Zero-forcing electrical filters for direct detection optical systems

Intersymbol interference in direct deIII. FABRY-PEROT INTERFEROMETER tection optical systems can limit the channel spacing We consider a Fabry-Perot filter as the optical demultiplexing in frequency division multiplexing and prevent multifilter [1]. The envelope of the impulse response is well approxlevel signaling. We investigate a zero-forcing electrical imated by e t / ' for t > 0. The envelope of the transmit filter to cancel intersymbol interference and compare pulse is equal to 1 in (0,1°7 ] and 0 otherwise, where M is its performance with a matched filter and. a rectangua the number of signaling levels and R is the bit rate. The symlar response Afilter, for M-ary amplitude modulation. bol set is equal to {V/ -T I 0 < i < M 1}. We keep the bit I. MODEL rate fixed and plot the normalized eye opening for different In our model, the receiver front-end is composed of an optical r', where -' = TR is a parameter proportional to bandwidth filter, a photodetector and a low-pass electrical filter as shown efficiency in units of (b/s/Hz). The normalized eye opening in Figure 1. The photodiode is modelled as a square-law deis the vertical opening of an eye-diagram when the electrical n(t) filter has unit energy. We consider a filter matched to the postdetection electrical pulse po,0 (t) (single pulse transmitted), the optical I electrical _ minimum noise variance zero-forcing filter, and a filter with filter filter rectangular impulse response on [0, 1.g]. For the rectanguFigure 1: The receiver front-end includes an optical filter, O a photodiode and a low-pass electrical filter followed by a baud rate sampler. . M 2 cm -4 vice whose output is proportional to the magnitude square M 6 of the received signal envelope. The thermal noise n(t) from * _. the electronics is assumed to be the dominant noise and is 5 -8 modelled as additive white Gaussian noise. ° -l II. ZERO-FORCING FILTERS -12 Z Without loss in generality, we design the filter for sampling Z; -14 time t = 0. The output of the electrical filter at this time is R216 R4 -16 R4 Z aiai j pi,(a) h(a)da + N, (1) -18 . -20 where h(t) is the time-reversed impulse response of the electrical filter, pi,j (t) = Re{p(t-iT)p(t jT)}, p(t) is the complex envelope of the received optical pulse taking into account the transmit pulse and the channel response, p (t) is the complex igure 2: The optical filter is a Fabry-Perot interferomconjugate of p(t), ak (taken as real here) is the kth transmit eter. The curves are labelled as follows: (M) matched amplitude, and N is the noise at the sampled output. We filter curve, (Z) zero-forcing filter, (R) rectangular filter, write the integral f pi,(t)h(t)dt as an inner product (pi,, h) and (2,4,8) level signaling. where (x, y) _ f (t)y(t)dt. We define the ISI space, I, as the space spanned by pi,j (t)'s lar filter, there is no advantage to multilevel signaling, same without po,o(t), the desired signal. The signal space is defined as in [1]. On the other hand the zero-forcing filter performs as the space spanned by I and po,o(t). A filter h(t) is a zeromuch better and there is advantage to four level signaling for forcing filter if the sampled output has no ISI, i.e. we want large values of r'. the output of h(t) to depend only on aoao at sampling time t = 0. A necessary and sufficient condition for h(t) to be a ACKNOWLEDGEMENTS zero-forcing filter is This work was supported by NSF grant 9206379-NCR. (pi,j, h) # O, if i = j = 0, andREFERENCES (pi,j, h) = 0, otherwise.(2) [1] L. Cimini and G. Foschini. Can multilevel signaling improve the For filters that satisfy (2), we are interested in the one that spectral efficiency of ask optical fdm systems? IEEE Tiransac-tions on Communications, 41(7):1084-1090, July 1993.minimizes the noise variance when (po,o, h) is set equal to aconstant. The time-reversed impulse response of this filter, ifit exists, is proportional to the component of po,o (t) orthogonalto I. Presented at 1994 IEEE International Symp. on I.T., June 27-July 1, 1994,