## 2 Nonlinear Mismatches in \(M\)-TIADC
### Model of M-TIADC Exploiting Hadamard Sequences  
The error signal \(E(e^{j\omega})\) can be interpreted as the sum of \(L-1\) gain-mismatched \(M\)-TIADC output, i.e., each \(\hat{E}_{l}\) corresponding to the output of an \(M\)-TIADC with gain \(\alpha_{0,l},\alpha_{1,l},\cdots,\alpha_{M-1,l}\) for the constituent channels where the input signal is \(\frac{1}{(2\pi)^{l-1}}X^{sl}(e^{j\omega})\), i.e., \(x^{l}[n]\) in the time domain. This interpretation is illustrated in Figure 2.  
The Hadamard transform is exploited to model \(M\)-TIADCs with gain and timing-skew mismatches in [8]. As the nonlinear mismatches are interpreted as a group of TIADCs with gain mismatches in this paper, we utilize the Hadamard transform to model \(M\)-TIADCs with nonlinear mismatches, as well. The binary Hadamard matrix \(F_{M}\) of order \(M\) exploited in this paper is defined by:  
\[F_{2}=\left[\begin{array}{cc}1&1\\ 1&-1\end{array}\right] \tag{10}\]  
Figure 2: Nonlinear mismatches in the \(M\)-time-interleaved analog-to-digital converter (TIADC) are interpreted as the sum of a series of gain-mismatched \(M\)-TIADCs.  
and:  
\[F_{2m}=\left[\begin{array}{cc}F_{m}&F_{m}\\ F_{m}&-F_{m}\end{array}\right]. \tag{11}\]  
Exploiting the binary Hadamard matrix \(F_{M}\), the error signal \(E(e^{j\omega})\) is rewritten as:  
\[E(e^{j\omega})=\sum_{l=2}^{L}\frac{1}{(2\pi)^{l-1}}\vec{X}^{l}\vec{U}\vec{c}_{ l}, \tag{12}\]  
where:  
\[\vec{c}_{l}=\frac{1}{M}\vec{F}_{M}\vec{\alpha}_{l} \tag{13}\]  
representing the nonlinearities' related coefficients and:  
\[\vec{U}=\frac{1}{M}\vec{W}\vec{F}_{M} \tag{14}\]  
denoting the discrete Fourier transform of \(F_{M}\). Two properties of the binary Hadamard matrix \(F_{M}\) are exploited in the above derivations, which are \(F_{M}\vec{F}_{M}^{T}=M\textbf{I}\) and \(F_{M}=\vec{F}_{M}^{T}\), where \(\textbf{I}\) is the identity matrix and \((\bullet)^{T}\) denotes the transpose of \((\bullet)\).  
By calculating the inverse discrete-time Fourier transform (IDTFT) of \(E(e^{j\omega})\), the error signal in the time-domain is expressed as:  
\[e[n]=\sum_{l=2}^{L}x^{l}[n]\sum_{k=0}^{M-1}T_{k}[n]c_{k,l}, \tag{15}\]  
where \(T_{k}[n]\) is the \(k\)-th binary Hadamard sequence of length \(M\), which is also the \(k\)-th row (or column) of the binary Hadamard matrix \(F_{M}\). Utilizing the time-domain expression of error signal \(e[n]\), a model based on the binary Hadamard sequences is presented in Figure 3. Although multipliers are still used to represent the modulation by the Hadamard sequences \(T_{k}[n]\) in Figure 3, they do not necessarily exist in the hardware implementation because modulation by the binary Hadamard sequences can be implemented by just changing the sign of the modulated signal \(x^{l}[n]\).  
The model shown in Figure 3, which is termed the multiplier Hadamard transform (MHT) structure in this paper, reveals that the nonlinearity-induced error signal \(e[n]\) can be regarded as the sum of the weighted Hadamard transform of the power terms \(x^{l}[n],l=2,...,L\). In order to calibrate the nonlinearity and nonlinear mismatches, a calibration structure is proposed in the next section based on the presented signal model based on the binary Hadamard sequences.  
Figure 3: (**a**) Model of \(M\)-TIADC with nonlinear mismatches; (**b**) the detail of the Hadamard transform (HT) block.