BPSK in Rayleigh fading

Diversity

PROF. MICHAEL TSAI 2014/4/28

BER Performance under Fading:

BPSK in Rayleigh fading

Without fading, 7-8 dB of SNR is good for P_ = 10^ . With fading, 20 dB of SNR is not enough!

BER Performance under Fading:

M-QAM in Rayleigh Fading

BER Performance under Fading:

BPSK in Nakagami fading

Intuition: how does fading affect average BER?

• Average BER:

• : Signal-to-Noise Ratio (SNR) per bit

•  : PDF of SNR (fading distribution divided by noise power)

•   : BER of a given SNR

 =   ^    



SNR

p(SNR)

 =   ^    



then ∫  (summation) Dominant Part

The width (variance)

dictates the average BER!

Concept: Diversity

Channel A Channel B

Low correlation (independent)

Receiver A Receiver B

     && !    

= "_# > "_ && "_% > "_

≈ "_# > "_ "_% > "_ ≪ ("_# > "_)

Space Diversity

……

RX antenna 1

RX antenna 2

RX antenna M

Each pair separated by at least half the wavelength (accurate version: 0.38 wavelength)

Low correlation
independent channels

Q: What’s the minimum required separation between 2 antennas? (for 802.11g and 802.11a)

A: 12.5 cm for 2.4 GHz 5.17 cm for 5.8 GHz (which is what you see for a typical router)

Directional (Angle) Diversity

• Split the 360 degree receiving angle into different “sectors”

• Each will receive a portion of multipath components (MPC)

• Extreme case: if the angle of each “sector” is very very small, then you only receive one MPC

no small scale fading

• Different sets of MPCs go through different paths
low correlation!

• Antenna design:

• Multiple sectors on the same antenna (switchable multiple antennas)

• Steerable directional antenna (mechanical)

New WiFi Access Points in the CSIE Building

• Ruckus Zoneflex 7962

• Currently in service in the CSIE building

• 802.11 a/b/g/n

• Over 4000 unique antenna patterns

• Many “sectors”, 3D too (from its appearance)

• Select multiple “good”

antennas for receiving

• Can be used to reduce interference too

Smart Antenna inside Ruckus Zoneflex 7962

Frequency Diversity

• Signals at two frequencies separated by at least one coherence bandwidth

low correlation!

independent!

• Small coherence bandwidth is sometimes good too

• For frequency diversity, two transmissions do not need to be too far apart in frequency

• OFDM utilize this property too

• Sub-carriers separated by at least one

coherence bandwidth can transmit redundant information for diversity (reliability)

• Sub-carriers within the same coherence

bandwidth can transmit different information for increasing the throughput

*₊ *_,

Separated by at least

one coherence bandwidth

freq.

Time Diversity

• Transmit the same packet (or a part of it) after -., -. >

/₀ (coherence time).

low correlation

independent

• How to do this?

• For channel with 1₂₃₄ < 1₆, coding techniques can utilize this

transmit redundant information in the same packet, separated by 1₆.

• Retransmission conceptually uses this too.

_{+ ,}

Separated by at least one coherence time

pkt 1 pkt 1’

Some related terms

• Micro-diversity:

to mitigate the effects of multipath fading (small-scale fading).

• Macro-diversity:

to mitigate the effects of shadowing from buildings and objects (large-scale fading).

• In this lecture, we will talk about micro-diversity.

A More Formal Representation for Receiver Diversity

7

= 



• ₈: amplification of signal

• ^9:;<: remove the phase of that branch (co-phasing)

Array Gain

• Array Gain:

Improvements from getting the signals from multiple antennas

• Usually refers to the gain without fading

• More formally, SNR of the combined signal can be calculated as:

₌ = ∑ ^?_{8@+ 8}₈ ^, A_ ∑ ^?_{8@+ 8}^, =

∑ B_C

A_

?8@+

,

A_ ∑ B_C A_

?8@+

= ME_F N_

Setting ₈ = ^H_I^<

J,  = 1, … , M

With fading,

what is the average BER?

• Diversity gain:

the performance advantage as a result of diversity combining (in fading).

• Average BER:

• Or we can express it as

m: the diversity order

• When m=M (the number of branches), we say that the system achieves full diversity order.

 =   ^   ₌  



 = ̅

Selection Combining (SC)

• Concept:

select the one branch with the best SNR and dump the rest.

• Advantage:

simple, no need to do co-phasing.

• Select the highest SNR: P_Q = ^R_T^QS

Q.

• In practice, SNR cannot be measured.

Since T_Q = T_U, ∀Q,

we can select the branch with the highest RSSI instead: R_Q^S + T_Q

Selection Combining (SC)

• The CDF of SNR after combining:

• No close form expression to obtain the average BER

Use simulation to obtain the result.

• Sometimes branch correlation is not 0

the performance will degrade

negligible when correlation < 0.5 ₌  = ₌ < 

= max ₊, _,, … , _? < 

= [ ₈ < 

? 8@+

BER Performance: BPSK with SC in Rayleigh fading

The biggest gain is from M=1 to M=2 (1
2 antennas)

Threshold Combining

• Concept:

Use one branch and dump the rest. When this one is not good anymore (SNR drops below a threshold),

randomly select another branch.

• Advantage:

Even simpler, no need to monitor the SNR of all branches.

• When there are only 2 branches, switch to the other branch when SNR is smaller than the threshold.

• This is called Switch-and-Stay Combining (SSC)

• SSC has the same performance (outage probability) as SC, when setting the threshold = the minimum required SNR

Switch-and-Stay Combining (SSC)

Maximal-Ratio Combining (MRC)

• Concept:

Use all branches. We amplify the branch more when its SNR is larger.

• Advantage:

Make use of all branches
best performance.

• Question:

How to set ₈ so that the SNR after combining is maximized?

₌ = ∑ ^?_{8@+ 8}₈ ^, A_ ∑ ^?_{8@+ 8}^,

Maximal-Ratio Combining (MRC)

• Answer:

\_Q^S should be proportional to the branch SNR ^RQS

T_U .

• After optimization, it turns out that

• And the SNR after combining becomes



= 

A

₌ = ^ ₈^, A_

? 8@+

= ^ ₈

? 8@+

Note that this is linear scale, not in dB!

BER Performance: BPSK with MRC in Rayleigh fading

MRC’s performance is significantly better!

(At the cost of more signal processing)

Can have better performance than without fading!

BER Performance: BPSK with SC in Rayleigh fading

MRC’s performance is significantly better!

(At the cost of more signal processing)

Equal-Gain Combining (EGC)

• Concept:

Use all branches, but combine them with equal weight=1.

• Advantage:

Use the signal from all branches, but in a simpler way.

• \_Q = _, ∀Q.

• The SNR after combining becomes

• EGC’s performance is quite close to MRC, typically only has less than dB of power penalty.

₌ = 1

A_M ^ ⁸

? 8@+

,