# High-Speed, High-Performance DQPSK Optical Links with Reduced Complexity VDEE Equalizers
## 2 DQPSK Equalization  
DQPSK is a four-level modulation format initially introduced to overcome spectral efficiency limitations of direct detection binary modulation formats [1]. Recently, DQPSK has been revisited together with other direct detection techniques for cost efficiency [13] or nonlinearity mitigation [14]. Today, processing the signal electronically at symbol rate speeds is very typical in coherent optical transponders, however, in applications restricted by cost, direct detection is revisited [15; 16; 17] and DQPSK is a very important candidate [2]. Let \(I_{1}(n)\in\{0,1\}\) and \(I_{2}(n)\in\{0,1\}\) represent the encoded transmitted sequences of the \(I\) and \(Q\) channels, which are encoded and modulated using a standard DQPSK transmitter setup (Figure 1). A typical optical communication link consists of numerous identical fiber spans where part of the accumulated dispersion in the single mode fiber (SMF) is compensated by means of DCF and losses by optical amplifiers. At the receiver side, after optical bandpass filtering, the signal is demodulated by means of two separate Mach Zehnder delay interferometers (MZDI). Each arm of the MZDI has a phase shift of \(\pi/4\) (in-phase component) and \(-\pi/4\) (quadrature component). Subsequently, each output of the MZDI devices is detected by a photodiode and is filtered by an electrical low-pass filter (ELPF), producing in this way a set of constructive and destructive signals for the in-phase, as well as for the quadrature component, denoted here by \(I_{c}(t)\), \(I_{d}(t)\), \(Q_{c}(t)\), and \(Q_{d}(t)\), respectively [18]. Although any pair of the combination \(\{I_{c}(t)\),\(I_{d}(t)\}\)\(\times\)\(\{Q_{c}(t)\),\(Q_{d}(t)\)may be utilized for the detection of the transmitted sequences, a scheme based on the differential output \(I(t)=I_{c}(t)\,-\,I_{d}(t)\) and \(Q(t)=Q_{c}(t)\,-\,Q_{d}(t)\), known as 'balanced detection', is usually used instead, offering in this case a 3 dB OSNR gain compared to the former approach. Electronic equalization is usually applied for the suppression of either all, or a part (residual) of the distortions introduced by the fiber. In the case when the DQPSK receiver is implemented by means of balanced detection, the equalization methods are referred to as 'balanced DQPSK equalization'. However, it has been demonstrated that the joint use of all the signals available at the receiver for equalization, provides improved results compared to those obtained if the differential output signals are used instead, for sequence, as well as for symbol-by-symbol equalization [9; 19]. The latter approach is known as 'joint DQPSK equalization'. Symbol-by-symbol electronic equalization of direct detection optical transmission by means of recursive Volterra nonlinear filters has drawn significant attention in the past, for NRZ, DPSK, and DQPSK signaling. The feasibility of the implementation of those methods on FPGA (Field-Programmable Gate Array) circuitry has recently been demonstrated [2; 12; 20], for rates up to 40 Gb/s. Due to the joint constructive/destructive processing, four signals are available as input to the equalizer. Further signal diversity is achieved by fractionally spaced sampling, as in this case the performance of the electronic equalizers becomes less sensitive to the sampling phase of the receiver [21]. We here adopt half rate spaced sampling, \(T_{s}/2\), where \(T_{s}\) is the symbol period. Thus, after digital to analog conversion (D2A), eight signals (depicted on Figure 2a) are eventually fed to the feed-forward part of the equalizer, namely:  
\[ \begin{array}{l}y_{1,1}(n)\triangleq I_{c}(nT_{s}),y_{2,1}(n)\triangleq I_{ c}(nT_{s}+T/2)\\ y_{1,2}(n)\triangleq I_{d}(nT_{s}),y_{2,2}(n)\triangleq I_{d}(nT_{s}+T/2)\\ y_{1,3}(n)\triangleq Q_{c}(nT_{s}),y_{2,3}(n)\triangleq Q_{c}(nT_{s}+T/2)\\ y_{1,4}(n)\triangleq Q_{d}(nT_{s}),y_{2,4}(n)\triangleq Q_{d}(nT_{s}+T/2)\\ \end{array}  \tag{1}\]  
Following [19], a symbol-by-symbol, joint DQPSK Volterra decision feedback (VDFE) equalizer, is described as \((l=1,2)\):  
\[