Lecture 
General broadcast channels
(Reading: NIT ., .)
∙
∙
∙
∙
Marton’s inner bound
Gaussian vector broadcast channel
Marton’s inner bound with common message
Nair–El Gamal outer bound
© Copyright – Abbas El Gamal and Young-Han Kim
Broadcast communication system
Yn
M , M , M
Encoder
Xn
p(y , y |x)
Yn
Decoder 
Decoder 
̂  , M
̂
M
̂  , M
̂
M
∙ DM broadcast channel (BC) (X , p(y , y |x), Y × Y )
∙ (nR , nR , nR , n) code, Pe(n) , achievability, C : Same as before
∙ Superposition coding is optimal for
󳶳
Degraded BC: X → Y → Y
󳶳
Less noisy BC: I(U; Y ) ≥ I(U; Y ) for every p(u, x)
󳶳
More capable BC: I(X; Y ) ≥ I(X; Y ) for every p(x)
󳶳
General BC with degraded message sets (R =  or R = ): Read NIT .
∙ This lecture: Inner bound on private-message capacity region of general BC
 / 
Marton’s inner bound
∙ A simple inner bound: (R , R ) is achievable for the DM-BC p(y , y |x) if
R < I(U ; Y ),
R < I(U ; Y )
for some pmf p(u )p(u ) and function x(u , u )
∙ Marton’s coding scheme: Allows U and U to be dependent
Theorem . (Marton )
(R , R ) is achievable if
R < I(U ; Y ),
R < I(U ; Y ),
R + R < I(U ; Y ) + I(U ; Y ) − I(U ; U )
for some pmf p(u , u ) and function x(u , u )
∙ Optimal for semideterministic BC (Y = y (X )) and Gaussian vector BC
 / 
Semideterministic BC
∙ Semideterministic BC (Y = y (X)): Set U = Y
The capacity region is the set of (R , R ) such that
R ≤ H(Y ),
R ≤ I(U; Y ),
R + R ≤ H(Y |U) + I(U; Y )
for some p(u, x)
∙ Deterministic BC (Y = y (X), Y = y (X)): Further set U = Y
The capacity region is the set of (R , R ) such that
R ≤ H(Y ),
R ≤ H(Y ),
R + R ≤ H(Y , Y )
for some p(x)
 / 
Example: Blackwell channel


Y

Y


X


X



Y


R
R

H(/)
R
/
/
/ /
∙
Proof
of achievability
∙
H󶀣  󶀳

C ()
C ()
 / 
=
= 󶁁(
)
≤ ( )
≤ ( / )
+
≤
Use multicoding and the mutual covering lemma
un (l ) 
R

C ()
̃  −R  )
n(R
C (m )
⋃
( )+
∈[/
/ ]󶁑
C (nR )
̃
n R 
un (l )

̃
n(R −R )
C ()
Tє(n)
󳰀 (U , U )
C (nR )
̃
n R 
 / 
Proof of achievability
∙ Codebook generation: Fix p(u , u ) and x(u , u )
󳶳
󳶳
̃
For each m ∈ [ : nR ] generate a subcodebook C (m ) consisting of n(R −R ) sequences
̃
̃
n
un (l ) ∼ ∏i= pU (ui ), l ∈ [(m − )n(R −R ) +  : m n(R −R ) ]
Similarly, generate C (m ), m ∈ [ : nR ]
C ()
un (l ) 
C ()
C ()
̃
n(R −R )
C (m )
C (nR )
̃
n R 
un (l )

̃
n(R −R )
C ()
C (nR )
̃
n R 
 / 
Proof of achievability
∙ Codebook generation: Fix p(u , u ) and x(u , u )
󳶳
󳶳
󳶳
For each (m , m ), find (l , l ) such that un (l ) ∈ C (m ), un (l ) ∈ C (m ),
(un (l ), un (l )) ∈ Tє(n)
󳰀
If no such pair exists, choose (l , l ) = (, )
Generate xn (m , m ) as xi (m , m ) = x(ui (l ), ui (l )), i ∈ [ : n]
C ()
un (l ) 
C ()

C ()
̃  −R  )
n(R
C (m )
C (nR )
̃
n R 
un (l )

̃
n(R −R )
C ()
Tє(n)
󳰀 (U , U )
C (nR )
̃
n R 
 / 
Proof of achievability
∙ Encoding:
󳶳
To send a message pair (m , m ), transmit xn (m , m )
C ()
un (l ) 
C ()
C ()
̃
n(R −R )
C (m )
C (nR )
̃
n R 
un (l )

̃
n(R −R )
C ()
C (nR )
̃
n R 
 / 
Proof of achievability
∙ Decoding:
󳶳
̂ j such that (unj (lj ), yjn ) ∈ Tє(n) for some unj (lj ) ∈ Cj (m
̂ j)
Decoder j = ,  finds unique m
C ()
un (l ) 
C ()

C ()
̃  −R  )
n(R
C (m )
C (nR )
̃
n R 
un (l )

̃
n(R −R )
C ()
C (nR )
̃
n R 
 / 
Analysis of the probability of error
∙ Consider P(E) conditioned on (M , M ) = (, )
∙ Let (L , L ) denote the pair of chosen indices
∙ Error events for decoder :
for all (Un (l ), Un (l )) ∈ C () × C ()󶁑,
E = 󶁁(Un (l ), Un (l )) ∉ Tє(n)
󳰀
E = 󶁁(Un (L ), Yn ) ∉ Tє(n) 󶁑,
̃
E = 󶁁(Un (l ), Yn ) ∈ Tє(n) (U , Y ) for some l ∉ [ : n(R −R ) ]󶁑
Thus, by the union of events bound,
P(E ) ≤ P(E ) + P(Ec ∩ E ) + P(E )
 / 
Mutual covering lemma (U = )
∙ Let (U , U ) ∼ p(u , u ) and є 󳰀 < є
∙ For j = , , let Ujn (mj ) ∼ ∏ni= pUj (uji ), mj ∈ [ : nrj ], be pairwise independent
∙ Assume that {Un (m )} and {Un (m )} are independent
Un
Un
Un (m )
Un (m )
 / 
Mutual covering lemma (U = )
∙ Let (U , U ) ∼ p(u , u ) and є 󳰀 < є
∙ For j = , , let Ujn (mj ) ∼ ∏ni= pUj (uji ), mj ∈ [ : nrj ], be pairwise independent
∙ Assume that {Un (m )} and {Un (m )} are independent
Lemma . (Mutual covering lemma)
There exists δ(є) →  as є →  such that
lim P󶁁(Un (m ), Un (m )) ∉ Tє(n) for all m ∈ [ : nr ], m ∈ [ : nr ]󶁑 = 
n→∞
if r + r > I(U ; U ) + δ(є)
∙ Proof: See NIT Appendix A
∙ This lemma extends the covering lemma:
󳶳
For a single Un sequence (r = ), r > I(U ; U ) + δ(є) as in the covering lemma
󳶳
Pairwise independence: linear codes for finite field models
 / 
Analysis of the probability of error
∙ Error events for decoder :
E = 󶁁(Un (l ), Un (l )) ∉ Tє(n)
for all (Un (l ), Un (l )) ∈ C () × C ()󶁑,
󳰀
E = 󶁁(Un (L ), Yn ) ∉ Tє(n) 󶁑,
̃
E = 󶁁(Un (l ), Yn ) ∈ Tє(n) (U , Y ) for some l ∉ [ : n(R −R ) ]󶁑
∙ By the mutual covering lemma (with r = R̃  − R and r = R̃  − R ),
̃  − R ) + (R
̃  − R ) > I(U ; U ) + δ(є 󳰀 )
P(E ) →  if (R
∙ Since Ec = {(Un (L ), Un (L ), X n ) ∈ Tє(n)
󳰀 },
by the conditional typicality lemma, P(Ec ∩ E ) → 
∙ By the packing lemma, P(E ) →  if R̃  < I(U ; Y ) − δ(є)
∙ Similarly, P(E ) →  if R̃  < I(U ; Y ) + δ(є)
∙ Using Fourier–Motzkin to eliminate R̃  and R̃  completes the proof
 / 
Relationship to Gelfand–Pinsker
∙ Fix p(u , u ) and x(u , u )
∙ Consider R = I(U ; Y ) − I(U ; U ) − δ(є) and R = I(U ; Y ) − δ(є)
∙ Marton scheme for communicating M is equivalent to G–P for p(y |u , u )p(u )
p(u )
Un
M
Encoder
Un
Xn
x(u , u )
p(y |x)
Yn
Decoder 
̂
M
∙ The corner point (R = I(U ; Y ), R = I(U ; Y ) − I(U ; U )) achieved similarly
∙ The rest of the pentagon region is achieved by time sharing
 / 
Application: Gaussian BC
Z
M
M -encoder
Xn
Yn
Xn
M
M -encoder
Yn
Xn
Z
̄
∙ Decompose X into the sum of independent X ∼ N(, αP) and X ∼ N(, αP)
̄
∙ Send M to Y = X + X + Z : R < C(αP/(αP
+ N )) (treat X as noise)
∙ Send M to Y = X + X + Z : R < C(αP/N ) (writing on dirty paper)
󳶳
Substitute U = X and U = βU + X , β = αP/(αP + N ) in Marton
∙ This coding scheme works even when N > N (unlike superposition coding)
 / 
Gaussian vector broadcast channel
Z
M , M
Encoder
Xn
Yn
G
Yn
G
̂
M
Decoder 
̂
M
Decoder 
Z
∙ Average power constraint: ∑ni= xT(m , m , i)x(m , m , i) ≤ nP
∙ Z , Z ∼ N(, Ir )
∙ Channel is not degraded in general (superposition coding not optimal)
∙ Marton coding (vector writing on dirty paper) is optimal, however
 / 
Capacity region
∙ R : (R , R ) such that
R
|G K GT + G K GT + Ir |

,
R < log

|G K GT + Ir |
R <

log |G K GT + Ir |

R
C
R
for some K , K ⪰  with tr(K + K ) ≤ P
∙ R : (R , R ) such that

R < log |G K GT + Ir |,

R
|G K GT + G K GT + Ir |

R < log

|G K GT + Ir |
for some K , K ⪰  with tr(K + K ) ≤ P
Theorem . (Weingarten–Steinberg–Shamai )
C is the convex closure of R ∪ R
 / 
Proof of achievability for R
Z
M
M -encoder
Xn
X
M
M -encoder
n
Xn
Yn
G
Yn
G
Z
∙ Decompose X into the sum of independent X ∼ N(, K ) and X ∼ N(, K )
∙
|G K GT + G K GT + Ir |

Send M to Y = G X + G X + Z : R < log

|G K GT + Ir |
∙ Send M to Y = G X + G X + Z : R <  log |G K GT + Ir | (vector WDP)

∙ R is achieved similarly
 / 
Marton’s inner bound with common message
Theorem . (Marton , Liang )
(R , R , R ) is achievable if
R + R < I(U , U ; Y ),
R + R < I(U , U ; Y ),
R + R + R < I(U , U ; Y ) + I(U ; Y |U ) − I(U ; U |U ),
R + R + R < I(U ; Y |U ) + I(U , U ; Y ) − I(U ; U |U ),
R + R + R < I(U , U ; Y ) + I(U , U ; Y ) − I(U ; U |U )
for some p(u , u , u ) and function x(u , u , u )
∙ Proof of achievability: Superposition coding + Marton’s coding
∙ Tight for all classes of BCs with known capacity regions
∙ Even for R = , larger than Marton’s inner bound with U =  (Theorem .)
 / 
Nair–El Gamal outer bound
Theorem . (Nair–El Gamal )
If (R , R , R ) is achievable, then
R ≤ min{I(U ; Y ), I(U ; Y )},
R + R ≤ I(U , U ; Y ),
R + R ≤ I(U , U ; Y ),
R + R + R ≤ I(U , U ; Y ) + I(U ; Y |U , U ),
R + R + R ≤ I(U ; Y |U , U ) + I(U , U ; Y )
for some p(u )p(u )p(u |u , u ) and function x(u , u , u )
∙ Tight for all BCs with known capacity regions that we discussed so far
∙ Does not coincide with Marton’s inner bound (Jog–Nair )
∙ Not tight in general (Geng–Gohari–Nair–Yu )
 / 
Summary
∙ Marton’s inner bound:
󳶳
Multidimensional subcodebook generation
󳶳
Generating correlated codewords for independent messages
∙ Mutual covering lemma
∙ Connection between Marton coding and Gelfand–Pinsker coding
∙ Writing on dirty paper achieves the capacity region of the Gaussian vector BC
∙ Nair–El Gamal outer bound
 / 
References
Geng, Y., Gohari, A. A., Nair, C., and Yu, Y. (). The capacity region for two classes of product broadcast
channels. In Proc. IEEE Int. Symp. Inf. Theory, Saint Petersburg, Russia.
Jog, V. and Nair, C. (). An information inequality for the BSSC channel. In Proc. UCSD Inf. Theory Appl.
Workshop, La Jolla, CA.
Liang, Y. (). Multiuser communications with relaying and user cooperation. Ph.D. thesis, University of
Illinois, Urbana-Champaign, IL.
Marton, K. (). A coding theorem for the discrete memoryless broadcast channel. IEEE Trans. Inf.
Theory, (), –.
Nair, C. and El Gamal, A. (). An outer bound to the capacity region of the broadcast channel. IEEE
Trans. Inf. Theory, (), –.
Weingarten, H., Steinberg, Y., and Shamai, S. (). The capacity region of the Gaussian multiple-input
multiple-output broadcast channel. IEEE Trans. Inf. Theory, (), –.
 / 