A Compensation Method for Magnitude Response Mismatches in Two-channel Time-interleaved Analog-to-Digital Converters Stefan Mendel
Christian Vogel
Christian Doppler Laboratory for Nonlinear Signal Processing Graz University of Technology, Austria Email:
[email protected]
Signal Processing and Speech Communication Laboratory Graz University of Technology, Austria Email:
[email protected]
Abstract— Analog-to-digital converters (ADCs) are critical components of signal processing systems and one of the bottlenecks of modern telecommunication systems. Time-interleaved ADCs (TI-ADCs), in which multiple ADCs are combined, are an effective way to achieve high sampling rates in order to comply with modern telecommunication standards. The drawback of such TI-ADCs are additional errors that are due to mismatches among the channels, which degrade the overall performance. Several qualified methods have been proposed to compensate offset, static gain, and timing (or linear-phase) mismatches. However, an effective method to compensate frequency-dependent magnitude response mismatches is still missing. In this paper we consider the compensation of frequency-dependent magnitude response mismatches for a two-channel time-interleaved sampling system. A single linear-phase finite impulse response (FIR) filter cascaded by a single time-varying multiplier provides the magnitude response compensation so that the performance of a TIADC is no longer limited by linear channel mismatches.
∞ q=−∞
y0 (t)
h0 (t) ADC0 x(t)
y(t) ∞ q=−∞
y[n]
y1 (t)
ADC1
In order to increase the sampling rate of an analog-to-digital converter (ADC) beyond a certain production process limit, time interleaving of ADCs has been proposed [1]. The analog input signal is successively sampled by different channel ADCs in a cyclic manner. Theoretically, the overall sampling rate of such a timeinterleaved ADC (TI-ADC) scales with the number of ADCs in contrast to a single ADC. Ideally, the characteristics of all ADCs should be identical; in practice, however, various technology dependent imperfections cause component mismatches. These mismatch errors become evident as fluctuation of the sampled values in the time domain and as an increase of spurious spectral components in the frequency domain. Therefore mismatches degrade for example the signal-to-noise and distortion ratio (SINAD) [2]. Offset, static gain, timing (or linear-phase), and frequency response mismatches are four significant types of mismatch errors [3]. Several techniques to compensate for timing mismatch errors have been introduced, e.g., [4] or [5]. A prototype implementation considering offset, gain, and timing mismatch calibration has been presented in [6]. In [7] it has been pointed out that magnitude response mismatches are the remaining factor limiting the performance, after having compensated for offset, static gain, and linear-phase mismatches. The influence of nonlinear-phase response mismatches is significantly smaller than the error due to magnitude response mismatches. Thus, an open issue to compensate frequency response mismatches is the compensation of magnitude response mismatches. The author in [8] shows a method to compensate frequency-dependent mismatches based on a discrete Fourier transform (DFT). This approach is limited to a finite number of samples due to the DFT calculation and it is not shown how to extend this method to continuous processing of samples. In [9] measurements of each channel transfer function are used to design finite-impulse response (FIR) filters, which need large tap sizes for compensating linear mismatches. An estimation
Impulse train to sequence converter
δ(t − (2 · q + 1) · Ts )
h1 (t)
I. I NTRODUCTION
1-4244-0395-2/06/$20.00 ©2006 IEEE.
δ(t − (2 · q + 0) · Ts )
TI-ADC Fig. 1. Time-interleaved ADC with two channels. Each ADC is modeled by a linear filter (h0 (t), h1 (t)) and a sampler, which has a sampling period of 2 · Ts and a constant time offset (0 · Ts , 1 · Ts ). The TI-ADC output y(t) can be interpreted as the sum of the signal y˜(t) and an error signal e(t) introduced by mismatches between h0 (t) and h1 (t).
equalizing technique to reduce the effect of linear and nonlinear mismatches in the transfer characteristic is applied in [10]. In this paper we propose a novel compensation structure for magnitude response mismatches in a two channel time-interleaved sampling system. Only a single FIR filter followed by a timevarying multiplier is used to compensate the magnitude response mismatches. Since many methods for compensating offset, static gain, and timing mismatches are known, we assume for the analysis that these errors have already been compensated and do not limit the TI-ADC performance. Nevertheless, in the simulation section we consider the influence of all mismatches. II. A NALYSIS Figure 1 depicts a linear two-channel time-interleaved ADC model, where the behavior of the channel ADCs is represented by linear filters with impulse responses h0 (t) and h1 (t). The output before impulse train to sequence conversion becomes y(t) =
1
x(t) ∗ hm (t) ·
m=0
= y˜(t) + e(t),
∞
δ(t − (2q + m) · Ts )
q=−∞
(1)
where δ(t) is the Dirac Delta function, Ts is the sampling period, and ∗ denotes the convolution operation. The TI-ADC output is y(t) =
712
α0 · X(jΩ)
y˜(t) + e(t), where y˜(t) represents the output without mismatches, i.e. for h0 (t) = h1 (t), and where e(t) is an error term introduced by mismatches between h0 (t) and h1 (t). The Fourier transform of (1) is Y (jΩ) = Y˜ (jΩ) + E(jΩ) (2)
α1 · X j Ω +
Ωs 2
α1 · X j Ω −
Ωs 2
Ω
with
1 α0 j Ω − qΩs Y˜ (jΩ) = Ts ∞
q=−∞
q=−∞
Ωs ×X j Ω − − qΩs 2
α1 α0
(4)
· α0 · X j Ω +
1 (|H0 (jΩ)| + |H1 (jΩ)|), 2 1 α1 (jΩ) = (|H0 (jΩ)| − |H1 (jΩ)|), 2
(6)
where α0 (jΩ) is the average and α1 (jΩ) is the weighted difference of the channel’s magnitude responses.
· α1 · X(jΩ)
− qΩs
0
Ωs
Ωs 2
α0 −
α21 α0
· X(jΩ)
Ω −Ωs
− Ω2s
0
Ωs
Ωs 2
(c) Spectrum of the compensated output ˆ Ycomp (jΩ) = Y (jΩ) − E(jΩ) Fig. 2. In (a) the output spectrum Y (jΩ) of a TI-ADC is depicted. Beside the input signal X(jΩ) (solid), additional spectral components, which are shifted (by Ω2s ) and weighted (by α1 (jΩ)) versions of the input spectrum ˆ X(jΩ) (dashed), are shown. In (b) the reconstructed error spectrum E(jΩ), which is a scaled and shifted version of the output spectrum in (a), is shown. In (c) the compensated output Ycomp (jΩ), obtained by subtracting (b) from (a) to delete the spurious component at Ω − Ω2s , is depicted. In the notation of this figure we have neglected the frequency dependence of the coefficients α0 and α1 .
∞ 1 Ycomp (jΩ) = Ts
α0 (j(Ω − qΩs )) · X(j(Ω − qΩs ))
q=−∞
α1 (j(Ω − qΩs ))2 · X(j(Ω − qΩs )) − α0 (j(Ω − qΩs ))
(8)
ˆ E(jΩ)−E(jΩ)
− qΩs
Ωs − qΩs 2
− Ω2s
The basic idea of the proposed compensation method is shown in Fig. 2. The TI-ADC output of (2) is shown in Fig. 2(a), where the spectrum Yˆ (jΩ) (3) is depicted as solid line and the spectrum E(jΩ), representing the magnitude response mismatches (4), is depicted as ˆ dashed line in Fig. 2(a). We obtain a reconstructed spectrum E(jΩ) of this mismatch spectrum E(jΩ) by scaling the TI-ADC output 1 (jΩ) and shifting it by Ω2s . That is, Y (jΩ) with α α0 (jΩ)
j Ω−
Ωs 2
and is illustrated in Fig. 2(b). The compensated output spectrum ˆ i.e. Ycomp (jΩ) is the difference between Y (jΩ) and E(jΩ),
III. C OMPENSATION M ETHOD
· α0 · X j Ω −
ˆ (b) Reconstructed spectrum E(jΩ) of the mismatch error spectrum E(jΩ)
(5)
α0 (jΩ) =
×Y
α1 α0
Ω −Ωs
where Ωs denotes the overall angular sampling frequency of the system and Hm (jΩ), X(jΩ), Y˜ (jΩ) and E(jΩ) are the Fourier transforms of hm (t), x(t), y˜(t) and e(t), respectively. The frequencydependent complex coefficients αk (jΩ) are given by the weighted DFT of the channel frequency responses Hm (jΩ). If both channel impulse responses h0 (t) and h1 (t) were equal, no mismatch errors would occur, i.e. e(t) and therefore E(jΩ) vanishes. In practice, however, the impulse responses are different and cause an additional spurious spectral copy of the input signal X(jΩ) at Ω2s , which is weighted with α1 (jΩ). Hence, as long as there are mismatches the performance of the TI-ADC is limited. In this paper we aim to compensate only mismatches of the magnitude responses |Hm (jΩ)|. Furthermore, we assume that all other mismatches have already been compensated. Consequently, we consider |Hm (jΩ)| instead of Hm (jΩ) and rewrite equation (5) as
q=−∞
α1 α0
m=0
Ωs 2 Ωs 2
Ωs 2
1 1 Hm (jΩ) · e−jkmπ , 2
α0 j Ω −
Ωs
Ωs 2
,
∞ Ωs 1 α1 j Ω − − qΩs E(jΩ) = Ts 2
∞ α1 j Ω − ˆ E(jΩ) =
0
(a) TI-ADC output spectrum Y (jΩ)
αk (jΩ) =
− Ω2s
(3)
×X j Ω − qΩs
and
−Ωs
(7)
and is shown in Fig. 2(c). The scaled and shifted spectral component due to magnitude response mismatches has been compensated. As a 2 , drawback, the weight of the input spectrum is influenced by αα10(jΩ) (jΩ) introduced by the difference of the actual error spectrum E(jΩ) and
713
90
z −D
y[n-D]
ycomp [n-D] 80
eˆ[n-D]
−
SINAD in dB
y[n]
g[n]
70 60 50
(−1)(n−D) 40
Magnitude Response Compensation 30 0
Fig. 3. The output of a two-channel TI-ADC y[n] = y˜[n] + e[n] is filtered with the linear phase FIR filter with an impulse response g[n] and a delay D. After modulating the signal with the time varying constant (−1)n−D the reconstructed error signal eˆ[n] is subtracted from the distorted, delayed signal y[n − D] to obtain ycomp [n − D] = y˜[n − D] + e[n − D] − eˆ[n − D].
ˆ its reconstruction E(jΩ). IV. I MPLEMENTATION Figure 3 shows the magnitude response compensation structure. The linear-phase FIR filter with impulse response g[n] is an approx1 (jΩ) . We reconstruct the error imation of the frequency response α α0 (jΩ) signal e[n] by filtering the TI-ADC output y[n] with the filter g[n] and modulating it, that is eˆ[n − D] = (y ∗ g)[n] · (−1)(n−D) ,
(9)
where D denotes the delay of the filter g[n]. Subtracting eˆ[n] from the TI-ADC output samples y[n − D] yields the compensated output ycomp [n−D]. For the design of g[n] we need to know the coefficient α1 (jΩ) . Therefore we have to identify α0 (jΩ) and α1 (jΩ) from the α0 (jΩ) frequency response of the TI-ADC output Y (jΩ). For this purpose we feed J sinusoids with different frequencies spread over the range of interest, e.g., the Nyquist range, sequentially into the TI-ADC, as shown in [9]. The frequencies Ωi with i = 0, 1, ..., J − 1 of the input signal and its shifted version at Ωi + Ω2s must satisfy the coherent sampling condition to prevent leakage effects. Furthermore, the shifted spectrum must not fall onto the same frequency bin with the input signal tone. For each frequency Ωi we identify α0 (jΩ) located at Ωi and α1 (jΩ) located at Ωi + Ω2s in the frequency response Y (jΩ). The frequency responses of the two channel ADCs H0 (jΩ) and H1 (jΩ) are calculated according to (5). V. S IMULATION R ESULTS In this section we show simulation results to demonstrate the effectiveness of the proposed method. For all simulations we have used a two-channel 14-bit TI-ADC, with static gains of 1.001 and 0.999 and clock skew deviations of −0.001 · Ts and 0.002 · Ts . We model the frequency response of the channels with a transfer function having first order low-pass characteristic: H0 (jΩ) = 1+j1 Ω and H1 (jΩ) =
Ωc
1 Ω 1+j (1+∆)Ω
c
, where ∆ is the difference of the magnitude
responses and Ωc is the nominal 3 dB cut-off frequency of the filters with Ωc = 3 · Ωs . The filter g[n] is designed with the firls method of MATLAB. For the following simulations we identify the coefficients α0 (jΩ) and α1 (jΩ) with J = 9 test frequencies uniformly spaced between ΩΩs = 0 and 0.5. Further, we calculate
0.1
0.2
0.3
0.4
0.5
normalized frequency (Ω/Ω ) s
Fig. 4. The SINAD of a 14-bit TI-ADC without compensation (dash dotted), after static gain and linear-phase compensation (solid with rectangles) and after additional magnitude response compensation with 5 (dashed) and 21 (dashed with circles) taps. The result is close to the SINAD of an ideal 14-bit TI-ADC (solid with stars)
the SINAD as defined in [2]. The linear phase mismatches consist of the clock skew deviations and the average linear-phase mismatches of the frequency responses and can be compensated with, e.g., the method in [5]. In Fig. 4 the SINAD for a fixed ∆ = 0.25 and different frequencies before (dash dotted) and after compensation is shown. The compensation of static gain and linear phase (solid) yields a SINAD improvement up to 40 dB. The SINAD is a maximum for the frequency, where the average gain and average phase fit best. After compensating the magnitude response mismatches, which is plotted for 5 (dashed) and 21 (dashed with circles) filter taps, the SINAD is further improved over the entire Nyquist range by up to 28 dB, even for only 5 taps. In Fig. 5, we vary the magnitude response mismatch coefficients ∆ from 0 to 2 for a fixed input frequency of ΩΩs = 0.1 and a filter order of 5. Even for small mismatches of the magnitude responses the SINAD is improved significantly. The SINAD is close to the SINAD of a TI-ADC without mismatches. In Fig. 6 the output power spectrum of a wide-band signal is depicted with (solid) and without (dash dotted) mismatches (∆ = 0.55) is shown. The SINAD is 28.52 dB (with mismatches) and 65.82 dB (without mismatches), respectively. After having compensated for static gain and linear-phase (solid) the spurious component is not completely compensated and the SINAD is 56.71 dB (cf. Fig. 7). After magnitude response mismatch compensation the SINAD is 63.23 dB, which is 2.59 dB close to the SINAD of a TI-ADC without mismatches. Thus, the proposed method is applicable for wide band signals as well. VI. C ONCLUSION A TI-ADC is used to achieve a high sampling rate with multiple ADCs switched in a time-interleaving manner. Unfortunately, additional errors due to mismatches are introduced, which degrade the performance. The limiting factor after compensating offset, static gain, and linear-phase mismatches are magnitude response mismatches. In this paper we have proposed a novel magnitude response mismatch compensation method for a two-channel TI-ADC. One advantage is that computationally cheap signal
714
−20
90
−30
Signal Power in dBc
SINAD in dB
80 70 60 50 40 30 0
spurious component due to mismatches
−40 −50
fundamental component
−60 −70 −80 −90
0.5
1
1.5
−100 0
2
Magnitude response mismatch coefficient ∆
no mismatches 0.1
0.2
0.3
0.4
0.5
normalized frequency Ω/Ω
s
Fig. 5. The SINAD without compensation (dash dotted), after static gain and linear-phase compensation (solid) and after additional magnitude response compensation (dashed).
Fig. 6. The power spectrum of a 14-bit TI-ADC with a wide-band input signal with (solid) and without (dash dotted) mismatches.
−20
processing operations, mainly a FIR filter with short tap size and a multiplier which just changes the sign of the signal are required. We have demonstrated the effectiveness of the proposed method for narrow band and wide band signals. In particular, for sinusoidal input signals we have further improved the SINAD by up to 28 dB after it has already been compensated for static gain and linear-phase mismatches. For a wide-band excitation the SINAD after compensation is only slightly smaller than the SINAD of a 14-bit TI-ADC without magnitude response mismatches.
Signal Power in dBc
−30
The proposed compensation method can be extended to an arbitrary number of channels, which is an ongoing project.
timing & static gain compensation
−40 −50
fundamental component
−60 −70 −80 −90
ACKNOWLEDGEMENT
+ magnitude response mismatch compensation
−100 0
Support of our research by Infineon Technologies Austria AG is gratefully acknowledged.
0.1
0.2
0.3
0.4
0.5
normalized frequency Ω/Ωs
R EFERENCES [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] “IEEE standard for terminology and test methods for analog-to-digital converters,” IEEE Std 1241-2000, June 2001. [3] N. Kurosawa, H. Kobayashi, K. Maruyama, H. Sugawara, and K. Kobayashi, “Explicit analysis of channel mismatch effects in timeinterleaved ADC systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 3, pp. 261–271, March 2001. [4] R. Prendergast, B. Levy, and P. Hurst, “Reconstruction of band-limited periodic nonuniformly sampled signals through multirate filter banks,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 8, pp. 1612–1622, August 2004. [5] Y. C. Jenq, “Perfect reconstruction of digital spectrum from nonuniformly sampled signals,” IEEE Transactions on Instrumentation and Measurement, vol. 46, no. 7, pp. 649 – 652, June 1997. [6] S. Jamal, F. Daihong, N. Chang, P. Hurst, and S. Lewis, “A 10-b 120-Msample/s time-interleaved analog-to-digital converter with digital background calibration,” IEEE Journal of Solid-State Circuits, vol. 37, no. 12, pp. 1618–1627, December 2002.
Fig. 7. The power spectrum after static gain and linear-phase compensation (solid) and after additional magnitude response mismatch compensation (dashed).
[7] 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. [8] K. Asami, “Technique to improve the performance of time-interleaved A-D converters,” in 2005 IEEE International Conference on Test, November 2005, pp. 851–857. [9] M. Seo, M. Rodwell, and U. Madhow, “Comprehensive digital correction of mismatch errors for a 400-Msamples/s 80-dB SFDR time-interleaved analog-to-digital converter,” IEEE Trans. on Microwave Theory and Techniques, vol. 53, no. 3, pp. 1072–1082, April 2005. [10] S. K. Hitoshi Sekiya, Masaaki Fuse, “Improvement in the performance of time-interleaved A-D converter by estimated equalizing technique,” Electronics and Communications in Japan (Part II: Electronics), vol. 88, no. 12, pp. 9–18, December 2005.
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