dspLog » Channel

Derivation of BPSK BER in Rayleigh channel

Krishna Sankar — Wed, 21 Jan 2009 21:56:26 +0000

This is a guest post by Jose Antonio Urigüen who is an Electrical and Electronic Engineer currently studying an MSc in Communications and Signal Processing at Imperial College in London. This guest post has been created due to his own curiosity when reviewing some concepts of BER for BPSK in Rayleigh channnel published in the dsplog.com

From the post on BER for BPSK in Rayleigh channnel, it was shown that, in the presence of channel , the effective bit energy to noise ratio is .

The bit error probability for a given value of is,

, where

The resulting BER in a communications sytem in the presence of a channel , for any random values of , must be calculated evaluating the conditional probability density function over the probability density function of .

where,

the probability density function of is

and .

First, we are going to derive the result for the definite integral using a different notation, and then we will apply the result to the concrete expression obtained for the BER.

Using the definitions for the ‘erfc’ and ‘erf’ functions, we can directly compute the first derivative:

And also the definite integral, simply calculating it by parts:

& = & x erfc(x) -\frac{1}{\sqrt{\pi}}e^{-x^2}\end{eqnarray}" border="0" alt="" align="absmiddle" />

In the following steps we will use the next also straightforward results:

In total, we want to derive the following integral, which can be solved by parts:

Lets find the last term applying the previous result for the ‘erf’ function, and doing a change of variable:

Finally, we conclude the expected result:

With the above demonstration, we can easily derive the BER for a Rayleigh channel using BPSK modulation:

This forms the proof for BER for BPSK modulation in Rayleigh channel.

MIMO with ML equalization

Krishna Sankar — Sun, 14 Dec 2008 09:38:17 +0000

We have discussed quite a few receiver structures for a 2×2 MIMO channel namely,

(a) Zero Forcing (ZF) equalization

(b) Minimum Mean Square Error (MMSE) equalization

(d) ZF-SIC with optimal ordering and

(e) MIMO with MMSE SIC and optimal ordering

From the above receiver structures, we saw that MMSE equalisation with optimally ordered Successive Interference Cancellation gave the best performance. In this post, we will discuss another receiver structure called Maximum Likelihood (ML) decoding which gives us an even better performance. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

2×2 MIMO channel

In a 2×2 MIMO channel, probable usage of the available 2 transmit antennas can be 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, as we now have 2 transmit antennas, we may 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, send and in the third time slot and so on.

4. Notice that as we are grouping two symbols and sending them in one time slot, we need only time slots to complete the transmission – data rate is doubled !

5. This forms the simple explanation of a probable MIMO transmission scheme with 2 transmit antennas and 2 receive antennas.

Figure: 2 Transmit 2 Receive (2×2) MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,

The received signal on the second receive antenna is,

where

, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

We assume that the receiver knows , , and . The receiver also knows and . The unknown s are and .

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

Equivalently,

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 to 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 experienced between each transmit to the receive antenna is independent and randomly varying in time.

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

with and .

7. The channel is known at the receiver.

Maximum Likelihood (ML)Receiver

The Maximum Likelihood receiver tries to find which minimizes,

Since the modulation is BPSK, the possible values of is +1 or -1 Similarly also take values +1 or -1. So, to find the Maximum Likelihood solution, we need to find the minimum from the all four combinations of and .

The estimate of the transmit symbol is chosen based on the minimum value from the above four values i.e

if the minimum is ,

if the minimum is and
if the minimum 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 and send two symbols in one time slot

(d) Find the minimum among the four possible transmit symbol combinations

(e) Based on the minimum chose the estimate of the transmit symbol

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

Click here to download Script for computing BER for BPSK in 2×2 MIMO Rayleigh channel with Maximum Likelihood Equalization

BER plot 2x2 MIMO Rayleigh channel with Maximum Likelihood equalisation

FIgure: BER plot 2×2 MIMO Rayleigh channel with Maximum Likelihood equalisation

Summary

1. The results for 2×2 MIMO with Maximum Likelihood (ML) equalization helped us to achieve a performance closely matching the 1 transmit 2 receive antenna Maximal Ratio Combining (MRC) case.

2. If we use a higher order constellation like 64QAM, then computing Maximum Likelihood equalization might become prohibitively complex. With 64QAM and 2 spatial stream we need to find the minimum from combinations ! In such scenarios we might need to employ schemes like sphere decoding which helps to reduce the complexity.

References

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

[WIRELESS-TSE, VISWANATH] Fundamentals of Wireless Communication, David Tse, Pramod Viswanath

MIMO with MMSE SIC and optimal ordering

Krishna Sankar — Sat, 06 Dec 2008 09:56:26 +0000

This post attempts to build further on the MIMO equalization schemes which we have discussed -

(a) Minimum Mean Square Error (MMSE) equalization,

(b) Zero Forcing equalization with Successive Interference Cancellation (ZF-SIC) and

We have learned that successive interference cancellation with optimal ordering improves the performance with Zero Forcing equalization. In this post, we extend the concept of successive interference cancellation to the MMSE equalization and simulate the performance. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Brief description of 2×2 MIMO transmission, assumptions on channel model and the noise are detailed in the post on Minimum Mean Square Error (MMSE) equalization.

MMSE equalizer for 2×2 MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,

The received signal on the second receive antenna is,

where

, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

We assume that the receiver knows , , and . The receiver also knows and . For convenience, the above equation can be represented in matrix notation as follows:

Equivalently,

The Minimum Mean Square Error (MMSE) approach tries to find a coefficient which minimizes the criterion,

Solving,

Using the Minimum Mean Square Error (MMSE) equalization, the receiver can obtain an estimate of the two transmitted symbols , , i.e.

Successive Interference Cancellation

(a) Simple

In classical Successive Interference Cancellation, the receiver arbitrarily takes one of the estimated symbols (for example the symbol transmitted in the second spatial dimension, ), and subtract its effect from the received symbol and . Once the effect of is removed, the new channel becomes a one transmit antenna, 2 receive antenna case and can be optimaly equalized by Maximal Ratio Combining (MRC).

(b) With optimal ordering

However, we can have more intelligence in choosing whether we should subtract the effect of first or first. To make that decision, let us find out the transmit symbol (after multiplication with the channel) which came at higher power at the receiver. The received power at the both the antennas corresponding to the transmitted symbol is,

The received power at the both the antennas corresponding to the transmitted symbol is,

If P_{x_2}" border="0" alt="" align="absmiddle" /> then the receiver decides to remove the effect of from the received vector and . Else if the receiver decides to subtract effect of from the received vector and , and then re-estimate .

Once the effect of either or is removed, the new channel becomes a one transmit antenna, 2 receive antenna case and the symbol on the other spatial dimension can be optimally equalized by Maximal Ratio Combining (MRC).

For detailed equations on the contruction of the new 2 x 1 channel using successive interference cancellation, please refer to the post on ZF-SIC with optimal ordering.

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 and send two symbols in one time slot

(d) Equalize the received symbols with Minimum Mean Square Error criterion

(e) Do successive interference cancellation by both classical and optimal ordering approach

(f) Perform Maximal Ratio Combining for equalizing the new received symbol

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

(h) 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 2×2 Rayleigh fading MIMO channel with MMSE-SIC equalization with and without optimal ordering

Figure; BER plot for 2×2 MIMO channel with MMSE-SIC equalization with and without optimal ordering

Observations

Compared to Minimum Mean Square Equalization with simple successive interference cancellation case, addition of optimal ordering results in around 5.0dB of improvement for BER of .

The performance is now closely matching with curve 1 transmit 2 receive antenna MRC case.

References

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

MIMO with ZF SIC and optimal ordering

Krishna Sankar — Sat, 29 Nov 2008 08:34:19 +0000

In previous posts, we had discussed equalization of a 2×2 MIMO channel with Zero Forcing (ZF) equalization and later, Zero Forcing equalization with successive interference cancellation (ZF-SIC). In this post, we will explore a variant of ZF-SIC called Zero Forcing Successive Interference Cancellation with optimal ordering. We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

Brief description of 2×2 MIMO transmission, assumptions on channel model and the noise are detailed in the post on Zero Forcing equalization with successive interference cancellation

Zero forcing equalizer for 2×2 MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,

The received signal on the second receive antenna is,

where

, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

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

Equivalently,

To solve for , The Zero Forcing (ZF) linear detector for meeting this constraint . is given by,

Using the Zero Forcing (ZF) equalization, the receiver can obtain an estimate of the two transmitted symbols , , i.e.

Successive Interference Cancellation with optimal ordering

In classical Successive Interference Cancellation, the receiver arbitrarily takes one of the estimated symbols, and subtract its effect from the received symbol and . However, we can have more intelligence in choosing whether we should subtract the effect of first or first. To make that decision, let us find out the transmit symbol (after multiplication with the channel) which came at higher power at the receiver. The received power at the both the antennas corresponding to the transmitted symbol is,

The received power at the both the antennas corresponding to the transmitted symbol is,

If P_{x_2}" border="0" alt="" align="absmiddle" /> then the receiver decides to remove the effect of from the received vector and and then re-estimate .

Expressing in matrix notation,

Optimal way of combining the information from multiple copies of the received symbols in receive diversity case is to apply Maximal Ratio Combining (MRC). The equalized symbol is,

Else if the receiver decides to subtract effect of from the received vector and , and then re-estimate

Expressing in matrix notation,

Optimal way of combining the information from multiple copies of the received symbols in receive diversity case is to apply Maximal Ratio Combining (MRC). The equalized symbol is,

Doing successive interference cancellation with optimal ordering ensures that the reliability of the symbol which is decoded first is guaranteed to have a lower error probability than the other symbol. This results in lowering the chances of incorrect decisions resulting in erroneous interference cancellation. Hence gives lower error rate than simple successive interference cancellation.

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 and send two symbols in one time slot

(d) Equalize the received symbols with Zero Forcing criterion

(e) Find the power of received symbol from both the spatial dimensions.

(f) Take the symbol having higher power, subtract from the received symbol

(f) Perform Maximal Ratio Combining for equalizing the new received symbol

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

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

Click here to download Matlab/Octave script for computing BER for 2×2 MIMO channel equalized by ZF-SIC with optimal ordering

Figure: BER plot for BPSK in 2×2 MIMO equalized by ZF-SIC with optimal ordering

Observations

Compared to Zero Forcing equalization with successive interference cancellation case, addition of optimal ordering results in around 2.0dB of improvement for BER of .

References

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

MIMO with Zero Forcing Successive Interference Cancellation equalizer

Krishna Sankar — Sun, 09 Nov 2008 13:42:59 +0000

The post on MIMO with Zero Forcing equalizer discussed a probable way of equalizing a 2×2 MIMO channel. The simulated results with the 2×2 MIMO system with zero forcing equalizer showed matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel. In this post, we will try to improve the bit error rate performance by trying out Successive Interference Cancellation (SIC). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

The background material on the MIMO channel has been described in the post on Zero Forcing equalizer. The text is repeated again for easy readability.

2×2 MIMO channel

In a 2×2 MIMO channel, probable usage of the available 2 transmit antennas can be 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.

4. Notice that as we are grouping two symbols and sending them in one time slot, we need only time slots to complete the transmission – data rate is doubled !

5. This forms the simple explanation of a probable MIMO transmission scheme with 2 transmit antennas and 2 receive antennas.

Figure: 2 Transmit 2 Receive (2×2) MIMO channel

Other Assumptions

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

4. The channel experienced between each transmit to the receive antenna is independent and randomly varying in time.

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

with and .

7. The channel is known at the receiver.

Zero forcing equalizer for 2×2 MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,

The received signal on the second receive antenna is,

where

, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

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

Equivalently,

To solve for , The Zero Forcing (ZF) linear detector for meeting this constraint . is given by,

To do the Successive Interference Cancellation (SIC), the receiver needs to perform the following:

Zero Forcing with Successive Interference Cancellation (ZF-SIC)

Using the Zero Forcing (ZF) equalization approach described above, the receiver can obtain an estimate of the two transmitted symbols , , i.e.

Take one of the estimated symbols (for example ) and subtract its effect from the received vector and , i.e.

Expressing in matrix notation,

The above equation is same as equation obtained for receive diversity case. Optimal way of combining the information from multiple copies of the received symbols in receive diversity case is to apply Maximal Ratio Combining (MRC).

The equalized symbol is,

This forms the simple explanation for Zero Forcing Equalizer with Successive Interference Cancellation (ZF-SIC) approach.

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 and send two symbols in one time slot

(d) Equalize the received symbols with Zero Forcing criterion

(e) Take the symbol from the second spatial dimension, subtract from the received symbol

(f) Perform Maximal Ratio Combining for equalizing the new received symbol

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

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

Click here to download Matlab/Octave script for simulating BER for BPSK modulation in 2×2 MIMO with Zero Forcing and Successive Interference Cancellation equalization (in Rayleigh channel)

Figure: BER plot for BPSK in 2×2 MIMO channel with Zero Forcing Successive Interference Cancellation equalization

Observations

Compared to Zero Forcing equalization alone case, addition of successive interference cancellation results in around 2.2dB of improvement for BER of .

The improvement is brought in because decoding of the information from the first spatial dimension () has a lower error probability that the symbol transmitted from the second dimension. However, the assumption is that is decoded correctly may not be true in general. We can discuss alternate approaches in future posts.

References

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

[WIRELESS-TSE, VISWANATH] Fundamentals of Wireless Communication, David Tse, Pramod Viswanath

MIMO with Zero Forcing equalizer

Krishna Sankar — Fri, 24 Oct 2008 00:47:58 +0000

We had discussed three Single Input Multiple Output (SIMO also known as receive diversity) schemes – Selection combining, Equal Gain Combining, Maximal Ratio Combining and a Multiple Input Single Output (MISO, also known as transmit diversity) scheme – Alamouti 2×1 STBC. Let us now discuss the case where there a multiple transmit antennas and multiple receive antennas resulting in the formation of a Multiple Input Multiple Output (MIMO) channel. In this post, we will restrict our discussion to a 2 transmit 2 receive antenna case (resulting in a 2×2 MIMO channel). We will assume that the channel is a flat fading Rayleigh multipath channel and the modulation is BPSK.

2×2 MIMO channel

In a 2×2 MIMO channel, probable usage of the available 2 transmit antennas can be 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.

4. Notice that as we are grouping two symbols and sending them in one time slot, we need only time slots to complete the transmission – data rate is doubled !

5. This forms the simple explanation of a probable MIMO transmission scheme with 2 transmit antennas and 2 receive antennas.

Having said this, some of you will wonder – the two transmitted symbols interfered with each other. Can we ever separate the two out? The rest of the post attempts to answer this question.

Figure: 2 Transmit 2 Receive (2×2) MIMO channel

Other Assumptions

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

4. The channel experienced between each transmit to the receive antenna is independent and randomly varying in time.

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

with and .

7. The channel is known at the receiver.

Zero forcing (ZF) equalizer for 2×2 MIMO channel

Let us now try to understand the math for extracting the two symbols which interfered with each other. In the first time slot, the received signal on the first receive antenna is,

The received signal on the second receive antenna is,

where

, are the received symbol on the first and second antenna respectively,

is the channel from transmit antenna to receive antenna,

, are the transmitted symbols and

is the noise on receive antennas.

We assume that the receiver knows , , and . The receiver also knows and . The unknown s are and . Two equations and two unknowns. Can we solve it? Answer is YES.

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

Equivalently,

To solve for , we know that we need to find a matrix which satisfies . The Zero Forcing (ZF) linear detector for meeting this constraint is given by,

This matrix is also known as the pseudo inverse for a general m x n matrix.

The term,

BER with ZF equalizer with 2×2 MIMO

Note that the off diagonal terms in the matrix are not zero (Recall: The off diagonal terms where zero in Alamouti 2×1 STBC case). Because the off diagonal terms are not zero, the zero forcing equalizer tries to null out the interfering terms when performing the equalization, i.e when solving for the interference from is tried to be nulled and vice versa. While doing so, there can be amplification of noise. Hence Zero Forcing equalizer is not the best possible equalizer to do the job. However, it is simple and reasonably easy to implement.

Further, it can be seen that, following zero forcing equalization, the channel for symbol transmitted from each spatial dimension (space is antenna) is a like a 1×1 Rayleigh fading channel (Refer Section 3.3 of [WIRELESS-TSE, VISWANATH]). Hence the BER for 2×2 MIMO channel in Rayleigh fading with Zero Forcing equalization is same as the BER derived for a 1×1 channel in Rayleigh fading.

For BPSK modulation in Rayleigh fading channel, the bit error rate is derived as,

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 and send two symbols in one time slot

(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 plot for 2×2 MIMO channel with ZF equalizer (BPSK modulation in Rayleigh channel)

Figure: BER plot for 2×2 MIMO channel with ZF equalizer (BPSK modulation in Rayleigh channel)

Summary

1. As expected, the simulated results with a 2×2 MIMO system using BPSK modulation in Rayleigh channel is showing matching results as obtained in for a 1×1 system for BPSK modulation in Rayleigh channel.

2. As noted in Section 3.3 of [WIRELESS-TSE, VISWANATH], the Zero Forcing equalizer is not the best possible way to equalize the received symbol. The zero forcing equalizer helps us to achieve the data rate gain, but NOT take advantage of diversity gain (as we have two receive antennas).

3. We might not be able to achieve the two fold data rate improvement in all channel conditions. It can so happen that channels are correlated (the coefficients are almost the same). Hence we might not be able to solve for the two unknown transmitted symbols even if we have two received symbols.

4. It is claimed that there can be receiver structures which enables us to have both diversity gain and data rate gain. In future posts, the attempt will be to discuss receiver structures which hopefully enables us to find out approaches which will help us to keep the data rate gain, but still move from the 1×1 curve to 1×2 MRC curve.

References

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

[WIRELESS-TSE, VISWANATH] Fundamentals of Wireless Communication, David Tse, Pramod Viswanath

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

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