dspLog » Diversity

Transmit beamforming

Krishna Sankar — Mon, 13 Apr 2009 15:53:26 +0000

In this post lets discuss a closed-loop transmit diversity scheme, where the transmitter has the knowledge of the channel. As there is a feedback path required from the receiver, to communicate the channel seen by the receiver to the transmitter, the scheme is called closed-loop transmit diversity scheme. Recall that the transmit diversity using Space Time Coding (Alamouti STBC) does not require the knowledge of the channel. In this post, we will restrict our discussion to a 2 transmit, 1 receive case. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Channel Model

1. We have 1 receive antennas and two transmit antenna.

2. The channel is flat fading – In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. For a more rigorous discussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3 Signal Time-Spreading from [DIGITAL COMMUNICATIONS: SKLAR]

3. The channel experienced by each receive antenna is randomly varying in time. For the receive antenna, each transmitted symbol gets multiplied by a randomly varying complex number . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of are Gaussian distributed having mean and variance .

4. The channel experience by each transmit antenna to receive antenna is independent from the channel experienced by other transmit antennas.

5. On each receive antenna, the noise has the Gaussian probability density function with

with and .

6. At each transmit antenna, the channel is known.

Figure: 2 transmit 1 receive beam steering

Transmit Beamforming

On the receive antenna, the received signal is,

where,

is the received symbol,
is the channel on the transmit antenna,
is the transmitted symbol and
is the noise on the receive antenna.

When transmit beamforming is applied, we multiply the symbol from each transmit antenna with a complex number corresponding to the inverse of the phase of the channel so as to ensure that the signals add constructively at the receiver. In this scenario, the received signal is,

where,

and

In this case, the signal at the receiver is,

For equalization, we need to divide the received symbol with the new effective channel, i.e,

BER Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Multiply the symbols with the beam steering matrics – corresponding to the phase of the channel

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulting BER for BPSK in flat fading Rayleigh channel with and without beamforming

Figure: BER plot for 2 transmit 1 receive beamforming for BPSK in Rayleigh channel

Observations

1. Sending the same information on multiple transmit antenna does not provide diversity gain. Intuituvely, this is due to the fact that the effective channel in a 2 transmit antenna case is again a Rayleigh channel; hence the BER performance is identical to 1 transmit 1 receive Rayleigh channel case.

2. If the transmit symbols are multiplied by a complex phase to ensure that the phases align at the receiver, there is diversity gain. However, the BER performance seems to be slighly poorer than the 1 transmit 2 receive MRC case. I guess, the noise is scaled by in the case of transmit beamforming, whereas the noise scaling is different in the case of Maximal Ratio Combining. I need to study bit more for a precise answer.

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Alamouti STBC with 2 receive antenna

Krishna Sankar — Sun, 15 Mar 2009 16:01:18 +0000

In the past, we had discussed two transmit, one receive antenna Alamouti Space Time Block Coding (STBC) scheme. In this post, lets us discuss the impact of having two antennas at the receiver. For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Alamouti STBC with two receive antenna

The principle of space time block coding with 2 transmit antenna and one receive antenna is explained in the post on Alamouti STBC. With two receive antenna’s the system can be modeled as shown in the figure below.

Figure: 2 Transmit 2 Receive Alamouti STBC

For discussion on the channel and noise model, please refer to the post on two transmit, one receive antenna Alamouti Space Time Block Coding (STBC) scheme.

The received signal in the first time slot is,

Assuming that the channel remains constant for the second time slot, the received signal is in the second time slot is,

where

are the received information at time slot 1 on receive antenna 1, 2 respectively,

are the received information at time slot 2 on receive antenna 1, 2 respectively,

is the channel from receive antenna to transmit antenna,

, are the transmitted symbols,

are the noise at time slot 1 on receive antenna 1, 2 respectively and

are the noise at time slot 2 on receive antenna 1, 2 respectively.

Combining the equations at time slot 1 and 2,

Let us define .

To solve for , we know that we need to find the inverse of .

We know, for a general m x n matrix, the pseudo inverse is defined as,

The term,

Since this is a diagonal matrix, the inverse is just the inverse of the diagonal elements, i.e

The estimate of the transmitted symbol is,

Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Group them into pair of two symbols

(d) Equalize the received symbols

(e) Perform hard decision decoding and count the bit errors

(f) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for computing BER for 2 transmit 2 receive Alamouti STBC for BPSK modulation in flat fading Rayleigh channel

Figure: BER plot for 2 transmi 2 receive Alamouti STBC

Observations

1. Can observe that the BER performance is much better than 1 transmit 2 receive MRC case. This is because the effective channel concatenating the information from 2 receive antennas over two symbols results in a diversity order of 4.

2. In general, with receive antennas, the diversity order for 2 transmit antenna Alamouti STBC is .

3. As with the case of 2 transmit, 1 receive Alamouti STBC, the fact that is a diagonal matrix ensured that there is no cross talk between , after the equalizer and the noise term is still white.

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

A Simple Transmit Diversity Technique for Wireless Communication Siavash M Alamouti, IEEE Journal on selected areas in Communication, Vol 16, No, 8, October 1998

Alamouti STBC

Krishna Sankar — Thu, 16 Oct 2008 01:16:32 +0000

In the recent past, we have discussed three receive diversity schemes – Selection combining, Equal Gain Combining and Maximal Ratio Combining. All the three approaches used the antenna array at the receiver to improve the demodulation performance, albeit with different levels of complexity. Time to move on to a transmit diversity scheme where the information is spread across multiple antennas at the transmitter. In this post, lets discuss a popular transmit diversity scheme called Alamouti Space Time Block Coding (STBC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Alamouti STBC

A simple Space Time Code, suggested by Mr. Siavash M Alamouti in his landmark October 1998 paper – A Simple Transmit Diversity Technique for Wireless Communication, offers a simple method for achieving spatial diversity with two transmit antennas. The scheme is as follows:

1. Consider that we have a transmission sequence, for example

2. In normal transmission, we will be sending in the first time slot, in the second time slot, and so on.

3. However, Alamouti suggested that we group the symbols into groups of two. In the first time slot, send and from the first and second antenna. In second time slot send and from the first and second antenna. In the third time slot send and from the first and second antenna.In fourth time slot, send and from the first and second antenna and so on.

4. Notice that though we are grouping two symbols, we still need two time slots to send two symbols. Hence, there is no change in the data rate.

5. This forms the simple explanation of the transmission scheme with Alamouti Space Time Block coding.

Figure: 2-Transmit, 1-Receive Alamouti STBC coding

Other Assumptions

1. The channel is flat fading – In simple terms, it means that the multipath channel has only one tap. So, the convolution operation reduces to a simple multiplication. For a more rigorous discussion on flat fading and frequency selective fading, may I urge you to review Chapter 15.3 Signal Time-Spreading from [DIGITAL COMMUNICATIONS: SKLAR]

2. The channel experience by each transmit antenna is independent from the channel experienced by other transmit antennas.

3. For the transmit antenna, each transmitted symbol gets multiplied by a randomly varying complex number . As the channel under consideration is a Rayleigh channel, the real and imaginary parts of are Gaussian distributed having mean and variance .

4. The channel experienced between each transmit to the receive antenna is randomly varying in time. However, the channel is assumed to remain constant over two time slots.

5. On the receive antenna, the noise has the Gaussian probability density function with

with and .

7. The channel is known at the receiver.

Receiver with Alamouti STBC

In the first time slot, the received signal is,

In the second time slot, the received signal is,

where

, is the received symbol on the first and second time slot respectively,
is the channel from transmit antenna to receive antenna,
is the channel from transmit antenna to receive antenna,
, are the transmitted symbols and
is the noise on time slots.

Since the two noise terms are independent and identically distributed,

For convenience, the above equation can be represented in matrix notation as follows:

Let us define . To solve for , we know that we need to find the inverse of .

We know, for a general m x n matrix, the pseudo inverse is defined as,

The term,

. Since this is a digonal matrix, the inverse is just the inverse of the diagonal elements, i.e

The estimate of the transmitted symbol is,

& = & (H^HH)^{-1}H^H\left(H\left[\begin{eqnarray}x_1 \\ x_2 \end{eqnarray}\right]+\left[\begin{eqnarray}n_1\\n_2^* \end{eqnarray}\right]\right)
\\&=&\left[\begin{eqnarray}x_1 \\ x_2 \end{eqnarray}\right] + (H^HH)^{-1}H^H\left[\begin{eqnarray}n_1\\n_2^* \end{eqnarray}\right]\\\end{eqnarray}" border="0" alt="" align="absmiddle" />.

If you compare the above equation with the estimated symbol following equalization in Maximal Ratio Combining, you can see that the equations are identical.

BER with Almouti STBC

Since the estimate of the transmitted symbol with the Alamouti STBC scheme is identical to that obtained from MRC, the BER with above described Alamouti scheme should be same as that for MRC. However, there is a small catch.

With Alamouti STBC, we are transmitting from two antenna’s. Hence the total transmit power in the Alamouti scheme is twice that of that used in MRC. To make the comparison fair, we need to make the total trannsmit power from two antennas in STBC case to be equal to that of power transmitted from a single antenna in the MRC case. With this scaling, we can see that BER performance of 2Tx, 1Rx Alamouti STBC case has a roughly 3dB poorer performance that 1Tx, 2Rx MRC case.

From the post on Maximal Ratio Combining, the bit error rate for BPSK modulation in Rayleigh channel with 1 transmit, 2 receive case is,

, where

With Alamouti 2 transmit antenna, 1 receive antenna STBC case,

and Bit Error Rate is

Key points

The fact that is a diagonal matrix ensured the following:
1. There is no cross talk between , after the equalizer.

2. The noise term is still white.

Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Group them into pair of two symbols

(d) Equalize the received symbols

(e) Perform hard decision decoding and count the bit errors

(f) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulating BER for 2 transmit, 1 receive Alamouti STBC coding for BPSK modulation in Rayleigh fading channel

Figure: BER plot for BPSK in Rayleigh channel with 2 Transmit and 1 Receive Alamouti STBC

Observations

Compared to the BER plot for nTx=1, nRx=2 Maximal ratio combining, we can see the Alamouti Space Time Block Coding has around 3dB poorer performance.

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

A Simple Transmit Diversity Technique for Wireless Communication Siavash M Alamouti, IEEE Journal on selected areas in Communication, Vol 16, No, 8, October 1998

Maximal Ratio Combining (MRC)

Krishna Sankar — Sun, 28 Sep 2008 06:11:26 +0000

This is the third post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. In the previous posts, we discussed selection diversity and equal gain combining (EGC).

In this post, we will discuss Maximal Ratio Combining (MRC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Background

We use the same constraints as defined in the Selection Diversity and Equal Gain Combining (EGC) post. Let me repeat the same.

1. We have N receive antennas and one transmit antenna.

4. The channel experience by each receive antenna is independent from the channel experienced by other receive antennas.

5. On each receive antenna, the noise has the Gaussian probability density function with

with and .

The noise on each receive antenna is independent from the noise on the other receive antennas.

6. At each receive antenna, the channel is known at the receiver.

7. In the presence of channel , the instantaneous bit energy to noise ratio at receive antenna is . For notational convenience, let us define,

Maximal Ratio Combining (MRC)

On the receive antenna, the received signal is,

where

is the received symbol on the receive antenna,
is the channel on the receive antenna,
is the transmitted symbol and
is the noise on receive antenna.

Expressing it in matrix form, the received signal is,

, where

is the received symbol from all the receive antenna

is the channel on all the receive antenna

is the transmitted symbol and

is the noise on all the receive antenna.

The equalized symbol is,

It is intuitive to note that the term,

i.e sum of the channel powers across all the receive antennas.

Note: The equations in the post refers the note on Receive diversity by Prof. RaviRaj Adve.

Effective Eb/No with Maximal Ratio Combining (MRC)

Earlier, we noted that in the presence of channel , the instantaneous bit energy to noise ratio at receive antenna is

Given that we are equalizing the channel with , with the receive antenna case, the effective bit energy to noise ratio is,

Effective bit energy to noise ratio in a N receive antenna case is N times the bit energy to noise ratio for single antenna case. Recall, this gain is same as the improvement which we got in Receive diversity for AWGN case

Click here to download Matlab/Octave script for plotting effective SNR with Maximal Ratio Combining in Rayleigh channel

Figure: Effective SNR with Maximal Ratio Combining in Rayleigh fading channel

Error rate with Maximal Ratio Combining (MRC)

From the discussion on chi-square random variable, we know that, if is a Rayleigh distributed random variable, then is a chi-squared random variable with two degrees of freedom. The pdf of is
.

Since the effective bit energy to noise ratio is the sum of such random variables, the pdf of is a chi-square random variable with degrees of freedom. The pdf of is,
.

If you recall, in the post on BER computation in AWGN, with bit energy to noise ratio of , the bit error rate for BPSK in AWGN is derived as

Given that the effective bit energy to noise ratio with maximal ratio combining is, the total bit error rate is the integral of the conditional BER integrated over all possible values of .

&=&\int_0^{\infty}\frac{1}{2}erfc\left(\sqrt{\gamma}\right) \frac{1}{(N-1)!(E_b/N_0)^N}\ \gamma^{N-1}e^{\frac{-\gamma}{(E_b/N_0)}}d\gamma\end{eqnarray}" border="0" alt="" align="absmiddle" />.

This equation reduces to

, where

Refer Equation 11.12 and Equation 11.13 in Section 11.3.1 Performance with Maximal Ratio Combining in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]. Again, I do not know the proof

BER Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Multiply the symbols with the channel and then add white Gaussian noise.

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulating BER for BPSK in Rayleigh channel with Maximal Ratio Combining

Figure: BER plot for BPSK in Rayleigh channel with Maximal Ratio Combining

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Receive diversity – Notes by Prof. Raviraj Adve

Equal Gain Combining (EGC)

Krishna Sankar — Fri, 19 Sep 2008 17:53:26 +0000

This is the second post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. In the previous post, we discussed selection diversity. In this post, we will discuss equal gain combining (EGC). For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Background

We use the same constraints as defined in the Selection Diversity post. Let me repeat the same.

1. We have N receive antennas and one transmit antenna.

4. The channel experience by each receive antenna is independent from the channel experienced by other receive antennas.

5. On each receive antenna, the noise has the Gaussian probability density function with

with and .

The noise on each receive antenna is independent from the noise on the other receive antennas.

6. At each receive antenna, the channel is known at the receiver.

7. In the presence of channel , the instantaneous bit energy to noise ratio at receive antenna is . For notational convenience, let us define,

From the discussion on chi-square random variable, we know that, if is a Rayleigh distributed random variable, then is a chi-squared random variable with two degrees of freedom. The pdf of is
.

Equal Gain Combining (EGC)

On the receive antenna, equalization is performed at the receiver by dividing the received symbol by the apriori known phase of . The channel is represented in polar form as . The decoded symbol is the sum of the phase compensated channel from all the receive antennas.

where
is the additive noise scaled by the phase of the channel coefficient.

For demodulation, we use the classical definition i.e.

0 \rightarrow 1" border="0" alt="" align="absmiddle" /> and

Note:

For PSK modulation schemes, the equalization by the phase of the channel coefficients suffice. However, for QAM case, we need to compensate for the amplitude also when equalizing. We are NOT discussing QAM case in this post.

Effective Eb/N0 with Equal Gain Combining

The equations listed below obtained from the article Receive diversity – Notes by Prof. Raviraj Adve.

In the presence of channel , the instantaneous bit energy to noise ratio at receive antenna is . For notational convenience, let us define,

The effective Eb/N0 with equal gain combining is the channel power accumulated over all receive chains, i.e.

The first term is chi-square random variable with degrees of freedom having mean value of . Hence the first term reduces to,

The second term is a product of two Rayleigh random variables. The mean of Rayleigh random variable with variance is . Hence the second term is,

Simplifying, the effective Eb/N0 with equal gain combining is,

Click here to download Matlab/Ocatve script for computing effective SNR with equal gain combining

Figure : Gain in Eb/N0 with equal gain combining

Bit Error Rate with Equal Gain Combining

The IEEE paper [ZHANG97] discuss the BER computation with Equal Gain Combining. Since the proof is tedious (and I did not understand) I am just noting the final results.

With two receive antennas, the BER with equal gain combining is,

Matlab model for simulating BER with Equal Gain Combining

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Multiply the symbols with the channel and then add white Gaussian noise.

(d) Accumulate the equalized symbols from all the receive paths

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave code for simulating BER for BPSK in Rayleigh fading channel with Equal Gain Combining

Figure: BER plot Receive Diversity with Equal Gain Combining

Can observe that the simulation results are in good agreement with the theoretical results.

Reference

[ZHANG97] Probability of error for equal-gain combiners over Rayleigh channels: some closed-form solutions Zhang, Q.T. Communications, IEEE Transactions on Volume 45, Issue 3, Date: Mar 1997, Pages: 270 – 273

Receive diversity – Notes by Prof. Raviraj Adve

Selection Diversity

Krishna Sankar — Sat, 06 Sep 2008 09:44:11 +0000

This is the first post in the series discussing receiver diversity in a wireless link. Receiver diversity is a form of space diversity, where there are multiple antennas at the receiver. The presence of receiver diversity poses an interesting problem – how do we use ‘effectively‘ the information from all the antennas to demodulate the data. There are multiple ways to approach the problem. The three typical approaches to be discussed are – selection diversity, equal gain combining and maximal ratio combining. In this post we will discuss selection diversity. For the discussion, we will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Background

1. We have N receive antennas and one transmit antenna.

4. The channel experience by each receive antenna is independent from the channel experienced by other receive antennas.

5. On each receive antenna, the noise has the Gaussian probability density function with

with and .

The noise on each receive antenna is independent from the noise on the other receive antennas.

6. At each receive antenna, the channel is known at the receiver. For example, on the receive antenna, equalization is performed at the receiver by dividing the received symbol by the apriori known i.e.

where
is the additive noise scaled by the channel coefficient.

7. In the presence of channel , the instantaneous bit energy to noise ratio at receive antenna is . For notational convenience, let us define,

From the discussion on chi-square random variable, we know that, if is a Rayleigh distributed random variable, then is a chi-squared random variable with two degrees of freedom. The pdf of is
.

What is selection diversity?

Consider a scenario where we have a single antenna for transmission and multiple antennas at the receiver (as shown in the figure below).

Figure: Receive diversity in a wireless link

At the receiver we have now N copies of the same transmitted symbol. Which then poses the problem – how to effectively combine them to reliably recover the data.

Selection diversity approach is one way out – With selection diversity, the receiver selects the antenna with the highest received signal power and ignore observations from the other antennas. The chosen receive antenna is one which gives .

Outage probability in Selection Diversity

The equations in the post refers the note on Receive diversity by Prof. RaviRaj Adve.

To analyze the bit error rate, let us first find the outage probability on the receive antenna. Outage probability is the probability that the bit energy to noise ratio falls below a threshold. The probability of outage on receive antenna is,

is the defined threshold for bit energy to noise ratio.

In N reveive antenna case, the probability that all bit energy to noise ratio on all the receive antenna are below the threshold is,

where

are the bit energy to noise ratio on the 1st, 2nd and so on till the Nth receive antenna.

Since the channel on each antenna is assumed to independent, the joint probability is the product of individual probabilities.

& = &\prod_{i=1}^{N}P[\gamma_i < \gamma_s]\\
& = & \left[1-e^{-\frac{\gamma_s}{(E_b/N_0)}} \right]^N\end{eqnarray}" border="0" alt="" align="absmiddle" />.

Note that the equation above defines the probability that the effective bit energy to noise ratio with N receive antennas (lets call ) is lower than the threshold . This is infact the cumulative distribuition function (CDF) of . The probability density function (PDF) is then the deriviate of the CDF.

Given that we know the PDF of , the average output bit energy to noise ratio is,

& = & \int_0^{\infty}\gamma \frac{N}{(E_b/N_0)}e^{-\frac{\gamma}{(E_b/N_0)}}\left[1-e^{-\frac{\gamma}{(E_b/N_0)}} \right]^{N-1} \\
& = & \frac{E_b}{N_0}\sum_{i=1}^N\frac{1}{i}\end{eqnarray}" border="0" alt="" align="absmiddle" />.

I do not know how to reduce the above integral to this simple sum.

This means that,

- with two receive antennas the effective bit energy to noise ratio is 1.5 times ,

- with three receive antennas, the effective bit energy to noise ratio is 1.833 times ,

- with four receive antennas, the effective bit energy to noise ratio is 2 times and so on.

If you recall the results from the AWGN with receive diversity case,

Effective bit energy to noise ratio in a N receive antenna case is N times the bit energy to noise ratio for single antenna case.

With selection diversity we are seeing that the effective SNR improvement is not a linear improvement with increasing the number of receive antennas. The returns diminish.

Figure: SNR gain with selection diversity

Click here to download Matlab/Octave script for computing the effective SNR in Ralyeigh channel with selection diversity

Bit Error probability with selection diversity

If you recall, in the post on BER computation in AWGN, with bit energy to noise ratio of , the bit error rate for BPSK in AWGN is derived as

Given that the effective bit energy to noise ratio with selection diversity is, the total bit error rate is the integral of the conditional BER integrated over all possible values of .

&=&\int_0^{\infty}\frac{1}{2}erfc\left(\sqrt{\gamma}\right)\frac{N}{(E_b/N_0)}e^{-\frac{\gamma}{(E_b/N_0)}}\left[1-e^{-\frac{\gamma}{(E_b/N_0)}} \right]^{N-1}d\gamma\end{eqnarray}" border="0" alt="" align="absmiddle" />.

This equation reduces to

Refer Equation 11.24 in Section 11.3.2 Performance with Selection combining in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]. Again, I do not know the proof

BER Simulation Model

The Matlab/Octave script performs the following

(a) Generate random binary sequence of +1’s and -1’s.

(b) Multiply the symbols with the channel and then add white Gaussian noise.

(d) Chose that receive path, equalize (divide) the received symbols with the known channel

(d) Perform hard decision decoding and count the bit errors

(e) Repeat for multiple values of and plot the simulation and theoretical results.

Click here to download Matlab/Octave script for simulating BER for BPSK in Rayleigh channel with selection diversity

Figure: BER plot for BPSK in Rayleigh channel with Selection Diversity

Observations

Around 16dB improvement at BER point by with two receive antenna selection diversity

References

Receive diversity – Notes by Prof. Raviraj Adve

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Happy learning.