UNIT II – DIVERSITY

CU7201 WIRELESS COMMUNICATION NETWORKS

Why Wireless?

•Mobility Anytime, Anywhere connectivity

Phone for people not for places

•Easy Installation Rapid deployment, reconfigurable, no cable, easy maintenance

•Digital Companion Voice, message, internet, multimedia

Hi! I’m on the prowl

Fading???

•Fading over time

•Fading over distance

Radio Wave Propagation

Street Sign

STOP

Line of Sight Reflection DiffractionScattering

TransmitterReceiver

Buildings

Earth surface

•Multiple replica of signal combines with random phase resulting in random amplitude attenuation and phase variation This is Fading

•Relative velocity causes further change with time.

Fading Envelope

Distance

Am

plitu

de

cosv

f D

FadingFading

Large Scale Fading

Large Scale Fading

Small Scale Fading

Small Scale Fading

Attenuation with distanceAttenuation

with distanceVariation

about meanVariation

about mean

Time delay spread /Multipath

Time delay spread /Multipath

Doppler shift / time variance of channelDoppler shift / time variance of channel

Flat FadingFlat Fading Frequency Selective Fading

Frequency Selective Fading

Fast FadingFast

FadingSlow

FadingSlow

Fading

Fading Mechanisms

FadingFading

Large Scale Fading

Large Scale Fading

Small Scale Fading

Small Scale Fading

Large Scale & Small Scale Fading

•Large Scale Fading

•Small Scale Fading

•Due to general terrain, density and height of buildings, vegetation •Variation occurs over very large distances (100m.-a few K.m.) •Important for predicting the coverage and availability of a particular service

•Due to local environment, nearby trees, buildings •Variation occurs over very short distances, on the order of the signal wavelength (<1 m.) •Important for design of modulation format and transmitter / receiver design

Fading● Large-scale fading (shadowing)

– Long term variation in the mean signal level caused by the mobile unit moving into the shadow of surrounding objects

● Small-scale fading (multipath)– Short term fluctuation in the signal amplitude

caused by the local multipath

Multipath fading● A small scale fading that describes short-term, rapid amplitude

fluctuations of the received signal during a short period of time● The actual power received over a much smaller distance vary

considerably due to the destructive/constructive interference of multiple signals that follow multiple paths to the receiver

● The direct ray is actually made up of many rays due to scattering multiple times by obstructions along its path, all travelling about the same distance

● Each of these rays appearing at the receiver will differ randomly in amplitude and phase due to the scattering

Multipath fading

SMALL SCALE FADING

Flat-fading (non-freq. Selective)

Flat fading– In a flat fading channel inter-symbol interference (ISI) is absent– The channel has a constant gain and a linear phase response over a

bandwidth that is greater than the bandwidth of the transmitted signal.

– The spectral characteristics of the transmitted signal are preserved at the receiver

– The channel does not cause any non-linear distortion due to time dispersion

– However, the strength of the received signal generally changes slowly in time due to fluctuations caused by multipath

Frequency selective fading

Small-scale fading due to movements: Doppler shift

● How rapidly the channel fades will be affected by how fast the receiver and/or transmitter are moving

● Motion causes Doppler shift in the received signal components

● the change in frequency of a wave for a receiver moving relative to the transmitter

Doppler shift● Say a mobile phone moving at velocity v km/hr in the x direction and the

radio wave impinging on the receiver at an angle βk● The motion introduces a Doppler frequency shift,

fk = vcos βk/λ

● If the ray is directed opposite to the mobile’s motion (β=0), then fk=v/λ● The frequency of the signal has increased by the Doppler spread, fk

Fast and Slow Fading

● Slow or fast fading depends on the coherence time, Tc● Coherence time is the measure of period over which the

fading process is correlated ● Tc is related to the delay spread, Tc=1/ds● The fading is said to be slow if the symbol duration, Ts is

smaller than the coherence time (or the bandwidth of the signal is greater than the Doppler spread.

Shannon Capacity and Mutual Information

• Shannon defined capacity as the maximum mutual information of channel.

• Maximum error-free data rate a channel can support.

• Mutual information :

• Mutual information measures the information that X and Y share: it measures how many knowing on of these variables reduces uncertainty about the other [1].

Capacity in AWGN

• Consider a discrete-time Additive White Gaussian Noise (AWGN) channel with channel input/output relationship.

• = + , where is the channel input at time , is the corresponding channel output and is a White Gaussian Noise random process.

• The channel SNR, the power in divided by the power in , is constant and given by =, P is transmit power, is the power spectral density of the noise with unit W/Hz.

• The capacity of channel by Shannon’s formula:

= , where the capacity units are bits per second (bps).

Channel and System Model- Flat Fading Channel

• Assume a discrete-time channel with stationary and ergodic time-varying gain 0 ≤ , and AWGN .

• The channel power gain follow a given distribution p(g) and independent of the channel input.

• The channel gain can change at each time i, either as an i.i.d process or with some correlation over time.

Channel and System Model (Cont…)

• In block fading channel g[i] is constant over some blocklength T after which time g[i] changes to a new independent value based on the distribution p(g).

• Let P denote the average transmit signal power, /2 denote the noise spectral density of n[i], and B denote the received signal bandwidth.

• The instantaneous received signal-to-noise ratio (SNR) is,

and its expected value over all time is .

Capacity of Flat-Fading Channels

We will consider 3 different scenarios :

1. Channel Distribution Information (CDI) : The distribution of g[i] is known to the transmitter and receiver.

2. 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].

3. Transmitter and Receiver CSI : The value of g[i] is known at the transmitter and receiver at time i, and both the transmitter and receiver know the distribution of g[i].

Channel Side Information at Receiver

• 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].

• In this case there are two channel capacity that are relevant to system design: Shannon (ergodic) capacity and capacity with outage.

• Shannon capacity- For AWGN channel ,it is the max. Data rate that can be sent over the channel with small error probability. The rate transmitter over the channel is constant.

• Relative to CSI, the transmitter cannot adopt its transmission strategy.

• Capacity with outage is defined as the maximum rate that can be transmitted over a channel with some outage probability corresponding to the probability that the transmission cannot be decoded with negligible error probability.

Shannon (Ergodic Capacity)

• Shannon capacity is equal to Shannon capacity for an AWGN channel with SNR , given by = , averaged over the distribution of .

• By Jensen’s inequality,

• Fading reduces Shannon capacity when only the receiver has CSI.

• Without transmitter side information, the code design must incorporate the channel correlation statistics, and the complexity of the maximum likelihood decoder will be proportional to the channel decorrelation time.

• In addition, if the receiver CSI not perfect, capacity can be significantly decreased

Capacity with Outage

• Capacity with outage allows bits sent over a given transmission burst to be decoded at the end of the burst with some probability that these bits will be decoded incorrectly.

• Specifically, the transmitter fixes a minimum receiver SNR and encodes for a data rate = . The data is correctly received if the instantaneous receiver SNR is greater than or equal to .

• If the receiver SNR is below then the bits received over that transmission burst cannot be decoded correctly with probability approaching one, and the receiver declares an outage. The probability of outage is =

• The average rate correctly received over many transmission bursts is,

Channel Side Information at the Transmitter and Receiver

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

• In this case there is no notion of capacity versus outage where the transmitter sends bits that cannot be decoded, since the transmitter knows the channel and thus will not send bits unless they can be decoded correctly.

• Derive Shannon capacity assuming optimal power and rate adaption relative to the CSI,

Shannon Capacity

• Consider the Shannon capacity when the channel power gain g[i] is known to both the transmitter and receiver at time i.

• Let us now allow the transmit power S() to vary with , subject to an average power constraint :

Time Diversity System

• Multiplexed coding and decoding to achieve the capacity given by

• The proof is “time diversity” system with multiplexed input and demultiplexed output as show below

Optimal Power Allocation

• To find the optimal power allocation s( ), we form the Lagrangian

• Next we differentiate the Lagrangian and set the derivative equal to zero:

• Solving for P(γ) with the constraint that P(γ) > 0 yields the optimal power adaptation that maximizes the capacity as

Optimal Adaptive Scheme

• Power Adaptation

• Capacity

• Note that the optimal power allocation policy (4.12) only depends on the fading distribution p(γ) through the cutoff value γ0.

This cutoff value is found from the power constraint.

Optimal Power Control

• The optimal power allocation policy only depends on the fading distribution p(γ) through the cutoff value γ0.

• This cutoff value is found from the power constraint.

• Specifically, by rearranging the power constraint and replacing the inequality with equality (since using the maximum available power will always be optimal) yields the power constraint.

• Note that this expression only depends on the distribution p(γ).

Zero-Outage Capacity and Channel Inversion

Zero-Outage Capacity

• Fading inverted to maintain constant SNR.

• Simplifies design (fixed rate).

• Since the data rate is fixed under all channel conditions and there is no channel outage.

Channel Inversion

• Suboptimal transmitter adaptation scheme where the transmitter uses the CSI to maintain a constant received power.

• This power adaptation, called channel inversion, is given by P(γ)/P = /γ, where equals the constant received SNR that can be maintained with the transmit power constraint. satisfies =1/E[1/γ].

• In Rayleigh fading E[1/γ] is infinite, and thus the zero-outage capacity given by is zero.

Outage Capacity and Truncated Channel Inversion

• The outage capacity is defined as the maximum data rate that can be maintained by suspending transmission in bad fading states.

• We can maintain a higher constant data rate in the other state and can increase the channel capacity.

• Outage Capacity is achieved with a truncated inversion policy for power adaption which only compensates for fading above a certain cutoff fade depth γ0:

• where γ0 is based on the outage probability: pout= p(γ<γ0). Since the channel is only used when γ >= γ0, =1/Eγ0 [1/γ], where

Outage Capacity with Truncation

• The outage capacity associated with a given outage probability pout and corresponding cutoff γ0 is given by

• We can also obtain the maximum outage capacity by maximizing outage capacity over all possible γ0:

Capacity of Time-Invariant Frequency-Selective Fading Channels

• Consider a time-invariant channel with frequency response H(f).

Assume a total transmit power constraint P.

• Assume H(f) is block-fading

Frequency is divided into subchannels of

Bandwidth B, where

H(f)=Hj is constant over each block.

Time-Invariant Frequency-Selective Fading Channel

• The frequency-selective fading channel consists of a set of AWGN channels in parallel with SNR on the jth channel, where P is the power allocated to the jth channel in this parallel set, subject to the power constraint jpj <= p.

• γ0 must satisfy

• The capacity becomes

Waterfilling Block FSF

• Waterfilling in frequency selective channels

Time-Variant Frequency-Selective Fading Channel

• For TI Channels, capacity achieved by water-filling in frequency.

• Capacity of time-varying channel is unknown.

• Approximate by dividing into subband.

Each subband has with Bc.

Independent fading in each subband.

Capacity is the sum of subband capacities.

where I is the capacity of the flat-fading subchannel with average power Pj