Second Order Taylor Polyphase Reconstruction of Periodic-Nonuniform Samples in Time-Interleaved ADCs

Abstract

This paper focuses on a filter bank approach for reconstructing periodic-nonuniform samples using second order Taylor polyphase decomposition for a four-channel time-interleaved ADC (TIADC). The proposed method is based on a parametric linear least squares approach to find the coefficients of synthesis filters used for signal reconstruction of periodic-nonuniform samples. The coefficients of synthesis filters found by linear least squares approach satisfy the general condition for an alias-free filter bank.

Appendix

The inverse DTFT of an all-pass filter with the frequency response $A_{i}(e^{j\omega })$ is

$$ a_{i}[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{j\omega (i+n)} d\omega $$

(43)

Computing the above integral is very simple and leads to

$$ a_{i}[n]=\frac{\sin\pi(i+n)}{\pi(i+n)} $$

(44)

where $n=0,...,N-1$, and N is the length of APF. Note, by defining

$$ n'=n-\frac{N}{2}, $$

(45)

we need to replace n by $n'$ in Eq. 44 to put the center of the gravity of APF , $a_{i}[n]$, in the center of a Kaiser window of the length N with the maximum value located at $\frac {N}{2}$. The inverse DTFT of first order differentiator $D_{i}(e^{j\omega })$ is

$$ d_{i}[n]=\frac{\cos\pi(i+n)}{(i+n)}-\frac{\sin\pi(i+n)}{\pi(i+n)^{2}} $$

(46)

Again, we need to replace n by $n'$ in Eq. 46 to put the center of the gravity of first order differentiator, $d_{i}[n]$, in the center of a Kaiser window of the length N with the maximum value located at $\frac {N}{2}$. Finally, the inverse DTFT of second order differentiator $D_{i}^{(2)}(e^{j\omega })$ is

$$ d_{i}^{(2)}[n]=\frac{\sin\pi(i+n)}{\pi(i+n)^{3}}-\frac{\cos\pi(i+n)}{(i+n)^{2}}-\frac{\pi \sin\pi(i+n)}{2(i+n)}$$

(47)

Like the previous cases, we need to replace n by $n'$ in Eq. 47 to put the center of the gravity of second order differentiator, $d_{i}^{(2)}[n]$, in the center of a Kaiser window of the length N with the maximum value located at $\frac {N}{2}$.