Capacity of Wireless Channels

(1)

Capacity of Wireless Channels

Wha Sook Jeon

Mobile Computing and Communications Lab.

(2)

Introduction

 Channel capacity limit

−

The maximum channel rates that can be transmitted over the wireless channel with asymptotically small error probability, assuming no constraints on the delay or complexity of the encoder/decoder

 Scope of this chapter

−

Capacity of a single-user wireless channel where the transmitter and/or receiver has a single antenna

■ a time-invariant additive white Gaussian Noise (AWGN) channel

■ a flat fading channel

■ a frequency selective fading channel

(3)

Capacity of AWGN Channel

(4)

 Shannon Capacity

−

■ Received signal-to-noise ratio (SNR)

■ P : the transmitted signal power

■ Nose power: 2 x two-sided noise PSD (N₀/2) x B or one-sided PSD (N₀) x B

−

Upper bound on the data rates that can be achieved under the real system constraints

−

On AWGN radio channel, turbo codes have come within a fraction of a decibel of Shannon capacity limit

) 1

(

log

₂

+ γ

= B C

B N P/ ₀ γ =

Capacity in AWGN

(5)

Wireless Networks Lab. 4

Capacity of discrete time-invariant channel

 Mutual information

−

The average amount of information received over the channel per symbol

−

■ H(X): the average amount of information transmitted per symbol (entropy)

■ H(X|Y): the average uncertainty about a transmitted symbol when a symbol is received, and the average

amount of information lost over noisy channel per symbol

■

−

Y)

| H(X H(X)

Y)

I(X; = −

)

| ( log 1 ) , ( Y)

| H(X ),

( log 1 ) ( H(X)

, p x y p x y

x x p

p

Y X

X x S y S

S

x

∑ ∑

∈

=

)

| ( log 1 ) , ) (

( log 1 ) ( Y)

I(X;

, p x y p x y

x x p

p

Y X

X x S y S

S

x

∑ ∑

∈

−

=

(6)

Channel Capacity of a Continuous Channel

 Entropy of X:

 Mutual Information I(X;Y)

x dx x p

p ( )

log 1 ) ( H(X) =

∫

−^∞∞

X)

| H(Y H(Y)

)

| ( log 1 ) , ) (

( log 1 ) (

)

| ( log 1 ) ( log 1 ) , (

) (

)

| log (

) , ( )

( ) (

) , log (

) , (

) (

)

| log (

) , ( )

| ( log 1 ) ( log 1 ) , (

)

| ( log 1 ) , ) (

( log 1 ) ( Y)

I(X;

−

=

−

=











 −

=

 =











 −

=

−

=

∫

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

∞

−

x dxdy y

y p x p y dy

y p p

x dxdy y

p y

y p x p

y dydx p

x y y p

x p y dxdy

p x p

y x y p

x p

x dxdy p

y x y p

x p y dxdy

x p x

y p x p

y dxdy x

y p x p x dx

x p p

(7)

Channel Capacity of a Continuous Channel

 Entropy of Z:

 Maximum entropy of Z, for a given E[Z

²

]

−

The maximum entropy is obtained when the distribution of Z is Gaussian

■

z dz z p

p ( )

log 1 ) ( H(Z) =

∫

−^∞∞

) ( where

2 , ) 1

( ² ²

2 2

2

∫

−^∞∞

− =

= e z p z dz

z

p σ π ^z σ

σ

) 2

2 log(

(Z) 1

H ²

max = πeσ

(8)

Capacity of a Band-limited AWGN Channel (1)



Channel capacity

−

Maximum amount of mutual information I(X;Y) per second

−

^{Two steps}

■ the maximum mutual information per sample

■ 2B samples (Nyquist’s sampling theory)



Maximum mutual information per sample

−

x, n, y: samples of the transmitted signal, noise, and received signal

−

^H(y|x)

■

■ Because y=x+n, for a given x, y is equal to n plus a constant. The distribution of y is identical to that of n except for a translation by x

■ ( | ) = ( − ), where (⋅)is thePDFof noisesample

n

n y x p

p x

y p

x dydx y

x p y p x

p

x dxdy y

y p x p

)

| ( log 1 )

| ( )

(

)

| ( log 1 ) , ( x)

| H(y

∫

∞

−

∞

−

∞

−

∞

−

=

(9)

Capacity of a Band-limited AWGN Channel (2)

−

I(x;y)=H(y) - H(n)

 Entropy of a band-limited white Gaussian noise with PSD N

₀

/2

−

Noise power: N= N₀B

−

 When the signal power is S and the noise power is N, and the signal s(t) and noise n(t) are independent, the mean square value of y is

) 2

2 log(

H(n) = 1 πeN

) n ) (

( log 1 ) (

) ( log 1 ) ) (

| ( log 1 )

| (

H u du

u p p

x dy y x p

y p x dy

y x p

y p

n n n n

=

− −

=

∫

∞

−

∞

−

∞

−

H(n) )

( H(n)

) ( H(n)

)

| ( log 1 )

| ( )

( x)

| H(y

=

∫

∞

−

∞

−

∞

−

∞

−

dx x p dx

x p

x dydx y x p

y p x

p

N S E[y²]= +

(10)

Capacity of a Band-limited AWGN Channel (3)

−

H(y) will be maximum if y is Gaussian

−



 Channel capacity:

 Reference :B. P. Lathi, Modern Digital and Analog Communication System, 3

^rd

Ed., Oxford. (Chapter 15)

) 1

log( N

B S

C = +

)]

( 2 2 log[

(y) 1

Hmax = πe S + N

y) (x;

I 2 × B × max

) 1

2 log(

1

) 2

2 log(

)] 1 (

2 2 log[

1

H(n) (y)

H y)

(x;

Imax max

S N

eN N

S e +

=

− +

=

−

=

π π

(11)

Capacity of Flat-Fading Channels

(12)

Capacity of Flat-Fading Channels

 The channel capacity depends on the information about g[i]

−

Channel distribution information (CDI) of g[i] known to the transmitter and receiver (**) and Channel side information (the value of g[i]) known to the receiver

−

CSI known to the receiver and transmitter, and (**)

(13)



The rate transmitted over the channel is constant (The transmitter cannot adapt its transmission strategy relative to the CSI)



Shannon (ergodic) capacity

−

Shannon capacity for AWGN channel averaged over the distribution of

−

Capacity-achieving code must be sufficiently long that a

received codeword is affected by all possible fading states =>

This can result in significant delay

−

If the receiver CSI is not perfect, capacity can be significantly decreased

[

^log ⁽¹ ⁾

]

^log ⁽¹ ^E[ ^])

E )

( ) 1 (

log ₂ ₂

0 2 +γ γ γ = +γ ≤ + γ

∫

^∞ ^B ^p ^d ^B ^B

γ

∫

^∞ ⁺

= 0 Blog2(1 γ )p(γ)dγ C

Shannon capacity of AWGN channel with the same average SNR

CSI at Receiver (1)

(14)

CSI at Receiver (2)

 Capacity with outage

−

The transmitter fixes a minimum received SNR and encodes for a fixed data rate

−

For the received SNR below , the received bits cannot be decoded correctly (outage)

−

Outage probability:

−

^Capacity:

−

The outage probability is a design parameter

γmin

) 1

(

log₂ +γ_min

= B C

γmin

) (γ <γ_min

= p P_out

) 1

( log )

1

( − ₂ +γ_min

= P B

C_out _out

(15)

CSI at Receiver (3)

dB 20 ] [ E γ =

Rayleigh fading channel with

(16)

CSI at Transmitter and Receiver (1)

 For fixed transmission power, the same capacity as when only receiver knows fading

∫

^∞ ⁺

= 0 Blog2(1 γ )p(γ)dγ C

(17)

CSI at Transmitter and Receiver (2)

 Transmission power as well as rate can be adapted.

 Adaptation of transmission power to the received SNR subject to an average power constraint

 average power constraint:

 The fading channel capacity with average power constraint

) (γ Pt

∫

0^∞P_t(γ )p(γ)dγ ^≤ ^Φ

γ

_Φ

∫

^∞

Φ

=





 



 + Φ

= ∫

0 2

) ( ) ( : ) (

) ) (

1 ( log

max γ γ γ γ

γ γ γ

γ

P p d

B

C

^t

d p P

P_t _t

* :

γ

the received SNR at transmission power Φ

(18)

 The range of fading values is quantized to a finite set

 For each , an encoder-decoder pair for an AWGN channel with SNR

 The input x_j for encoder has average power and data rate C_j where

C_j is the capacity of time-invariant AWGN channel with received SNR Pt(γ j)γ j ^Φ

A time diversity system

} 1

:

{γ _j ≤ j ≤ N γ j

) ( _j Pt γ

γ j

(19)

CSI at Transmitter and Receiver (4)

 Optimal power allocation

−

Lagrangian

−

Differentiate the Lagrangian and set the derivate to zero

−

Solve for with the constraint that



 −Φ

 +



 



 + Φ

=

∫

0^∞ 2 ( ) ( )

∫

0^∞ ( ) ( )

1 log )

), (

( γ λ P γ γ p γ dγ λ P γ p γ dγ

B P

J _t ^t _t

0 ) ) (

( 1

2 ln / )

( ) ), (

(  =



 



 −

Φ



 





Φ

= +

∂

∂ γ λ γ

γ γ γ

λ

γ p

P B P

P J

t t

t





<

≥

= −

Φ ₀

0 0

0 1 ) 1

(

γ γ

γ γ γ γ γ

Pt

) (γ

Pt P_t(γ) > 0

(20)

 Capacity

− Time-varying data rate : the rate corresponding to the instantaneous SNR is

− Transmission power adaption

■

Optimal power allocation (Water filling)

∫

^∞ ^_^ ^_^

=

0

) ( log

0

γ 2 γ γ

γ

γ p d

B C

(

₀

)

log2 γ γ

γ

B





<

≥

= −

Φ ₀

0 0

0 1 ) 1

(

γ γ

γ γ γ γ γ

Pt

(21)

 Water filling

Φ ) (

γ

Pt

the better channel, the more power and the higher data rate

cutoff value

1 )

1 ( such that 1

0 0

0  =



 



 −

∫

γ^∞ γ γ γ γ

γ p d

(22)

 Channel inversion and zero outage

−

The transmitter controls the transmission power using CSI so as to maintain a constant received power (inverts the channel fading)

−

The channel appears to the encoder and decoder as a time- invariant AWGN channel

−

transmission power:

■

−

Fading channel capacity with channel inversion is equal to the AWGN channel capacity with SNR

)

(γ Φ =σ γ Pt

∫

^∞ ⁼

= from 0 ( ) ( ) 1 ]

1

1E[ γ σ γ γ γ

σ p d



 



 +

= +

= E[1 ]

1 1 log )

1 (

log₂ ₂

σ B γ B

C

σ

(23)

 Channel inversion and zero outage

−

A fixed data rate regardless of channel condition

−

Encoder and decoder are designed for an AWGN channel with SNR => the simplest scheme to implement

−

zero outage:

■ Should maintain a constant data rate in all fading states

■ Zero outage capacity is significantly smaller than Shannon capacity on fading channel

−

In Rayleigh fading, the zero outage capacity is zero

−

Channel inversion is common in spread-spectrum system with near-far interference imbalances

σ

(24)

 Truncated channel inversion

−

Suspending transmission in bad fading states

−

Truncated channel inversion

■ Power adaptation policy that compensates only for fading above a cutoff

■

■ Outage capacity for a given and corresponding cutoff

−

■ Maximum outage capacity

−

( )

¹ ¹

0 0 1 ( )

] 1 [ E

where

∞ −

− 



 



=

= γ

∫

γ γ γ

γ γ

σ p d

) ] (

1 [ E 1 1 log )

( ₂ ₀

0

γ γ γ

γ

 ≥











 +

= B p

P C _out

γ0





<

= ≥

Φ ₀

0

0 )

(

γ γ

γ γ γ γ σ

Pt

) (

y probabilit

Outage P_out = p γ <γ₀

γ0

Pout

) ] (

1 [ E 1 1 log

max ₂ ₀

0 0

γ γ γ

γ γ  ≥









 +

= B p

C

(25)

Capacity Comparison

(26)

Capacity of

frequency-selective fading channel

(27)

Time-invariant Channel (1)

 Total power constraint: P

 A time-invariant channel with frequency response H(f) that is known to both transmitter and receiver

 Block fading

−

Frequency is divided into

subchannels of bandwidth B with constant frequency response H_j over each subchannel

−

P_j: Tx power on the jth subchannel

−

A set of AWGN channels in parallel with SNR (|H_j|²P_j/N₀B) on the jth channel

−

Power constraint:

∑

j Pj ^≤ P

 Analysis is like that of flat fading channel with frequency axis

(28)

 Capacity under block fading

−















 +

=

∑

≤ N B

P B H

C ^j ^j

P P

Pj j j 0

2

2 :

max

1 log





<

≥

= −

0 0 0

0 1 1

γ γ

γ γ γ γ

j j j j

P P

1 1 such that 1

0 0  =



 



 −

∑

j γ γ j

γ



 



=

∑



≥ 0

2 :

log

0 γ

γ

γ γ

j j

B C

j

Water-filling P

P_j B

N P H_j

j

0 2

γ =

(29)

 Continuous H(f)

−

N df f P f C H

P df f P f

P 









 +

= ∫ _≤

∫

0 2 ) 2

( : ) (

) ( ) 1 (

log max





<

≥

= −

0 0 0

) ( 0

) ( ) ( 1 ) 1

(

γ γ

f f f

P f P

) 1 1 (

such that 1

) 0 ( : 0

0

 =





 −

∫

≥

f df

f

f γ γ

γ

γ γ 0

)2

) (

( N

P f f = H

γ

f df C

f f



 



=

∫



≥ 0

2 )

( :

) log (

0 γ

γ

γ γ

(30)

Time-varying Channel (1)

Channel division

Time-varying flat fading channel

(31)

Time-varying Channel (2)







<

≥

= −

Φ 0

0 0

0 1 ) 1

(

γ γ

γ γ γ γ γ

j j j j

Pj

1 )

1 ( such that 1

0 0

0

 =



 



 −

∑∫

^∞ _j ^j ^j

j

d p γ γ γ

γ γ

γ

:

0 2

c j

j N B

H Φ γ =

j j j

c j

d p

B

C γ γ

γ γ

γ log ( )

0 2

0 



 



=

∑ ∫

^∞ 

∑ ∫

^∞

Φ

≤ 

 



 + Φ

= ∑ ∫

0 2

) ( ) ( :

) (

) ) (

1 ( log

max _c ^j ^j ^j _j _j

d j p P

P

d P p

B C

j j j j j

j j

γ γ γ

γ

γ γ γ γ

the instantaneous SNR on the jth subchannel

assuming the total power Φ is allocated to the frequency