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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
50kb
/sec
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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.
good Excellent b V. good
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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
good Excellent b V. good time
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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