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Hybrid Free-Space Optical and Radio-Frequency

Communications: Outage Analysis

Nick Letzepis∗, Khoa D. Nguyen∗, Albert Guill´ en i F` abregas†, and William G. Cowley∗

∗Institute for Telecommunications Research, University of South Australia, Mawson Lakes SA 5095

†University of Cambridge, Department of Engineering, Cambridge CB2 1PZ UK

nick.letzepis@unisa.edu.au; dangkhoa.nguyen@unisa.edu.au;

guillen@ieee.org; bill.cowley@unisa.edu.au.

Abstract—We study hybrid free-space optical (FSO) and

radio-frequency (RF) communications, whereby information is

conveyed simultaneously using both optical and RF carriers.

We consider the case where both carriers experience scintil-

lation, which is a slow fading process compared to typical

data rates. A parallel block-fading channel model is proposed,

that incorporates differences in signalling rates, power scaling

and scintillation models between the two carriers. Under this

framework, we study the outage probability in the large signal-to-

noise ratio (SNR) regime. First we consider the case when only the

receiver has perfect channel state information (CSIR case) and

obtain the SNR exponent for general scintillation distributions.

Then we consider the case when perfect CSI is known at both the

receiver and transmitter, and derive the optimal power allocation

strategy that minimises the outage probability subject to peak and

average power constraints. The optimal solution involves non-

convex optimisation, which is intractable in practical systems.

We therefore propose a suboptimal algorithm that achieves the

same diversity as the optimal one and provides significant power

savings (on the order of tens of dBs) over uniform allocation.

I. INTRODUCTION

Free-space optical (FSO) communication has the potential

to provide fiber-like data rates with the advantages of quick

deployment times, high security and no frequency regulations.

Unfortunately such links are highly susceptible to atmospheric

effects. Scintillation induced by atmospheric turbulence causes

random fluctuations in the received irradiance of the optical

laser beam [1]. Numerous studies have shown that perfor-

mance degradation caused by scintillation can be significantly

reduced through the use of multiple-lasers and multiple-

apertures, creating the well-known multiple-input multiple-

output (MIMO) channel (see e.g. [2]–[5]). However, it is the

large attenuating effects of cloud and fog that pose the most

formidable challenge. Extreme low-visibility fog can cause

signal attenuation on the order of hundreds of decibels per

kilometre [6]. One method to improve the reliability in these

circumstances is to introduce a radio frequency (RF) link

to create a hybrid FSO/RF communication system [6]–[8].

When the FSO link is blocked by cloud or fog, the RF link

maintains reliable communications, albeit at a reduced data

rate. Typically a millimetre wavelength carrier is selected for

the RF link to achieve data rates comparable to that of the

This work has been supported by the Sir Ross and Sir Keith Smith Fund,

Cisco Systems as well as the Australian Research Council under ARC grants

RN0459498, DP0558861 and DP088160.

FSO link. At these wavelengths, the RF link is also subject to

atmospheric effects, including rain and scintillation [9], [10],

but is less affected by fog. The two channels are therefore

complementary: the FSO signal is severely attenuated by fog,

whereas the RF signal is not; and the RF signal is severely

attenuated by rain, whereas the FSO is not. Both, however,

are affected by scintillation.

Lacking so far in the literature on hybrid FSO/RF chan-

nels is the development of a suitable channel model and its

theoretical analysis to determine the fundamental limits of

communication. This is the central motivation for our paper.

We propose a hybrid channel model based on the well known

parallel channel [11], and take into account the differences

in signalling rate, and the atmospheric fading effects present

in both the FSO and RF links. These fading effects are slow

compared to typical data rates, and as such, each channel is

based on a block-fading channel model. First we examine the

case when perfect CSI is known at the receiver only (CSIR

case), then we consider the case when CSI is also known at

the transmitter (CSIT case), and power allocation is employed

to reduce the outage probability subject to power constraints.

For the CSIR case we calculate the SNR exponents of the

hybrid channels for general scintillation distributions in each

of the channels. Then for the CSIT case, we derive the optimal

power allocation algorithm subject to both peak and average

power constraints. This optimal solution involves non-convex

optimisation, which has prohibitive complexity in practical

systems. We therefore propose a suboptimal solution and prove

that it has the same SNR exponent as optimal power allocation.

The remainder of this paper is organised as follows. In

Section II we present our channel model and assumptions.

Section III presents our main results for the CSIR-only case

while Section IV discusses power allocation and SNR expo-

nents for the CSIT case. Section V draws final concluding

remarks. Detailed proofs of our results can be found in [12].

II. CHANNEL MODEL AND ASSUMPTIONS

Consider a hybrid FSO communication system where a

binary data sequence is binary encoded into parallel FSO and

RF bit streams. The RF link modulates the encoded bits and

up-converts the baseband signal to a millimetre wavelength RF

carrier frequency. The FSO link employs intensity modulation

and direct detection, i.e. information is modulated using only

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the irradiance of a laser beam. The RF and FSO signals are

transmitted simultaneously through an atmospheric channel.

The received RF signal is then downconverted to baseband and

sent to the decoder. At the same time, the received irradiance

is collected by an aperture, converted to an electrical signal via

photodetection and sent to the decoder. The received signals

are jointly decoded to recover the transmitted message.

We define a hybrid channel symbol, (x, ˆ x) ∈ Xn

consisting of component FSO and RF symbols, which are

transmitted in parallel with perfect synchronism and have the

same symbol period Ts. The RF component symbol, denoted

by ˆ x, is drawn uniformly from a complex signal set Xrf⊂ C

of size |Xrf| = M = 2m, with unit average energy, i.e.

E[|ˆ x|2] = 1. Since the FSO link employs a much higher carrier

frequency than the RF link, we assume the FSO component

consists of n symbols drawn uniformly from a constellation

Xfso representing a pulse type modulation scheme, i.e. it

consists of n symbols, which are further composed of Q pulse

intervals of duration Tp, where Ts = nQTp. The signal set

Xfso⊂ (0,1)Qis a set of Q length binary vectors, where a

binary 1 at index i indicates a pulse of duration Tp in time

slot i. We assume each Tpsecond ‘on’ pulse is normalised to

have unit energy and denote the average FSO symbol energy

by γ = E

. Let q ? log2(|Xfso|), hence the total

bits per hybrid channel symbol is m + nq bits.

Both FSO and RF channels are affected by scintillation [1],

[9], [10], which is a slow fading process compared to typi-

cal data rates. We therefore propose a parallel block-fading

channel model, whereby the component channels are divided

into a finite number of blocks of symbols, and each block

experiences an i.i.d. fading realisation. The scintillation ex-

perienced by each component channel is also assumed to be

independent.1Typically, the RF scintillation has a coherence

time on the order of seconds [9], [10], whereas the FSO

scintillation is much faster, having a coherence time on the

order of tens of milliseconds [1]. We therefore decompose the

FSO and RF components of the codeword into A and B blocks

of K and L symbols respectively, where A ≥ B.2Note that the

total number of symbols in each FSO/RF component codeword

is the same, i.e. AK = BL. We assume that the number of

symbols in each block tends to infinity, but the ratio remains

a fixed constant, i.e. limK,L→∞

FSO/RF component channels are modelled by independent

additive white Gaussian noise (AWGN) channels.3Hence we

write the received FSO and RF signals as

fso× Xrf,

??nQ

i=1xi

?

L

K=

A

B. We assume both

ya[k] = paρhaxa[k] + za[k]

?

1This will be true over short time intervals, but over longer time scales

meteorological variations will result in correlated channel fades.

2Given that the coherence time of the RF scintillation is on the order of

seconds, the most realistic scenario is B = 1. However, for generality we

will assume B is an arbitrary positive integer.

3Note that this assumption for the FSO channel may not be accurate under

certain conditions [13].

(1)

ˆ yb[l] =ˆ pbˆ ργˆhbˆ xb[l] + ˆ zb[l],

(2)

for l = 1,...,L, k = 1,...,K a = 1,...,A and b =

1,...,B, where: ya[k] ∈ RnQand ˆ yb[l] ∈ C are the noisy

received symbols for the FSO and RF channels respectively;

xa[k] ∈ Xn

za[k] ∈ RnQis a i.i.d. vector of zero mean unit variance

real Gaussian noise, and zb[l] ∈ C is unit variance complex

Gaussian noise (CN(0,1)); ha > 0 andˆhb > 0 are inde-

pendent random power fluctuations due to scintillation, each

i.i.d. drawn from distributions fH and fˆ Hrespectively, with

normalisation E[ha] = E[ˆhb] = 1; paand ˆ pbdenotes the power

of block a and b for the FSO and RF channels respectively.

The γ parameter in (2) ensures both FSO and RF symbols

have the same energy. The parameters 0 < ρ, ˆ ρ < 1 in (1)

and (2) model differences in the relative strengths of the two

parallel channels, e.g. it reflects long-term fading effects due

to rain, fog or cloud as well as other parameters such as

aperture/antenna gains and propagation loss.4

In this paper, we consider two CSI scenarios: (1) CSI

at receiver only, uniform power allocation is employed; (2)

perfect CSI at both receiver and transmitter, transmit power

allocation is employed to minimise outages subject to long-

term and individual peak (or short-term) power constraints

constraints, i.e.

fsoand ˆ xb[l] ∈ Xrf denote the transmitted symbols;

E[?p?] + E[?ˆ p?] ≤ Pav,

?p? ≤ Pfso

1

A

(3)

(4)

peak

and

?ˆ p? ≤ Prf

1

B

peak,

where ?p? =

the FSO channel (1), the amplitude of the received electrical

signal is directly proportional to the transmitted optical power,

due to the photodetection process [14]. As we shall see

later, this scaling will significantly affect the design of power

allocation strategies.

?A

a=1paand ?ˆ p? =

?B

b=1ˆ pb. Note that for

III. ASYMPTOTIC OUTAGE ANALYSIS: CSIR CASE

The information outage probability of the hybrid system is

?

where h

=

(p1,...,pA), ˆ p = (ˆ p1,..., ˆ pB), R is the target rate of the

system in bits per hybrid channel use and,

Pout(Pav,R) ? Pr

Itot(p, ˆ p,h,ˆh) < R

?

,

(5)

(h1,...,hA), ˆh

= (ˆh1,...,ˆhB), p

=

Itot(p, ˆ p,h,ˆh) =

?

is the instantaneous input-output mutual information [11],

where Iawgn

X

tual information of the AWGN channel with input constellation

X and SNR u.5

4Although in practise ρ and ˆ ρ are randomly varying with time (and are also

most likely correlated random variables), we assume they remain unchanged

over many codeword time intervals and therefore are fixed constants.

5Note that the achievable rate (6) implicitly assumes joint encoding and

decoding across FSO and RF channels.

n

A

A

a=1

Iawgn

Xfso(h2

aρ2p2

a) +1

B

B

?

b=1

Iawgn

Xrf

(ˆhbˆ ργˆ pb),

(6)

(u) ∈ (0,log2|X|) denotes the input-output mu-

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Rather than assuming a specific fading distribution model,

we instead assume it is characterised by each channel’s single

block transmission SNR exponent, defined as

d(i)

fso? lim

u→∞−logPr{Iawgn

u→∞−logPr{Iawgn

fso

(h2u2) < Rfso}

(logu)i

(ˆhu) < Rrf}

(logu)j

(7)

d(j)

rf? lim

rf

,

(8)

for given component channel rate constraints Rfso and Rrf,

where i,j ∈ {1,2}.6

Suppose that perfect CSI is known only at the receiver

(CSIR case). The transmitter allocates power uniformly across

all blocks, i.e. p1= ... = pA= ˆ p1= ... = ˆ pB= p = Pav.

Then we have the following result.

Theorem 3.1: Define component d(i)

and (8) respectively. Suppose ρ, ˆ ρ > 0 and i = j = k. Then,

Pav→∞−logPout(Pav,R)

?

0 ≤ κ1≤ A,0 ≤ κ2≤ B

where Rc? R/(m + nq) and δ ?

bits to total transmitted bits.

From Theorem 3.1, we see that the overall SNR exponent

depends on Rc, δ, A, B and the individual SNR exponents dfso

and drfin a non-trivial way. However, for the most basic and

interesting scenario, A = B = 1, the solution to (9) reduces

to a simple intuitive form.

Corollary 3.1: Suppose A = B = 1. The solution to (9) is

divided into two cases as follows.

1) If δ ≤1

2) If δ ≥1

In most practical systems, δ ≥

period, the number of transmitted FSO bits will be greater

than the number of RF transmitted bits. From (12), we see

that the highest diversity is achieved if the binary code rate

Rcis set to be less than 1 − δ = m/(m + nq), i.e. the total

information rate is less than the maximum information rate of

the stand-alone RF channel. If 1 − δ < Rc≤ δ, the exponent

is the same as a single FSO link. For high binary code rates,

δ < Rc< 1, the asymptotic performance is dominated by the

worst of the two exponents.

fsoand d(j)

rf

as in (7)

d(k)?

lim

logPav

=inf

K(δ,Rc)

?

d(k)

fsoκ1+ d(k)

rfκ2

(9)

?

,

K(δ,Rc) ?

κ1,κ2∈ Z : δκ1

A+ (1 − δ)κ2

B> 1 − Rc,

?

,

(10)

nq

m+nqis the ratio of FSO

2, then

d(k)=

d(k)

d(k)

rf

min(d(k)

fso+ d(k)

rf

0 < Rc≤ δ

δ < Rc≤ 1 − δ

1 − δ < Rc< 1.

fso,d(k)

rf)

(11)

2, then

d(k)=

d(k)

d(k)

fso

min(d(k)

fso+ d(k)

rf

0 < Rc≤ 1 − δ

1 − δ < Rc≤ δ

δ < Rc< 1.

fso,d(k)

rf)

(12)

1

2, i.e. in a hybrid symbol

6SNR exponents for typical scintillation distributions can be found in [2].

Theorem 3.2: Define component channel SNR exponents

d(i)

rf

as in (7) and (8) respectively. Suppose i > j

then the SNR exponent is

?

d(j)= d(j)

rf

1 +

1 − δ(1 − Rc))

Otherwise, if i < j then the SNR exponent is

?

?

fsoand d(j)

d(i)= d(i)

fso

1 +

?

?A

?

δ(δ − Rc)

B

??

0 < Rc≤ δ

(13)

??

δ < R < 1.

(14)

d(j)= d(j)

rf

1 +

?

?A

B

1 − δ(1 − δ − Rc)

??

0 < Rc≤ 1 − δ.

(15)

d(i)= d(i)

fso

1 +

δ(1 − Rc)

??

1 − δ < Rc< 1

(16)

Theorem 3.2 shows how the overall performance of the hybrid

channel will be affected when one of the component channels

has an asymptotic outage probability that decays with SNR

much faster than the other. In particular, we see that the overall

SNR exponent will be dominated by the worst of the two

component channel SNR exponents unless the binary code

rate is below a certain threshold dependent on δ.

IV. ASYMPTOTIC OUTAGE ANALYSIS: CSIT CASE

Suppose both the transmitter and receiver have perfect

knowledge of the CSI and the transmitter adapts the power to

reduce the outage probability subject to constraints (3) and (4).

The optimal power allocation strategy requires the solution to

the following minimisation problem.

Theorem 4.1: The solution to problem (17) is given by

?

(0,0)

where (℘, ˆ ℘) is the solution to

In(18),

s∗

isa threshold

sup

s : E(h,ˆh)∈R(s)[?℘? + ?ˆ ℘?] ≤ Pav

?

Unfortunately, (19) is a non-convex optimisation prob-

lem [15], since in general Iawgn

X

in p. Thus (19). Instead of solving (19), we propose a

suboptimal algorithm that, as we shall see, exhibits the same

asymptotic behaviour as the optimal solution. In this direc-

tion, first consider the following lemmas. The proofs follow

Minimise:

Subject to:

Pr

E[?p?] + E[?ˆ p?] ≤ Pav,

?p? ≤ Pfso

?

Itot(p, ˆ p,h,ˆh) < R

?

peak, ?ˆ p? ≤ Prf

peak

.

(17)

(℘∗, ˆ ℘∗) =

(℘, ˆ ℘)

?℘? + ?ˆ ℘? ≤ s∗

otherwise,

(18)

minimise

subject to

?p? + ?ˆ p?

Itot(p, ˆ p,h,ˆh) ≥ R

?p? ≤ Pfso

p, ˆ p ? 0

peak,?ˆ p? ≤ Prf

peak

(19)

determined

?

.

by

s∗

=

?

?

, where R(s)

(h,ˆh) ∈ RA+B: ?℘? + ?ˆ ℘? ≤ s

?

(p2) is not a concave function

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straightforwardly via the Karush-Kahn-Tucker conditions [15]

and using the relationship between mutual information and

the minimum mean-square error (MMSE) for Gaussian chan-

nels [16].

Lemma 4.1: Define the minimisation problem,

where ?p2? ?

?

nλ

minimise

subject to

?p2? + ?ˆ p2?

Itot(p, ˆ p,h,ˆh) ≥ R

p, ˆ p ? 0,

a. The solution to (20) is

?

(20)

1

A

?A

?

a=1p2

p∗

a=ΥXfso

h2

aρ2,

1

and

ˆ p∗

b= ΨXrf

?

ˆ ρˆhbγ,λ

?

(21)

??,

,

where

ΨX(u,t) is the solution x to the equation mmseX(xu) =2x

mmseX(p) denotes the MMSE of a Gaussian channel

with discrete input constellation X, mmse−1

inverse MMSE function, and λ is chosen such that

Itot(p, ˆ p,h,ˆh) = R.

Lemma 4.2: Define the minimisation problem

Let p∗and ˆ p∗be the solution to (20) in Lemma 4.1, and ℘

and ˆ ℘ be the solution to (22). The solution to (22) is separated

into four cases depending on p∗and ˆ p∗.

1) If p∗and ˆ p∗satisfy the constraints in (22). Then ℘ = p∗

and ˆ ℘ = ˆ p∗.

2) If

?

where λ2is chosen such that

is chosen such that

ΥX(u,t)

?

1

ummse−1

X

?min?mmseX(0),t

u

tu,

X(u) is the

minimise

subject to

?p2? + ?ˆ p2?

Itot(p, ˆ p,h,ˆh) ≥ R

??p2? ≤ Pfso

peak,

?

?ˆ p2? ≤ Prf

peak

p, ˆ p ? 0

(22)

??(p∗)2? ≤ Pfso

℘a=ΥXfso

peakand

??(ˆ p∗)2? > Prf

and ˆ ℘b= ΨXrf

?

B

?

peak. Then

?ˆhbγˆ ρ,λ2

peakand λ1

?

h2

aρ2,

1

nλ1

??

,

?ˆ ℘2? = Prf

n

A

A

?

a=1

Iawgn

Xfso(ρ2h2

a℘2

a) = R −1

B

b=1

Iawgn

Xrf

(ˆ ℘bˆhbˆ ργ).

If??℘2? > Pfso

solution to (22) is the same as the previous case, with

the roles of rf and fso interchanged.

4) If

solution to (22) is infeasible.

Comparing (19) with (22), we see that (22) is minimising

the sum of the mean-square power of the FSO and RF

channels, subject to individual short-term root mean-square

(RMS) power constraints. By applying Jensen’s inequality [11]

to these constraints, we see that a solution to (22) will also

satisfy the constraints in (19) and hence can be considered a

suboptimal solution to (19). Therefore to find a suboptimal

peakthen the solution to (22) is infeasible.

??(p∗)2? > Pfso

??(p∗)2? > Pfso

3) If

peakand

??(ˆ p∗)2? ≤ Prf

??(ˆ p∗)2? > Prf

peak. Then the

peakand

peak, then the

solution to the original minimisation problem (17) we use the

solutions in Lemma 4.2 for (℘, ˆ ℘) instead of solving (19).

The asymptotic outage performance of optimal power al-

location for discrete-input block-fading AWGN channels was

analysed by Nguyen et al. in [17], [18]. In particular, from [17,

Prop. 3], if the peak-to-average power ratios αfsoand αrfare

finite, then the SNR exponent will be the same as the CSIR

case given in Theorems 3.1 and 3.2. When there are no peak-

to-average power constraints then the SNR exponent of the

optimal power allocation strategy is [18, Th. 2]

where d(1)

Theorem 4.2: Suppose αfso,αrf → ∞. Then the SNR

exponent of the suboptimal power allocation scheme described

by (17) with (22) is given by (23).

The implications of (23) are described as follows. When

d(1)

at a certain threshold of average power, i.e. the hybrid system

is able to maintain a constant level of instantaneous input-

output mutual information. The threshold at which this occurs

is referred to as the delay-limited capacity of the system [19].

Note that if d(1)

outage curve will not go vertical, nor will it converge to a

constant slope when plotted on a log-log scale [17]. When

the peak-to-average power ratios are finite, the peak power

constraints introduce an error floor with a slope equal to the

CSIR case. The height of the error floor is dependent on αfso

and αrf [17].

To demonstrate the implications of our asymptotic results,

we conducted a number of Monte Carlo simulations. Whilst

our results cover a wide range of hybrid system parameters

and fading distributions, due to space limitations, we will

focus on one particular set of specifications, i.e.: an RF

carrier employing 64QAM; an FSO carrier employing 4PPM;

A = B = 1, m + nq = 24 bits, δ =

scintillation on both channels (modelling very strong turbu-

lence). Fig. 1(a) shows the hybrid outage performance with

our suboptimal power allocation strategy compared to uniform

power allocation (cross marked curve) when Rc= 1/4. Note

that d(1)

rf

and from (23), the SNR exponent is d(1)

there are no peak power constraints, the curve will go vertical

at a certain average power threshold. This can be seen in

Fig. 1(a) (thick solid curve), for Pav > 7 dB outages are

completely removed. We see that there is a power saving of

more than 20 dB compared to uniform power allocation to

achieve 10−4outage probability. When peak power constraints

are introduced, as expected, we see that an error floor is

introduced with the same slope as the CSIR case. The floor

shifts down in probability as the peak-to-average power ratio

increases. Fig. 1(b) shows the case when Rc = 5/6. Since

d(1)

d(1)

csit=

∞

d(1)

d(1)

csir> 1

csir< 1,

d(1)

csir

1−d(1)

csir

(23)

csiris the SNR exponent for the CSIR case.

csit= ∞, then the outage probability curve will be vertical

csir= 1 in (23) then d(1)

csit= ∞, however, the

3

4; and exponential

fso= d(1)

= 1. Thus, from Corollary 3.1, d(1)

csir= 2,

csit= ∞, i.e. when

csir= 1, when there are no peak power constraints, the

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Page 5

?50510152025

10

?4

10

?3

10

?2

10

?1

10

0

Pav(dB)

(a) Rc=1

Pout(Pav, R)

CSIR

α = 0

α = 5

α = 15

α = ∞

4.

0510 15202530

10

?4

10

?3

10

?2

10

?1

10

0

Pav(dB)

(b) Rc=5

Pout(Pav, R)

CSIR

α = 0

α = 5

α = 15

α = 25

α = ∞

6.

Fig. 1. Outage performance of the hybrid FSO/RF channel with CSIT (solid) and uniform power allocation (dashed). System parameters included: ρ = ˆ ρ = 0.5,

A = B = 1, n = 9, 4PPM FSO and 64QAM RF with peak and average power constraints, and peak-to-average power ratios αfso= αrf= α in decibels.

Exponential distributed fading on both channels.

outage curve will no longer go vertical (thick solid curve).

As expected we see an error floor is introduced when the

peak-to-average power ratio is finite.

V. CONCLUSIONS

We proposed a simple hybrid FSO/RF channel model based

on parallel block fading channels. This hybrid model takes

into account differences in signalling rates and fading effects

typically experienced by the component channels involved.

Under this framework, we examined the information theo-

retic limits of the hybrid channel. In particular, we studied

its asymptotic high SNR outage performance by analysing

the outage diversity or SNR exponents. When CSI is only

available at the receiver, in the general case, the exponent

is not available in closed form. Instead, we derived simple

expressions from which it can be computed numerically. When

CSI is also available at the transmitter, we derived the optimal

power allocation scheme that minimises the outage probability

subject to peak and average power constraints. Due to the

power scaling of the FSO channel, this requires the solution

to a non-convex optimisation problem, which is intractable in

practical systems. We proposed a suboptimal power allocation

strategy, which is much simpler to implement and has the same

SNR exponent as the optimal power allocation.

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URL:

Convex Optimization,Cambridge

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