Adaptive Compensation of Frequency Response Mismatches in High-Resolution Time-Interleaved ADCs using a Low-Resolution ADC and a Time-Varying Filter Shahzad Saleem
Christian Vogel
Signal Processing and Speech Communication Laboratory Graz University of Technology, Austria Email:
[email protected]
Signal Processing and Speech Communication Laboratory Graz University of Technology, Austria Email:
[email protected]
Abstract— This paper investigates the adaptive compensation of frequency response mismatches in an M -channel timeinterleaved analog-to-digital converter (TI-ADC) using an extra low-resolution ADC and a time-varying FIR filter. The introduced compensation structure may be used to compensate any linear frequency response mismatches including time skew mismatches. The coefficients of the time-varying filter are adapted using the least-mean square (LMS) algorithm. The performance of the proposed compensation structure is demonstrated through numerical simulations.
(nM + 0)Ts ˆ 0 (jΩ) H
QH
ADC0 (nM + m)Ts x(t)
ˆ m (jΩ) H
QH
MUX
ADCm
digital output y[n] fs =
1 Ts
(nM + (M − 1))Ts ˆ M −1 (jΩ) H
I. I NTRODUCTION As the functionality of communication systems is moving more and more into the digital domain in order to provide increased flexibility and more precision [1], the requirements on the data converters increase in terms of higher speed and larger bandwidth. In this regard, a TI-ADC can be a reasonable solution to achieve higher sampling rates for medium-to-high resolution applications. By using a TI-ADC one can increase the throughput of a data converter by using several ADCs in parallel and sampling the data in a time-interleaved manner. As shown in Fig. 1, in an M -channel TI-ADC each individual ˆ m (jΩ), m = ADC with an analog frequency response H 0, 1, ..., M − 1 operates with a sampling rate of fs /M , thus the overall sampling frequency is fs . The performance of a TI-ADC however, suffers from mismatch errors among the subconverters [2]. These mismatch errors uncover the time-varying behavior of a TI-ADC and cause modulated spurious images of the input spectrum in the output spectrum leading to a significant decrease in the performance. Beside the calibration of gain and timing mismatches [3]– [6], the calibration of frequency response mismatches can lead to a further improvement in the overall performance of a TIADC. A hybrid filter-bank model of a two-channel TI-ADC for semi-blind calibration of bandwidth mismatches is presented in [7]. In [8] a compensation technique for frequency response mismatches using a least-square design for an M -periodic
978-1-4244-5309-2/10/$26.00 ©2010 IEEE
QH
ADCM −1 TI-ADC reference output d[n]
nTs QL
R(jΩ)
ADCR
fs
Fig. 1: Model of a TI-ADC comprising M linear highˆ m (jΩ), m = 0, 1, ..., M − resolution time-invariant channels H 1 with quantizers QH and an extra low-resolution ADC R(jΩ) with quantizer QL .
time-varying filter is presented. A flexible and scalable structure to compensate frequency response mismatches in TIADCs using an M -periodic time-varying compensation filter is presented in [9]. An adaptive technique for an all-digital communication receiver using feed-forward equalizer (FFE) to correct mismatches in a TI-ADC is presented in [10]. In this paper, we present a structure comprising an M channel high resolution TI-ADC and an extra low-resolution ADC. The extra ADC and the time-varying FIR filter are used to calibrate frequency response mismatches introduced by the TI-ADC, where output of the extra ADC acts as reference input for the M -periodic least-mean square (LMS) algorithm that estimates the coefficients sets for the M -periodic time-
561
x[n]
hn [l]
y[n]
x[n]
y[n]
hn [l]
yc [n]
gn [l] fn [l]
Fig. 2: Discrete-time system model of a TI-ADC.
varying compensation filter. For a certain power budget, the quantization accuracy limits the throughput in a high-resolution ADC. On the one hand, to increase the throughput, we can either use the timeinterleaved structure or can limit the quantization accuracy. On the other hand, the use of an extra-low resolution ADC does not significantly increase the power consumption of a high-resolution TI-ADC. For example, compared to the power dissipation of 33 mW for a 10-bit 100 MS/s pipeline converter a 4-bit 1.25 GS/s flash converter has a power dissipation of only 2.5 mW [11]. Furthermore, technology-scaling has helped to improve the power efficiency of low-resolution ADCs over time. On average, the power dissipation for flash and pipeline ADCs has halved every 2.5 years over the past ten years [11].
Fig. 3: Cascade of the two M -periodic time-varying filters hn [l] and gn [l] results in a new time-varying filter fn [l]. where the starting value of l is 0 since a TI-ADC is a causal system. From (6) it can be concluded that the output y[n] of an M -channel TI-ADC can be generated by passing the input sequence x[n] through a discrete-time time-varying filter with impulse response hn [l] which is M -periodic, i.e. hn [l] = hn+M [l], as shown in Fig. 2. In other words, for each time instant n we get a different response of the M -channel TIADC system. This time-varying behavior accounts for the frequency response mismatches in a TI-ADC. The output d[n] of an extra low-resolution ADC quantized to QL bits can be written as
II. S YSTEM M ODEL When a bandlimited analog input signal x(t) is passed through an M -channel TI-ADC to digitize it, the discrete-time Fourier transform (DTFT) of output y[n] is given by [9] M −1
Y (ejω ) =
2π ˘ k (ej(ω−k 2π M ) )X(ej(ω−k M ) ), H
(1)
k=0
where x(t) is assumed to be bandlimited, i.e. X(jΩ) = 0 for |Ω| ≥ Ωb , Ωb ≤ Tπ and X(ejω ) = T1 X(j Tω ) for |ω| < π. ˘ k (ejω ) are given by The coefficients H M −1 2π ˘ k (ejω ) = 1 H Hn (ejω )e−jkn M M n=0
(2)
with Hn (ejω ) representing the linear and time-invariant chanˆ n (j ω ) nel frequency responses defined by Hn (ejω ) = H T for |ω| < π and n = 0, 1, ..., M − 1. Taking the inverse DTFT of (1) gives y[n] =
M −1 +∞
˘ k [l]x[n − l]ejkn 2π M h
(3)
k=0 l=−∞
where
M −1 2π ˘ k [l] = 1 h hn [l]e−jkn M M n=0
and hn [l] =
M −1
(4)
(5)
˘ k [l] and hn [l] form From (4) and (5) it can be noted that h discrete-time Fourier series (DTFS) pairs [12]. Using (5) we can simplify (3) as ∞
+∞
r[l]x[n − l]
(7)
l=−∞
where r[n] is the impulse response of the low-resolution ADC with the linear and time-invariant frequency response R(jΩ). III. C ASCADED T IME -VARYING F ILTERS In order to compensate the frequency response mismatches, we propose the usage of an M -periodic time-varying causal FIR filter with impulse response gn [l] where for each time instant n an appropriate set of coefficients is used. This FIR filter is cascaded with the time-varying filter hn [l]. The cascade of the two time-varying filters results in a new M -periodic time-varying filter fn [l] as shown in Fig. 3. Such a cascaded structure has earlier been proposed in [8], [10] for the compensation of frequency response mismatches in an M -channel TI-ADC. In [8] authors have presented a least-square design for an M -periodic time-varying compensation filter. In [10] an M -periodic time-varying adaptive FFE is used to compensate sample-time and bandwidth mismatches. However in this paper, we use an extra low-resolution ADC and an adaptive technique (LMS) to obtain the coefficients sets for the time-varying compensation filter. The output of the M -periodic time-varying FIR filter gn [l] is L gn [l]y[n − l]. (8) yc [n] = l=0
˘ k [l]ejkn 2π M . h
k=0
y[n] =
d[n] =
hn [l]x[n − l],
Substituting (6) in (8) and after a few simple manipulations, we can rewrite (8) as [8] ∞ fn [l]x[n − l] (9) yc [n] = l=0
where
(6)
fn [l] =
L p=0
l=0
562
gn [p]hn−p [l − p].
(10)
reference ADC x[n]
r[n]
xr [n]
QL
TABLE I: Simulated gain, relative timing and frequency offsets Values
ε[n]
d[n]
LMS
ADC ADC0 ADC1 ADC2 ADC3
yc [n]
compensation filter
TI−ADC hn [l]
QH
y[n]
yc [n]
gn [l]
fn [l]
Compensated output
Fig. 4: Adaptive compensation structure comprising of an M periodic time-varying causal FIR filter fn [l] and filter r[n] representing an extra low-resolution ADC generating the desired input d[n] for the M -periodic LMS algorithm that is used to adapt the coefficients sets of gn [l].
Equation (10) represents the cascade of the two M -periodic time-varying filters hn [l] and gn [l] as shown in Fig. 3. The resulting cascaded time-varying filter fn [l] is also M -periodic, i.e. fn [l] = fn+M [l]. IV. A DAPTIVE C OMPENSATION Based on the system model of a TI-ADC developed in Section II and a cascaded time-varying filters structure presented in Section III, we now present a compensation structure comprising of an M -periodic time-varying cascaded FIR filter fn [l] in combination with the output of the extra low-resolution ADC d[n] as shown in Fig. 4. This structure represents the typical channel-equalization problem where the response of an M -channel TI-ADC, i.e. hn [l] replaces the communication channel and gn [l] is the M -periodic equalizer with M coefficients sets. The coefficients sets of gn [l] are estimated by using the M -periodic LMS algorithm. The output of the extra low-resolution ADC d[n] acts as the desired input to the LMS algorithm. The adaptation error ε[n] is given by ε[n] = d[n] − yc [n].
(11)
Substituting yc [n] and d[n] from (9) and (7) in (11) and rearranging ε[n] =
+∞
(r[l] − fn [l]) · x[n − l].
(12)
l=−∞
From (12) it can be concluded that ε[n] ≈ 0 if fn [l] ≈ r[l].
(13)
The condition given by (13) can be only achieved if we can minimize ε[n] in some sense. In our case, ε[n] is minimzed by using an M -periodic LMS algorithm thus resulting in the correct estimates for the coefficients sets of gn [l] as gn+M [l] = gn [l] + μ · y[n − l] · ε[n], where μ is the adaptation step size.
(14)
αm 1.01 0.98 0.99 1.02
rm −0.007Ts +0.002Ts −0.003Ts +0.008Ts
βm +0.10 −0.02 −0.05 +0.05
Once the correct estimates for the coefficients sets of gn [l] have been found, then these estimates can be used to suppress the time varyingness of an M -channel time-interleaved ADC to finally get the frequency response mismatch compensated output yc [n]. V. S IMULATION R ESULTS Simulations were performed to investigate the performance of the compensation structure with a four-channel TI-ADC suffering from gain, timing, and frequency response mismatches. The channel frequency responses being used for the simulations were αm ˆ m (jΩ) = H ejΩT rm (15) 1 + j (1+βΩm )Ωc where Ωc is the 3-dB cutoff frequency of the first order response, αm are the gain mismatches, βm are the relative frequency offsets from Ωc , and rm are the relative timing offsets from the ideal sampling instants. We have simulated a 16-bit four-channel TI-ADC with sampling rate Ωs = Ωc . The simulated gain, relative timing and frequency offsets values are shown in Tab. I. The input signal was multi-sine with random amplitudes, phases and a bandwidth of 0.8Ωs /2. The frequency response of the low-resolution ADC was taken as the first order frequency response given by R(jΩ) =
1.03 . 1 + j ΩΩc
(16)
For the adaptation, the step size μ was computed based on the variance σx2 of the input signal and the order of the time-varying compensation filter gn [l]. The performance after compensation was measured by computing the value of the signal-to-noise ratio (SNR) as N −1 2 |x [n]| r SNR = 10log10 N −1n=0 (17) 2 n=0 |xr [n] − yc [n]| where yc [n] denotes the corrected samples after compensation and xr [n] is the convolution of x[n] and r[n] without any quantization as shown in Fig. 4. Figure 5 shows the power spectrum of the uncompensated output signal y[n] with QL = 6 bits. The SNR is 28.6 dB. The power spectrum of the compensated output signal yc [n] using a compensation filter gn [l] with 29 taps is shown in Fig. 6. The computed value of the SNR is now 57.2 dB which is an approximate improvement of 29 dB as compared to the uncompensated output.
563
0
60 55 50
−40
SNR [dB]
Signal Power [dBc]
−20
−60
Initial SNR 20 2 Samples 221 Samples 222 Samples 223 Samples
45 40 35
−80 30
−100 0
0.1
0.2 0.3 Normalized frequency Ω/Ω
0.4
25 1
0.5
s
Fig. 5: Power spectrum of the uncompensated output y[n] with QH = 16 bits and QL = 6 bits (last 4096 samples). The computed SNR is 28.6 dB.
2
3
4 QL
5
6
7
Fig. 7: QL vs SNR for a compensation filter gn [l] with 29 taps (QH = 16 bits).
R EFERENCES 0
Signal Power [dBc]
−20
−40
−60
−80
−100 0
0.1
0.2 0.3 Normalized frequency Ω/Ω
0.4
0.5
s
Fig. 6: Power spectrum of the compensated output yc [n] using a compensation filter with 29 taps (last 4096 samples). The computed SNR is 57.2 dB thus leading to an improvement of 29 dB compared to the uncompensated output.
A comparison of QL against the SNR using the different number of samples for a compensation filter gn [l] with 29 taps is shown in Fig. 7. By increasing the number of reference bits, i.e. QL and the number of samples, the SNR increases. For QL = 3 bits, an SNR improvement up to 22 dB was observed using 223 samples while for QL = 6 bits, an SNR improvement up to 29 dB was observed using 223 samples. VI. C ONCLUSIONS In this paper we have presented a compensation structure for an M -channel TI-ADC suffering from frequency response mismatches. The compensation structure comprises of an M periodic time-varying FIR filter and the output of an extra lowresolution ADC acting as the desired input for an M -periodic LMS algorithm that estimates the coefficients sets for the timevarying filter. We have shown through simulations that by using an extra low-resolution ADC in combination with a timevarying filter the frequency response mismatches in an M channel TI-ADC can be compensated and it considerably improves the SNR.
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