Binary symmetric channel review

Binary Symmetric Channel

1. INTRODUCTION TO CHANNEL MODEL AND CHANNEL CAPACITY

2. PRESENTED BY- SHILPA DE(35000116019)

3. CONTENTS: BINARY SYMMETRIC CHANNEL DISCRETE MEMORYLESS CHANNEL TYPES OF CHANNEL EXAMPLES OF CHANNEL CHANNEL CAPACITY CROSSOVER PROBABILITY

4. BINARY SYMMETRIC CHANNEL A Binary Symmetric Channel is viewed as , a binary bit stream enters at the transmitting end and another bit stream comes out at the receiving end.

5. Representation of a Binary Symmetric Channel This binary Discrete-input ,Discrete –output channel is characterized by the possible input set X={0,1} and possible output set Y={0,1} and a set of conditional probabilities that relate X and Y.

6. Let the noise in the channel cause independent errors with average probability of error p. P(Y=0|X=1) = P(Y=1|X=0) = p P(Y=1|X=1) = P(Y=0|X=0) = 1-p A BSC is a special case of Discrete Memoryless Channel.

7. Representation of DMC Discrete: implies when input set X and output set Y gives us finite values. Memory less : Implies when current o/p value depends on current i /p value , not previous i /p value. X Y x1 x2 x3 . ; xm y 1 y 2 y 3 . . y m P( Yi|Xj )

8. DEFINITION OF DMC When channel accept a input symbol X, and in respond generate output symbol Y, this input and output along with a conditional probability called DMC. The conditional probability- P(Y= y i | X= x j ) = P( y i | x j ) that characterized a DMC is arranged in the matrix form called the Probability transition matrix.

9. TYPE OF CHANNELS: 1)Single Input Single Output (SISO) 2) Single Input Multiple Output (SIMO) 3) Multiple Input Single Output (MISO) 4) Multiple Input Multiple Output (MIMO)

10. single input single output In the diagram, S is input , Y is output, X T is Transmitting antenna and Y R is the Receiving antenna. Where C is the capacity. B is the Bandwidth of the signal and S/N is the signal to noise ratio.

11. In the diagram, S is input,Y 1 and Y 2 are two output from two receiving antenna , X T is transmitting antenna. Y R1 and Y R2 are two receiving antenna. The channel capacity of the SIMO system is Single Input Multiple Output

12. In the diagram,S 1 and S 2 are inputs from Transmitting antenna. X T1 and X T2 are two Transmitting antenna. Y R receiving antenna The capacity of this system is C is the capacity, M T is the number of antennas used at transmitter side, B is Bandwidth Of the signal and S/N is the signal to noise ratio. Multiple Input Single Output

13. Multiple Input Multiple Output

14. SOME EXAMPLE OF CHANNEL RELAY CHANNEL: In Relay channels there is a source ,a destination and intermediate relay nodes. This relay nodes facilitate communicate between source and destination. There is two way to facilitate the transfer of information- 1)Amplify-and-Forward 2)Decode-and-Forward

15. RELAY CHANNEL Amplify-and-Forward: Each relay node simply amplifies the received signal and forward it to the next relay node , maintaining a fixed average transmit power. Decode-and-Forward: The relay node can first decode the received signal and then re-encodes the signal before forwarding it to the next relay node .

16. MULTIPLE ACCESS CHANNEL In Multiple Access Channel, Suppose M transmitters wants to communicate with a single receiver over a common channel.

17. BROADCAST CHANNEL In Broadcast Channel a single transmitter wants to communicate with M receivers over a common channel.

18. CHANNEL CAPACITY: The channel capacity of a discrete memoryless channel is defined as- The maximum average mutual information in any single use of the channel,where the maximization is over all possible input probabilities .

19. C=max I(x;y) p(xj) Where average mutual information provided by the output y about input x is given by- q-1 r-1 I(x;y)=∑ ∑ p(xj)p(yi|xj) log(yi|xj)/p(yi) j=0 i=0 where, P(xj)=input symbol probability P(yi)=output symbol probability P(yi|xj)=channel transition probability(determined by channel characteristics)

20. So, C=max I(x;y) p(xj) q-1 r-1 = max∑ ∑p(xj)p(yi|xj) logp(yi|xj)/p(yi) j=0 i=0 The maximization of I(x;y) is performed under the constraints P(xj)>=0 and q-1 ∑p(xj)=1 j=0

21. Units: The units of channel capacity is bits/channel use(where base of logarithm is 2) If base of the logarithm is e,the units of channel capacity will be nats/channel use(coming from natural logarithm)

22. Crossover probability : In case of BSC with channel transition probability p(0|1)=p(1|0)=p Thus,the transition probability matrix is given by P= 1-p p p 1-p Here,P is reffered to as crossever probability.

23. Now by symmetry,the capacity- C=max I(x;y) P(xj) Is achieved for input probabilities p(0)=p(1)=0.5 So,we obtain the capacity of a BSC as C=max I(x;y) =max H(yi)-H(yi|xj) =1-(+plog(1/p)+(1-p)log(1/1-p)) =1+plogp+(1-p)log(1-p)

24. Let us define the entropy function, H(yi|xj)=H(p)=plog(1/p)+(1-p)log(1/1-p) =-plogp-(1-p)log(1-p) Hence,we can rewrite the capacity of a binary symmetric channel as C=1-H(p)

25. The capacity of a BSC is-

26. Now,from previous equation of channel capacity- For p=0(noise free channel),the capacity is 1bit/channel use.Here we can successfully transmit 1 bit of information. For p=0.5,the channel capacity is 0.Output gives no information about input.it occurs when the channel is broken. For 0.5<p<1,the capacity increases with increasing p. For p=1,again channel capacity is 1 bit/channel use.

27. REFFERENCE: INFORMATION THEORY,CODING AND CRYPTOGRAPHY(RANJAN BOSE) www.geeksforgeeks.com www.wikepedia.com

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