# High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDEE Equalizers
## 2 DQPSK Equalization
\begin{array}{l}u_{l}(n)=\sum\limits_{i=1}^{2}\sum\limits_{k=1}^{4}\sum \limits_{m=0}^{M_{l}-1}f_{i,m}^{k,l}y_{i,k}(n-m)+\sum\limits_{i=1}^{2}\sum \limits_{k=1}^{4}\sum\limits_{m_{1}=m_{2}=m_{1}}^{M_{l}-1}f_{i,m_{1},m_{2}}^{ k,l}y_{i,k}(n-m_{1})y_{i,k}(n-m_{2})+\\ \sum\limits_{i=1}^{2}\sum\limits_{m_{1}=1}^{M_{l}}\sum\limits_{m_{2}=m_{1}}^{ M_{l}}b_{i,m_{1},m_{2}}^{l}\hat{I}_{i}(n-m_{1})\hat{I}_{i}(n-m_{2})\\ \end{array}  \tag{2}\]  
where \(u_{1}(n)\) and \(u_{2}(n)\) are the output signals of equalizer for the in-phase and quadrature part, respectively. Parameters designated by symbol \(f\) correspond to the feed-forward (FF) part of the equalizer which compensates for the precursor distortions. Parameters designated by symbol \(b\) correspond to the feedback (FB) part of the equalizer which compensates for the post cursor distortions correspondingly. Integers \(M_{f}\) and \(M_{b}\) represent the memory of the FF and the FB part of the equalizer, respectively, and are both related to the amount of distortions that affect the transmission. Figure 1: Block diagram of a NRZ-DQPSK set exhibiting N transmission spans and a single wavelength transmitter together with a typical single-ended receiver. Signals \(\hat{I}_{1}(n)\triangleq D[u_{1}(n)]\) and \(\hat{I}_{2}(n)\triangleq D[u_{2}(n)]\) represent the recovered in-phase and quadrature bit streams, with D denoting the decision device, as it is explained also in [7; 9] and [22]. Hence, the VDFE equalizer described by Equation (2), hereafter will be denoted as VDFE[\(M_{f}\),\(M_{b}\)], and its structure is summarized in Figure 2b. The mathematical formulae that describes the VDFE can be expressed in a linear regression form as \(u_{l}(n)=\theta_{l}^{T}\varphi_{M}(n),\ l=1,2\) where \(\varphi_{M}(n)\) is a vector that depends on the input signals and on the recovered bit streams, \(\theta_{l},\ l=1,2\) represents the vectors that carries the equalizer coefficients for the in-phase and the quadrature part of the equalizer, with \(T\) denoting the matrix transpose. Integer \(M\) denotes the number of coefficients for each part of the equalizer. It provides a metric of complexity of the implementation of the recursive scheme, as the number of multiplications and additions required for the computation of \(u_{1}(n)\) and \(u_{2}(n)\) is \(C=2\,M\), in our case given by:  
\[C_{VDFE(M_{f},M_{b})}=16M_{f}+8M_{f}\Big{(}M_{f}+1\Big{)}+2M_{b}(M_{b}+1) \tag{3}\]  
The equalizer coefficients \(\theta_{l},\ l=1,2\) are estimated by minimizing a properly chosen cost function between the output signals and a desired response signal [22]. The least squares (LS) estimator is perhaps the most popular design approach, where the parameters sought are estimated minimizing the sum of the squared error between the output of the equalizer and a set of N training data, \(I_{1}(0),I_{1}(1),\ldots,I_{1}(\text{N}-1)\) and \(\ I_{2}(0),I_{2}(1),\ldots,I_{2}(\text{N}-1)\), in our case. The optimum parameter vectors \(\theta_{l},\ l=1,2\) correspond to the solution of a linear system of equations, the so called 'normal equations', which is obtained either explicitly by means of a direct system solver, or implicitly engaging an iterative or a time recursive estimation scheme. In the case of time varying distortions, the estimated values may be updated either periodically or on a sample-by-sample basis, blindly working in a decision-directed mode. A significant reduction in the computational complexity of the equalization devices in DQPSK signaling may be obtained by processing the differential signals using balanced detection at the receiver.