ENGI 4557 Digital Communications Practice Problems 207 (Part 2) H = n p i log 2 ( ) p i C = W log 2 ( + S N ) SNR = 6m + 0 log 0 ( 3σ2 x V ) 2 SNR = 6m 0 ( ) n n! = k k!(n k)! x = σ 2 = + + x p(x)dx (x x) 2 p(x)dx ( p) n exp( np) η = n o /n f ( P f ) + n a /n f + (2t p R)/n f η = n o /n f ( P f ) + (W s )P f η = ( P f )( n o /n f ) S max = G exp( 2G) or G exp( G) ρ max = E(X ) E(X ) + (2e )t p. Consider a (7,4) Hamming code with the following check matrix: 0 0 0 H = 0 0 0 0 0 0 Justify your answers for full credit: (a) Show a triple error vector that is not detected. (b) Suppose the following vector is received: (00) T. What bit is most likely to be in error? (c) Estimate the probability of error correction failure if the error rate in the channel is 0 6. 2. A linear code consists of codewords nine bits long, four information bits and five check bits. Check bits are calculated as follows: b 5 = b + b 2 b 6 = b 2 + b 3 b 7 = b 3 + b 4 b 8 = b + b 4 b 9 = b + b 3 + b 4.
(a) Show the check matrix (H) for this code. (b) Show the list of codewords. (c) What is the minimum distance of this code? How many errors can be forward-corrected? 3. Consider a (8,4) linear code with the following check matrix: Justify your answers for full credit: H = 0 0 0 0 0 0 0 0 0 0 0 0 (a) Suppose the following vector is received: (0000) T. Is any error detected? Justify. (b) List all undetected error vectors. (c) Estimate the probability of error detection failure if the bit error rate in the channel is 0 6. 4. A linear code has the following check matrix: H = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (a) How many information and check bits are there in this code? (b) Write the equation to calculate b 6 from the information bits. (c) What is the minimum distance of this code? Can this code be used for forward error correction? Justify your answer. (d) What is the probability of error-detection failure if the bit error rate is 0 5? 5. An asynchronous TDM is fed by three channels. Each channel has a nominal bit rate of 5 Mbps. Output frames have 26 bits: 40 data bits plus one flag and one data/stuff bits per each input channel. (a) What nominal bit rate is required at the output to use the stuff bits half of the time when input bit rates are nominal? (b) What are the absolute maximum and minimum input bit rates that can be accommodated, assuming that the output bit rate is nominal. 6. An asynchronous TDM is fed by three channels. Each channel has a nominal bit rate of 0 Mbps. Output frames have 69 bits: 9 bits from each input plus two flag and two data/stuff bits per each input channel. The output bit rate is 35 Mbps (a) Assume input bit rates are nominal. How many data bits per channel are used on average? (b) What are the absolute maximum and minimum input bit rates that can be accommodated? 7. In ADSL, why is the total available channel bandwidth divided in many sub-channels? 2
8. Explain what is the difference between frequency- and wavelength- division multiplexing. 9. Briefly explain what is SONET. 0. What is the main advantage of synchronous multiplexing used in SONET networks compared to the asynchronous multiplexing used in the traditional digital multiplexing hierarchy?. A cellular phone network has total allocated bandwidth of 25 MHz. Each one-way radio channel needs 40 khz and can carry 4 digital channels. Assume 5% of channels are reserved for setup and control and the reuse factor is 7. The expected load in each cell is 00 Erlangs and the average call duration is 5 minutes. (a) How many two-way channels are available for user traffic in each cell? (b) Is the number of available channels enough for a low blocking probability? Explain why. (c) How many calls per minute are received on average in one cell? 2. Explain why a frame sequence number is necessary for stop-and-wait ARQ. 3. Briefly explain how the ENQ (enquire) frame can help to improve efficiency of stop-and-wait ARQ. 4. Data is transmitted using an ARQ protocol over a channel with a nominal bit rate of 3 Mbps, a propagation time of 2 ms and a bit error rate of 0 6. Assume information frames have 2500 information bytes and 32 bytes overhead, ACK frames are 32 bytes long and processing time is negligible. (a) Calculate the effective bit rate with stop-and-wait ARQ. (b) Is there any way to improve the stop-and-wait ARQ efficiency? Explain. (assume the physical channel can not be changed). (c) Calculate the effective bit rate achieved with go-back-n ARQ. What window size should be used? 5. Derive the formula for the effective bit rate with stop-and-wait ARQ protocol. Assume the following parameters are known: frame failure probability (P f ), propagation time (t p ), number of bits and overhead in information frame (n f, n o ), number of bits in ACK frame (n a ), physical bit rate in channel (R). Assume processing time is negligible and consider the following results valid for α < : iα i = ( α) 2, α i = α ( α) 6. A channel has a bit rate of 0 Gbps and distance of 0 km. Propagation speed in the link is 3 0 8 m/s, processing time at both ends is 0. µs, header overhead and ACK frames are 20 bytes long. Information frames (including overhead) are 2048 bytes long. If Go-back-N ARQ is used. (a) What is the minimum window size for maximum effective bit rate? Justify your answer. (b) Calculate the effective bit rate in the line if the bit error rate is 0 4 and the window size is set to 00. 3
7. Derive the formula for the effective bit rate with selective repeat ARQ protocol. Assume the following parameters are known: window size (W s ), frame failure probability (P f ), propagation time (t p ), number of bits and overhead in information frame (n f, n o ), number of bits in ACK frame (n a ), physical bit rate in channel (R). Assume processing time is negligible and consider the following results valid for α < : iα i = ( α) 2, α i = α ( α) 8. Data is transmitted using an ARQ protocol over a channel with a nominal bit rate of 30 Mbps, a propagation time of 2 ms and a bit error rate of 0 6. Assume information frames have 2500 information bytes of which 64 bytes are overhead. ACK frames are 64 bytes long. Processing time is 0. ms. (a) Calculate the effective bit rate (in bps) achieved with go-back-n ARQ. What window size should be used? (b) Suppose go-back-n with a window size of 6 is used. How many bits on the header should be used to represent the frame sequence? 9. Data is transmitted using an ARQ protocol over a channel with a nominal bit rate of 00 Mbps, a propagation time of 2 ms. Assume the frame overhead (header, CRC, etc.) is equal to 64 bytes, independently of the frame length. ACK frames are 64 bytes long and processing time is negligible. (a) How long (in bytes) should the frame length be set to achieve an efficiency of 80% with stop-and-wait ARQ? Assume there are no transmission errors. (b) Calculate the effective bit rate achieved with selective repeat ARQ, a bit error rate of 0 4 and a frame lenght of 52 bytes. What window size should be used and why? 20. A communication link uses selective-repeat ARQ. Information frames have a fixed overhead of 256 bits and the bit error rate in the line is 0 8. Assume that the window size is large enough to keep the channel busy at all times. What is the optimum frame length that maximizes efficiency? 2. Workstations are connected in a LAN with a star topology using the CSMA-CD protocol. The centre of the star is a hub that repeats incoming frames to all ports. The distance between workstations and the hub is 00 m and the propagation speed is 2.3 0 8 m/s. The bit rate in the bus is 00 Mbps. What is the frame size (in bits) that results in the same throughput as the Slotted Aloha protocol? 22. A wireless LAN uses polling to communicate 30 stations to a base station. Assume all stations are 80 m away from the base station. The bit rate in the channel is 0 Mbps, propagation speed is 3 0 8 m/s, information frames are 024 bytes long. Polling messages (and responses if station has no data to transmit) are 64 bytes long. Stations are allowed to transmit 2 frames/poll. (a) What is the maximum possible data frame arrival rate? (b) What is the maximum data frame arrival rate if only one station has frames to transmit? (c) Suppose that the LAN protocol is changed to Slotted Aloha. Is the maximum possible frame arrival rate improved? 4
23. Forty terminals are connected to a bus. The maximum distance between terminals is 2.5 km (assume a propagation velocity of 2.5 0 8 m/s). The bit rate in the bus is 00 Mbps. Frames are 500 bytes (not bits) long. (a) What protocol would you choose to maximise the frame throughput (Slotted Aloha or CSMA- CD)? Justify your answer. (b) If you can adjust the frame size from a minimum of 500 bytes to a maximum of 4500 bytes, what size and protocol would you choose to to maximise the frame throughput? What is the value of the maximum throughput in bits per second? 24. Explain the multi token operation in token ring networks. Can a collision occur in a token ring network? 25. Twenty terminals are connected to a bus. The maximum distance between terminals is 25 km (assume a propagation velocity of 2.5 0 8 m/s). The bit rate in the bus is 00 Mbps. (a) What protocol and what frame size would you choose to maximise the throughput (Slotted Aloha or CSMA-CD)? Justify your answer. Note: assume any improvement above 95 % is considered negligible. (b) If you can adjust the frame size from a minimum of 200 bytes to a maximum of 000 bytes, what size and protocol would you choose to to maximise the throughput? What is the value of the maximum throughput in bits per second? 26. Briefly describe how reservation systems work. What can be done to improve the throughput if there are many stations? 27. A wireless network uses a reservation system over a 20 Mbps radio channel. The frame is composed of 500-byte data slots and a 200-byte reservation slot. Reservation mini-slots are 8-byte long. The Slotted Aloha protocol is used during the reservation period and each station can reserve data slot per frame. (a) What is the maximum normalized throughput that the system can achieve? (b) What is the normalized throughput when only two stations are active? (c) If stations are allowed to reserve 4 data slots per frame. What is the maximum normalized throughput? 28. Explain the difference between a datagram and a connection-oriented packet network. Mention at least one pro and con for each type. 29. Ten packets are transmitted over 3 hops of a store-and-forward datagram network. Each packet is 800 bytes long and the bit rate in all lines is 00 Mbps. The delay in each hop is 2 ms. Calculate the total elapsed time from the transmission start to the time that the last bit is received. 30. Briefly explain why the IP network uses hierarchical addressing. 3. What is the purpose of a switch/router? 32. Give one example of a connection-oriented packet network 33. Give one example of a connectionless packet network 5