Fig' 2' Equivalent Receiver Structure MF and MWD respectively denote the electronic matched filter a

1
Multiuser Capacity of Nonlinear Fiber Optics with
Single- and Multi-Wavelength Detection
Mohammad H. Taghavi, George C. Papen, and Paul H.
Siegel Center for Wireless Communications, UCSD
Abstract Previous results 1 suggest that the
crosstalk produced by the fiber nonlinearity in a
WDM system imposes a severe limit to the capacity
of fiber optics, since the nonlinear crosstalk
limits the maximum achievable signal to
interference plus noise ratio (SINR). We show
that by optimally using the information from all
the wavelengths for detection, the change in the
capacity region due to the nonlinear effect is
minimal. However, if the receiver only uses the
output of one wavelength-channel, the capacity is
significantly reduced due to the nonlinearity,
and saturates as the interference power becomes
comparable to the noise, which is consistent with
the earlier results. The results hold in channels
with or without memory. Channel Model For a
single mode optical fiber with chromatic
dispersion and a Kerr nonlinearity, the slowly
varying complex envelope or low-pass equivalent
of the optical field, A(t,z), at time t and
distance z from the transmitter is described by
the nonlinear Schrödinger equation where ? is
the nonlinearity parameter, and
.We neglect all terms of O(d2) and
higher. We consider a WDM system with K users
transmitting in different wavelengths where
each user has an average power constraint As a
result of the nonlinearity, the index of
refraction changes instantaneously with signal
intensity. Hence, each symbol is modulated by the
symbols in all other wavelengths (Fig,
1). Fig. 1. Nonlinear mixing of the
wavelength channels. It can be shown that in the
weakly nonlinear channel, a bank of matched
filters followed by symbol-rate samplers shown in
Fig. 2 provides all the useful information, to
the first order. The outputs of the i-th branch,
i.e. yiR and yiI, can be written as the real and
imaginary parts of the complex variable (1) wher
e ni is the optical amplifier noise, modeled as a
complex Gaussian random variable with variance s2
per dimension.

• Note The transmitter doesnt need to know the
  crosstalk properties to achieve the capacity!
• Single-Wavelength Detection
• If we have access to the output at only one
  wavelength, then the model is an interference
  channel, similar to Fig. 3.
• Fig. 3. A two-user interference channel.
• Since, the interference is weak, the best that we
  can do is to treat the interference as noise. The
  crosstalk terms in (1) are not independent
  however they form a weighted U-statistic, which
  obeys a central limit theorem 3, and hence
  approaches a Gaussian random variable. Therefore,
  for large numbers of users, the maximum rate of
  each user with single-wavelength detection is
• where piltPi is the transmit power at the i-th
  wavelength.
• Note It is not necessarily best to transmit with
  the maximum power, since it also increases the
  interference power.
• Comparison and Conclusion
• We compare the capacity per user of the two
  schemes for a channel with 32 wavelength-channels.
  The uncertainty due to the weak nonlinearity
  approximation is 1 at P5 dBm/user and 5 at P9
  dBm/user.

• Fig. 2. Equivalent Receiver Structure (MF and MWD
  respectively denote the electronic matched filter
  and the multi-wavelength detector.)
• Multi-Wavelength Detection
• When the receiver has access to all the
  wavelengths, the fiber becomes a multiple-access
  channel.
• Proposition 1 To the first order of
  approximation on the nonlinearity, the capacity
  region with MWD is (the same as the nonlinear
  channel!)
• (2)
• Outline of the proof This is a multiple-access
  channel. For each subset S of the users, , we
  have 2
• Upper Bound Using a Gaussian bound, we have
• with equality if Y is a Gaussian K-vector. Also,
  using the properties of the crosstalk, we can
  upper-bound the determinant by
• and conclude that (2) is an outer bound to the
  capacity region.
• Achievability Send i.i.d. complex Gaussian
  symbols with maximum power. Then use a simple
  interference cancellation scheme to achieve (2)
  as follows
• Step 1. Neglect the crosstalk and noise, and
  estimate each xi by yi..
• Step 2. Use these estimates to cancel the
  crosstalk in (1) for all channels i.e. form the
  test-statistics