# Calibration for Sample-And-Hold Mismatches in \(M\)-Channel TIADCs Based on Statistics
## 2 The Model  
The sampling rate of the TIADC is denoted as \(f_{s}\). The analog input signal \(x(t)\), is assumed to be band-limited to \(f_{s}/2\), indicating that \(x(t)\) can be perfectly recovered from the uniform-sampling samples \(x[n]=x(nT)\), \(T_{s}=1/f_{s}\). As the focus of this paper is the bandwidth mismatch, it is assumed that other types of mismatches have been removed by the methods mentioned in Section 1. The S/H circuit is usually modeled by a first-order RC filter which is illustrated in Figure 1[18; 38]. The circuit is essentially a low-pass filter with the \(-3\) dB bandwidth \(\Omega_{c}=1/RC\). (\"\(R\)\" is the equivalent resistance of the circuit, and \"\(C\)\" is the equivalent capacitance.) In reality, the values for \(R\) and \(C\) differ between channels owing to variations in the manufacture, and they change slowly due to the fluctuation in temperature and voltage. Figure 1: RC model of sample-and-hold circuits. The transfer function of the \(m\)-th channel's \(S/H\) circuit can be expressed as (1)  
\[H_{m}\left(j\Omega\right)=\frac{1}{1+j\Omega\frac{1+\beta_{m}}{\Omega_{c}}}\quad, \tag{1}\]  
where \(\beta_{m}\) is the mismatch coefficient and \(\Omega_{c}\) is the \(-3\) dB bandwidth of the reference channel. The \(M\)-channel TIADC's model can be described as Figure 2.  
To investigate the influence which bandwidth mismatch has on the dynamic performance of the TIADC, the discrete-time Fourier transform (DTFT) of downsampled signals \(v_{m}[Mn]\) (defined in Figure 2) are  
\[V_{m}\left(e^{jM\omega}\right)=\frac{1}{M}\sum_{k=0}^{M-1}H_{m}\left(e^{j\left( \omega-\frac{2k\pi}{M}\right)}\right)e^{j\left(\omega-\frac{2k\pi}{M}\right)}X \left(e^{j\left(\omega-\frac{2k\pi}{M}\right)}\right), \tag{2}\]  
where  
\[H_{m}(e^{j\omega})=H_{m}(j\Omega T_{s}),\quad-\pi<\omega<\pi, \tag{3}\]  
and \(X\left(e^{j\omega}\right)\) is the DTFT of signal \(x[n]\). Then the DTFT of the TIADC's output signal is  
\[Y\left(e^{j\omega}\right)=\sum_{m=0}^{M-1}Y_{m}\left(e^{j\omega}\right)=\sum_ {m=0}^{M-1}V_{m}\left(e^{jM\omega}\right)e^{-jm\omega}, \tag{4}\]  
where \(Y_{m}\left(e^{j\omega}\right)\) is the DTFT of \(y_{m}[n]\) defined in Figure 2.  
Taking 2-channel TIADC for instance, (4) is reduced to  
\[ \begin{split} Y\left(e^{j\omega}\right)&=Y_{0} \left(e^{j\omega}\right)+Y_{1}\left(e^{j\omega}\right)=V_{0}\left(e^{j2\omega} \right)+V_{1}\left(e^{j2\omega}\right)e^{-j\omega}\\ &=\frac{1}{2}\left[H_{0}\left(e^{j\omega}\right)+H_{1}\left(e^ {j\omega}\right)\right]X\left(e^{j\omega}\right)+\frac{1}{2}\left[H_{0}\left( e^{j\left(\omega-\pi\right)}\right)-H_{1}\left(e^{j\left(\omega-\pi\right)}\right) \right]X\left(e^{j\left(\omega-\pi\right)}\right),\end{split}  \tag{5}\]  
where the first term is the linear distortion of the input spectrum, and the second term is the error spectrum. The two spectra are symmetric around frequency \(\pi/2\), which is shown in Figure 3.  
Figure 2: The model of an \(M\)-channel time-interleaved analog-to-digital converters (TIADC) with bandwidth mismatches. (The “\(\downarrow M^{\prime}\) module is a down-sampler of factor \(M\), while the “\(\uparrow M^{\prime}\) module is an up-sampler of factor \(M\). The “\(Q^{\prime}\) module is the quantizer.) Like in a single ADC, there's no need to explicitly equalize the frequency response, and it is not obligatory here for a TIADC [21,22]. So Channel 0 can be chosen as the reference channel (\(\beta_{0}=0\)), and the filter \(H_{m}(j\Omega)\) (\(m\
eq 0\)) can be divided into two cascaded filters as is depicted in Figure 4. The bandwidth mismatch error can be regarded as the effect of an error generator whose transfer function is \(H_{m}\left(j\Omega\right)/H_{0}\left(j\Omega\right)\). To simplify the compensation, we try to use polynomials to approximate the transfer function of the error generator.