ELEC1200: A System View of Communications: from Signals to Packets Lecture 3

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1 ELEC2: A System View of Commuicatios: from Sigals to Packets Lecture 3 Commuicatio chaels Discrete time Chael Modelig the chael Liear Time Ivariat Systems Step Respose Respose to sigle bit Respose to geeral iput ELEC2

2 Commuicatio System Trasmitter ecodes iformatio as a physical waveform. This physical waveform travels over a medium (e.g. air), which h modifies the waveform as it passes. This medium is called the chael. The receiver must take this modified waveform ad try to figure out the origial iformatio x(t) Source iformatio to sed Trasmitter physical waveform y(t) Chael Dest iformatio received Receiver physical waveform ELEC2 2

3 Discrete Time Chael Moder commuicatio systems are built from computers that work i discrete time, but most chaels are cotiuous time. Thus, moder trasmitters create discrete time sigals, ad the geerate cotiuous time sigals to sed. Moder receivers receive cotiuous time sigals, which they sample to obtai discrete time sigals. Source Dest set bits received bits Bits to Discrete-time Waveform Discrete-time waveform to Bits x() y() Trasmitter Receiver Hold circuit Sampler ELEC2 3 Chael x(t) y(t)

4 Discrete Time Chael For this course, we will ot worry about the details of the discrete/cotiuous coversio. We treat the mappig from the trasmitted samples x() to the received samples y() as a black box, called a discrete time chael. Defiitio: A black box is somethig we caot see the isides of, just what comes i ad out, like a magicia s hat. Source Dest set bits received bits Bits to Discrete-time Waveform Trasmitter Discrete-time waveform to Bits Receiver x() y() Hold circuit Sampler ELEC2 4 Chae el Discret te Time Chael x(t) y(t)

5 Systems For this class, a system is somethig that takes a waveform x() ad produces a output waveform y() (e.g. a chael) x() system y() ELEC2 5

6 Mathematical Models A mathematical model describes the operatio of a system usig mathematical formulas. x() Physical Chael y() a (actual measured respose) Mathematical Model y () (model respose) m We have a good model if The model ad actual resposes are similar, y m () y a () The relatioship betwee y m () ad x() is simple. Egieers use mathematical models to Uderstad the operatio of the system Predict the performace of a system i differet situatios Develop modificatios to the system that improve performace. Desig ew systems for ew applicatios. ELEC2 6

7 Possible effects of the chael The chael may cause the received sigal y() to differ from the trasmitted sigal x() i several ways, icludig Atteuatio (decrease i amplitude) Delay we study oly Offset these four ow Blurrig of trasitios Noise x() chael y() ELEC2 7

8 Modellig atteuatio, delay, ad offset x() d k c y() Disc crete Ti ime Cha ael k = atteuatio (k < ) d=delay delay c = offset Mathematical Model: What are possible physical causes for these? y() = kx( d) + c ELEC2 8

9 Blurrig of trasitios The bit sequece is ecoded by a waveform that makes istataeous chages from oe value () to aother (). Due to physical limitatios it ti i The trasducer that creates the physical waveform The electroics that drive the trasducer The physical medium that carries the waveform The sesor that seses the physical waveform The electroics that process the sesor sigal the actual received waveform caot make istataeous chages. I egieerig terms, we say that the chael is badlimited. x(t) () y(t) () Chael ELEC2 9

10 Effect of badlimited chael (SPB=2) iput (blue) output (black) Discre ete Time Cha el Assumes o atteuatio, delay, offset or oise. ELEC2

11 Effect of badlimited chael (SPB=) iput (blue) output (black) Discre ete Time Cha el ELEC2

12 Effect of badlimited chael (SPB=5) iput (blue) output (black) Discre ete Time Cha el ELEC2 2

13 Developig a badlimited chael model Ca we develop a mathematical model that eables us to predict the output of a badlimited chael to ay iput? Yes! If we assume that t the chael is liear ad time ivariat. We use the fact that ay iput ca be expressed as the sum of uit step fuctios. b = b = b 2 = b 3 = b 4 = b 5 = b 6 = b 7 = x() Time Chael y() iscrete D ELEC2 3

14 Liear Fuctios A fuctio y = f(x) ca be viewed as somethig that takes a umerical iput (x) ad produces a umerical output (y) x f(x) y A liear fuctio is a fuctio of the form y = ax where a is a costat. y a Note: y = ax+b is ot liear (uless b=). x ELEC2 4

15 Properties of Liear Fuctios Homogeeity: If you multiply the iput by a costat, c, you get the output multiplied by c. x ax y cx ax cy Additivity: If you put the sum of two umbers ito the fuctio, the output is the sum of the outputs to each umber applied (put i) to the fuctio idividually. x ax y ax x +x 2 y +y 2 x 2 ax y 2 ELEC2 5

16 Nice thig about liear fuctios If you kow a fuctio is liear, ad the output for ay ozero iput the, you ca compute the output for ay other iput. Example: The output of a liear fuctio for x = 2 is y = 4. What is the output if x = 4? x = 6? x=85? 8.5? y a ELEC2 x 6

17 Liear Systems A liear system is a system that satisfies the same two properties as a liear fuctio. Homogeeity: x() system y() cx() system cy() Additivity x () x 2 () system system y () y 2 () x ()+x 2 () system y ()+y 2 () ELEC2 7

18 Homogeeity If you scale (multiply) the iput by c times, the output is also scaled by c times. x() system y() system cx() cy() ELEC2 8

19 Additivity The output to the sum of two iputs is the sum of the outputs to each iput cosidered idividually. x () system y () x 2 () system y 2 () x ()+x 2 () system y ()+y 2 () ELEC2 9

20 Time Ivariat Systems A time ivariat system is oe where if you delay the iput by d, you get the exact same output, just delayed by the same amout d. x() system y() x(-d) system y(-d) d d ELEC2 2

21 Liear Time Ivariat (LTI) Systems A system that is both liear ad time ivariat is kow as a LTI system Is the chael with atteuatio, delay ad offset a LTI system? y() = kx( d) + c c d k x() y() ELEC2 2 Time Ch hael screte Di

22 Output of LTI Systems If a system is LTI, the you ca fid the output just by kowig the output to a uit step fuctio. How? Express the iput as the sum/differece of uit step fuctios ad use liearity ad time ivariace u()-u(-5) system s()-s(-5) u() system s() -u(-5) system -s(-5) ELEC2 22

23 Step Respose s() step respose = the output of the system to a iput uit step u() system s() The expoetial step respose (ext page) is a good model for the step resposes of the commuicatio chael i our lab. It ca model the followig effects: chages i amplitude (k) propagatio delay (d) blurrig of trasitios (a) ELEC2 23

24 Expoetial Step Respose d+ s() = k( a )u( d) s() k a =.8 d = s() k a=8.8 d = ELEC2 24

25 Expoetial Step Respose d+ s() = k( a )u( d) s() k a =.8 d = s() k a=5.5 d = ELEC2 25

26 Respose to sigle bit (SPB = 32) y( ) -s(- SPB) s( ) ue = iput ack = outpu ut Bl Bl ELEC2 s() = Expoetial approach, a =.8, k =, D = 26

27 Respose to sigle bit (SPB = 6) y( ) -s(- SPB) s( ) ue = iput ack = outpu ut Bl Bl ELEC2 s() = Expoetial approach, a =.8, k =, D = 27

28 Respose to sigle bit (SPB = 8) y( ) -s(- SPB) s( ) ue = iput ack = outpu ut Bl Bl ELEC2 s() = Expoetial approach, a =.8, k =, D = 28

29 Respose to sigle bit (SPB = 4) y( ) -s(- SPB) s( ) ue = iput ack = outpu ut Bl Bl ELEC2 s() = Expoetial approach, a =.8, k =, D = 29

30 Respose to sigle bit (SPB = 2) y( ) -s(- SPB) s( ) ue = iput ack = outpu ut Bl Bl ELEC2 s() = Expoetial approach, a =.8, k =, D = 3

31 Respose to sigle bit (SPB = ) y( ) -s(- SPB) s( ) ue = iput ack = outpu ut Bl Bl ELEC2 s() = Expoetial approach, a =.8, k =, D = 3

32 Respose to more geeral iput b = b = b 2 = b 3 = b 4 = b 5 = b 6 = b 7 = x() = u() u( 5) + u( )... y() = s() s( 5) + s( )... x() y() Discre ete Time Cha el Importat caveat (thig to be careful about) If we add a offset or oise, the the chael is o loger LTI However, we ca still deal with this by cosiderig the output to be the sum of the output of a LTI system plus the offset/oise. ELEC2 32

Web Appendix O - Derivations of the Properties of the z Transform

Web Appendix O - Derivations of the Properties of the z Transform M. J. Roberts - 2/18/07 Web Appedix O - Derivatios of the Properties of the z Trasform O.1 Liearity Let z = x + y where ad are costats. The ( z)= ( x + y )z = x z + y z ad the liearity property is O.2