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Consecutive adaptive blind estimation of timing offsets for arbitrary channel time-interleaved ADCs

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Abstract

In this paper an estimation method for timing offset in time-interleaved analog-to-digital convertors (TIADCs) is proposed. The method is based on the estimation of relative timing offset between two consecutive channels. The least-mean-squares algorithm is used for estimating relative timing offsets. The convergence analysis is accomplished using frequency domain representation of the signals. The method can be used for arbitrary channel TIADCs due to using an anti-aliasing filter. Also, a non-sequential estimation method based on estimated differential delays is proposed at the end.

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Abbreviations

\(x_{c}(t)\) :

Continuous time signal

\(x{[n]}\) :

Discrete time signal

\(\tilde{x}_{i}{[n]}\) :

The output signal in channel \(i\)

\(f_{i}{[n]}\) :

Impulse response of the analysis filter

\(Lp(e^{j\omega })\) :

Anti-aliasing filter

\(F_{i}(e^{j\omega })\) :

Frequency response of the analysis filter

\(\varDelta \delta _{i+1}\) :

Relative timing offset

\(h{[n,\varDelta \hat{\delta }_{i+1}]}\) :

Fractional delay filter impulse response

\(L(e^{j\omega })\) :

Low-pass filter frequency response

\(D(e^{j\omega })\) :

Differentiator filter frequency response

\(e{[n,\varDelta \hat{\delta }_{i+1}]}\) :

Error signal

\(J{[\varDelta \hat{\delta }_{i+1}]}\) :

Objective function

\(P\) :

Power of differentiated signal

\(\mu \) :

Step size

\(\epsilon _{i}\) :

Estimation error of relative timing offset

\(\gamma _{i}\) :

Estimation error of timing offset

\(\theta _{i}\) :

Time delay in channel \(i\)

\(p(i,j)\) :

Differential delay between channel \(i\) and \(j\)

\(\varvec{A}\) :

Sparse matrix

\(\hat{\varvec{\theta }}\) :

Estimated delay vector

References

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Authors and Affiliations

  1. Department of Electrical and Telecommunication Engineering, Norwegian University of Science and Technology, N 7491 , Trondheim, Norway

    Arash Shahmansoori

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  1. Arash Shahmansoori
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Correspondence to Arash Shahmansoori.

Appendix: The derivation of convergence analysis

Appendix: The derivation of convergence analysis

The proof for the convergence analysis proposed in Sect. 3 is presented in this part. The first expectation term presented in (26) is

$$\begin{aligned} -E[\tilde{x}_{i+1}[n](d[n]*\tilde{x}_{i}[n])] \end{aligned}$$
(61)

Replacing \(\tilde{x}_{i+1}[n]\) by

$$\begin{aligned} \tilde{x}_{i+1}[n]=\tilde{x}_{i}[n]*h[n,\varDelta \delta _{i+1}] \end{aligned}$$
(62)

where \(h[n,\varDelta \delta _{i+1}]\) is defined in (20), we obtain

$$\begin{aligned}&-\,E[\tilde{x}_{i+1}[n](d[n]*\tilde{x}_{i}[n])]\nonumber \\&\quad =-\,E[(\tilde{x}_{i}[n]*l[n])(\tilde{x}_{i}[n]*d[n])]\nonumber \\&\;\qquad -\varDelta \delta _{i+1}E[(\tilde{x}_{i}[n]*d[n])(\tilde{x}_{i}[n]*d[n])] \end{aligned}$$
(63)

Based on the general Parsevals’ identity for real-time sequences \(a[n]\) and \(b[n]\)

$$\begin{aligned} E[a[n]b^{\star }[n]]=\frac{1}{2\pi }\int \limits _{-\pi }^{\pi }A(\omega )B^{\star }(\omega )d\omega \end{aligned}$$
(64)

Note that because \(b[n]\) is a real-time sequence, its conjugate is equal to itself or \(b^{\star }[n]=b[n]\). In (62), \(A(\omega )\) and \(B^{\star }(\omega )\) are the DTFT of \(a[n]\) and the conjugate of the DTFT of \(b[n]\), respectively. Applying (64) to (61) leads to

$$\begin{aligned} -E[\tilde{x}_{i+1}[n](d[n]*\tilde{x}_{i}[n])]=-\varDelta \delta _{i+1}\times P \end{aligned}$$
(65)

That is because the first expectation term in (63) can be written as

$$\begin{aligned}&-E[(\tilde{x}_{i}[n]*l[n])(\tilde{x}_{i}[n]*d[n])]\nonumber \\&\quad =-\frac{1}{2\pi }\int \limits _{-\pi }^{\pi }\tilde{X}_{i}(e^{j\omega })L(e^{j\omega })\tilde{X}^{\star }_{i}(e^{j\omega })D^{\star }(e^{j\omega })d\omega \end{aligned}$$
(66)

Replacing \(L(e^{j\omega })\) and \(D(e^{j\omega })\) from (13) and (14), respectively, and a simple calculation, we find

$$\begin{aligned}&-E[(\tilde{x}_{i}[n]*l[n])(\tilde{x}_{i}[n]*d[n])]\nonumber \\&\quad =\frac{j}{2\pi }\int \limits _{-\pi }^{\pi }\omega |\tilde{X}_{i}(e^{j\omega })|^{2}d\omega =0 \end{aligned}$$
(67)

Note that in (67) the integrand is an odd function, so the result is zero. The second expectation term in (61) can be written as

$$\begin{aligned}&E[(\tilde{x}_{i}[n]*d[n])(\tilde{x}_{i}[n]*d[n])]\nonumber \\&\quad =\frac{1}{2\pi }\int \limits _{-\pi }^{\pi }\tilde{X}_{i}(e^{j\omega })D(e^{j\omega })\tilde{X}^{\star }_{i}(e^{j\omega })D^{\star }(e^{j\omega })d\omega \end{aligned}$$
(68)

Replacing \(D(e^{j\omega })\) from (14) and a simple calculation, we obtain

$$\begin{aligned}&E[(\tilde{x}_{i}[n]*d[n])(\tilde{x}_{i}[n]*d[n])]\nonumber \\&\quad =\frac{1}{2\pi }\int \limits _{-\pi }^{\pi }\omega ^{2}|\tilde{X}_{i}(e^{j\omega })|^{2}d\omega \end{aligned}$$
(69)

or equivalently,

$$\begin{aligned} \begin{array}{l} E[(\tilde{x}_{i}[n]*d[n])(\tilde{x}_{i}[n]*d[n])]\\ \quad =\frac{1}{\pi }\int \limits _{0}^{\pi }\omega ^{2}|\tilde{X}_{i}(e^{j\omega })|^{2}d\omega =P \end{array} \end{aligned}$$
(70)

Replacing (67) and (70) in (63) results (65).

The second expectation term presented in (26) is

$$\begin{aligned} E[(\tilde{x}_{i}[n]*l[n])(\tilde{x}_{i}[n]*d[n])]=0 \end{aligned}$$
(71)

Note that like in (67) the result is zero because the integrand is an odd function. Finally, the third expectation term in (26) is

$$\begin{aligned} E[(\tilde{x}_{i}[n]*d[n])(\tilde{x}_{i}[n]*d[n])]=P \end{aligned}$$
(72)

Note that like in (70) the result is \(P\). Substituting (65), (71), and (72) in (26) results in (27).

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Shahmansoori, A. Consecutive adaptive blind estimation of timing offsets for arbitrary channel time-interleaved ADCs. SIViP 9, 45–55 (2015). https://doi.org/10.1007/s11760-012-0405-2

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