## 2 System Structure and Problem Formulation  
We consider SFBC OFDM broadcasts with \(N\) subcarriers, \(\
u\) cyclic prefix (CP) samples, \(F\) broadcast antennas, and a receiver antenna. Figure 1 depicts the notional schematic diagram of a broadcaster. An error control encoder and an interleaver work together to secure a binary stream. Figure 1: Schematic of an SFBC OFDM transmitter. An interleaver spreads out the sequence of bits in a bit stream to offer time diversity by minimizing the effect of transmission errors such as bursts. The encoded information is converted into data symbols by a digitized modulator, with each symbol being selected from an \(M\)-point signal pattern, \(\Omega\). A vector \(\mathbf{D}=[d_{0},d_{1},\cdots,d_{U-1}]\) of length \(U\) is formed by combining a number of training symbols with data information. There are no limitations placed on the channel coding, interleaver, mapping technique, pilots' allocation, or constellation type that may be used in this scenario. Separate chunks of length \(P\) are created from the components of vector \(\mathbf{D}\). The encoder of SFBC signals utilizes a predetermined \(F\times Q\) code matrix \(C\left(\mathbf{d}^{(p)}\right)\) to distribute the \(p\)th chunk, \(\mathbf{d}^{(p)}\), over \(Q\) consecutive subcarriers for transmission through \(F\) antenna elements. For clarification, the broadcast matrices of SFBC1 (\(P=2,Q=1,F=2\)), SFBC2 (\(P=2,Q=2,F=2\)), SFBC3 (\(P=3,Q=4,F=3\)), and SFBC4 (\(P=4,Q=8,F=4\)) are [38]:  
\[C^{(\text{SFBC1})}(\mathbf{d}=[d_{0}\ d_{1}])=[d_{0}\ d_{1}]^{T}, \tag{1}\]  
\[C^{(\text{SFBC2})}(\mathbf{d}=[d_{0}\ d_{1}])=\left[ \begin{array}{ccc}d_{0} &d_{1}\\ -d_{1}^{*}&d_{0}^{*}\\ \end{array} \right]^{T}, \tag{2}\]  
\[C^{(\text{SFBC3})}(\mathbf{d}=[d_{0}\ d_{1}\ d_{2}])=\left[ \begin{array}{ cccc}d_{0}&d_{1}&d_{2}\\ -d_{1}^{*}&d_{0}^{*}&0\\ d_{2}^{*}&0&-d_{0}^{*}\\ 0&-d_{2}^{*}&d_{1}^{*}\\ \end{array} \right]^{T}, \tag{3}\]  
\[C^{(\text{SFBC4})}(\mathbf{d}=[d_{0}\ d_{1}\ d_{2}\ d_{3}])=\left[ \begin{array}[] {cccc}d_{0}&d_{1}&d_{2}&d_{3}\\ -d_{1}&d_{0}&-d_{3}&d_{2}\\ -d_{2}&d_{3}&d_{0}&-d_{1}\\ -d_{3}&-d_{2}&d_{1}&d_{0}\\ d_{0}^{*}&d_{1}^{*}&d_{2}^{*}&d_{3}^{*}\\ -d_{1}^{*}&d_{0}^{*}&-d_{3}^{*}&d_{2}^{*}\\ -d_{2}^{*}&d_{3}^{*}&d_{0}^{*}&-d_{1}^{*}\\ -d_{3}^{*}&-d_{2}^{*}&d_{1}^{*}&d_{0}^{*}\\ \end{array} \right]^{T}, \tag{4}\]  
where \(T\) and \(*\) are the matrix -transpose and complex conjugate, respectively. In (1)-(4), we remove the block index \(p\) for ease of notation. It is important to keep in mind that \(P\) and \(F\) can have different values. In each broadcast arm, the outputs of the SFBC encoder are combined to produce a sequence \(\mathbf{u}_{x}^{(f)}\) of size \(N\), where \(N=\frac{QN_{d}}{P}\). We have \(F\) broadcast arms, each of which is made up of various radio frequency circuits coupled to a transmitting antenna. We append the symbol \(\alpha\) to the vector \(\mathbf{u}_{\alpha}^{(f)}\) to underline that the conveyed vector is dependent on the used SFBC signal \(\alpha\in\{\text{SFBC1},\text{SFBC2},\text{SFBC3},\text{SFBC4}\}\). Accordingly, an OFDM symbol is produced by applying an \(N\)-point inverse discrete Fourier transform (IDFT), and the last \(\upsilon\) samples are included as a CP. The \(n\)th sample sent from the \(f\)th broadcast antenna is written as:  
\[s_{\alpha}^{(f)}(n)=\frac{1}{\sqrt{N+\
u}}\sum_{k=0}^{N-1}u_{\alpha}^{(f)}(k)e ^{i2\pi nk/N},n=0,\cdots,N+\
u-1 \tag{5}\]  
where \(u_{\alpha}^{(f)}(k)\) is the \(k\)th element of sequence \(\mathbf{u}_{\alpha}^{(f)}\). The broadcast sequence \(\mathbf{s}_{\alpha}^{(f)}=\left[s_{\alpha}^{(f)}(0),\cdots,s_{\alpha}^{(f)}(N +\
u-1)\right]^{T}\) is vulnerable to the negative effects of the IQD of the \(f\)th branch. We describe \(\theta^{(f)}\) and \(\rho^{(f)}\) as the phase and amplitude discrepancies between I and Q segments of the \(f\)th arm at the broadcaster.