A No e on Message Passing Decoding O e
Non-Gaussian Channels
A. Chawla
Oc obe 6, 2025
Abs ac
In hese no es, he au ho s wo k ou he ou line o he analysis o
SPARC codes when he channel model is ee space op ical (FSO). In his
s udy, he ading coe icien is assumed o be a known cons an . The key
con ibu ion is an applica ion o Shannon’s s anda d noisy channel coding
heo em o he a o emen ioned channel model.
1 Fading Coe icien s a e Known Cons an s: Chan-
nel Model and Ini ial Time
In his sec ion we se up he case whe e he ading coe icien o he FSO channel
is known. Then we inpu a SPARC codewo d [1] in o he channel. Then we s ep
o wa d in ime using he AMP algo i hm and ob ain he decoding es ima e o
he inpu codewo d coo dina e (see Equa ion 26 in Subsec ion 1.2). Wo king
his ou explici ly se s he s age o an applica ion o he Shannon heo em in
Sec ion 2.
x=Coe F SO ·Aβ0(1)
whe e Coe F SO is known and Aβ0is a single codewo d coo dina e. We ha e
he channel equa ion,
Y=Hx +I h (2)
whe e His ea ed as a cons an a p esen . Nex we le
J=Y
H=x+I h
H.(3)
whe e J ep esen s a single numbe and H ep esen s a single ading coe icien .
We assume His known. The e o e,
Ji=Coe F SO(Aβ0)i+I h
Hi
.(4)
1
Since Coe F SO is a known cons an ,
Ki≡Ji
Coe F SO
= (Aβ)i+I h
Coe F SO ·Hi
(5)
whe e Ais a known, spa se, design ma ix, βis a spa se message ec o and he
las e m is he noise e m. In ec o o m,
K=→
−→
A−→
β0+−→
n(6)
which belongs o Rnwi h −→
β0∈RN. He e →
−→
Ais he design ma ix o he SPARC
code, −→
β0is he message ec o and −→
nis he i.i.d. noise ec o . The a iance o
each i.i.d. . . is ¯
I2
h
(Coe F SO Hi)2whe e Coe F SO = 2RP√Tand Hiis a known
cons an . Gi en −→
Kand →
−→
A1, we wish o econs uc −→
β0.
Nex , we will ad ance h ough he AMP algo i hm s eps, indexed by he
ime . We s a wi h S ep 1, a ime = 0.We guess β0
0= 0. The e o e,
z0=−→
K−→
−→
Aβ0
0+1
δz−11
N
N
X
i=1
η′
−1(→
−→
A†z−1+β−1
0)(7)
whe e δ=n
LM . Fu he , any a iable wi h a nega i e index is assumed o be 0,
as pe Mon ana i. Thus,
z−1≡0(8)
η′
−1≡0(9)
and
β−1
0≡0.(10)
The e o e,
z0=−→
K−→
−→
Aβ0
0+ 0 (11)
wi h β0
0= 0.The e o e,
z0=−→
K. (12)
1No e ha his design ma ix is n×LM, wi h N≡LM.
2
1.1 Fi s Time S ep
Nex ,
β1
0=η0(→
−→
A†z0+β0
0)(13)
whe e
z0=−→
K(14)
and β0
0= 0. Fu he ,
η0≡η0
i|i=0(s)(15)
and
η0
i=pnPl
esi√nPl
τ2
0
Pj∈sec(i)esj√nPl
τ2
0
(16)
whe e 1≤i≤N. E alua ing he exp ession a i= 0, we ge
η0=pnPl
es0√nPl
τ2
0
Pj∈sec(0) esj√nPl
τ2
0
(17)
We de ine,
τ2
0≡i h
2
(2RP√THi)2+E[β2
0]
δ(18)
whe e β0is a andom a iable whose dis ibu ion coincides wi h he empi ical
dis ibu ion o he en ies o −→
β0. This is based on Equa ion (1.4) and adjoining
ex om Mon ana i’s pape [2], and Equa ion [8] o he e e ence DMM09 om
he same pape . He e δ=n
LM and Plis he alloca ed powe .
We ind,
z1=−→
K−→
−→
Aβ0
0+1
δz0⟨η′
0(→
−→
A†z0+x0)⟩(19)
whe e z0=−→
Kand x0= 02. Recall ha η′
0was speci ied ea lie . The e o e,
z1=−→
K+1
δ−→
K⟨η′
0
→
−→
A†−→
K⟩.(20)
2No e ha : β0≡x,β0
0≡x0,. . .,β
0≡x .
3
1.2 Second Time S ep
Then
β2
0=η1(→
−→
A†z1+β1
0).(21)
He e we ask he ques ion as o wha is τ2
1, which is needed in η1? We e e he
eade o Equa ion (1.4) om Mon ana i’s pape . The e,
τ2
1=i h
2
(2RP√THi)2+1
δE[η0(β0+τ0z)−β0]2(22)
whe e η0is speci ied in Ⅎ,β0has a pd which ma ches he empi ical dis ibu ion
o he en ies o −→
β0and z∼N(0,1) is independen o β0. Fu he , Equa ion
(18) speci ies τ0. Nex ,
z2=−→
K−→
−→
Aη1(→
−→
A†z1+β1
0) + 1
δz1⟨η′
1(→
−→
A†z1+β1
0)⟩(23)
and so he e o e,
z2=−→
K−→
−→
Aη1(→
−→
A†z1+β1
0) + 1
δz1⟨η′
1(→
−→
A†z1+β1
0)⟩.(24)
The e o e,
z2=−→
K−→
−→
Aη1(→
−→
A†(−→
K+1
δ−→
K⟨η′
0
→
−→
A†−→
K⟩) + β1
0). . . (25)
. . . +1
δ(−→
K+1
δ−→
K⟨η′
0
→
−→
A†−→
K⟩)⟨η′
1(→
−→
A†(−→
K+1
δ−→
K⟨η′
0
→
−→
A†−→
K⟩) + β1
0)⟩.(26)
and so on, o he highe indices.
Thus, we ha e a nes ed o i e a i e z om which we econs uc β +1
0us-
ing Mon ana i’s Equa ion (1.1), and ha gi es us, o la ge enough , a good
app oxima ion o β0.
2 Analysis Ou line
In his sec ion, ou goal is o cha ac e ize he pe o mance o he SPARC-FSO-
AMP sys em. We b eak down he cha ac e iza ion in o a sequence o s eps.
1. We ix some la ge
2. Ge he equa ion o β +1
0, as in he pa s abo e. I will be in e ms o he
ecei ed ec o , −→
k, he empi ical dis ibu ion o he message ec o ( ia
τ), g
P−→
β0, he powe alloca ion Pl, he SPARC s uc u e (si, sj, sec(j)e c.)
and FSO channel-noise and modula ion cons an s.
4
3. Assume a uni o m dis ibu ion on he messages and de i e g
P−→
β0.
4. Fix a powe alloca ion Pl, o example an exponen ial powe alloca ion.
5. Fix he SPARC s uc u e si, sj, sec(j)e c.
6. Fix he channel noise and modula ion pa ame e s as cons an s.
7. Thus we ha e,
β +1
0∼ (−→
K, g
P−→
β0, Pl, si, sj, I h, H).(27)
De ine R′≡(g
P−→
β0, Pl, si, sj, I h, H). Tha is, β +1
0∼g(−→
K), since R′is
known, whe e −→
Kwas ecei ed and β +1
0is ou app oxima ion o he mes-
sage ec o o he pseudo-channel.
8. Finally, using g(−→
K)as he decoding, we a e done.
9. Then we compa e g(−→
K)wi h β0 o de e mine whe he he decoding is
co ec and hus he o e all ansmission goal is me o no .
10. So we wan he ollowing quan i y:
P {β0=g(−→
K)}(28)
11. Fo di e en , we could say ha since he decoding s uc u e was di e en ,
he code is di e en .
12. As → ∞,we ha e he e a sequence o codes.
13. Recall he noisy channel coding heo em: o e e y a e R < C, he e
exis s a sequence o codes whose P oemax →0as → ∞.3
14. Thus, ix a a e Rand so a SPARC s uc u e (L, n, N)and hen we ha e
he e a cus omized ( ha is, (L, n, N)-dependen ) code o each . Tha
is, we ha e a code sequence. Find he maximum alue o Rsuch ha
Equa ion (28) con e ges o 0 as → ∞. This would be he “capaci y” o
maximum achie able a e Rmax on an FSO, using he SPARC-AMP code.
15. Finally, we may compa e his Rmax wi h LDPC-FSO-Belie -P opaga ion
code’s maximum achi eable a e o decide whe he SPARCs a e “be e ”
is-a- is LDPC codes.
3No e ha Equa ion (1) in G eig-Venka a amanan s a es ha
R=Llog(M)
n=L
n(log N
L)(29)
whe e Lis he numbe o sec ions and n, N a e dimensional pa ame e s.
5
3 Conclusion
In his no e we ha e se up he exac analysis ou line o de e mine he com-
pa a i e pe o mance o di e en codes such as LDPC and SPARCs o e he
ee space op ical channel wi h app oxima e message passing decoding. No e
ha he channel model o he FSO sys em is i sel unde de elopmen , wi h
e ec s such as u bulence no ully accoun ed o ye in he li e a u e (see he
ex o And ews and Phillips o example), and he e o e he s eps p esen ed
in his wo k, i used wi h he basic model should be aken wi h some deg ee o
unce ain y.
I such a coding/decoding sys em is o be used in he con ol se ing o
say synch oniza ion, hen Sahai’s any ime amewo k will need o be in eg a ed
wi hin his se up. One imagines a sequen ial semi-o hogonal epea ed pulse
posi ion modula ion sys em, bu wi h AMP decoding which i sel plays ou
online. The coding side will need o be e o mula ed as well. Some s eps in his
di ec ion we e aken in he ollowing p ep in : [3].
Re e ences
[1] Kuan Hsieh and Ramji Venka a amanan. Modula ed spa se supe posi ion
codes o he complex awgn channel. IEEE T ansac ions on In o ma ion
Theo y, 67(7):4385–4404, 2021.
[2] Mohsen Baya i and And ea Mon ana i. The dynamics o message passing on
dense g aphs, wi h applica ions o comp essed sensing. IEEE T ansac ions
on In o ma ion Theo y, 57(2):764–785, 2011.
[3] Aman Chawla. Any ime eliabili y analysis o ini e s a e slow ading ee
space op ical channels. June 2025.
6
