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Midterm 1, due by 10.00 am, Wednesday 07.25.18 (This is an individual test, worth 150 points. You may consult any printed/web-based resource you want, but you may not discuss this test with anybody.) July 23, 2018 1. [worth 40 points total] LetDbe the bitstream received by the DLC layer from the IP layer. The bitstream Dmaps to the equivalent polynomial D(x ) and vice versa ; D $ D(x ). Also, let G(x ) be the generator polynomial of a CRC code and Gits equivalent bit-domain representation; G$ G(x ). Suppose G(x ) = x4 + x3 + x2 + 1 and D= 1000001.

(a) Find G. (2 points) (b) Find D(x ). (2 points) (c) Let be the degree of G(x ). What is ? (2 points) (d) Let A(x ) = D(x )x . Find A(x ). (2 points) (e) Let Abe the bit-domain representation of A(x ). What is A? (2 points) (f ) Let Q(x ) be the quotient when A(x ) is divided by G(x ) modulo 2 and R(x ) the remainder. Find Q(x ) and R(x ). (5 + 5 = 10 points) (g) Let Qand Rbe the equivalent bit-domain representations of Q(x ) and R(x ) re- spectively. Find Qand R. (2 + 2 = 4 points) (h) Let P(x ) = A(x ) R(x ), where ` ' denotes subtraction using modulo 2 arithmetic.

Find P(x ). (2 points) (i) Let Pbe the bit domain representation of P(x ). This is the transmitted frame after error control coding. What is P? (2 points) (j) The received polynomial P0 ( x ) can be written as the sum of the transmitted poly- nomial P(x ) and an error polynomial E(x ). That is:

P 0 ( x ) = P(x ) + E(x ) Suppose E(x ) = x5 + x4 + x3 + x+ 1. Will this error pattern be detected by the receiver? (12 points) 1 2.

[worth 20 points total] A 8-bit checksum is to be appended to the following sequence of four 8-bit words [11011000 ;10001111 ;00010010 ;10101001 ;checksum].

(a) Compute the checksum (in binary form). You must show all steps leading to your answer, as shown on slide 41of Part- 1of Chapter- 3notes. (5 points) (b) Suppose the channel ips the 13 th and 16 th bits of the transmitted frame (i.e., the fth and eighth bits of the second 8-bit word). Will the receiver be able to detect this error? Explain your answer. (5 points) (c) Suppose the channel ips the 1 st , 2 nd , 5 th , 7 th , 17 th , 18 th , 21 st , and 23 rd bits of the transmitted frame. Will the receiver be able to detect this error? Explain your answer.

(5 points) (d) Suppose the channel ips the rst bit of the transmitted frame. Will the receiver be able to correctthis error? Provide proper explanation if your answer is yes. If your answer is no, you can provide a counterexample to prove your point. Of course, you can assume that the receiver does not know how many errors the channel made.

(5 points) 2 3.

[worth 20 points total] Suppose the English alphabet only contained the vowels a, e , i, o, and u. All possible messages can use only these 5 characters. You want to map each of these characters to 4-bit codewords. Assume that there is no parity check bit.

(a) How would you map the ve alphabets to 4-bit codewords such that the Hamming Distance of the dictionary is maximized (i.e, you want to separate out the ve characters in Hamming distance space as much as possible)? (12 points) (b) What is the Hamming Distance of your dictionary? (4 points) (c) How many total characters can you have in your dictionary (properly mapped to codewords) so that you could correct1-bit errors, assuming that each character is still represented by 4 bits? (4 points) 3 4.

[worth 30 points] Two hostsAand Bare connected by a 200 km. long 100 Kbps link.

Assume that the transmission medium is a copper cable (speed of light through copper is roughly 2 10 8 m/s). The hosts have adopted a Stop-and-Wait DLC protocol. Host A is the transmitter (sends data frames) and host Bis the receiver (sends only ACK frames). The following additional parameters are given:

all data frames sent by host Aare 1000 bits each all ACK frames sent by host Bare 100 bits each processing delays at both hosts are negligible.

(a) Suppose host Ahas 10 data frames to send to B. The transmission process starts at time t= 0. At what time can the communication process be considered complete, as- suming that the channel does not make any error? You must show all intermediate work which leads up to your answer. (20 points) (b) Repeat part (a), except, consider that each frame has a probability of error of P f = 0 :1. That is, what is the expected time at which the communication process can be considered complete? (10 points) 4 5.

[worth 20 points total] Suppose you have a 5 5 data array which you have planned to protect with row and column oddparity bits (which means that the receiver is expecting odd parity checks), as shown in eqn. (1). You are processing the array row by row, starting with the rst row and computing the check bit c 16 , then the second row and computing the bit c 26 , etc . While computing the sixth row of check bits, you inadvertently make an error in your code (or involuntarily su er from a lapse in concentration if you prefer hand calculations) and, instead of providing odd parity on the columns, you implement even parity, including the sixth column.

0 0 0 0 0 c 16 0 1 0 1 0 c 26 0 1 1 1 0 c 36 1 0 1 0 1 c 36 1 1 1 1 1 c 46 c 61 c 62 c 63 c 64 c 56 c 66 (1) (a) Compute all the check bits. (5 points) (b) How will the receiver behave if it gets the frame correctly? Of course, the receiver does not know about the goof-up at the transmitter end. (5 points) (c) Find an example of an error pattern which will maintain odd parity along both dimensions. You can assume that the check bits are also susceptible to errors, but you cannot alter the sixth row of the array you computed in part(a) above .

Be sure to put a box around the errored bits so that I can clearly identify your error pattern . (10 points) 5 6.

[worth 20 points total] In class, we talked about bit stu ng when the ag is the special bit stream 01 6 0 (zero, then six ones, then zero). Instead of 01 6 0, suppose we use the ag 10 6 1 (one, then six zeros, then one).

(a) What would be the stu ng rule at the transmitting DLC? (5 points) (b) Suppose the payload eld is 100000110000011. Show the payload eld after stu ng, using the stu ng rule of (a). Be sure to put a box around the stu ed bits so that I can clearly identify those . (10 points) (c) Show the bit pattern after stu ng and framing. You can assume that no other headers/trailers are used. (5 points) 6