Chapter # 4 : Capacity of Wireless Channelssite.iugaza.edu.ps/musbahshaat/files/master_chapter_4...Capacity of Flat-Fading Channels Channel Side Information at Receiver/Shannon (Ergodic)

EELE 6333: Wireless Commuications

Chapter # 4 : Capacity of Wireless Channels

Spring, 2012/2013

EELE 6333: Wireless Commuications - Ch.4 Dr. Musbah Shaat 1 / 18

Outline

1 Capacity in AWGN

2 Capacity of Flat-Fading Channels

3 Capacity of Frequency-Selective Fading Channels


Capacity in AWGN ... 1

System Capacity:

The maximum data rates that can be transmitted over wireless channelswith asymptotically small error probability, assuming no constraints ondelay or complexity of the encoder and decoder.

Consider a discrete-time additive white Gaussian noise (AWGN)channel with channel input/output relationship y [i ] = x [i ] + n[i ].

x [i ] is the channel input at time i .y [i ] is the corresponding channel output.n[i ] is a white Gaussian noise random process.

The capacity of this channel is given by Shannon’s well-knownformula

C = B log2(1 + γ) bits/second (bps)

B is the channel bandwidth.γ is the channel SNR, the ratio between the transmitted power P andthe power of the noise, i.e. γ = P/(N0B) where N0 is the powerspectral density of the noise.


Capacity in AWGN ... 2

Shannon’s coding theorem proves that a code exists that achievesdata rates arbitrarily close to capacity with arbitrarily smallprobability of bit error.

The converse theorem shows that any code with rate R > C has aprobability of error bounded away from zero.

Shannon capacity is generally used as an upper bound on the datarates that can be achieved under real system constraints.

On AWGN radio channels, turbo codes have come within a fractionof a dB of the Shannon capacity limit.


Capacity of Flat-Fading ChannelsChannel and System Model ... 1

Assume a discrete-time channel with stationary and ergodic (itsstatistical properties (such as its mean and variance) can be deducedfrom a single, sufficiently long sample (realization) of the process)time-varying gain

√gi and AWGN n[i ].

The channel power gain g [i ] follows a given distribution p(g), e.g.for Rayleigh fading p(g) is exponential.

In a block fading channel, g [i ] is constant over some blocklengthT after which time g[i] changes to a new independent value basedon the distribution p(g).

Let P̄ denote the average transmit signal power, N0/2 denote thenoise power spectral density of n[i ], and B denote the receivedsignal bandwidth.

The instantaneous received signal-to-noise ratio (SNR): γ[i ] = P̄g [i ]N0B

.

The distribution of g [i ] determines the distribution of γ[i ] and viceversa.


Capacity of Flat-Fading ChannelsChannel and System Model ... 2

The channel gain g [i ], also called the channel side information (CSI).

The capacity of this channel depends on what is known about g [i ]at the transmitter and receiver.

Channel Distribution Information (CDI): The distribution of g [i ]is known to the transmitter and receiver.Receiver CSI: The value of g [i ] is known at the receiver at time i ,and both the transmitter and receiver know the distribution of g [i ].Transmitter and Receiver CSI: The value of g [i ] is known at thetransmitter and receiver at time i , and both the transmitter andreceiver know the distribution of g [i ].


Capacity of Flat-Fading ChannelsChannel Side Information at Receiver ... 1

Consider the case where the CSI g [i ] is known at the receiver attime i ⇒ γ[i ] is known at the receiver at time i .

Also assume that both the transmitter and receiver know thedistribution of g [i ].

In this case there are two channel capacity definitions that arerelevant to system design: Shannon capacity, also called ergodiccapacity, and capacity with outage.

Capacity with outage is defined as the maximum rate that can betransmitted over a channel with some outage probabilitycorresponding to the probability that the transmission cannot bedecoded with negligible error probability.

The probability of outage characterizes the probability of data lossor, equivalently, of deep fading.


Capacity of Flat-Fading ChannelsChannel Side Information at Receiver/Shannon (Ergodic) Capacity ... 1

Shannon capacity is equal to Shannon capacity for an AWGNchannel with SNR γ, given by Blog2(1 + γ), averaged over thedistribution of γ (probabilistic average).

Since the probabilistic average E [x ] is given byE [x ] =

∫∞−∞ xp(x)dx , hence,

C =∫∞

0 B log2(1 + γ)p(γ)dγ

By Jensens inequality E (ϕ(x)) ≤ ϕ(E (x)), Hence,

E (B log2(1 + γ)) ≤ B log2(1 + E (γ)) = B log2(1 + γ̄)

where γ̄ is the average SNR on the channel.

The Shannon capacity of a fading channel with receiver CSI only isless than the Shannon capacity of an AWGN channel with the sameaverage SNR.

Fading reduces Shannon capacity when only the receiver has CSI.


Capacity of Flat-Fading ChannelsChannel Side Information at Receiver/Shannon (Ergodic) Capacity ... 2

Ex. 4.2: Consider a flat-fading channel with i.i.d. channel gain g [i ] which cantake on three possible values: g1 = .05 with probability p1 = .1, g2 = .5 withprobability p2 = .5, and g3 = 1 with probability p3 = .4. The transmit power is10 mW, the noise spectral density is N0 = 10−9 W/Hz, and the channelbandwidth is 30 kHz. Assume the receiver has knowledge of the instantaneousvalue of g [i ] but the transmitter does not. Find the Shannon capacity of thischannel and compare with the capacity of an AWGN channel with the sameaverage SNR.

The channel has 3 possible received SNRsγ1 = Ptg1/(N0B) = (0.01× (0.05)2)/(30000× 10−9) = 0.8333By the same way: γ2 = 83.333 and γ3 = 333.33The Shannon capacity is given byC =

∑3i=1 B log2(1 + γi )p(γi ) = 199.26 Kbps

The average SNR for this channel isγ = .1(.8333) + .5(83.33) + .4(333.33) = 175.08The capacity of an AWGN channel with this SNR isC = B log2(1 + 175.08) = 223.8 Kbps


Capacity of Flat-Fading ChannelsChannel Side Information at Receiver/Capacity with Outage

Capacity with outage allows bits sent over a given transmissionburst to be decoded at the end of the burst with some probabilitythat these bits will be decoded incorrectly.

The transmitter fixes a minimum received SNR γmin and encodes fora data rate C = B log2(1 + γmin).

The data is correctly received if the instantaneous received SNR isgreater than or equal to γmin.

If the received SNR is below γmin then the bits received over thattransmission burst cannot be decoded correctly with probabilityapproaching one, and the receiver declares an outage.

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

The average rate correctly received over many transmission bursts isCo = (1− pout)B log2(1 + γmin) since data is only correctly receivedon 1− pout transmissions.

The value of γmin is a design parameter based on the acceptableoutage probability.


Capacity of Flat-Fading ChannelsChannel Side Information at Transmitter and Receiver

When both the transmitter and receiver have CSI, the transmittercan adapt its transmission strategy relative to this CSI.

Since the transmitter knows the channel and thus will not send bitsunless they can be decoded correctly.

The transmitter side information does not increase capacity unlesspower is also adapted.

The maximizing power adaptation policy under the average powerconstraint is a ”water-filling”. WHAT!!


Capacity of Flat-Fading ChannelsChannel Side Information at Transmitter and Receiver ... 2

Zero-Outage Capacity and Channel Inversion

The transmitter can use the CSI to maintain a constant receivedpower, i.e. it inverts the channel fading.

The channel then appears to the encoder and decoder as atime-invariant AWGN channel.

This power adaptation is called channel inversion.

Fading channel capacity with channel inversion is just the capacityof an AWGN channel with constant SNR σ

C = B log2[1 + σ]

What is the advantage and disadvantage of this scheme?.

The channel capacity of this scheme is called zero-outagecapacity, since the data rate is fixed under all channel conditionsand there is no channel outage.

Truncated channel inversion: can be achieved by suspendingtransmission in particularly bad fading states.


Capacity of Flat-Fading ChannelsCapacity Comparisons


Capacity of Frequency-Selective Fading ChannelsTime-Invariant Channels ... 1

Consider a time-invariant channel with frequency response H(f ) andassume a total transmit power constraint P.

Assume that H(f ) is block-fading, so that frequency is divided intosubchannels of bandwidth B, where H(f ) = Hj is constant overeach block.

The frequency-selective fading channel thus consists of a set ofAWGN channels in parallel with SNR |Hj |2Pj/(N0B) on the j th

channel, where Pj is the power allocated to the j th channel.



The capacity of this parallel set of channels is the sum of ratesassociated with each channel with power optimally allocated over allchannels

C =∑

maxPj :∑

j Pj<P

B log2

(1 +

|Hi |2Pj

N0B

)The optimal power allocation is found via the same Lagrangiantechnique used in the flat-fading case, which leads to thewater-filling power allocation.

Pj =[λ− N0B

|Hi |2

]+

λ = 1K0

[PT +

∑K0j=1

N0B|Hi |2

]



Ex. 4.7: Consider a time-invariant frequency-selective block fading channelconsisting of three subchannels of bandwidth B = 1 MHz. The frequencyresponse associated with each channel is H1 = 1,H2 = 2 and H3 = 3. Thetransmit power constraint is P = 10 mW and the noise PSD is N0 = 10−9

W/Hz. Find the Shannon capacity of this channel and the optimal powerallocation that achieves this capacity.

Assume that all the channel are working,

λ = 13

[0.01 +

∑3j=1

1×10−3

|Hi |2

]= 0.0037867

P1 = [0.003787− 1× 10−3] = 2.787 mW, P2 = 3.538 mW, andP3 = 3.675 mW

C = 10−6 [log2(1 + 2.786) + log2(1 + 3.536× 4) + log2(1 + 2.786× 9)]

C = 10.93 Mbps

Homework: Repeat the Ex. with P = 4 W, |Hj | = {1.64, 2.02, 1.22, 0.3},B = 1 MHz and N0 = 0.4× 10−6 and check your answer by a computersimulation


Capacity of Frequency-Selective Fading ChannelsTime-Varying Channels

It is difficult to determine the capacity of time-varyingfrequency-selective fading channels, even when the instantaneouschannel H(f , i) is known perfectly at the transmitter and receiver.

We can approximate channel capacity in time-varyingfrequency-selective fading by taking the channel bandwidth B ofinterest and divide it up into subchannels the size of the channelcoherence bandwidth Bc . Then, the waterfilling is the optimalsolution.


Homework

The homework assignment will be available tomorrow’s night on thecourse webpage. The homework is due in one week.


Documents

Chapter # 4 : Capacity of Wireless Channelssite.iugaza.edu.ps/musbahshaat/files/master_chapter_4...Capacity of Flat-Fading Channels Channel Side Information at Receiver/Shannon (Ergodic)