Wireless Communication Technologies_Diversity Systems

Wireless Communication Technologies 16:332:559 (Advanced Topics in Communications) Lectures #14 and #15 (March 21, 26, 2001) Instructor Nayaran Mandayam Summarized by Rong Zhang

1 Diversity Systems Diversity is a powerful communication receiver technique that provides wireless link improvement at relatively low cost. The receiver is provided with multiple copies of information, which are transmitted over 2 or more dependent channels. If one radio path undergoes a deep fade, another independent path may have a strong signal. In this way, diversity improves the performance of data transmission over fading channel. There are 2 types of diversity: •

Macroscopic diversity: intended to combat the effect of long term/large scale fading. Signals that are received from a mobile station by two or more geographically separated base stations are combined together.

•

Microscopic diversity: intended to combat the effect of short scale fading. Works well when different signals (branches) experience independent fading

There are wide variations of diversity implementations. They generally fall into five categories: •

Frequency diversity: transmitting signals on two or more carriers that are separated by at least the coherence bandwidth of the channel.

•

Time diversity: transmitting signals repeatedly with intervals of at least the coherence time. If fading rate is too slow, it need long time interval between transitions.

•

Polarization diversity: orthogonal polarization.

•

Angle diversity: using directional antennas to create independent copies of transmitted signal through multiple paths.

•

Space diversity: placing receiving antennas at different locations and resulting in different and positively independent signals. By selecting the best signal at all times, the receiver can mitigate small-scale fading effect.

Manipulating

electric

and

magnetic

field

to

obtain

The frequency and time diversity require more bandwidth and the other methods usually involve more complex antenna system.

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We know that the signal correlation, ρ, is a function of distance, d, between antenna 2πd  as: ρ = J 02   where J0 (x) is the Bessel function of zero order. Figure 1 shows the  λ  relationship between the correlation and antenna distance.

Figure 1. Receiver Antenna Distance vs. Correlation If we want the correlation to be zero, the distance between two antennas should be approximately proportional to λ/2. So approximately half wavelength separation is required to get zero correlation. For example, to provide uncorrelated signals with f c= 500MHz, d should be 30cm, while if f c=5MHz, the distance increases to 30m.

2 Selection Diversity 2.1 CDF of SNR for Selection Diversity Suppose there are M independent copies of signals available from M independent paths in a diversity system. Each path is called a diversity branch. Let us assume the received signal on each branch is independent and Rayleigh distributed with mean power 2σ2 . Thus the SNR achieved on each branch γi, i=1, 2,…, M, is distributed exponentially as:

P(γ i ) =

γ 1 exp( − i ) γ0 γ0

___(1)

Eb is the mean of the received power on the i-th branch, Eb /N0 is the N0 SNR achieved without fading effect, and Eb is the bit energy. The CDF for i-th branch is:

where γ 0 = 2σ 2

γ

P(γ i ≤ γ ) = ∫ P(γ i ≤ γ )dγ i = 1 − exp( − 0

γ ) γ0

___(2)

Selection diversity picks up the best signal among M branches in terms of SNR. So the selection criterion is defined as:

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{

}

P(γ ) = Pr max{γ i } ≤ γ = P{γ 1 < γ ,..., γ M i

 γ  < γ } = 1 − exp( − )  γ0  

M

___(3)

Figure 2. CDF for selection diversity with M=1,2,3,6 So the probability of very low SNRs decreases rapidly when diversity is used. When SNR falls far below γ0 , i.e., γ/γ0 is small, we can approximate Equation 3 as: M

 γ  γ  P(γ ) ≈ 1 − (1 − )  =   γ0   γ0 

M

___(4)

2.2 Average SNR of Selection Diversity We can obtain the pdf of γ by taking derivative of equation (3) as:  γ   γ  d M p (γ ) = P(γ ) = exp − 1 − exp −  dr γ0  γ 0   γ 0 

M −1

___(5)

So the average SNR due to selection diversity is:

[

]

M

E max {γ i } = ∫ γp(γ )dγ = γ 0 ∑ i

∞

0

1 k =1 k

___(6)

 γ (1 + 1 / 2)   =1.8dB. Similarly, if If we increase M=1 to M=2, the average gain is 10 log  0 γ  o  we increase M=2 to M=3, there is 0.9dB improvement. For most practical applications, M=2 is always chosen as the optimal value.

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3 Maximal Ratio combining (MRC) 3.1 CDF of SNR for MRC Selection diversity always picks the best one among all the signals. Implicitly pick the best one is always possible as long as the signals are monitored at a rate higher than rate of fading. However, it does not use all the possible branches simultaneously. In MRC method, all brunches are used simultaneously. Each branch signal is weighted with a gain factor proportional to tis own SNR. MRC results in a ML receiver channel, therefore it gives the best possible performance among the diversity combining techniques. Figure 3 shows the basic idea of the MRC method.

……… g1 TX

g2

gM

Co-phasing and Summing Figure 3. MRC Implementation diagram

Where the gain of the i-th branch is defined to be proportional to its SNR, i.e., gi ∝ (S/N)i for all i = 1, 2, … , M. If ai is the signal envelop at the i-th brunch, the combined signal M

envelop for M branches is: a = ∑ g i ai . Further, we assume the noise components are i =1

M

i.i.d. in each brunch, so the total noise power is estimated as: N t = N 0 ∑ g i2 . The i =1

resulting SNR is

M   ∑ gi ai  E E  γ = a 2 b = b  i=1M Nt N0 ∑ g i2

2

i =1

____ (7)

Using Schwarz inequality, we can obtain the upper bound for SNR as: 2

M   ∑ g i ai  E  ≤ Eb γ = b  i=1M Nt N0 ∑ g i2 i =1

M

M

i =1

i =1

∑ g i2 ∑ ai2 M

∑g i =1

2 i

=

Eb N0

M

∑a i =1

2 i

_____(8)

Equality is obtained in Schwarz inequality when gi=kai, where k is a constant. So the maximum value of output SNR is given as

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γ =

Eb N0

M

M

i =1

i =1

∑ ai2 = ∑

Eb 2 M a = γi N0 i ∑ i =1

____(9)

This means the resulting SNR is the sum of individual SNR of each branch. Further, in order to obtain the pdf of combined signal, we observe that SNR in the i-th branch is given by γi =

Eb 2 Eb 2 ai = ( xi + y i2 ) N0 N0

____(10)

where x i and yi are independent Gaussian random variables with zero mean and equal variance σ2 . Thus we know that γi ~ χ2 with 2 degree of freedom. Since we have M M

branches, γ = ∑ γ i ~ χ2 with 2M degree of freedom. Then the pdf of γ is given as: i =1

p (γ ) =

where γ 0 = 2σ 2

1 γ M −1 γ exp( − ) M ( M − 1)! γ 0 γ0

γ ≥0

___(11)

Eb is the mean SNR on each branch. Finally, the CDF is: N0

P(γ ) = ∫

γ

0

γ M 1 γ  p( x) dx =1 − exp( − )∑ γ 0 i=1 (i − 1)! γ 0

  

i-1

Figure 4. CDF of MRC For example, with M = 2, the probability of SNR is 10dB below γ0 , i.e., γ/γ0 = 0.1, can be calculated as: PMRC = 1 - e-0.1 * [1 + 0.1] = 1 – 0.995 = 0.005 PSD = (1 - e-0.1)2 = (0.095)2 = 0.009 5

We can see that MRC is better than selection diversity.

3.2 Mean SNR for MRC The average SNR for MRC is relatively easy to be obtained by summing the average SNR of individual branch as: M

E[ ∑ γ i ] = Mγ 0

___ (12)

i =1

So, the mean SNR increase linearly in M. MRC increases performance over selection diversity compares to the mean of SNR for SD in equation (6).

4 Equal Gain Combining 4.1 SNR for Equal Gain Combining Equal gain combining is a special case of MRC by simply setting gi=1 for i = 1,…, M. This results in: M

a = ∑ ai i=1

a 2 Eb E M  γ = = b  ∑ ai  MN 0 MN 0  i =1 

2

where γ is the resulting SNR. We can see that SNR is the summation of M Rayleigh random variables. There is no close form available for the CDF of γ when M > 2. But we can see that the performance is a little worse than MRC, and better than selection diversity.

4.2 Average SNR for Equal Gain Combining E[γ ] =

Eb  M  E E (∑ Ai ) 2  = b MN 0  i=1  MN 0

M

M

∑∑ E[ A A ] i =1 j =1

i

j

___(13)

For uncorrelated branches, we have E[AiAj]= E[Ai]E[Aj]. If the braches are uncorrelated, π 2 and with Rayleigh signals of E[Aj2 ]=2σ2 and E[ Ai ] = σ , the mean of the SNR for 2 equal gain combining is given as: E[ γ ] =

Eb  π 2 2 π 2 σ )  = γ 0 [1 + ( M − 1) ] 2 Mσ + M ( M − 1)( MN 0  2 4 

If M is large, the mean can be further simplified as: E[ γ ] ≈

π Mγ 0 . 4

From Figure 4, we can see that the slope of Equal Gain Combing is little worse than MRC.

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Figure 4 Comparison of MRC Equal Gain Combing and Selective diversity

5 Switched Diversity In switched diversity, only one branch at a time is used. The strategy is to remain using the current branch until the signal envelope drops below pre-determined threshold value x. When signal falls below x, two strategies could be used •

“switch & stay”: select other signal/branch always

•

“switch & examine” select other signal/branch only if the other signal/branch is above threshold.

This method has simplicity in implementation as only one receiver (front end) is required in constant with selection diversity where all channels are to be multiple coherency. Figure 5 demonstrate the idea of switched diversity where the dark line is the combined process.

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a2 (t)

Switch & Stay r(t)

Switch & Exam

x a2 (t)

time Figure 5. Switched Diversity Technique There is one problem in switched diversity technique that discontinuous signal may be obtained at switching instants. Note that a1 (t) and a2 (t) are independent Rayleigh processes. Assume they have identical mean and autocorrelation r(t) is a combined process composed of a1 (t) and a2 (t) in according to switching level x. Let γx denote SNR corresponding to level x. Then the probability that the SNR is one branch is below the threshold is

q = P{γ a 1or

a2

< γ x}

Let the mean of γ ≡γ0 . Since SNR in each branch is exponential, we can obtain:

q (γ ) = 1 − exp( −

γx ) γ0

The SNR in the combined signal can be shown to have the pdf

(1 − q) PΓ (γ ) γ ≥ γ x p c (γ ) =  γ >1, we have (1+µ)/2≅1, (1-µ)/2≅1/(4Mr). So for large SNR (γ0 >10dB)  2 M − 1 1   Pd ≈    M  4γ 0 

M

We can see Pd decreases with M power g increases mean SNR, while without diversity the decreasing is linear.

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Figure 8. Bit error rate for BPSK with MRC diversity 7.1.2 coherence FSK with MRC

 2M − 1 1  Pd ≈     M  2γ 0 

M

 2M − 1 1   Pd ≈    M  2γ 0 

M

7.1.3 DPSK with MRC

7.1.4 non-coherence orthogonal FSK

 2 M − 1 1  Pd ≈     M  γ 0 

M

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Figure 9 Bit Error Rate for BFSK with MRC diversity References 1. Narayan B. Mandayam, class notes. 2. Gordon L Stüber, Principles of Mobile Communications, Kluwer Academic Publishers, 1996 3. John G. Proakis, Digital communications, 3rd edition, McBraw-Hill Inc., 1996

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