Adaptive blind calibration of timing offsets in a two-channel time-interleaved analog-to-digital converter through Lagrange interpolation

Abstract

This paper introduces an adaptive blind calibration structure for the compensation of timing offsets in a two-channel time-interleaved analog-to-digital converter through Lagrange interpolation filters. By using Lagrange interpolation filters as reconstruction filters, we can exploit slight over-sampling to estimate timing offsets by using the least-mean-square algorithm. Utilizing the estimated timing offsets, we can compensate timing offset mismatches. We show the efficiency of the calibration structure by the simulations, where we compare the results with the other calibration methods from the literature.

Appendix: First-order approximation of discrete-time Lagrange reconstruction filters

The first-order approximations for discrete-time Lagrange reconstruction filters presented in (7) and (8), in channels zero and one, respectively, are computed in this part. Starting from (5), and assuming

$$\begin{aligned}&\cos \frac{\pi }{2}\delta _{1}\approx 1\end{aligned}$$

(39)

$$\begin{aligned}&\sin \frac{\pi }{2}\delta _{1}\approx \frac{\pi }{2}\delta _{1}, \end{aligned}$$

(40)

leads to

$$\begin{aligned} \frac{\sin \frac{\pi }{2}(n-1-\delta _{1})}{\sin \frac{\pi }{2}(-1-\delta _{1})}\approx \cos \frac{\pi }{2}(n-\delta _{1}). \end{aligned}$$

(41)

Simplifying (41), and using (39) and (40), we obtain

$$\begin{aligned} \cos \frac{\pi }{2}(n-\delta _{1})\approx \cos \frac{\pi }{2}n+\frac{\pi }{2}\delta _{1}\sin \frac{\pi }{2}n. \end{aligned}$$

(42)

Replacing (42) instead of the second term of (5), presented in (41), results in (7). Note that the first term in (5) represents a LPF, $l_{p}[n]$, defined in (9). The first term in (6) is a low-pass fractional delay filter, with the cutoff frequency at $\omega _{c}=\frac{\pi }{2}$, which can be approximated by a first-order Taylor expansion. To do so, we start from the discrete-time Fourier transform (DTFT) of the first term in (6) as follows.

$$\begin{aligned} \hbox {DTFT}[2\frac{\sin \frac{\pi }{2}(n-1-\delta _{1})}{\pi (n-1-\delta _{1})}]=2e^{-j\omega (1+\delta _{1})}, \end{aligned}$$

(43)

for $|\omega |\le \frac{\pi }{2}$. Therefore, the DTFT of the first term in (6) can be approximated using first-order Taylor expansion

$$\begin{aligned} 2e^{-j\omega (1+\delta _{1})}\approx 2e^{-j\omega }-j\omega 2\delta _{1}e^{-j\omega }, \end{aligned}$$

(44)

for $|\omega |\le \frac{\pi }{2}$. Taking the inverse DTFT, IDTFT, of the two terms in (44) leads to

$$\begin{aligned}&\hbox {IDTFT}[2e^{-j\omega }]=l_{p}[n-1]\end{aligned}$$

(45)

$$\begin{aligned}&\hbox {IDTFT}[-j\omega 2\delta _{1}e^{-j\omega }]=\delta _{1}d[n-1] \end{aligned}$$

(46)

where $l_{p}[n]$ and $d[n]$ are defined in (9) and (10), respectively. Therefore, the first term in (6) can be approximated as

$$\begin{aligned} 2\frac{\sin \frac{\pi }{2}(n-1-\delta _{1})}{\pi (n-1-\delta _{1})}\approx l_{p}[n-1]+\delta _{1}d[n-1]. \end{aligned}$$

(47)

Finally, replacing (47) instead of the first term in (6) and replacing the denominator of the second term in (6), $\sin \frac{\pi }{2}(1+\delta _{1})=\cos \frac{\pi }{2}\delta _{1}$, using (39) result in (8).