MAE143A Signals & Systems

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1 MAE43A Signals & Systems Winter 206 MAE43A Signals & Systems Tu, Th: 09:30-0:50 Pepper Canyon 06 We 09:00-09:50 Pepper Canyon 09 Professor Bob Bitmead Jacobs 609, TAs: Martin Sehr Eric Sihite 206 Classes, breaks, homeworks, midterms, final Week Tuesday Wednesday Thursday Jan 5 Jan 6 Jan 7 2 Jan 2 Jan 3 Jan 4 H 3 Jan 9 Jan 20 Jan 2 H2 4 Jan 26 Jan 27 Jan 28 H3, M 5 Feb 2 Feb 3 Feb 4 H4 6 Feb 9 Feb 0 Feb H5 7 Feb 6 Feb 7 Feb 8 H6 8 Feb 23 M2 Feb 24 Feb 25 H7 9 Mar Mar 2 Mar 3 H8 0 Mar 8 Mar 9 Mar 0 H9 Mar 5 Final

2 MAE43A Signals & Systems Winter Classes, breaks, homeworks, midterms, final Week Tuesday Wednesday Thursday Jan 5 Jan 6 Jan 7 2 Jan 2 Jan 3 Jan 4 H 3 Jan 9 Jan 20 Jan 2 H2 4 Jan 26 Jan 27 Jan 28 H3, M 5 Feb 2 Feb 3 Feb 4 H4 6 Feb 9 Feb 0 Feb H5 7 Feb 6 Feb 7 Feb 8 H6 8 Feb 23 M2 Feb 24 Feb 25 H7 9 Mar Mar 2 Mar 3 H8 0 Mar 8 Mar 9 Mar 0 H9 Mar 5 Final

3 MAE43A Signals & Systems Winter MAE43A Signals & Systems?!?!? What is this class about? Signals real-valued scalar functions of time x(t) Often represents a physical quantity over time Voltage, current, pressure, speed, heart rate Could also represent economic quantities over time Employment, value, account balance Could even represent psychological quantities Opinions, approval ratings, confidence, satisfaction Systems devices, processes, algorithms which operate on an input signal x(t) to produce an output signal y(t) A system with memory is called a dynamic system

4 MAE43A Signals & Systems Winter Signals & systems t [a, b] t belongs to a real interval (possibly infinite),, then Signals x(t), y(t) are continuous-time signals System linking the two is a continuous-time system t belongs to the natural numbers, t Ν, then Signals x(t), y(t) are discrete-time signals Time t counts the number of sampling times, Δ System linking the two is a discrete-time system Continuous-time dynamic systems often described by differential equations Discrete-time dynamic systems often described by difference equations Memory is captured by the initial conditions

MAE43A Signals & Systems Winter 206 5 Text Book: Michael J.

Other related texts are fine too Including Ulaby & Yagle,

stick to the book where sensible A helpful and cheap book might

5 MAE43A Signals & Systems Winter Text Book: Michael J. Roberts, Fundamentals of Signals & Systems, McGraw Hill, 2008 Other related texts are fine too Including Ulaby & Yagle, Chaparro, etc Some homework will refer to this exact book We will stick to the book where sensible A helpful and cheap book might be the Schaum Outline Signals and Systems by Hwei Hsu, 20 Lots of worked problems

6 MAE43A Signals & Systems Winter Signals & Systems (rough) planned schedule Matlab throughout - discrete samples of continuous signals Continuous signals and their properties week Continuity, boundedness, periodicity, Discrete time signals and their properties week Continuous systems and their analysis 2 weeks Linearity, causality, time-invariance, stability Discrete time systems and their analysis.5 weeks Continuous system frequency analysis Frequency response.5 weeks Discrete system frequency analysis.5 weeks Sampling week Chapters, 2, 5 3, 6 4, 6 5,7 0, 2, 3 4

7 MAE43A Signals & Systems Winter Prerequisites what we assume you know Math 20D Introduction to differential equations ODEs, solutions, Laplace transforms, complex numbers Math 20E Vector calculus Green s theorem, Taylor series Math 20F Linear algebra Matrices and vectors, bases, eigenvalues and eigenvectors MAE05 Introduction to mathematical physics Fourier series, integral transforms

MAE43A Signals & Systems Winter 206 8 Office Hours and other assistance Bob Bitmead Tuesdays 3:30-5:00

8 MAE43A Signals & Systems Winter Office Hours and other assistance Bob Bitmead Tuesdays 3:30-5:00 EBU2 305 Martin Sehr Mondays 08:30-0:00 EBU2 305 Eric Sihite Wednesdays 3:00-4:00 EBU2 05 or by appointment

9 MAE43A Signals & Systems Winter Homework, midterm and exam Homework will be set weekly except Week 0 and due in class on the following Thursday The midterms will take place in class Thursday January 28 and Tuesday February minutes each The final will take place Tuesday, March 5, 08:00-:00 probably in the class room Pepper Canyon 06

10 MAE43A Signals & Systems Winter Grading and passing with flying colors Final score is the maximum of the following two numbers.00 x Final % 0.5 x Final % x Midterm % x Homework % [Secret: the two numbers are almost always the same] To succeed: avail yourself of all the help including other books, past students, friends, the web do the homework and matlab yourself seek assistance early and as necessary

11 MAE43A Signals & Systems Winter 206 Policy If you send me an , I will read it If it takes longer to deal with the than it took you to write it, then dealing with the is entirely discretionary First use class and then use the office hour sessions to ask questions

MAE43A Signals & Systems Winter 206 2 A speech signal

12 MAE43A Signals & Systems Winter A speech signal 0.3 Voice recording sample value (units) Now is the time for all good men to come to the aid of their country time (s) x 0 4

13 MAE43A Signals & Systems Winter Speech signal Voice recording Seven seconds of speech sampled at Hertz digitized at 6 bits sample value (units) time (s) A discrete-time signal representing samples from a continuous-time voltage signal which, in turn, is the output from a piezoelectric transducer of air pressure (a microphone) x 0 4 Because we have fairly rapid sampling we can consider (for the moment) this a continuous-time signal We will return to this later

14 MAE43A Signals & Systems Winter Speech Signal zoomed Now is spoken Growing amplitudes Decaying amplitudes Low power High power Periodic high frequency low frequency Noisy/unpredictable speech sample (units) N ow i s Time (s)

15 MAE43A Signals & Systems Winter Speech signal Clearly the signal is segmented (over time) into phonemes Some parts have high amplitude and therefore power Voiced speech - most of this piece is voiced Vocal cords vibrating Strong periodic behavior Very predictable sample-to-sample Unvoiced speech (mostly just the s sound) Vocal cords not vibrating Mouth, lips and tongue affect the moving air Noisy looking Not predictable sample-to-sample We see different rates of attack and decay Can you identify the Australian accent?

16 MAE43A Signals & Systems Winter Familiar signals the constant signal The constant signal constant over all time x(t) = c, t (, ) The Laplace transform of this constant signal is L x (s) = Z 0 x(t)e st dt = 0.7 s e st 0 = 0.7 s Note that the Laplace transform ignores the t<0 part

17 MAE43A Signals & Systems Winter Familiar signals the step function The unit step function or Heaviside function ( 0, t < 0 (t) =, t 0 The Laplace transform of the unit step This is the same as for a constant function of value because they are identical for t 0 This function is discontinuous at t=0 L (s) = Z 0 (t)e st dt = s

18 MAE43A Signals & Systems Winter Continuous approximation to a unit step Here are some approximations to (t) which are continuous ˆ k (t) = erf(kt) Here erf(x) is the error function erf(x) = p 2 Z x exp( z 2 ) dz The red curve is k=0 The black curve is k=20

19 MAE43A Signals & Systems Winter Familiar signals the ramp function Ramp function ( 0, t < 0 r(t) = t, t 0 This function is unbounded but it is continuous Laplace transform Notice that the ramp function is the integral of the step L r (s) = Z 0 r(t) = r(t)e st dt = Z t Z 0 (z) dz te st dt = s 2

20 MAE43A Signals & Systems Winter Familiar signals the impulse function The impulse function or Dirac delta function (t) =0, for t 6= 0 R (t) dt =, for > 0 The impulse function is neither continuous nor bounded Laplace transform L (s) = The step is the integral of the impulse (t) = Z 0 Z t (t)e st dt = (z) dz Z (t) dt =

21 MAE43A Signals & Systems Winter Bounded continuous approximation of the impulse Continuous and bounded approximations of δ(t) sin kt ˆk(t) =k t red is 25 ˆ20(t) sin 20t = 500 t black is 7.5 ˆ40(t) sin 40t = 700 t The impulse function has a sampling property for any function f(t) continuous at t=0 Z b a f(z) (z) dz = f(0) if 0 2 (a, b)

22 MAE43A Signals & Systems Winter Familiar signals real exponentials Red Blue Black Laplace e 0.5t e t e 2t s 0.5, s, s 2 All of these signals are unbounded Red Blue Black Laplace e 0.5t e t e 2t s +0.5, s +, s +2

23 MAE43A Signals & Systems Winter Red Blue Familiar signals one-sided real exponentials e 0.5t (t) e t (t) Black e 2t (t) Laplace s +0.5 poles -0.5, -, -2 s + s +2 Red Blue Black Laplace poles (0,0.5), (0,), (0,2) s 0.5 e 0.5t e t e 2t (t) (t) (t) s, s s, s 2 s

24 MAE43A Signals & Systems Winter Familiar signals - sinusoids Red Blue Black sin(5t) sin(3t) sin(7t) Red Black sin(5t) cos(5t)

25 MAE43A Signals & Systems Winter One-sided sinusoids Sinusoids sin(0t)(t) sin(5t)(t) sin(20t)(t) Laplace transforms 0 5 s s Poles ±j0, ±j5, ±j20 20 s Sinusoid and cosinusoid sin(0t)(t) cos(0t)(t) Laplace transforms 0 s s s Poles ±j0

26 MAE43A Signals & Systems Winter Complex exponentials Blue Red e 2t sin(20t)(t) ±e 2t (t) Laplace transforms 20 (s + 2) = 20 s 2 +4s s +2 Poles -2±j20, -2 The (upper) red curve is called the envelope of the blue curve

27 MAE43A Signals & Systems Winter More complex exponentials Blue Red e 2t sin(20t)(t) e 2t (t) Laplace transform 20 (s 2) = 20 s 2 4s s 2 Poles 2±j20, 2 in the right half of the complex plane that is, the real part is positive These signals are unbounded

28 MAE43A Signals & Systems Winter Periodic signals Periodic signals repeat x(t + kt) =x(t) for k 2 Z The minimal cycle time T is called the period Here it is one second For a periodic signal we only need to specify it over one period and we know it everywhere Sinusoids, cosinusoids and constants are periodic One-sided variants are not, k above can be negative

29 MAE43A Signals & Systems Winter Even and odd signals Even signals x( t) =x(t) such as cos(t) Odd signals x( t) = x(t) such as sin(t) x(t) = x even (t)+x odd (t) x even (t) = [x(t)+x( t)] 2 x odd (t) [x(t) x( t)] 2

30 MAE43A Signals & Systems Winter Real world industrial signals Macknade bulk sugar dryer, Queensland Australia m-long rotating drum evaporative cooling and drying Hot wet sugar in the top (left), cool dry air in the bottom (right) Hot moist air out the top, cool dry sugar out the bottom Sugar InSugarOutAir OutAir In

31 MAE43A Signals & Systems Winter Macknade rotary bulk sugar dryer - experiments

32 MAE43A Signals & Systems Winter Input sugar temperature signal

33 MAE43A Signals & Systems Winter Input air temperature signal

34 MAE43A Signals & Systems Winter Input air humidity signal

35 MAE43A Signals & Systems Winter Output sugar temperature signal A quantized signal

36 MAE43A Signals & Systems Winter Macknade sugar dryer is a 3-input -output system

37 MAE43A Signals & Systems Winter Mathematical model of the sugar dryer system s a v E i ( k τ) = ma τ( P sugar ([ k - ] τ, T i ) - P air ([ k - ] τ, M i, M i )) M i s ( k τ) = M i s ([ k - ] τ) - Ei ( k τ), M i a ( k τ) = M i a ([ k - ] τ) + E i ( k τ) s a T a i ( k τ) =T a ha τ+ C pv E i ( k τ ) i ([ k - ] τ ) + [ ]T i ([ k - ] τ) - T i ([ k - ] τ) a v C pa M i ( k τ) + C pv M i ( k τ) [ ] T s i ( k τ) =T s L H 2 O E i ( k τ) + ha τ[ s a T i ([ k - ] τ) - T i ([ k - ] τ) ] i ([ k - ] τ )- s w C ps M i ( k τ ) + C pw M i ( k τ) m M i ( k τ) = [ - α] M m i ( k τ) + αm m i -( k τ), M v i ( k τ) = M v i + ( k τ) This is a set of nonlinear difference equations in (state) variables E i, M s i, Ma i, Mm i, Mv i,ta i, Ts i The blue quantities are parameters of the model

38 MAE43A Signals & Systems Winter Properties of signals Domain region of times under consideration Continuous-time, discrete-time, finite time interval Support region of time over which they are nonzero One-sided functions, impulses, limited extent Amplitude maximal magnitude Boundedness, norm, energy Smoothness degree of continuity, differentiability, etc. Everywhere, piecewise, Periodic, deterministic, random, etc. Generally connected with the ability to predict the signal

39 MAE43A Signals & Systems Winter Signal energy and power Quantifying the size of a signal is important in many applications: How much electricity can be used in a defibrillator? How much energy should an audio signal have to be heard? The energy of the signal x(t) is E x = x t ( ) 2 dt

40 MAE43A Signals & Systems Winter Signal energy and power Some signals have infinite energy. In that case, we may use the concept of average signal power For a periodic signal, signal power is x(t), with period T, the average P x = T T x( t) 2 dt If the signal is not periodic, then P x = lim T T T /2 T /2 x( t) 2 dt

41 MAE43A Signals & Systems Winter Signal norms a measure of signal size General L p signal norms kxk p 4 = applez x(t) p dt p applez 4 L 2 Euclidean signal norm kxk 2 = often related to signal energy voltage, current, velocity signals x(t) 2 dt 2 L norm kxk 4 = Z x(t) dt L norm or sup norm kxk 4 = sup x(t) t2(,)

42 MAE43A Signals & Systems Winter Signal transforms Laplace and Fourier Expression of the signal in a different domain Laplace transform for signals defined on domain [0 -, ] X(s) 4 = Z 0 x(t)e st dt Z c+j x(t) = X(s)e st ds, for t 0 2 j c j Fourier transform for signals defined on the domain [-, ] X(!) 4 = Z x(t)e j!t dt, x(t) = 2 Z X(!)e j!t d!, for all t Since the transforms are invertible no information is lost in using them instead of the original time-domain description

MAE143A Signals & Systems?!?!?

MAE143A Signals & Systems?!?!? 7//4 MAE43A Signals & Systems 24 Winter MAE43A Signals & Systems http://oodgeroo.ucsd.edu/~bob/signals Tu, Th 8:-9:2 Pepper Canyon 6 Mo 6:-6:5 Pepper Canyon 6 Professor Bob Bitmead rbitmead@ucsd.edu Jacobs