AUC Course Project

1. ECNG 436/4312 Course Project
Part I : MIMO Detection
Mina Yonan
m.yonan@cu.edu.eg
m.yonan@aucegypt.edu

2. Single User SISO decoder
Going back to the simple system model
Example:
• Assume the sender transmits a BPSK signal, i.e. The transmitted
symbols are chosen from the set {𝑥0, 𝑥1} .
• Consider a discrete model where we have 𝑁 observations
• Consider the hypotheses :
ℋ0
∶ 𝑦 𝑛 = 𝑥0(𝑛) + 𝑤0 𝑛 𝑤ℎ𝑒𝑟𝑒 𝑛 = 0, … , 𝑁 − 1
ℋ1
∶ 𝑦 𝑛 = 𝑥1(𝑛) + 𝑤1 𝑛 𝑤ℎ𝑒𝑟𝑒 𝑛 = 0, … , 𝑁 − 1
where 𝑤 𝑛 is AWG noise 𝑁 0, 𝜎2
How is the receiver designed?

3. Single User SISO decoder
• This is a single-user detection problem in communication
theory
• Objective: Minimize the error probability
𝑃𝑒 = 𝑃 ℋ0
ℋ1
) 𝑃 ℋ1
+ 𝑃 ℋ1
ℋ0
) 𝑃 ℋ0
• Solution (Statistical Decision Theory):
– We first construct the hypothesis tests
– Decide an optimum threshold:
• Bayesian approach : The detector that minimizes the error
probability is the conditional likelihood ratio test

4. Single User SISO decoder
- The optimal detector (using Bayesian approach)
𝑃 ℋ0
𝒚) > 𝑃 ℋ1
𝒚)
• If the prior probabilities are known
𝑃 𝒚 ℋ0
)𝑃(ℋ0
)
𝑃(𝒚)
>
𝑃 𝒚 ℋ1
)𝑃(ℋ1
)
𝑃(𝒚)
• If the prior probabilities are equal (unknown)
𝑃 𝒚 ℋ0
) > 𝑃 𝒚 ℋ1
)
MAP
a posteriori probability
Maximum Likelihood
For more reading : ch3: S.M. Kay “Detection”

5. Multiple Input Multiple Output (intro.)
𝑳 𝒚 =
𝑃 𝒚 ℋ0
)
𝑃 𝒚 ℋ1
)
>
𝑃(ℋ1
)
𝑃(ℋ0
)
= γ
ML
𝑃 𝒚 ℋ0
) =
1
(2𝜋𝜎2)
𝑁
2
exp
−1
2𝜎2
෍
𝑛=0
𝑁−1
(𝑦 𝑛 − 𝑥0(𝑛))2
𝑃 𝒚 ℋ1
) =
1
(2𝜋𝜎2)
𝑁
2
exp
−1
2𝜎2
෍
𝑛=0
𝑁−1
(𝑦 𝑛 − 𝑥1(𝑛))2
=
−1
2
෍
𝑛=0
𝑁−1
𝑥0
2
(𝑛) +
−1
2
෍
𝑛=0
𝑁−1
𝑥1
2
(𝑛) − ෍
𝑛=0
𝑁−1
𝑦 𝑛 𝑥0 𝑛 + ෍
𝑛=0
𝑁−1
𝑦 𝑛 𝑥1 𝑛 > 𝜎2ln(γ)
𝑳 𝒚 = exp
−1
2𝜎2 ෍
𝑛=0
𝑁−1
(𝑦 𝑛 − 𝑥0(𝑛))2
− ෍
𝑛=0
𝑁−1
(𝑦 𝑛 − 𝑥1(𝑛))2
> γ
Assume real signals
−1
2
𝐸0
1
2
𝐸1

6. • Neyman-Pearson detector using matched-filter
- Matched-filter receiver (ℎ0(𝑛) = 𝑥0
∗
𝑁 − 1 − 𝑛 , ℎ1(𝑛) = 𝑥1
∗
𝑁 − 1 − 𝑛 , where ()∗
denotes conjugate)
- For real signals the conjugate operation ()∗ is replaced by the transpose operation () 𝑇
Single User SISO decoder

7. Multiple Input Multiple Output (intro.)
• Rayleigh Flat Fading VS AWGN
AWGNFading Channel
𝑦 = 𝑥 + 𝑛𝑦 = ℎ𝑥 + 𝑛System
Model
- BPSK : 𝑥 = ±𝑎
- 𝑛~ 𝑁(0,1)
- BPSK : 𝑥 = ±𝑎
- ℎ is Rayleigh distribution
ℎ ~ 𝐶𝑁(0,1) and 𝑛~ 𝑁(0,1)
Assumptions
𝑃(𝑒) = 𝑄 2 𝑆𝑁𝑅
𝑄 𝑥 <
1
𝑥
𝑒−𝑥2
𝑃 𝑒 ≤
1
2 𝑆𝑁𝑅
𝑒−2𝑆𝑁𝑅
𝑃 𝑒 ℎ = 𝑄 2 |ℎ|2 𝑆𝑁𝑅
𝑃(𝑒) =
1
2
(1 −
𝑆𝑁𝑅
1+𝑆𝑁𝑅
)
At high SNR 𝑃(𝑒) ≅
1
4𝑆𝑁𝑅
Probability of
error

8. Multiple Input Multiple Output (intro.)
• Communication System over Fading Channels :

9. Multiple Input Multiple Output (Diversity)
Diversity : Sending signals that carry the same information through
different paths, time ,frequency, multiple independently faded
replicas of data symbols are obtained at the receiver end and more
reliable detection can be achieved
Types :
- Repetition coding
- Time diversity
- Frequency diversity
- Space diversity

10. Multiple Input Multiple Output (Diversity)
• Can we improve the error probability to
1
𝑆𝑁𝑅 𝐿
• Yes, we will see some diversity techniques to achieve this.
• The error exponent at high SNR can be roughly defined as the
diversity (the slope of the error curve at high SNR)
Formally, the diversity can be defined as

11. Multiple Input Multiple Output (Diversity)

12. Multiple Input Multiple Output (Diversity)
- MIMO technique :
• spatial diversity :
- Purpose : improve reliability
- Type : Receive diversity
- Method : Combiner (Selective Combining , Equal gain combining,
Maximal Ratio Combing )

13. Multiple Input Multiple Output (Diversity)
- MIMO technique :
• Spatial diversity :
- Purpose : improve reliability
- Type : transmit diversity
- Method : Space-Time Coding

14. Space-Time Coding (Alamouti scheme)

15. For more reading : David Tse, wireless communication Ch:4
Space-Time Coding (Alamouti scheme)

16. For more reading : David Tse, wireless communication Ch:4
Space-Time Coding (Alamouti scheme)

17. For more reading : David Tse, wireless communication Ch:4
Space-Time Coding (Alamouti scheme)

18. Multiple Input Multiple Output (Space-Time
Coding)
For more reading : David Tse, wireless communication Ch:4

19. Multiple Input Multiple Output (Space-Time
Coding)
For more reading : David Tse, wireless communication Ch:4

20. Multiple Input Multiple Output (Spatial
multiplexing)
- MIMO technique :
Spatial multiplexing : SM refers to transmitting multiple data
streams over a multipath channel by exploiting multipath. By so doing,
multiple data channels are able to be transmitted simultaneously over
the same frequency band.
- Purpose : increase throughput

21. Multiple Input Multiple Output (Spatial
multiplexing)

22. Multiple Input Multiple Output (Spatial
multiplexing)
• Let us start with MIMO deterministic channel (i.e. fixed
channel)
• Assume 𝑛 𝑡transmit antennas and 𝑛 𝑟 receive antennas
𝑦 𝑛 𝑟×1 =
ℎ11 … ℎ1×𝑛 𝑡
⋮ ⋱ ⋮
ℎ 𝑛 𝑟×1 ⋯ ℎ 𝑛 𝑟×𝑛 𝑡
𝑥1
⋮
𝑥 𝑛 𝑡
+ 𝑁 𝑛 𝑟×1

23. Multiple Input Multiple Output (Spatial
multiplexing)
• Spatial Multiplexing
What’s Q ?

24. Multiple Input Multiple Output (Receiver
Structure (MLD) )

25. Multiple Input Multiple Output (Receiver
Structure (MLD) )

26. Multiple Input Multiple Output (Zero-Forcing)