EE450 Discussion 5 Shannon s Theorem Modulations Multiplexing Shannon s Theorem C B log2 1 SNR Theoretical Maximum Capacity that can be obtained on a line Sets an Upper Bound on the capacity given the conditions Used for Calculating the Signal to Noise Ratio Given the Bandwidth and capacity of the channel Bandwidth Given the SNR and Channel Capacity Capacity Given the SNR and the Bandwidth Problem 1 What SNR is needed to put a T 1 carrier on a 50 khz line What do we know T 1 Capacity 1 544 Mbps Bandwidth 50 KHz Move them around and Solve 1 544 000 50 000 log 2 1 SNR 2 30 88 1 SNR Continued So SNR 1976087931 SNR is typically measured in DB 10 SNR In this case Use SNR dB 10 log SNR dB 10 log 10 1976087931 SNR aprox 92 9 dB However you must NOT plug SNR into Shannon s theorem in dB format Problem 2 Calculate the maximum rate supported by a telephone line with BW of 4 KHz When the signal is 10 volts the noise is 5 milivolts SNR Signal power Noise Power Power is proportional to square of the voltage S N 10 2 0 005 2 4000000 B 4000 Hz C B log 2 1 S N Reminder log 2 x ln x ln 2 C 4000 log 2 1 4000000 87726 bps Review on Modems Modem Stands for MOdulator DEModulator Uses Sine wave As the carrier Signal Digital to Analog Encoding WCB McGraw Hill The McGraw Hill Companies Inc 1998 Modulation Need to Encode Digital Data in an Analog Signal In modem transmission we use different techniques for modulation Amplitude Modulation Frequency Modulation Phase Shift Amplitude Modulation Varies the Amplitude of the Signal Amplitude Modulation Same Signal Greater Amplitude Amplitude 1 Amplitude 2 Amplitude Modulation Amplitude 2 1 Amplitude 1 0 This signal Represents 0011010 ASK WCB McGraw Hill The McGraw Hill Companies Inc 1998 FSK WCB McGraw Hill The McGraw Hill Companies Inc 1998 Phase Shift Modulation Start with our normal sine wave The sine wave has a period of P P may be denoted as T instead in the equations Phase Shift Modulation Shift the Phase of the Sine Wave Shifted diagram shows that the cycle starting at 1 vs starting at 0 PSK WCB McGraw Hill The McGraw Hill Companies Inc 1998 PSK Constellation WCB McGraw Hill The McGraw Hill Companies Inc 1998 4 PSK WCB McGraw Hill The McGraw Hill Companies Inc 1998 4 PSK Constellation WCB McGraw Hill The McGraw Hill Companies Inc 1998 8 PSK Constellation WCB McGraw Hill The McGraw Hill Companies Inc 1998 PSK WCB McGraw Hill The McGraw Hill Companies Inc 1998 Combining Both Modulation used in Modern Modems Uses Amplitude Modulation Phase Shift Keying QAM Quadrature Amplitude Modulation Big Name Simple Concept 4 QAM and 8 QAM Constellation WCB McGraw Hill The McGraw Hill Companies Inc 1998 8 QAM Signal WCB McGraw Hill The McGraw Hill Companies Inc 1998 16 QAM Constellation WCB McGraw Hill The McGraw Hill Companies Inc 1998 Bit Rate and Baud Rate WCB McGraw Hill The McGraw Hill Companies Inc 1998 Problem 3 A modem uses an 8 PSK modulation scheme supporting data rate of 4800 bps What is the signaling rate aka baud rate 8 PSK Phase Shift Keying 8 different encoding levels Each encoding has log2 8 3 bits 4800 3 1600 Baud Rate Ways of Multiplexing Demultiplexing Time Division Multiplexing TDM You have n input lines coming in The same of lines going out Only one line interconnecting How Packing the Data In Multiplexing A way of aggregating data onto a single line without compromising the rate at which original data is sent We are not limiting anyone s channel capacity We are simply sending there signal through a shared line Sharing by time slots Sample the Line 1 Place its value in slot 1 Sampling the Line and Send its value on its way Line1 Line2 Line3 Line4 Slot1 More Time Slots Line 2 Places its Sample on the Line And it goes on Line1 Line2 Line3 Line4 Slot2 Slot1 Other Side Demultiplexing Similar to Multiplexing just the reverse Line1 Slot2 Slot1 Line2 Line3 Line4 Line1 Slot2 Line2 Line3 Line4 TDM WCB McGraw Hill The McGraw Hill Companies Inc 1998 Synchronous TDM WCB McGraw Hill The McGraw Hill Companies Inc 1998 TDM Multiplexing WCB McGraw Hill The McGraw Hill Companies Inc 1998 TDM Demultiplexing WCB McGraw Hill The McGraw Hill Companies Inc 1998 Data Rate WCB McGraw Hill The McGraw Hill Companies Inc 1998 TDM Example T 1 Lines Carries the equivalent of 24 voice lines Each analog voice line is sampled at 8000 times a second Digital Sample is thrown on the Digital Carrier Line On the other side Digital samples are used to reconstruct Analog Signal STDM Asynchronous TDM How does it Work Checks to see if there is data to transmit on input line If there is transmit data If not move on to next input line Asynchronous TDM WCB McGraw Hill The McGraw Hill Companies Inc 1998 Frames and Addresses a Only three lines sending data WCB McGraw Hill The McGraw Hill Companies Inc 1998 TDM vs STDM Synchronous TDM Gray Slots are actually carrying data Slot 1 Slot 2 Slot 3 Slot 4 Slot 1 Slot 2 Slot 3 Statistical TDM Uses empty time slots but does add some overhead Slot 1 Slot 2 Slot 1 Slot 2 Slot 3 Slot 1 Notes on TDM Sampling Occurs very quickly Applicable to fixed number of flows Requires Precise Timing Resources are guaranteed STDM Statistical Time Division Multiplexing Similar to regular TDM but different in this If we are no longer guaranteed a time slot why use it Traffic is sent on demand Only if there is data on line 1 will slot 1 be occupied by line 1 Resources are not guaranteed We the carrier can take advantage of one of the input lines not being busy Important distinction STDM is used mainly for Digital Lines TDM 1 2 3 4 5 vs Resources are guaranteed to the users Sampling Occurs very quickly Applicable to fixed number of flows Requires Precise Timing Wastes valuable carrier space STDM 1 2 3 4 Traffic is sent on demand utilizing unused time slots so it benefits the Carrier Resources are not guaranteed so when time slots are busy the users suffer In real life there is some overhead Speedup isn t as obvious Statistical TDM Parameters I Number of Input Sources R Data rate of each source bps Alpha mean fraction of time each source is transmitting M Effective capacity of multiplexed line K M I x R Ratio of multiplexed line capacity to total input rate lambda x I x R Average Arrival time Ts 1 M Service time in seconds Line Utilization Fraction of total link capacity being used Many different forms to express line utilization Ts x I x R M K M Sample Problem 5 Ten 9600 bps lines are multiplexed using TDM Ignoring overhead bits what is the total capacity required for
View Full Document