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Ch4. Capacity of

Wireless Channelsfrom Andrea Goldsmith book

Instructor:

• Mohammed Taha O. El Astal

Introduction

In communications, the capacity of channel dictate the maximum data

rate that can be transmitted over wireless channels with small error

probability, assuming no constraints on delay or complexity of the encoder

and decoder.

Wireless Channel

4.1 Capacity in AWGN channel

Shannon's coding theorem proves that a code exist that achieve R arbitrarily

close to C with small BER.

The converse theorem shows that any code with R>C has a BER bounded

away from zero.

The proofs of the coding theorem and converse place no constraints on the

complexity or delay of the system, so it use as upper bound of R that can be

achieved in real communications systems.

+AWGN channel

n[i]

x[i] y[i]

where :

•B is the channel BW in Hz.

•P is the transmitted power in watt.

•C is the capacity of channel in bps.

CONT.

At the time that Shannon developed his theory of information:

•R over standard telephone lines =100 bps.

•By Shannon, R = 30 kbps

•It was not a useful bound for real systems.

However, breakthroughs in hardware, modulation,

and coding techniques have brought commercial

modems of today very close to the speeds predicted by Shannon in the

1940s.

In fact, current modems can exceed this 30-kbps limit, why ?

CONT.

EXAMPLE 4.1:

Consider a wireless channel where power falloff with distance follows the formula Pr(d )

= Pt(d0/d )3 for d0 = 10 m. Assume the channel has bandwidth B = 30 kHz and AWGN

with noise PSD N0/2, where N0 = 10−9 W/Hz. For a transmit power of 1W, find the

capacity of this channel for a transmit–receive distance of 100 m and 1 km.

SNR1=33=15dB

C1=152.6Kbit/sec

SNR2=33=15dB

C2=1.4Kbit/sec

Note the significant decrease in capacity at greater distances due to the path-loss

exponent of 3, which greatly reduces received power as distance increases.

4.2 Capacity of Flat Fading Channels :

The channel gain g[i] follows a given distribution p(g).

g[i] is independent of the channel input.

g[i] can change at each time i, either as i.i.d process or with some correlation

over time.

In a block fading channel, g[i] is constant over some block length T, after which

time g[i] changes to a new independent value based on the distribution p(g).

Let Ṗ denote the average Tx. power, N0 is PSD of the n[i], and B is the received

BW, then determine SNR at Rx?? and the average SNR at Rx.??

CONT.

Since Ṗ/N0B is a constant term , then the distribution of g[i]

determines the distribution of [i] and vice versa.

The channel gain g[i], also called CSI (Channel Side Information).

So, the Capacity of flat fading channel , depend on what known in TX

and RX :

1. Channel distribution Information (CDI) at Tx. & Rx.

2. Receiver CSI.

3. Tx. & Rx. CSI.

sim

ple

desig

n s

yste

m

mo

re C

effic

ient

4.2.2 CDI known:

The capacity is given by:

is a quite complicated depending on the nature

of the fading distribution.

Moreover, fading correlation introduces

channel memory, and this makes finding the

solution even more difficult.

For these reasons, this case remains an open

problem for almost all channel distributions.

4.2.3 CSI at Rx:

This case of the CSI (g[i]) is known at Rx. and also we assume that both

the Tx. and Rx. know the distribution of g[i].

Here, two capacity definition arise :

1. Shannon capacity .

which define the maximum data rate that

can be transmitted over the channel with

small error probability .

In this case, the Tx. must deal with the poor

channel states which greatly reduce the Shannon Capacity.

Also, this case have a constant rate since the Tx. cannot adapt its strategy

relative to CSI.

good Excellent b V. good

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sec

time

C

CONT.

2. Capacity with outage :

is defined as the maximum rate that can

be transmitted over the channel with

some outage probability corresponding

to the probability that the transmission

can not be decoded with negligible error

probability.

In this case, a high data rate can be sent over the channel and decoded

correctly except when the channel is in a deep fade.

The probability of outage characterizes the probability of data loss or

equivalently of deep fading.

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C

pout

pout

Shannon “Ergodic” Capacity:

Shannon capacity of a fading channel with receiver CSI for an average power

constraint Ṗ can be obtained by :

In particular, it is incorrect to interpret it as the average capacity of the

instantaneous SNR, since only the receiver knows the instantaneous SNR.

The data rate transmitted over the channel is constant, regardless of γ.

Shannon capacity of a fading channel with receiver CSI only is less than the

Shannon capacity of an AWGN channel with the same average SNR.

The Fading will reducing the Shannon Capacity when only the Rx. has the CSI.

CONT.

EXAMPLE 4.2:

Consider a flat fading channel with i.i.d. channel gain √g[i], which can take on

three possible values: √g1 = .05 with probability p1 = .1, √g2 = .5 with

probability p2 = .5, and √g3 = 1 with probability p3 = .4. The transmit power is

10 mW, the noise power spectral density N0/2 has N0 = 10−9 W/Hz, and the

channel bandwidth is 30 kHz.

Assume the receiver has knowledge of the instantaneous value of g[i] but the

transmitter does not. Find the Shannon capacity of this channel and compare

with the capacity of an AWGN channel with the same average SNR.

SNR1=.8333=-.79dB

SNR2=83.333=19.2dB

SNR3=333.33=25dB

C=199.22Kbps

average SNR=175.08=22.43dBC=223.8kbps

Note that this rate is about 25 kbps larger than that of the flat fading channel with

receiver CSI and the same average SNR.

Capacity with outage :

Capacity with outage applies to slowly varying channels.

Since TX. does not know γ value, it must fix a R (corresponding γmin)

to independent of the instantaneous received SNR.

The probability of outage is thus Pout = p (γ < γmin).

The average rate correctly received is Cout = (1−Pout)Blog2(1+ γmin)

since data is only correctly received on 1 − Pout transmissions.

specify γmin

Tx will transmit using

C correspond to γmin

At Transmitter : At Receiver :

γ

declare

an

outage

Everyth

ing is

fine

CONT.

The value of γmin is a design parameter.

Capacity with outage is typically

characterized by a plot of capacity

versus outage.

Here, the normalized capacity C/B as a

function of outage probability Pout for a

Rayleigh fading channel is shown.

In general :

γmin ++pout++C++

γmin --pout--C--

CONT.

for Pout < .1C= 26.23 kbps.C0=26.23Kbps

For .1 ≤ Pout < .6C=191.94kbps C0= 172.75 kbps.

For .6 ≤ Pout < 1C=251.55kbps C0= 125.78 kbps.

EXAMPLE 4.3:

Assume the same channel as in the previous example, with a bandwidth of 30

kHz and three possible received SNRs: γ1 = .8333 with p(γ1) = .1, γ2 = 83.33

with p(γ2 ) = .5, and γ3 = 333.33 with p(γ3) = .4.

Find the capacity versus outage for this channel, and find the average rate

correctly received for outage probabilities Pout < .1, Pout = .1, and Pout = .6.

4.2.4 CSI at TX. & RX. :

When both the Tx. & Rx. have CSI, the

Tx. can adapt its strategy relative to this

CSI.

Since the Tx. knows the channel , there

is no need to send bit unless the RX. can

decode it correctly. good Excellent b V. good

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C

CONT.

This case originally considered by Wolfowitz.

The derivation leads to the following results:

the optimal power adaption that maximize C will be as follow :

for some cutoff value

the cutoff value γ0 must satisfy

If γ [i] is below this cutoff then no data is transmitted over the ith time

interval, so the channel is used at time i only if γ0 ≤ γ [i] < ∞.

The capacity will be by :

CONT.

Note that this multiplexing strategy is not only the way to achieve that

capacity ,it can also be achieved by adapting the Tx. power and sending

at fix rate

Zero-outage capacity & channel Inversion :

Alternatively, do inversion for the fading

effect to have a fixed Rx. SNR (also fixed R).

Use at the transmitter, where

σ is the required constant Rx. SNR.

The channel then appears to the encoder

and decoder as a time-invariant AWGN

channel.

this value σ must satisfy :

So

and

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C

Additional capacity due

to additional Tx. power.

CONT.

It is called zero-outage capacity, since the data rate is fixed under all

channel conditions and there is no channel outage.

It has the advantage of maintaining a fixed data rate over the channel

regardless of channel conditions.

It exhibit a large data-rate reduction relative to Shannon capacity in

extreme fading environments.

for example , in Rayleigh fading,, E[1/γ ] is infinite and thus the zero-outage

capacity is zero

CONT.

E[1/SNR]=.1272C=94.43 kbps.

Note that this is less than half of the Shannon capacity with optimal water-filling

adaptation.

EXAMPLE 4.5:

Assume the same channel as in the previous example, with a bandwidth of 30

kHz and three possible received SNRs: γ1 = .8333 with p(γ1) = .1, γ2 = 83.33

with p(γ2 ) = .5, and γ3 = 333.33 with p(γ3) = .4.

Assuming transmitter and receiver CSI, find the zero-outage capacity of this

channel.

Outage Capacity and Truncated Ch. Inversion:

zero-outage capacity

must maintains a

constant R in all fading

states

this cause

zero-outage capacity

significantly smaller

than Shannon capacity

states

The problem

By suspending

transmission in bad

fading states ,so we

maintain a higher

constant R in the

other states and

thereby significantly

increase capacity

The solution

outage capacity (through

truncated channel

inversion)

CONT.

The outage Capacity is defined as the maximum R that can be

maintained in all non outage channel state times.

Outage capacity is achieved with truncated channel inversion policy

for power adaption that only compensate for fading above a certain

cutoff fade depth γ0.

The outage capacity associated with a given outage probability Pout

and corresponding cutoff γ0 is given by

CONT.

E[1/SNR]= .0072C=192.457 kbps.E[1/SNR]=.0012C= 116.45 kbps.

The outage capacity is larger when the channel is used for SNRs γ2 and γ3. Even

though the γ3 is significantly larger than γ2 , the fact that this larger SNR occursonly 40% of the time makes it inefficient to only use the channel in this best state.

EXAMPLE 4.6:

Assume the same channel as in the previous example, with a bandwidth of 30

kHz and three possible received SNRs: γ1 = .8333 with p(γ1) = .1, γ2 = 83.33

with p(γ2 ) = .5, and γ3 = 333.33 with p(γ3) = .4.

Find the outage capacity of this channel and associated outage probabilities for

cutoff values γ0 = .84 and γ0 = 83.4. Which of these cutoff values yields a larger

outage capacity?

4.2.5 Capacity with receiver diversity :

The prop. effects lead to depolarization .

Thus, receiving both polarizations using a dual-polarized antenna, and

processing the signals separately, offers diversity.

But the average Rx. signal strength in the two diversity branches is not

identical, this lead to decrease the effectiveness this scheme.

Various antenna arrangements have been proposed in order to mitigate

this problem.

4.3 Capacity of frequency selective fading channels :

When the channel is time invariant , it is

typically assumed that H(f) is known at

both the Tx. and Rx.

Here, we will assume that the fading is

block fading, so the channel will appear as

a set of AWGN channel in parallel with

SNR : in the jth channel

C of F.S. fading Ch.

C of time varying-F.S.

fading ch.C of TI-F.S. fading ch.

CONT.

it is same as the case of water filling but with frequency instead of

time, so :

for some cutoff value γ0 which must satisfy :

Then the capacity become as follow :

CONT.

γ1 = 10, γ2 = 40, and γ3 = 90.γ0 = 2.64C= 10.93 Mbps.

EXAMPLE 4.7:

Consider a time-invariant frequency-selective block fading channel that has

three sub channels of bandwidth B = 1 MHz. The frequency responses

associated with each sub channel are H1 = 1, H2 = 2, and H3 = 3, respectively.

The transmit power constraint is P = 10 mW and the noise PSD N0/2 has N0 =

10−9 W/Hz.

Find the Shannon capacity of this channel and the optimal power allocation that

achieves this capacity.

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