a time-interleaved ADC modeled with a nonlinear hybrid filter bank.
where X˜m is the output of the nonlinearity Fm (.) for the sampled input signal ∞ Ωs 1 Xm ( jΩM) = ∑ Xa j Ω − p M · MTs p=−∞ Ωs , (4) · Hm j Ω − p M where the overall sampling frequency is Ωs = 2π Ts . The sampling process introduces aliased spectral components shifted by integer s multiples of the channel sampling frequency Ω M. One possibility to characterize the input-output relation of a static nonlinearity is to use a Maclaurin series, which is a Taylor series around x0 = 0, ∞
∑
f (x) =
k=0
f (k) (0) k x . k!
(5)
The argument x is determined by the input signal xm (t) sampled at t = nMTs . Thus, we can rewrite Eq. (5) for the output of the mth channel nonlinearity Fm (.) as x˜m (t) =
∞
∑
k=0
(k)
fm (0) k xm (t) k!
∞
∑
k=0
+
1 Ts
+
1 Ts
v p,k ( jΩ) =
(7)
(∗k)
where Xm ( jΩ) is defined as the (k-1)-fold convolution of Xm ( jΩ) with itself: ⎧ ⎪ 2πδ (Ω) for k = 0 ⎪ ⎪ ⎪ ⎨Xm ( jΩ) for k = 1 (∗k) Xm ( jΩ) = 1 k−1 ⎪ (X ∗ X ∗ ... ∗ X ) ( jΩ) for k ≥ 2 m m m ⎪ ⎪ 2π ⎪ ⎩
α p ( jΩ) =
βp =
A 2j
k
Y e
jΩTs
To show the connection to existing analyses of gain (gm ), off(0) set (om ) and timing mismatches (∆tm ) [9, 2], we set f m (0) = om
(13)
k ∑ n (−1)k−n e jΩ0 (2n−k)t n=0 k
(14)
= ·
1 A k k k · ∑ ∑ ∑ p=−∞ k=0 k! 2 j n=0 n Ωs k−n · δ Ω − Ω0 (2n − k) − p (−1) M 2π Ts
∞
∞
2π 1 M−1 (k) ∑ fm (0) e− jpm M . M m=0
(k)
(9)
2π 1 M−1 ∑ om e− jpm M . M m=0
and obtain for the Fourier transform k k A k (∗k) Xa ( jΩ) = ∑ n (−1)k−n 2πδ (Ω − Ω0 (2n − k)) . 2 j n=0 (15) Substituting Eq. (15) in Eq. (9) leads to
·
2π 1 M−1 − j(Ω−p Ωs )∆tm (k) M fm (0) e− jpm M . ∑e M m=0
(12)
From Eq. (9,10) we recognize that additional spectral compo(k) nents occur, when the derivatives of the nonlinearity f m (0) or the timing deviations ∆tm do not match. In order to find out where these spectral components appear, we study a sinusoidal input signal xa (t) = A sin (Ω0 t) and assume vanishing timing deviations ∆tm . We can write the kth power of the input signal xa (t) as
(8)
·
Ωs 2π 1 M−1 ∑ gm e− j(Ω−p M )∆tm e− jpm M , M m=0
(11)
and
(k−1)−convolutions
After inserting Eq. (7) in Eq. (3) we obtain (∗k) ∞ Xa ∞ j Ω − p ΩMs 1 = Y e jΩTs · ∑ ∑ Ts p=−∞ k! k=0
k! Ωs j Ω − p α ( jΩ) X a ∑ p M p=−∞ ∞ Ωs ∑ β p 2πδ Ω − p M p=−∞ p=−∞ k=2 ∞
Ωs 2π 1 M−1 (k) fm (0) e− j(Ω−p M )∆tm e− jpm M , ∑ M m=0
(6)
(k)
j Ω − p ΩMs
where
xka (t) =
fm (0) (∗k) Xm ( jΩ) , k!
∑ ∑ v p,k ( jΩ)
Xa
(10)
and its Fourier transform as X˜m ( jΩ) =
1 Ts
=
(∗k)
∞
∞
(16)
When all derivatives f m (0) are identical the last row in Eq. (16) (k) evaluates to f m (0) and we obtain the case of a single ADC, where the harmonics appear at integer multiples of Ω0 k. As (k) soon as the derivatives f m (0) differ, we get additional spectral Ωs lines at ±Ω0 k + p M , where the values are weighted by the discrete Fourier transform (DFT) of the corresponding kth deriva(k) tives f m (0). Hence, the harmonic tones produced for p = 0 are (k) weighted by the average of the M derivatives f m (0).
,
(k)
ADC0
tion over all f sn (0) has to be close to zero in order to remove the harmonics related to the input frequency. A solution exploiting this property is to design channel ADCs, which have pairwise complementary nonlinearities, so that in the ideal case (k) (k) fm (0) = − f m+1 (0) holds for k = 1. Channel ADCs designed in this way and the method of randomization finally distribute the error energy of the nonlinearities over the entire frequency band, even for the harmonics of the p = 0 baseband.
ADC1
4. SIMULATION RESULTS
ADC1 ADC0 ADC0
ADC1 ADC2
1
2
3
ADC2
4
Sampling instant (nTs) Fig. 3. Randomization of the channel ADCs with M = 2 and R = 1. The dashed line shows one possible way to choose the random sequence.
3. COMPENSATION OF NONLINEARITY MISMATCHES It has been shown that channel randomization is one method to decrease the SFDR in a TIADC [10, 11]. For that purpose, we either need additional channel ADCs or we have to decrease the sampling rate. If we have a redundant array of (M + R) channel ADCs, operating with a sampling interval of MTs , and we would like to build a TIADC with a sampling interval of Ts , we can choose at each time instant among R + 1 channel ADCs without violating the sampling constraints. This is illustrated in Fig. 3, where 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. If we use a periodic pseudo random sequence with the period N, fulfilling the above requirements, where one period of the sequence is defined as {sn }0N−1 with sn ∈ {0, 1, ..., M + R − 1} to determine the sequence of channel ADCs, we can rewrite Eq. (9) as (∗k) Ωs ∞ ∞ j Ω − p X a N 1 · = Y N e jΩTs ∑ ∑ Ts p=−∞ k! k=0 ·
2π 1 N−1 − j(Ω−p Ωs )∆tsn (k) N fsn (0) e− jpn N . ∑e N n=0
In order to show the effects of nonlinearity mismatches we use a behavioral MATLAB simulation environment [12]. We have simulated a TIADC with 10bit resolution in three different channel configurations. For each configuration we have taken 1024 samples of a sinusoidal input signal and plot the output spectrum. In Fig. 4 we see a TIADC with four channels and nonlinearity mismatches. Beside the harmonics which are related to the input signal, we s notice additional aliased harmonics centered around p Ω M ± k f0 = [0 ± k f 0 , 0.25 ± k f 0 , 0.5 ± k f 0 , 0.75 ± k f 0 ]. In Fig. 5 we see a TIADC with the same nonlinearities as in Fig. 4 but with two additional channels (and nonlinearities) to allow randomization of the channels without decreasing the sampling rate. The non-aliased harmonics of the input signal are still there, as we would expect it from our analysis. In Fig. 6 we combine randomization of a six channel TIADC with pairwise complementary nonlinearities and get a nearly flat noise floor. There are still small harmonics in the baseband, since we did not simulate perfectly complementary pairs of nonlinearities, as this would not have reflected real ADC channels. Nevertheless, for the three configurations the SFDR has increased from 59.8dB to 72dB and the total harmonic distortion (THD) has decreased from -59.7dB to -69.7dB. 5. CONCLUSION We have analyzed nonlinearity mismatches in TIADCs. Therefore, we have extended the theory of linear hybrid filter banks by static nonlinearities and have applied it to TIADCs. We have developed a mathematical framework and have investigated the consequences of nonlinearity mismatches. Similar to the well-known linear mismatch effects, baseband frequencies are aliased in upper frequency bands. Based on our analysis, we have developed a randomization strategy for redundant time-interleaved ADC arrays, which significantly increases the SFDR and also decreases the THD.
(17) Through randomization we virtually increase the number of channels to N and, consequently, the number of aliased spectra. Nevertheless, the amplitude of the DFT coefficients, except for p = 0, decreases due to the factor N1 in Eq. (17) and, therefore, the distortion energy is distributed more evenly over the frequency band. For p = 0, the DFT returns the average value, which is advantageous if we only consider gain and timing mismatches, since we amplify the input spectra by the average gain and the average timing mismatch tends to zero. However, the other harmonics caused by the nonlinearities behave like in a single ADC. Thus, the randomization process distributes all mismatch depending harmonics over the frequency band, but harmonics related to the input frequency are still there and limit the SFDR. From Eq. (17) we see that for p = 0 the averaged summa-
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6. REFERENCES [1] W. C. Black Jr. and D. A. Hodges, “Time-interleaved converter arrays,” IEEE J. Solid-State Circuits, vol. 15, no. 6, pp. 1024–1029, December 1980. [2] C. Vogel, “Comprehensive error analysis of combined channel mismatch effects in time-interleaved ADCs,” in Proc. IMTC 2003., May 2003, vol. 1, pp. 733–738. [3] “IEEE standard for terminology and test methods for analogto-digital converters,” IEEE Std 1241-2000, June 2001. [4] J. B. Sim˜oes, J. Landeck, and C. M. B. A. Correia, “Nonlinearity of a data-acquisition system with interleaving/multiplexing,” IEEE Trans. Instrum. Meas., vol. 46, no. 6, pp. 1274–1279, December 1997.
1 INL in LSB
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Fig. 4. TIADC with four channels. The nonlinearities, characterized as integral nonlinearity (INL), of all channels are different but have similar general characteristics.
Fig. 5. TIADC with six channels to allow randomization without decreasing the sampling rate. The aliased frequencies are randomly spread over the frequency band but the harmonics of the input signal are still there.
[5] N. Kurosawa, H. Kobayashi, and K. Kobayashi, “Channel linearity mismatch effects in time-interleaved ADC systems,” in Proc. ISCAS 2001, May 2001, vol. 1, pp. 420–423.
INL in LSB
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[6] P. L¨owenborg, H. Johansson, and L. Wanhammar, “On the frequency-response of M-channel mixed analog and digital maximally decimated filter banks,” in Proc. European Conf. Circuit Theory Design, Stresa, Italy, August 1999.
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[7] S. R. Velazquez, T. Q. Nguyen, and S. R. Broadstone, “Design of hybrid filter banks for analog/digital conversion,” IEEE Trans. Signal Processing, vol. 46, no. 4, pp. 956–967, April 1998.
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[8] A. Petraglia and S.K. Mitra, “Analysis of mismatch effects among A/D converters in a time-interleaved waveform digitizer,” IEEE Trans. Instrum. Meas., vol. 40, no. 5, pp. 831– 835, October 1991.
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[9] M. Gustavsson, J. J. Wikner, and N.N. Tan, CMOS Data Converters for Communications, Kluwer Academic Publishers, 2000. [10] H. Jin, E. K. F. Lee, and M. Hassoun, “Time-interleaved A/D converter with channel randomization,” in Proc. ISCAS ’97, 1997, vol. 1, pp. 425–428.
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[11] M. Tamba, A. Shimizu, H. Munakata, and T. Komuro, “A method to improve SFDR with random interleaved sampling method,” in Proc. International Test Conference, 2001, October 2001, pp. 512–520. [12] C. Vogel and H. Koeppl, “Behavioral modeling of timeinterleaved ADCs using MATLAB,” in Proc. Austrochip 2003, October 2003, pp. 45–48.
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Fig. 6. TIADC with six channels, pairwise complementary nonlinearities, and randomization. Most of the error energy is distributed over the frequency band.
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