2. Diversity - · PDF file5 2: Diversity Wireless Communication Systems Rayleigh Flat Fading Channel BPSK: x = ±a Coherent detection. Conditional on h, Averaged over h, at high SNR

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2: Diversity

Wireless Communication Systems

2. Diversity

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2: Diversity


Diversity In Wireless Radio

• Communication over a flat fading channel has poor performance due to significant probability that channel is in a deep fade.

•• PerformancePerformance is increased by providing more independent looksindependent looks at the signal paths that fade independently.

• Diversity can be provided across timetime, frequencyfrequency and spacespace (Relate this to the three spreadsthree spreads that we talked about in the channel modeling section)

• Our goal here is how to exploit the added diversity in an efficient manner.

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2: Diversity


Baseline: AWGN Channel

BPSK modulation: x = ± a

Error probability decays exponentially with SNR.

y x w= +

( )0

2

0

2/ 2

SNR

SNR =

eap Q Q

N

aN

⎡ ⎤= = ⋅⎢ ⎥

⎢ ⎥⎣ ⎦

Received Signal Transmitted Symbol Noise

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2: Diversity


Gaussian Detection

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2: Diversity


Rayleigh Flat Fading Channel

BPSK: x = ± a Coherent detection.

Conditional on h,

Averaged over h,

at high SNR.

(0,1)y h x wh= ⋅ +

CN:

( )2 2 SNRQ h⋅

1 112 4

SNR1+SNR SNRep

⎛ ⎞= − ≈⎜ ⎟⎜ ⎟ ⋅⎝ ⎠

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2: Diversity


Rayleigh vs AWGN

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2: Diversity


Conditional on h,

When error probability is very small.When error probability is large:

Typical error event is due to channel being in deep faderather than noise being large.

Typical Error Event

( )2 2 SNRQ h⋅

1/2 SNRh ?1/2 SNRh <

2

2

2

1 1

~ exp(1), . . ( )

SNR SNRe

xh

p P h

h i e f x e−

⎛ ⎞≈ < ≈⎜ ⎟⎝ ⎠

=

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2: Diversity


BPSK, QPSK and 4-PAM

• BPSK uses only the I-phase.The Q-phase is wasted.• QPSK delivers 2 bits per complex symbol.• To deliver the same 2 bits, 4-PAM requires 4 dB more transmit power.• QPSK exploits the available degrees of freedom in the channel better.

• A good communication scheme exploits all the available d.o.f. in the channel.

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2: Diversity


Time Diversity• Time diversity can be obtained by interleavinginterleaving and codingcoding over

symbols across different coherent timecoherent time periods.

Coding alone is not sufficient!. Not effective over slow fading channel

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2: Diversity


Example: GSM

• A frequency division duplex system with 25 MHz for the UL and 25MHz for the DL– Bands are divided into 200 KHz sub-channel– Each sub channel is shared by 8 users in a time division

fashion.

• • • 125 sub-channels • • •

25 MHz

200 KHz

TS0 TS1 TS2 TS3 TS4 TS5 TS6 TS7

8 users per sub-channel

4.615 ms

577 µs

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2: Diversity


Example: GSM ….

• Voice is divided into frames of length 20 ms – Each frame is compressed into 228 bits.– A rate ½ convolutional code (generator polynomials are D4+D3+1 and

D4+D3+D+1) is used to encode the data → 456 bits.• To achieve time diversity, the coded bits in two such 20 ms speech frame are

interleaved across 8 consecutive time slots assigned to a specific user (114 bits per slot)

– A delay of roughly 40 ms • Amount of time diversity limited by delay constraint and how fast channel

varies.• To get full diversity of 8, needs v > 30 km/hr at fc = 900Mhz.

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2: Diversity


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2: Diversity


Simplest Code: Repetition

After interleaving over L coherence time periods,

Repetition coding:Repetition coding: xl = x for all l

This is classic vector detection in white Gaussian noise.

1, ,l l l ly h x w l L= ⋅ + = K

1 2 1 2 1 2[ , , , ] , [ , , , ] , [ , , , ]T T TL L L

xy y y h h h w w w

= ⋅ +

= = =

y h wy h wL L L

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2: Diversity


Geometry

For BPSK: x = ± a

Is a sufficient statistic (match filtering).

Reduces to scalar detection problem:

*

,A Ba a

y

= + = −

=

u h u h

h yh

%

y x w= ⋅ +h% %

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2: Diversity


Deep Fades Become Rarer

( )2 1!

LPL

ε ε< ≈h

2 1

1 1! L

SNR

SNR

ep P

L

⎛ ⎞≈ <⎜ ⎟⎝ ⎠

≈

h

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2: Diversity


Performance

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2: Diversity


Beyond Repetition Coding

• Repetition coding gets full diversity, but sends only one symbol every L symbol times.

• Does not exploit fully the degrees of freedom in the channel. (analogy: PAM vs QAM)

• How to do better?

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2: Diversity


Example: Rotation code (L=2)

where d1 and d2 are the distances between the codewords in the two directions.

x1, x2 are two BPSK symbols before rotation.

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2: Diversity


Rotation vs Repetition Coding

Rotation code uses the degrees of freedom better!

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2: Diversity


Product Distance

product distanceChoose the rotation angle to maximize the worst-case product distance to all the other codewords:

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Antenna Diversity

Receive Transmit Both

Multiple antennas are used for transmission and or reception of the signal.Key parameter here is the separation between the antennas (coherencecoherencedistancedistance).

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2: Diversity


Receive Diversity

• Use a number of receive antennas that are well separate (> coherence distancecoherence distance ) to generate independent receptions of the transmitted signal

• Same as repetition coding in time diversity, except that there is a further power gain.

• Optimal reception is via match filtering ( also known optimal optimal beamformingbeamforming).

x= ⋅ +y h w

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2: Diversity


Receive Diversity ….

•• Selection Diversity:Selection Diversity: choose received signal with the largest received power, SNR, etc.

•• Switched Diversity:Switched Diversity: choose an alternate receive antenna if the signal level falls below a certain threshold.

•• Linear Combining:Linear Combining: linearly combine a weighted copy of all received siganls

• There is a dramatic improvement even with just two branchs

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2: Diversity


DataDetector 1

DataDetector 1

Rx: Selection and Switched Diversity

FrontEnd

FrontEnd

SelectorData

FrontEnd

FrontEnd

DataDetector

DataSwitch

SelectionDiversity

SwitchedDiversity

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2: Diversity


Rx Diversity: Linear Combining

FrontEnd

FrontEnd

∑DataData

Detector

y1

α2

α1

y2

1

2

1 1 1 1 1

2 2 2 2 2

j

j

y h a n h A e

y h a n h A e

φ

φ

= ⋅ + = ⋅

= ⋅ + = ⋅

1 2

1 2

1 2

1 2

1 1 2 22

1 2 , 1 1 2 2

1 1 2 2

,,

( , ) arg min | |

( )

Equal Gain:Maximal Ratio:

MMSE Training:Decoding:

j j

j j

e eA e A e

y y a

a y y

φ φ

φ φ

α α

α αα αα α α α

α α

− −

− −

= == ⋅ = ⋅

= + −

= +S

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2: Diversity


Transmit Diversity

• Provide a diversity benefit to a receiver without having multiple receive antennas by using multiple transmit antennasmultiple transmit antennas.

• If the transmitter know the channel then transmit x = x. h /||h|| . This will maximizes the received SNRmaximizes the received SNR by in-phase addition of signals at the receiver (transmit beamforming).

• Reduces to scalar channel: y = ||h|| x + w, same as Rx beamforming.• What happens if transmitter does not have explicit knowledge of the

channel? Two kind of transmit diversity techniques: – Transmit diversity with feedback from the receiver– Transmit diversity without feedback from the receiver

Blind, i.e. no trainingWith feedforward information, i.e. with training.

•• Note:Note: transmitting the same symbol from all antennas does not work! Why?.

*y = ⋅ +h x w

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2: Diversity


Transmit Diversity with Feedback

• w1 and w2 are varied such that |y(t)|2 is maximizedmaximized.• w1 and w2 are adapted based on feedback information from the

receiver.

Mod

w1

w2

S1(t)

S2(t)

Tx 1

Tx 2 y(t)Demod

y1(t)

y2(t)

y(t) y1(t)

y2(t)

2 21 2 1w w+ =

1 2,variations

w w

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2: Diversity


Transmit Diversity with Frequency Weighting

• Use frequency weighting to mitigate the harm of scenario B.• Simulate fast fadingfast fading → can use conventional channel coding and channel coding and

intereleavingintereleaving techniques.• Suitable for slow fading or staticslow fading or static channels.

ChannelEncoder

ejθ(t)

S1(t)

S2(t)

Tx 1

Tx 2 y(t) ChannelDecoder

y1(t)

y2(t)

( ) 2 mkT f kTθ π=

Rx

y1(t)y2(t)

(A) Constructive Interference

(B) Destructive Interference

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2: Diversity


Tx Diversity with Antenna Hopping

• At time i, 1≤ i ≤ N, transmit s from antenna i.• Achieves a diversity orderdiversity order of N.•• Bandwidth efficiencyBandwidth efficiency is 1/N.

RepetitionCode

Tx 1

MLDetection

Rx

s 1 2, ,..., Ns s s

Tx 2

Tx N

T

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2: Diversity


Tx Diversity with Channel coding

• The channel code has a minimum Hamming distanceHamming distance dmin≤ N.• At time i, transmit code symbol ci from antenna i.• After receiving the N code symbols, the receiver performance ML

decoding to decode the received code word.

Channel Code

Tx 1

MLDetection

Rx

1 2, ,..., ks s s 1 2, ,..., Nc c c

Tx 2

Tx N

T

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2: Diversity


Tx Diversity via Delay Diversity

• Provides a diversity benefit by introducing intentional multipath (simulating a multipath channel).

• Receiver uses an equalizer or MLSEequalizer or MLSE for detection.• Provides diversity order of N, no loss in BW efficiency• Provides very little diversity benefit, if the channel if multipath to begin with.

Tx 1

Equalizer

Rx

( )s tTx 2

Tx N

DelayT

Delay(N-1)T

( )s t

( )s t T−

( ( 1) )s t N T− −

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2: Diversity


Space-time Codes

• Space-time codes are designed specifically for the transmit diversity scenario.

• For each input symbol, the space-time encoder chooses the constellation points to simultaneouslysimultaneously transmit from each antenna so that codingcoding and diversitydiversity gains are maximizedmaximized.

•• No cheating!No cheating! . Total transmitted powertransmitted power is still the samesame as single transmit antennas case.

• Both flavors: trellistrellis codes & blockblock codes.

‘Space-Time Codeword’

QAMSymbol

Multi-Element Receiver

TxEn

code

r...

.

.

.

.

.

1

NT

A

BC

MIMOChannel Matrix, H

t0 t1 t2

Rx

Dec

oder

.

.

.

1

NR

A’

B’C’

.

.

.

.

.

TxA

nten

na

Space-TimeSymbol

Multi-Element Transmitter

‘Space-Time Codeword’

-

TxEn

code

r...

.

.

.

.

.

.

.

.

1

T



t0 1 t2

Rx

Dec

oder

.

.

.

.

.

.

R

’

’’

.

.

.

.

.

--

2: Diversity


Space-Time Block Codes: Alamouti Scheme

•• Idea:Idea:

•• Assumption:Assumption: channel is quasiquasi--staticstatic.

c cc cc c1 2

1 2

2 1

→−L

NMOQP

∗

∗

Burst 1

Burst 2

ConstellationMapperInformation

Source

ST Block Code

[ ]*

1 21 2 *

2 1

c cc c

c c⎡ ⎤−

→ ⎢ ⎥⎣ ⎦

timetime

Ant

enna

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2: Diversity


Decoding of STBC

• Received Signal:

• H is orthogonal:orthogonal:

– Projecting onto the two columns of the H matrix yields:

r h c h c n

r h c h c n1 1 1 2 2 1

2 1 2 2 1 2

= + +

= − + +* *

r H c n=LNM

OQP= −

LNM

OQPLNM

OQP+

LNM

OQP= ⋅ +

rr

h hh h

cc

nn

1

2

1 2

2 1

1

2

1

2* * * *

H H I h I* h h12

22 2e j

~ ~*r H r h c n2

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2: Diversity


Decoding of STBC ….

• The noise term is still whitewhite c1 and c2 are detected independentlyindependentlylower complexitylower complexity.

• Double the symbol rate of repetition coding

• 3dB loss of received SNR compared to transmit beamforming.

• With M receive antennas:

• With M receive antennas, a diversity order of 2M is achieved.• Only simple linear processingsimple linear processing at the receiver is required.• Complete CSICSI is required at the receiver. In practice channel channel

estimationestimation is used to obtain CSI.

~ ~*r H r h c nFHG

IKJi i

i

M

ii

M

1

2

1

~n

2: Diversity


Decoding of STBC ….

Linear Combiner

soft decision for c2

soft decision for c1

c1

ML Decision

c2

ML Decision

H2

H1

r1 r1 c1

c2r2r2

~ * *r H Hrr

= M P1 21

2

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2: Diversity


Space-time Code Design

A space-time code is a set of matrices

Full diversity is achieved if all pairwise differences have full rank.

Coding gain determined by the determinants of

Time-diversity codes have diagonal matrices and the determinant reduces to squared product distances.

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2: Diversity


Frequency Diversity

•• Resolution of Resolution of multipathsmultipaths provides diversity.• Full diversity is achieved by sending one symbol every

L symbol times.– Sounds like repetition coding →this is inefficient

• Sending symbols more frequently may result in intersymbol interference (ISI).

• Challenge is how to mitigate the ISI while extracting the inherent diversity in the frequency-selective channel.

1

0[ ] [ ] [ ]

L

ll

y m h x m l w m−

=

= − +∑

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2: Diversity


Approaches

• Time-domain equalization (eg. GSM)

• Direct-sequence spread spectrum (eg. IS-95 CDMA)

• Orthogonal frequency-division multiplexing OFDM (eg. IEEE 802.11a/g, Flash-OFDM, IEEE 802.16, IEEE 802.20)

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2: Diversity


ISI Equalization

• Suppose a sequence of uncoded symbols are transmitted. Can full diversity be achieved?

• Answer is YES!YES!• Maximum likelihood sequence detection (MLSD) is the

optimal solution:– Performed using the Viterbi algorithm.– Complexity might be quite high

• Other suboptimal equalization techniques:– Linear equalizers: zero-forcing and MMSE– Decision feedback equalizers (DFE)

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2: Diversity


MLSD: Reduction to Transmit Diversity

0[1] [1]y x h= ⋅

0 1[2] [2] [1]y x h x h= ⋅ + ⋅

0 1 2[3] [3] [2] [3]y x h x h x h= ⋅ + ⋅ + ⋅

0 1 2[4] [4] [3] [2]y x h x h x h= ⋅ + ⋅ + ⋅

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2: Diversity


MLSD: Reduction to Transmit Diversity …

• Consider the transmitted sequence

[ ][ ] [ ]0 1

[1], [2], , [ 1]

, , , , [1], [2], , [ 1]

[1] [2] [ ] [ 1]0 [1] [2] [ ] [ 2]0 0 [1] [2]

0 0 [1] [2] [ ]

T T T

T

T TL

y y y N L

h h h w w w N L

x x x N x N Lx x x N x N L

x x

x x x N

= ⋅ +

= + −

= = + −

⋅ ⋅ ⋅ ⋅ ⋅ + −⎡ ⎤⎢ ⎥⋅ ⋅ ⋅ + −⎢ ⎥⎢ ⎥= ⋅ ⋅ ⋅ ⋅ ⋅⎢ ⎥⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⎢ ⎥⎢ ⎥⋅ ⋅ ⋅ ⋅⎣ ⎦

y h X wy

h w

X

L

L L

[ ][1], [2], , [ 1]T x x x N L= + −x L

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2: Diversity


MLSD: Reduction to Transmit Diversity …• Maximum Likelihood detection of the sequence x based on the

received sequence y (MLSD). The probability of confusing xA with xB , when xA is transmitted is:

• The probability of error decays like SNR whenever the differenceXA-XB is of rank L (i.e. full rank).

• Can show that is XA-XB indeed of full rank (see details in text). Hence:

{ } ( )( )* *A B A B

A B

21 l

SNRPr

2

11 SNR / 4

L

l

E Q

λ=

⎧ ⎫⎛ ⎞⋅ − −⎪ ⎪⎜ ⎟→ = ⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

≤+ ⋅∏

h X X X X hx x

MLSD achieves full diversity on symbol x[N] using the observations up to time N+L-1, i.e. a delay of L-1 symbol times.

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2: Diversity


Direct Sequence Spread Spectrum

• Data Rate R is than the transmission bandwidth W (also known as chip rate) .

Processing Gain: Processing Gain: n n == W/RW/R• Signal-to-noise ratio per chip is low.• Example: IS-95 CDMA system (used by Verizon, Sprint):

ChannelEncoder

MP channel

Pseudo RandomPattern Generator

Pseudo RandomPattern Generator

ChannelDecoder

Modulator DemodulatorInformationbits

Outputbits

W = 1.288 MHz, R=9.6 kbits/sec, G =128

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2: Diversity


Direct Sequence Spread Spectrum ….

• Symbol duration Tb >> delay spread Td → ISI is not a ISI is not a problemproblem (as compared to interference from other users).

• Channel is constant over one symbol. i.e. symbol duration Tb=n/W << coherence time Tc.

[ ]s m

[ ]u i

Data

PNSequence

1τ 2τ 3τ 4τ

ChannelIR

[ ]h m

[ ]0h m[ ]1h m

[ ]2h m [ ]3h m

1sT

W=

bTSymbol duration

Chip duration

dTDelay Spread

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2: Diversity



• PN Sequences are chosen such that shifted versions of the same sequence are nearly orthogonal, i.e. :

( ) ( ) ( ) ( )* * 2

1

, , [ ]n

l k l l

i

u i l k=

= ≠∑u u u u=

[ ]uR m

m

n

1−

PN Sequence Autocorrelation

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2: Diversity



Σi=1

n

( )*0u

*0h

( )*1u

*1h

Σi=1

n

( )*Lu

*1Lh −

Σi=1

n

Σ Decision

correlator

correlator

correlator

y

0y

1y

1Ly −

0,..., 1l l ly h s w l L= ⋅ ⋅ + = −u

RAKE Receiver

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2: Diversity



• Signal at the output of the correlators:• The orthogonality ul of implies that wl are i.id ΧΝ(0,N0) . • This looks exactly the same as the L-branch diversity model for the

repetition code interleaved over time that we have seen before. Hence, the RAKE receiver in this case is nothing more a ML ratio combinerML ratio combiner of the signals from the L multipath components (LL RAKE fingersRAKE fingers)

• Probability of error:

• Assume a Rayleigh fading model such that hl are i.i.d ΧΝ(0,1/L), i.e. the energy is split among the L taps (and normalizing such that E{||hl||2}=1). The SNR per branch is SNR= ||u||2/(N0L)=(1/L). Es /N0

• As L→∞, ||hl||2 → 1 and the probability of error becomes:

0,..., 1l l ly h s w l L= ⋅ ⋅ + = −u

12 2

00

2 /L

e ll

p E Q h N−

=

⎧ ⎫⎛ ⎞⎪ ⎪= ⎜ ⎟⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∑u

( )02 /e sp Q E N→

i.e. the performance of AWGN with the same i.e. the performance of AWGN with the same EEss /N/N00 is asymptotically achievedis asymptotically achieved

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2: Diversity


ISI and Frequency Diversity

• In narrowband systems, ISI is mitigated using a complex receiver.

• In asynchronous CDMA uplink, ISI is there but small (compared to interference from other users).

• But ISI is not intrinsic in a channel with frequency diversity.

• The transmitter needs to do some work too!

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2: Diversity


OFDM: Basic Concept

• Most wireless channels are underunder--spreadspread (Td << Tc) .• Can be approximated by a linear time invariantlinear time invariant channel

over a long time scale.• Complex sinusoidssinusoids are the only eigenfunctions of linear

time-invariant channels.• Should signal in the frequency domainfrequency domain and then transform

to the time domain.

Frequency f

( )H f

1f 2f 3f 4f

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2: Diversity


OFDM Modulation ….1

0

, , ,

1

1

1

1 1 1

1

1

( ) ( ) ( - ) ( )

0 00 00 00 0

0

0 0

L

l

k k k

o L

o L L

k o L

L o L

L

N N N N N

o

y i h l x i l w i

h h hh h h

h h hh h h

h h h

+

=

−

×

−

× × ×

= +

= ⋅ +

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑y H X n

H

L LO L

O O O O OL L

L L LM O O O O O M

L L

• The channel Matrix H is circulantcirculant, hence it will have the eigenvaluedecomposition Hk=Q*ΛkQ.

• FFT of received signal*

k k k

k k k k

⋅ = ⋅ ⋅ + ⋅

= ⋅ +

Q y Q Q Λ Q X Q wY Λ S N

IFFT P/S CyclicPrefixS/P

( )ks i

kS kX

( )kx i ( )kx i%

52

2: Diversity


OFDM Modulation ….

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 01

2

3

4

5

6

7

8

9

's sT T Tν= +

TνOFDM symbol

Cyclic prefix

Data Symbols

'1 1/ sf T=

'2 2 / sf T=

'/N sf N T=

'sT

53

2: Diversity


Tone Orthogonality

Frequency f

54

2: Diversity



55

2: Diversity



OFDM transforms the communication problem into the frequency domain:

a bunch of nonnon--interfering interfering (parallel)(parallel) sub-channels, one for each sub-carrier.

Can apply time-diversity techniques.

0,..., 1i i i iy h s w i N= ⋅ + = −%

( ) 0,..., 1ii Wh H H i f i NN⋅⎛ ⎞= = ⋅Δ = −⎜ ⎟

⎝ ⎠%

56

2: Diversity


Cyclic Prefix Overhead

• OFDM overhead = length of cyclic prefix / OFDM symbol time

• Cyclic prefix dictated by delay spreaddelay spread.• OFDM symbol time limited by channel coherence timecoherence time.• Equivalently, the inter-carrier spacing should be much

larger than the Doppler spread.• Since most channels are underspread, the overhead can

be made small.

57

2: Diversity


Diversity Summary

• Fading makes wireless channels unreliable.• Diversity increases reliability and makes the channel

more consistent.• Smart codes yields a coding gain in addition to the

diversity gain.• Different diversity schemes for different channel

conditions:–– Time Diversity:Time Diversity: exploit the diversity inherent in the

time varying nature of the channel–– Frequency Diversity:Frequency Diversity: exploit the multipath

(frequency) in the channel–– Space Diversity:Space Diversity: exploits the spatial variation in the

channel

58

2: Diversity


The Big Picture So Far

• In wireless communications, information are sent over wireless channels

• Impairments of Wireless channels:

–– Path Loss and Shadowing:Path Loss and Shadowing:

Link budget and cell planningLink budget and cell planning

–– MultipathMultipath Fast Fading:Fast Fading:

Use diversity (time, frequency, space) techniquesUse diversity (time, frequency, space) techniques.

2. Diversity - · PDF file5 2: Diversity Wireless Communication Systems Rayleigh Flat Fading Channel BPSK: x = ±a Coherent detection. Conditional on h, Averaged over h, at high SNR

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