Optical communication through the turbulent atmosphere
with transmitter and receiver diversity, wavefront control,
and coherent detection
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Citation
Puryear, Andrew L., and Vincent W. S. Chan. “Optical
communication through the turbulent atmosphere with transmitter
and receiver diversity, wavefront control, and coherent
detection.” Free-Space Laser Communications IX. Ed. Arun K.
Majumdar & Christopher C. Davis. San Diego, CA, USA: SPIE,
2009. 74640J-17. © 2009 SPIE--The International Society for
Optical Engineering
As Published
http://dx.doi.org/10.1117/12.824842
Publisher
The International Society for Optical Engineering
Version
Final published version
Accessed
Thu May 26 07:55:42 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/52726
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
Optical communication through the turbulent atmosphere
with transmitter and receiver diversity, wavefront control,
and coherent detection
Andrew L. Puryeara and Vincent W.S. Chanb
a Massachusetts
b Massachusetts
Institute of Technology, 77 Mass Ave, Room 36-545, Cambridge, MA 02139
Institute of Technology, 77 Mass Ave, Room 36-549, Cambridge, MA 02139
ABSTRACT
Net-centric warfare in todays dynamically changing military environments and the need for low-cost gigabit
intra-city communication present severe challenges for current free-space optical systems. Enabled by high-speed
electronics and advances in wavefront control, we develop an architecture that provides free-space coherent
optical links with information capacity, security, network robustness and power management performance that
exceed the current state-of-the-art, including commercially deployed systems, R&D test-beds, and alternative
theoretical architectures proposed. The deleterious effects of the turbulent atmosphere are mitigated with several
diversity transmitters and receivers. We allow the phase and the amplitude of each transmitter to be controlled
independently and assume, through coherent detection, that the phase and amplitude of the received wave
is measured. Thus we can optimally allocate transmit power into the diffraction modes with the smallest
propagation losses to increase channel capacity and mitigate turbulence-induced outages. Additionally, spatial
mode modulation and rejection provides robust communication in the presence of denial of service via interference
by adversaries with a priori knowledge of the system architecture. Some possible implementations of this system
are described. New results, including asymptotic singular value distribution, expected bit error rate, interference
performance, and performance in the presence of inhomogeneous turbulence, are given. Finally, performance of
this system is compared with the performance of optical diversity systems without wavefront control and optical
systems without diversity, both current state-of-the-art systems.
Keywords: Optical communication, turbulent atmosphere, wavefront distortion, coherent detection
1. INTRODUCTION
The atmosphere is a challenging environment for transmission of data, with optical turbulence limiting data rate.
Microscale fluctuations in index of refraction causing deep fades of 20-30dB of typical duration of 1-100ms.2 This
paper focuses mitigating this turbulence using diversity at both the transmitter and receiver in conjunction with
wavefront pre-distortion and coherent field recombination. We assume independent control of the phase and
magnitude of each transmitter. Through coherent detection, we assume the receiver can measure the phase and
magnitude of the received wave. Thus we can optimally allocate transmit power into the spatial modes with the
smallest propagation loss in order to decrease bit errors and mitigate turbulence-induced outages. Additionally,
spatial mode modulation and rejection provides robust communication in the presence of interference by other
users with knowledge of the system architecture.
Previous work on sparse aperture coherent detection systems has not included wavefront predistortion, and
assumes the turbulence is in the nearfield of only the receiver. Average bit error rate (BER) was studied in
Rosenberg 1973,3 Hoversten 1970.4 Outage probability was studied in Lee 20045 and Lee 2007.6 In contrast,
we present performance results for sparse aperture coherent detection systems including wavefront predistortion.
We present closed-form expressions for turbulence average BER that apply when there are many transmit and
receive apertures. Because the instantaneous BER can vary significantly around the average BER, we also
Further author information: (Send correspondence to A.L.P.)
A.L.P.: E-mail: puryear@mit.edu, Telephone: 1 617 715 2747
V.W.S.C.: E-mail: chan@mit.edu, Telephone: 1 617 258 8222
Free-Space Laser Communications IX, edited by Arun K. Majumdar, Christopher C. Davis,
Proc. of SPIE Vol. 7464, 74640J · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.824842
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Figure 1. Sparse Aperture System Geometry - A field is transmitted from ntx transmitters in the ρ-plane to nrx receivers
the ρ -plane.
present new outage probability expressions. We quantify the effect of an optimal interferer and transmitter
power allocation. We conclude that spatial mode control significantly improves communication performance,
especially for nearfield operation. Finally, we provide recommendations for the design of such a system.
In Sec. 2, we describe the system geometry and discuss the applicability of the architecture in the atmospheric
environment. Additionally, we describe the optimal modulation, detection, and channel measurement scheme in
terms of the singular value decomposition (SVD) of the instantaneous stochastic Green’s function. In Sec. 3, we
derive the asymptotic distribution of the system singular values and present a detailed discussion of the relevance
of the theory to a wide range of real-world applications. Next, we show that the detected interference power
decreases proportional to the inverse of the number of receivers. Using the theories describing asymptotic singular
value distribution and interference power as a function of receivers, we derive an analytic closed-form expression
for optimal system performance in terms of BER. We also prove the modulation and detection scheme provided
in Sec. 2 achieves the optimal performance as the number of receive apertures approaches infinity. Further,
we show very good agreement between a Monte Carlo simulation and theoretical results. This is followed by a
conclusion in Sec. 6.
2. PROBLEM FORMULATION
For this paper, we parallel the problem formulation given in Puryear 200910 except we build more richness into
the model by allowing for spatial dispersion of the received waveform. We assume transmitters are arranged in
the ρ-plane and receivers are arranged in ρ -plane. The ρ- and ρ -planes are assumed to be parallel, as shown
in Fig. 1. The distance from the ρ-plane origin to transmitter i is denoted ρi . Similarly, the distance from the
ρ -plane origin to receiver j is denoted ρj .
We assume a coherent monochromatic scalar field of wavelength λ is transmitted from ntx apertures in the
ρ-plane. The field propagates z meters through a linear, isotropic, statistically homogeneous medium to the ρ plane where it is detected with nrx apertures. Each aperture’s diameter is assumed to be less than a correlation
length. We assume the area of each transmit aperture is equal to the area of all other transmit apertures.
Further, we assume the area of each receive aperture is equal to the area of all other receive apertures. The
transmit apertures, however, are not assumed to have the same area as the receive apertures. Additionally,
the minimum distance between each transmitter is at least a correlation length, far enough apart so that the
statistics of all ntx nrx links are uncorrelated. We refer to this geometry as a sparse aperture system. In contrast,
we refer to a system with one large transmit aperture and one large receive aperture as a single aperture system.
We define a pair of ancillary variables, nmin = min(ntx , nrx ) and nmax = max(ntx , nrx ). We will see that the
system performance depends on ntx and nrx only though these ancillary variables.
We can model the field propagation from the transmit plane to the receive plane as:
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α(d)
α(1)
α(m)
y(t) = √
H(d) x(t) + √
H(1) x(t − Δ1 ) + √
H(m) x(t − Δm ) + w(t)
nmax
nmax
nmax
(1)
= y(d) (t) + y(1) (t) + ... + y(m) (t) + w(t)
where x(t) ∈ Cntx is a vector representing the transmitted field at time n, y(t) ∈ Cnrx is a vector representing the
received field at time n, and w(t) ∈ Cntx represents additive white circularly symmetric complex Gaussian noise
received at time n. α(d) ∈ R is the system power gain for the direct line-of-sight path; including deterministic
path attenuation and transmitter and receiver area power gain. Without loss of generality, we assume the system
power gain α(d) to be unity. α(k) ∈ R is the system path gain for the k th multipath component. The direct-path
(d)
channel is represented by H(d) ∈ Cnrx ×ntx : element hij of H(d) is the direct path gain from the ith transmit
aperture to the jth receive aperture. The matrices H(k) , k ∈ {1, ..., m} represent the multipath interference:
(k)
element hij is the gain of the kth reflected (or refracted) path from the ith transmit aperture to the jth receive
aperture. Δi is the time of arrival of multipath component i measured from the time of arrival of the direct
path. The path delay may be positive or negative because a multipath ray can travel at a higher speed than the
direct ray. While multipath, as modeled, is rarely a limiting phenomena for free-space laser communication, we
retain the terms for full generality. The terms of the second equality in (1) are defined as:
α(d)
y(d) (t) = √
H(d) x(t)
nmax
α(1)
H(1) x(t − Δ1 )
y(1) (t) = √
nmax
..
..
..
.
.
.
(2)
α(m)
H(m) x(t − Δm )
y(m) (t) = √
nmax
where y(d) (t) is the received field due to the direct path and y(k) (t) is the received field due to the k th multipath
component.
We have implicitly invoked Taylor’s frozen turbulence hypothesis8 by assuming H(d) is time invariant. This
is true if the bit period is much less than the atmospheric correlation time, which is easily satisfied for gigabit
communication.
The normalization (nmax )−1/2 is chosen to ensure:
• additional apertures will not increase the system power gain and
• that symmetry is satisfied.
For free-space propagation, the extended Huygens-Fresnel principle applied to the path from the ith transmit
aperture to the jth receive aperture reduces to:
ρi − ρ 2
j2πz
1
j
(d)
1+
(3)
hij = exp
j
λ
2z 2
We can model the atmospheric turbulence as piecewise constant over each aperture because we assumed each
1
term in the system power gain
aperture is smaller than a correlation length. Note that we have included the λz
(d)
α .
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Assuming the effects of the turbulence-induced index of refraction variation on the propagating field can be
modeled using Rytov’s method, the fading along the path from a single transmit aperture to a single receive
aperture is a multiplicative factor of the form eχ+jφ .
ρi − ρ 2
j2πz
1
j
(d)
1+
(4)
× exp χ ρi , ρj + jφ ρi , ρj
hij = exp
2
j
λ
2z
where the log-amplitude and phase fluctuations, χ(ρi , ρj ) and φ(ρi , ρj ), are independent jointly Gaussian random
scalars. That is χ ∼ N (mχ , σχ2 ) where mχ is the log-amplitude mean and σχ2 is the log-amplitude variance. By
conservation of energy, mχ = −σχ2 . φ ∼ N (mφ , σφ2 ) where mφ is the phase mean and σφ2 is the phase variance.
We assume σφ2 2π so that the phase probability distribution function is approximately uniform from zero to
2π, φ ∼ U [0, 2π].
We decouple the input-output relationship of H(d) with a singular value decomposition (SVD):
√
1
H(d) = UΓV†
nmax
(5)
where the ith column of U is an output eigenmode, the ith row of V is an input eigenmode, and the i, ith
entry of the diagonal matrix Γ is the singular value, or diffraction gain, associated with the ith input/output
eigenmode. We use (·)† as the conjugate transpose. For the context of this paper, an eigenmode is a particular
spatial field distribution, or spatial mode. We will denote the diffraction gain associated with the ith eigenmode
as γi so that:
1/2
ỹi = γi x̃i + w̃i
(6)
where x̃, ỹ, and w̃ are related to x, y, and w through the usual SVD, such as in.7 Note w̃i retains its circularly
symmetric complex Gaussian distribution. We have not included the multipath components in (6).
We assume instantaneous channel state information measured by the receiver is available to the transmitter.
Implied in this assumption is a feedback path from the receiver to the transmitter of sufficient rate and delay to
allow for some minimum set of channel information to be received at the transmitter before the atmospheric state
has changed. The delay is required to be less than an atmospheric correlation time, on the order of 1-100ms,
which is reasonable for communication links on the order of tens of kilometers.
The channel state can be measured by transmitting an impulse from each transmitter sequentially and
recording the time variation of the received field at each receiver. From this measurement, we can build up
a channel transfer matrix as a function of time. We can then assign the channel transfer matrix at the time
instance with the largest power gain to H(d) . In practice, we do not need to calculate the multipath transfer
matrices. They can be measured, if desired, by selecting the channel transfer matrices at the time instances with
the m + 1 largest power gains, corresponding to the direct propagation path and m multipath components, and
creating H(d) and H(k) . The need to compute the SVD and feed back some information regarding the direct
path transfer matrix H(d) limits the distance from the transmit plane to the receive plane. Time to compute the
transfer matrix is small compared to the propagation time, an important consideration for implementation.
We have presented an idealized channel measurement scheme. However, other methods are likely more feasible
for implementation. For instance, a system would establish an initial channel state estimate via a full channel
measurement and subsequently track channel state perturbations with time, thus taking advantage of current
information to reduce the number of channel measurements. This method is effective provided the tracking
bandwidth is much higher than atmospheric temporal correlation bandwidth. Because the apertures are sparse
(transmitters and receivers in each place are at least a correlation length apart), nrx × ntx channel measurements
are needed to fully characterize the atmospheric state (i.e. populate H(d) ).
The received field is detected coherently with either a heterodyne or a homodyne receiver. Wavefront predistortion is implemented by controlling the amplitude and phase at each transmit aperture independently.
However, we do not allow the amplitude or phase to vary within an individual aperture. We will limit the total
power transmitted at a given instant to Pt , independent of the number of transmitters:
nmin
2
Pt
(7)
E x =
ntx
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Throughout this paper, we will often assert that the number of transmit and receive apertures is large. For
mild turbulence and a range of 10km, the correlation length is about 1cm: 100 transmitters can be placed in a
10cm-by-10cm patch. Thus, the assertion that the number of apertures is large is valid for useful systems.
3. SINGULAR VALUE DISTRIBUTION
(d)
hij
The transfer function
describing propagation through random media can be represented as the deterministic
S
free-space Green’s function hF
ij modified by amplitude χij and phase φij perturbation terms:
(d)
S
iφij
hij = hF
ij χij e
(8)
In the random field of the atmospheric optical channel, χij and φij are random variables. Assuming the entries
of H(d) are independent and identically distributed (i.i.d.), we can find the probability density function (pdf)
governing the distribution of the system singular values.
Theorem 1. Assuming the entries of H(d) are i.i.d., the pdf of diffraction gains, γi2 , for a single ocean state,
as the number of transmit apertures and receive apertures asymptotically approaches infinity, converges almost
surely to the Marcenko-Pastur density:
+
√ + √
x − (1 − β)2
(1 + β)2 − x
+
(9)
fβ (x) = (1 − β) δ(x) +
2πx
min(nrx , ntx )
and (x)+ = max(x, 0).
max(nrx , ntx )
Proof. A nice proof of the Marcenko Pastur density for i.i.d. matrices can be found in Bai.9 where β =
Fig. 2 shows the Marcenko Pastur density for various system geometries.
As stated in the theorem, the entries of H(d) must be i.i.d. for the Marcenko Pastur density to describe
the singular value distribution. We now include a discussion that reasons that the entries of H(d) are in fact
approximately i.i.d. for many realistic applications.
Because each ray traverses a different path from transmitter i to receiver j, the entries of H(d) will be
uncorrelated provided transmit/receive apertures are separated by at least a correlation length. uncorrelated
implies independent because the underlying distribution is Gaussian.
The transmission range is much larger than optical wavelengths wavelengths for this system (on the order of
1μm for optical systems), so the far field approximation is valid and the difference between amplitude variations
among paths is negligible. Additionally, the transmission range is much larger than the distance between transmit
and receive apertures. If all path lengths are within 10%, we estimate these path lengths as being the same,
so that the amplitude statistics should also be the same for each path. Consequently, χij has approximately
identical statistics for all (i, j) pairs.
φij is a Gaussian distribution with zero mean and variance σφ2 . For channels with sufficient turbulence, σφ2
is much greater than 2π. Because the phase is purely real, it can always be represented as a value from 0 to
2π. Because the variance is very large compared to this range, the distribution can be approximated as uniform
from 0 to 2π for all (i, j) pairs. Thus, φij can also be approximated as identically distributed.
S
Additionally, hF
ij can be absorbed into the phase term, as it is a complex number which can be represented
FS
S
as eiφij so that the transfer function now becomes hij = χij eiφij where φij = φij + φF
ij . Without loss of
generality, the expression is shown for λz = 1. Again, because of the periodicity of the complex exponential ei(·) ,
this additive term does not change the distribution and it remains uniform from 0 to 2π. Thus, hij is modeled
as a function of i.i.d. random variables.
Thm. 1 assumes a large number of apertures. This is true for almost any physically realizable system. For
instance, if we have a link range of 10km, we can easily place up to 100 transmitters and 100 receivers and still
meet all of the restrictions necessary for the theorem to apply. Therefore, the theory is applicable for any practical
system. Fig. 3 presents a comparison simulation verses theory for approximately 100 transmitters/receivers. To
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1
β=0.2
β=0.5
β=1
0.8
fβ(γ)
0.6
0.4
0.2
0
0
1
2
γ
3
4
5
Figure 2. Marcenko Pastur Density - Probability density function of diffraction gain γ under different transmitter/receiver
aperture configurations.
β=0.2
0.8
β=1.0
β=0.5
1
1
Empirical Distribution
Theoretical Distribution
0.8
1
Empirical Distribution
Theoretical Distribution
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
1
γ
2
3
0
0
1
2
γ
3
Empirical Distribution
Theoretical Distribution
4
0
0
1
2
γ
3
4
5
Figure 3. Comparison of Marcenko Pastur Density with Simulation - Probability density function of diffraction gain γ
under different transmitter/receiver aperture configurations along with simulated singular value distribution.
create the histogram, many atmospheric states were simulated and the empirical distribution calculated from
the SVD of these states. We see, even for the case with 100 transmitters and 100 receivers, that the simulated
distribution has very nearly converged to the Marcenko Pastur density.
As a corollary to Thm. 1, the number of eigenmodes corresponding to non-zero singular values converges,
2
=
almost
to min(ntx , nrx ). Additionally, the maximum singular value converges, almost surely, to γmax
√ surely,
2
(1 + β) .
4. PERFORMANCE OF SPARSE APERTURE SYSTEMS
In this section we present the performance of sparse aperture systems with wavefront control and coherent
detection.
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4.1 Multipath Dispersion Rejection
Here, we use the term multipath loosely to mean the dispersion induced from the atmospheric channel. We show
here that the sparse aperture system effectively eliminates multipath induced intersymbol interference (ISI) when
there are many transmit and receive apertures. This will be an important result when we calculate the system
performance. As a note, this system does not collect power from multipath components, the system only rejects
it.
Here, we show that multipath interference can be reduced to any desired level by increasing the number
of transmitters and receivers. Further, we show that the detected interference power is proportional to 1/nrx .
Formally, we wish to select a spatial modulation x[n] and receiver spatial recombination scheme S to minimize
the instantaneous BER (the term instantaneous is used to imply the BER for a single atmospheric state) or,
equivalently, maximize the instantaneous signal to interference noise ratio:
2E (d)
∗
m
x [n] = arg min Q
2 +
(k)
x[n],S
σw
k=1 E
(10)
E (d)
m
= arg max
2 +
(k)
x[n],S
σw
k=1 E
2
is the energy due to the noise w[n], E (d) is average detected signal energy and E (k) is the average
where σw
detected energy in the k th multipath component. Q(·) is the q-function. We have assumed the waveform
transmitted through a multipath channel contributes as noise through ISI. The optimization in (10) is very
general. Specifically, x∗ [n] would have nulls in the direction of multipath components if such a waveform was
optimal. Further, the optimization will find the waveform that propagates most efficiently through the turbulent
atmosphere. Thm. 2 gives the solution to (10) for the case with many transmit and receive apertures.
Theorem 2. As the number of transmit and receive apertures grows large, the spatial field distribution that
minimizes instantaneous BER, as formulated in (10), is given by:
x∗ [n] = a[n]vmax
(11)
where we have used SVD, H(d) = UΓV† . Thus, vmax is the input eigenvector associated with the maximum
singular value γmax of H(d) . Data is encoded by time variation of a[n] ∈ C, which is spatially constant at a
particular time. √For example, to transmit a bit C[n] ∈ {0, 1} with energy Et using binary phase shift keying
(BPSK): a[n] = Et eiπC[n] . A sufficient statistic for optimum detection is:
φ[n] = Re{u†max y[n]}
where φ[n] is a sufficient statistic at time n. The associated optimal instantaneous probability of error is:
2α(d) γmax Et
Perror,i = Q
m
2 + Et
(k)
σw
k=1 α
nrx
(12)
(13)
Proof. To prove this theorem, we show that this scheme maximizes the signal energy E (d) while at the same time
minimizing the interference energy E (k) ∀k for a large number of transmit and receive apertures. We assume,
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without loss of generality, that the transmit power is unity, Pt = 1. First, we show that (11) maximizes E (d) :
E (d) =
=
=
=
=
max
E[(φ[n])2 ]
max
E[(Re{u†max y(d) [n]})2 ]
y(d) [n]:x[n]2 =1
y(d) [n]:x[n]2 =1
1
nmax
max
x[n]:x[n]2 =1
1
nmax
1
ζi :
max
2
i
ζi =1
E[(Re{u†max H(d) x[n]})2 ]
E[(Re{u†max H(d)
n
min
ζi vi a[n]})2 ]
(14)
i=1
E[(Re{u†max H(d) vmax a[n]})2 ]
nmax
1
(Re{u†max H(d) vmax })2 Et
=
nmax
= γmax Et
where we have used that, because the input eigenmodes form a complete orthonormal set, any input field x[n]
can be written as a linear combination of input eigenmodes.
x[n] =
ntx
ζi vi
(15)
i=1
We have also used that the energy in the transmit waveform can be expressed as:
2
n
ntx
tx
2
ζi vi = 1 =⇒
ζi2 = 1
x = i=1
(16)
i=1
Thus we have shown that this scheme maximizes the signal energy E (d) . We now show that (12) asymptotically
(nrx → ∞) drives the interference power E (k) to zero, regardless of our choice of input field:
E (k) = E[(Re{u†max y(k) [n]})2 ]
α(k)
E[(Re{u†max H(k) vmax })2 ]
nmax
= α(k) E[(Re{u†max ṽ})2 ]
=
−1/2
where we have defined ṽ = nmax H(k) vmax . So, we can write:
⎡ 2 ⎤
nrx
⎦
u†max,i ṽi
E (k) = α(k) E ⎣ Re
i=1
⎡
⎤
nrx
2
⎦
= α(k) E ⎣
Re u†max,i ṽi
i=1
=α
(k)
E ξ2
where umax,i is element i of umax and ṽi is element i of ṽ. We have defined ξ =
†
Re
u
ṽ
. The
i
max,i
i=1
nrx
entries of H(k) are identically distributed, independent of each other, and independent of the entries of the
direct channel matrix H(d) . The entries of H(k) are independent of the entries of H(d) because the multipath
components traverse a different path from the direct component and therefore experience different fading. The
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central limit theorem can be applied to the summation because the terms in the summation, Re u†max,i ṽi , are
independent. As the number of receive apertures grows large, the central limit theorem implies:
1
ξ ∼ N 0,
nrx
And as a result:
E (k) =
α(k)
nrx
(17)
Thus, we have proved that (12) asymptotically (nrx → ∞) drives the interference power E (k) to zero, regardless
of our choice of input field. Finally, we prove (13) by substituting (17) and (14) into (10). In essence the receiver, shown in (12), is tuned to detect only the direct mode. As we increase the number
of receive apertures, the tuner becomes increasingly narrow. As a result multipath components become less and
less likely to couple into the direct mode tuned receiver. Nice intuition arises from an abstract vector space
interpretation. The nrx -dimensional vector space is the closed set of all possible receive fields. We can view the
received field vector as the summation of the direct field vector, the multipath field vectors, and the noise vector.
The sufficient statistic in (12) is interpreted as an inner product between the received field vector and the direct
field vector. The multipath field vectors have random orientation with respect to the direct field vector. As a
result, the expected inner product between the multipath field vectors and the direct field vector becomes small
as the vector space dimension increases. Thus, the multipath interference is rejected.
Given that multipath interference is rejected by the receiver, we are free to select the transmit spatial waveform
that propagates most efficiently. Equation (15) gives the expression for the most efficient propagation, provided
the instantaneous Green’s function is known.
While an equation for instantaneous probability of error is provided, we have not presented a closed-form
solution for the maximum singular value γmax and, therefore, have not yet presented a closed-form expression
for probability of error. In the next section we calculate the maximum singular value and provide a closed-form
expression for turbulence averaged bit error rate.
4.2 Asymptotic Bit Error Rate
Binary phase shift keying (BPSK) is a common modulation scheme for optical communication systems with
coherent detection. Though results developed here may easily be applied to other modulation schemes (e.g.
binary on-off keying, frequency shift keying, etc.), we will limit ourselves to the discussion of BPSK systems for
brevity.
For this modulation scheme, the turbulence average BER is:
∞
Perror,i fβ (γmax )dγmax
Pr(error) =
0
∞ Eb
=
Q
2γmax 2 fβ (γmax )dγmax
σ
0
(18)
m
2
(k)
+ nErxt
where fβ (γmax ) is the pdf of the largest singular value. We defined Eb = α(d) Et and σ 2 = σw
k=1 α
for notational convenience. This result is general for any BPSK sparse aperture optical communication system,
but depends on an unknown pdf, fβ (γmax ). However the pdf for the largest singular value is known in the
asymptotic case. As the number of apertures grows large, the pdf of the largest singular value converges, almost
surely, to:
2
lim fβ (γmax ) = δ γmax − 1 + β
(19)
nmin →∞
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where δ(·) is the Dirac delta. Using (19) to evaluate (18) provides a closed-form expression for the probability
of error:
lim
nmin →∞
=
0
∞
Pr(error)
Q
Eb
2γmax 2
σ
2
δ γmax − 1 + β
dγmax
2 Eb
=Q
2 1+ β
σ2
(20)
While this result is only exact in the asymptotic case, it provides a very good approximation for a finite
√ 2
but large number of apertures. The 1 + β term is the power gain over a system without the wavefront
predistortion. This power gain term results from the ability to allocate all of the system transmit power into the
spatial mode with the best propagation performance. Essentially, we select the mode with the best constructive
interference for the particular receiver aperture geometry and atmospheric state.
Using the monotonicity of the Q-function, it is easy to prove that the optimum system is balanced, using
the same number of transmit and receive apertures. For a balanced system, setting β = 1 yields a power gain
of four. This result is intuitively satisfying. Consider two scenarios: first, if there are more transmit apertures
than receive apertures, the system can form more spatial modes than it can resolve and degrees of freedom are
unused. Second, if there are more receive apertures than transmit apertures, the system can resolve more spatial
modes than it can form and, again, degrees of freedom are unused. As a result, our intuition suggests a balanced
system is optimal.
As the number of receive apertures becomes much larger than the number or transmit apertures, β → 0, the
system performance approaches that of the system without wavefront predistortion. This is also an expected
result. As the system becomes very asymmetric, the ability to predistort the wavefront is lost.
The gain, in terms of probability of error, of moving to a diversity system with wavefront predistortion is:
Pr(error|sparse aperture)
= e−3SN R
Pr(error|no diversity)
(21)
At high SNR, using the sparse aperture system provides a large gain in BER compared to the no diversity system.
At low SNR, the advantage of the more sophisticated system is less pronounced.
It is clear, in the asymptotic case, that the average BER does not depend on turbulence strength. Effectively,
the many apertures act to average out the spatial variation induced by the atmospheric turbulence. Turbulence
strength does factor into the system design; in stronger turbulence, apertures may be placed closer together while
in weaker turbulence, they must be placed farther apart. Further, stronger turbulence causes slower convergence
to the Marcenko-Pastur density; which means more apertures are required for (20) to be valid.
Lastly, as the total aperture size increases for a single aperture system, the power gain saturates as the
aperture size approaches the correlation length. We have shown that the sparse aperture system, however, does
not saturate with total aperture size. Indeed, the number of apertures used is only limited by form factor
constraints.
A Monte-Carlo simulation was performed to validate the theory presented in (20). In the simulation, we
assumed that the instantaneous atmospheric state was available at the transmitter. For a single atmospheric state,
an equiprobable binary source was encoded according to (11), transmitted through the simulated atmosphere,
detected coherently, and the number of raw bit errors recorded. This process was repeated many times with
independent realizations of the atmosphere to arrive at the average BER presented in Fig. 4. In the figure, we
show theory and simulation versus SNR, Eb /σ 2 . The number of transmitters was 100, 100, 200 and the number
of receivers was 100, 50, 20 giving β = 1, β = 0.5, and β = 0.1.
From the figure, we see very good agreement between theory and simulation. As we stated earlier, the theory
provides an approximate solution to any system with a large but finite number of apertures. Here, we see the
approximation is very close to the theory.
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0
10
−1
Probability of Error
10
−2
10
−3
10
−4
10
−20
Theory (β = 1)
Simulation (β = 1)
Theory (β = 0.5)
Simulation (β = 0.5)
Theory (β = 0.1)
Simulation (β = 0.1)
−15
−10
−5
Eb/σ2
0
5
10
Figure 4. BER vs SNR - A comparison of Monte-Carlo simulation and theory for binary phase shift keying with σχ2 = 0.1.
5. SYSTEM PERFORMANCE IN THE PRESENCE OF AN INTERFERER
Any deployed system will be affected by interference. In a densely populated urban area, other systems might
inadvertently couple power into the receiver. There are other situations where a hacker might couple power into
the receiver and deny service. In this section, we review results given in Puryear10 for the effects of a worst-case
interferer. Based on these results, we provide fuller analysis and insight into the system performance in the
presence of an interferer.
To conduct a worst-case analysis, we will assume the interferer has instantaneous knowledge of channel state,
i.e. knowledge of the channel eigenmodes with their associated singular values. Further, we will assume the
interferer has knowledge of the modulation scheme and is synchronized with the transmitter.
In this section, we will again choose BPSK as the modulation scheme. For an optical system, transmitting
using BPSK, with transmit energy in a bit period of Eb,t and interference energy in a bit period of Ib,t the
average worst-case BER is:
2γi Eb,t
min E Q
(22)
E[P (error)] =
max
σ 2 + Ib,i
Ib,i :
Ib,i =Ib,t pi :
pi =1
where the expectation is over the atmospheric turbulence and the transmitter power allocation pdf. Ib,i is the
interference energy per bit allocated to the ith eigenmode. pi is the probability that Eb,t energy is allocated
to the ith eigenmode. In this formulation the interferer is able to shape its waveform to couple an arbitrary,
but limited, amount of energy into each eigenmode. Similarly, the transmitter is able to couple an arbitrary,
but limited amount of energy into each eigenmode. By including the expectation we have assumed that the
transmitter can change its spatial mode much faster than the interferer can measure the transmitter’s spatial
mode and adapt. Alternatively, we have assumed that the interferer does not have knowledge of the transmitter’s
power allocation scheme, which may be changed dynamically. This assumption is required for convergence. Thus
the optimization can be interpreted as follows. For a given distribution of interference power, Ib,i , the transmitter
allocates power to minimize BER. For a given average distribution of transmit power Eb,t , the interferer allocates
power to maximize BER.
The solution (22) is presented in Theorem 3.
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Theorem 3. For the problem setup in (22), the optimal interference power allocation is:
Ib,i =
γi
− σ2
μ
+
where μ is chosen to satisfy the total power constraint:
γi
μ
−σ
2
+
= Ib,t
The optimal transmitter power allocation is then uniform:
1
, ∀i ∈ S
|S|
γk
S = i i = arg max 2
k σ + Ib,k
pi =
where |S| is the cardinality of the set S. The associated BER is then:
2γi∗ Eb,t
E[P (error)] = Q
σ 2 + Ii∗
Proof. The proof is completed by enforcing the Karush-Kuhn-Tucker conditions, but is not included here for
brevity. The interference power allocation stated in Theorem 3 is much like water-filling; we plot the values of σ 2 /γi
versus the singular value index, i, and imagine the line traced out as a vat which may hold water. Interference
power is allocated to eigenmodes such that the water level on the graph (which represents the inverse of the
signal-to-interference noise ratio) is 1/μ. Interference power is first allocated to the eigenmodes with the largest
singular value. As interference power is increased, it is allocated to weaker and weaker eigenmodes. Thus, as
expected, the optimal weak interferer will degrade the channel by allocating its total power to the strongest
eigenmode. A strong interferer will allocate power to all eigenmodes, effectively creating a channel where all
non-zero eigenmodes are equal. Fig. 5 shows the affect of a moderate interferer. In the top graph, we show
the signal to noise ratio (red histogram) for the singular value distribution of some atmospheric state. The blue
histogram shows the resulting signal to interference noise ratio after the interferer has optimally selected its
interference power allocation according to Thm. 3. We see that the resulting maximum SNIR is constant over
many eigenmodes at a level of μ. The optimal transmitter power allocation pfd, shown as the green histogram,
is uniform over the eigenmodes with an SNIR of μ.
The transmit power hops randomly among the eigenmodes with maximum signal to interference noise power
γi /(σ 2 + Ib,i ). The frequency at which the transmit power hops eigenmodes is governed by the ability of the
interferer to measure the transmit waveform. If the interferer can measure the waveform quickly, the transmitter
must mode hop faster. In the limiting case where the interferer cannot measure the transmit waveform, the
transmitter does not need to mode hop.
The Marcenko-Pastur density of singular values has not been used in the formulation or proof of Theorem 3;
in fact, the theorem is valid for even a small number of apertures. Unfortunately, μ in Theorem 3 must be solved
for numerically. In the case of a strong interferer with a large number of apertures however, we can evaluate
BER in the presence of an interferer in closed-form.
Theorem 4. For a sparse aperture system with a large number of apertures, the expected BER in the presence
of a strong interferer is:
2nmin βEb,t
E[P (error)] = Q
Ib,t + nmin σ 2
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Figure 5. Moderate Interference - Signal to noise ratio and signal to interference noise ratio vs eigenmode number and
optimal transmitter eigenmode hopping pdf.
Provided the interferer has sufficient total power:
Ib,t ≥ nmin σ 2
(1 −
β
√
β)2
−1
Proof. To prove this theorem, we begin with the optimal interference power allocation given in Theorem 3 and
assume the interferer has enough power to interfere with each non-zero eigenmode:
γi
2
−σ
Ib,t =
μ
γi
− nmin σ 2
=
μ
nmin β
− nmin σ 2
=
μ
where we have used, for a large number of apertures, the fact that the number of non-zero singular values
converges, almost surely, to nmin and the average singular value converges, almost surely, to β. Solving for μ
gives:
nmin β
μ=
Ib,t + nmin σ 2
We assumed at the outset of this proof that the interferer has enough power to interfere with each non-zero
eigenmode. For this to be true, the power allocated to the minimum eigenmode must be non-negative for the μ
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Figure 6. Stong Interference - Signal to noise ratio and signal to interference noise ratio vs eigenmode number and optimal
transmitter eigenmode hopping pdf.
we just derived:
γmin
2
−σ ≥0
μ
√ 2 Ib,t + nmin σ 2
1− β
− σ2 ≥ 0
nmin β
√ 2
where we have used that the minimum non-zero singular value is, almost surely, 1 − β . Solving this expression for Ib,t gives the minimum power constraint in the theorem. Finally, substituting μ into the optimal power
allocation and expected BER given in Theorem 3 completes the proof. This result is intuitively satisfying. As the number of system apertures is increased, the interferer must spread
its power among more spatial modes thus reducing its impact. Indeed, as the number of apertures becomes large
the interferer is completely rejected. Physically, the condition on the interference power guarantees that some
interference power is allocated to each non-zero eigenmode. Put into water-filling terms, the interferer has enough
water to completely fill the vat.
Fig. 6 shows the SNR, SNIR and transmitter power allocation pdf for the case of a strong interferer. In
this case, the interferer has enough power to effectively interferer with all non-zero eignemodes. As a result, the
transmitter allocates power equally to all eigenmodes.
Again, we see that β = 1 yields the best performance.
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Figure 7. Weak Interference - Signal to noise ratio and signal to interference noise ratio vs eigenmode number and optimal
transmitter eigenmode hopping pdf.
Corollary 1. For the case of the strong interferer, a balanced system, β = 1, provides the best interference
rejection. Proof. Its clear from the BER expression given in Theorem 4 that setting β = 1 gives the lowest
BER. Finally, fig. 7 shows the system performance in the presence of a weak interferer. As expected, the interferer
and transmitter allocate power to only the strongest eigenmode. The resulting average probability of error is:
Eb,t
2
2(1 + β) 2
E[P (error)] = Q
σ + Ib,t
5.1 System Performance in Inhomogeneous Turbulence
Thus far, we have been concerned with system performance in homogeneous turbulence. This implies a horizontal
link or a limited vertical link. For the case of a vertical link over kilometers, such as an aircraft or satellite terminal
to a ground terminal, turbulence will be inhomogeneous. The limiting Marcenko Pastur distribution is still valid
however provided a large number of transmitters and receivers and the path from each transmitter to each
receiver is independent. We define power gain as the SNR improvement resulting from wavefront predistortion:
2
Pgain = 1 + β
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(23)
4.5
Power Gain
Predator Drone
Commercial A/C
LEO Sat
GEO Sat
4
Diversity Power Gain
3.5
3
2.5
2
1.5
1
0
10
2
10
4
10
Vertical Link Range (meters)
6
10
8
10
Figure 8. Power Gain in Inhomogeneous Turbulence - Power gain for clear air vertical turbulence as a function of altitude
above sea level.
If we assume a transmitter and receiver of linear extent L, the power gain becomes:
⎛
⎞2
f loor ρLg + 1
⎠
Pgain = ⎝1 +
f loor ρLs + 1
(24)
where ρg is the correlation length measured on the ground and ρs is the correlation length measured altitude.
As the extent L grows very large, the function approaches:
2
ρs
(25)
lim Pgain = 1 +
L→∞
ρg
It is apparent that the power gain given in (24) is a discritized version of the limiting power gain given in (25).
Therefore we will use (25) as the metric for inhomogeneous turbulence power gain.
Fig. 8 shows (25) for a vertical clear air path where ρs and ρg are calculated using:
ρo =
2.91k
2
0
z0
Cn2 (z)
z
z0
−3/5
5/3
dz
(26)
where Cn2 (z) is the atmospheric structure constant as a function of altitude z. For the figure, we used the
atmospheric structure constant profile provided in Shapiro 1978.2 The green line denotes the altitude of an
operational predator drone, the red line denotes the altitude of a commercial aircraft, the cyan line denotes a low
earth satellite altitude and the purple line denotes the altitude of a geosynchronous satellite. Thus, the system
provides power gain for altitudes below the commercial aircraft altitude.
6. CONCLUSION
Optical communication over the turbulent atmosphere has the potential to provide reliable rapidly-reconfigurable
multi-gigabit class physical links over tens of kilometers. Such systems, however, are prone to long (up to 100ms)
and deep (10-20dB) fades and are susceptible to interference. In this paper, we have shown that a sparse
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aperture system with spatial mode control provides protection against fading and interference in addition to
better performance (average BER) compared with a single aperture system if the transmitter and receiver are
both in the atmosphere.
We proved that, given fairly benign conditions on the placement and size system transmit and receive apertures, the singular value decomposition converges almost surely to the Marcenko Pastur distribution. From a
system design perspective, we showed that multipath interference can be reduced to any level desired by increasing the receiver diversity. Additionally, we explored the affects of an interferer on communication performance.
Power gain was defined and calculated for a clear vertical path to geosynchronous satellite altitude. These results
all showed significant performance gains result from spatial mode control.
Optical communication generally provides such high data rates that the added complexity involved in implementing a system that communicates over multiple spatial modes simultaneously is not typically justified by
the added rate. As such, this paper has focused on metrics related to communicating on one spatial mode at
any given instant, such as BER and outage probability. The results presented for expected BER and expected
BER in the presence of an interferer can be easily extended to permit the use of expected channel capacity as
the performance metric. Closed-form results for outage probability defined in terms of channel capacity have yet
to be discovered.
ACKNOWLEDGMENTS
This work was supported by the Defense Advanced Research Projects Agency (DARPA) under the Technology
for Agile Coherent Transmission Architecture (TACOTA) program and by the Office of Naval Research (ONR)
under the Free Space Heterodyne Communications Program.
REFERENCES
[1] V.W.S. Chan,“Free-Space Optical Communications,” Journal of Lightwave Technology (Invited Paper), Volume 24, Issue 12, pp.4750-4762, Dec. 2006.
[2] J. H. Shapiro, “Imaging and optical turbulence through atmospheric turbulence,” in Laser Beam Propagation
in the Atmosphere, J. W. Strohbehn, Ed. New York: Springer-Verlag, 1978.
[3] S. Rosenberg and M. C. Teich, “Photocounting array receivers for optical communication through the lognormal atmospheric channel 2: Optimum and suboptimum receiver performance for binary signaling,” Appl.
Optics, vol. 12, pp. 2625-2634, Nov. 1973.
[4] E. V. Hoversten, R. O. Harger, and S. J. Halme, “Communication theory for the turbulent atmosphere,”
Proc. IEEE, vol. 58, pp. 1626-1650, Oct. 1970.
[5] E. J. Lee and V. W. S. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel
using diversity,” IEEE J. Select. Areas Commun., vol. 22, Nov. 2004.
[6] E. J. Lee and V. W. S. Chan, “Diversity Coherent Receivers for Optical Communication over the Clear
Turbulent Atmosphere,” IEEE ICC2007, 2009.
[7] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. New York, NY: Cambridge University
Press, 2005.
[8] G.I. Taylor, “The spectrum of turbulence,” Proc. R. Soc. Lond., Ser A 164, pp. 476490, 1938.
[9] Z. D. Bai, B. Miao, and J. F. Yao, “Convergence Rates of Spectral Distributions of Large Sample Covariance
Matrices,” SIAM J. Matrix Anal. Appl., vol 25, 2003.
[10] A. Puryear and V.S.W. Chan, “Coherent Optical Communication over the Turbulent Atmosphere with
Spatial Diversity and Wavefront Predistortion,” IEEE Globecom, 2009.
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