MAE143A Signals & Systems?!?!?

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1 7//4 MAE43A Signals & Systems 24 Winter MAE43A Signals & Systems Tu, Th 8:-9:2 Pepper Canyon 6 Mo 6:-6:5 Pepper Canyon 6 Professor Bob Bitmead rbitmead@ucsd.edu Jacobs 69, TAs: Chun-Chia (Ben) Huang hhunhia@gmail.com Minh Hong Ha h5ha@ucsd.edu Classes, breaks, homeworks, midterms, final Week Monday Tuesday Thursday Jan 6 No work Jan 7 Jan 9 2 Jan 3 Jan 4 Jan 6 H 3 Jan 2 MLKJ Jan 2 Jan 23 H2 4 Jan 27 Jan 28 M Jan 3 H3 5 Feb 3 Feb 4 Feb 6 H4 6 Feb Feb Feb 3 H5 7 Feb 7 Pres Day Feb 8 Feb 2 H6, M2 8 Feb 24 Feb 25 Feb 27 H7 9 Mar 3 Mar 4 Mar 6 H8 Mar Mar Mar 3 H9 Final Mar 2 8-am MAE43A Signals & Systems 24 Winter 2 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

7//4 MAE43A Signals & Systems 24 Winter 3 Signals & systems t belongs to a real interval (possibly infinite),, then Signals x(t), y(t) are continuous-time signals System linking the two is a

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 t

2 7//4 MAE43A Signals & Systems 24 Winter 3 Signals & systems 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 t [a, b] MAE43A Signals & Systems 24 Winter 4 Text Book: Luis F. Chaparro, Signals & Systems using MATLAB, Academic Press, 2 Other related texts are fine too Some homework will refer to this exact book Web site: We will stick to the book except for its close treatment of communications and control A helpful and cheap book might be the Schaum Outline Signals and Systems by Hwei Hsu, 2 Lots of worked problems 2

3 7//4 MAE43A Signals & Systems 24 Winter 5 Signals & Systems (rough) planned schedule Continuous signals and their properties Continuity, boundedness, periodicity, week Continuous systems and their properties 3 weeks Linearity, causality, time-invariance, stability, state Continuous signals and systems analysis 2 weeks Convolution, Fourier and Laplace transforms Sampling & discrete time signals and systems 2 weeks Sampling, reconstruction, discrete Fourier transform (DFT) Random signals 2 weeks Expectation, correlation, prediction, spectrum Chapters, 2, 6 3, 4, MAE43A Signals & Systems 24 Winter 6 Prerequisites what we assume you know Math 2D Introduction to differential equations ODEs, solutions, Laplace transforms, complex numbers Math 2E Vector calculus Green s theorem, Taylor series Math 2F Linear algebra Matrices and vectors, bases, eigenvalues and eigenvectors MAE5 Introduction to mathematical physics Fourier series, integral transforms 3

7//4 MAE43A Signals & Systems 24 Winter 7 Office Hours and other assistance Bob Bitmead Wednesdays 4:-6: EBU2 35 Ben Huang Mondays 7:-8:3 EBU2 35 Minh Ha Tuesdays 5:-7: EBU2 35 or by appointment

4 7//4 MAE43A Signals & Systems 24 Winter 7 Office Hours and other assistance Bob Bitmead Wednesdays 4:-6: EBU2 35 Ben Huang Mondays 7:-8:3 EBU2 35 Minh Ha Tuesdays 5:-7: EBU2 35 or by appointment MAE43A Signals & Systems 24 Winter 8 Homework, midterm and exam Homework will be set weekly except Week and due in class on the following Thursday The midterms will take place in class Tuesday, January 28 and Thursday February 2 sixty minutes each The final will take place Thursday, March 2, 8:-:am probably in the class room Pepper Canyon 6 4

2 x Homework % [Secret: the two numbers are almost always the same] To succeed: avail yourself of all the help including other books, past students,

5 7//4 MAE43A Signals & Systems 24 Winter 9 Grading and passing with flying colors Final score is the maximum of the following two numbers. x Final %.5 x Final % +.3 x Midterm % +.2 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 MAE43A Signals & Systems 24 Winter A speech signal.3 Voice recording.2. sample value (units)..2.3 Now is the time for all good men to come to the aid of their country time (s) x 4 5

6 7//4 MAE43A Signals & Systems 24 Winter Speech signal Voice recording.3.2. Seven seconds of speech sampled at 225 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 4 Because we have fairly rapid sampling we can consider (for the moment) this a continuous-time signal We will return to this later MAE43A Signals & Systems 24 Winter 2 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) 6

7 7//4 MAE43A Signals & Systems 24 Winter 3 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? MAE43A Signals & Systems 24 Winter 4 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 x(t)e st dt =.7 s e st =.7 s Note that the Laplace transform ignores the t< part 7

8 7//4 MAE43A Signals & Systems 24 Winter 5 Familiar signals the step function The unit step function or Heaviside function (, t < (t) =, t The Laplace transform of the unit step L (s) = Z (t)e st dt = s This is the same as for a constant function of value because they are identical for t This function is discontinuous at t= MAE43A Signals & Systems 24 Winter 6 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= The black curve is k=2 8

9 7//4 MAE43A Signals & Systems 24 Winter 7 Familiar signals the ramp function Ramp function (, t < r(t) = t, t This function is unbounded but it is continuous Laplace transform L r (s) = Z r(t)e st dt = Z te st dt = s 2 Notice that the ramp function is the integral of the step r(t) = Z t (z) dz MAE43A Signals & Systems 24 Winter 8 Familiar signals the impulse function The impulse function or Dirac delta function (t) =, for t 6= R (t) dt =, for > The impulse function is neither continuous nor bounded Laplace transform L (s) = The step is the integral of the impulse (t) = Z Z t (t)e st dt = (z) dz Z + (t) dt = 9

10 7//4 MAE43A Signals & Systems 24 Winter 9 Bounded continuous approximation of the impulse Continuous and bounded approximations of δ(t) sin kt ˆk(t) =k t red is 25 ˆ2(t) sin 2t = 5 t black is 7.5 ˆ4(t) sin 4t = 7 t The impulse function has a sampling property Z b a f(z) (z) dz = f() if 2 (a, b) for any function f(t) continuous at t= MAE43A Signals & Systems 24 Winter 2 Familiar signals real exponentials Red Blue Black Laplace e.5t e t e 2t s.5, s, s 2 All of these signals are unbounded Red Blue Black Laplace e.5t e t e 2t s +.5, s +, s +2

11 7//4 MAE43A Signals & Systems 24 Winter 2 Red Blue Familiar signals one-sided real exponentials e.5t (t) e t (t) Black e 2t (t) Laplace s +.5 poles -.5, -, -2 s + s +2 Red Blue Black Laplace s.5 e.5t e t e 2t (t) (t) s, s poles (,.5), (,), (,2) (t) s, s 2 s MAE43A Signals & Systems 24 Winter 22 Familiar signals - sinusoids Red Blue Black sin(5t) sin(3t) sin(7t) Red Black sin(5t) cos(5t)

12 7//4 MAE43A Signals & Systems 24 Winter 23 One-sided sinusoids Sinusoids sin(t)(t) sin(5t)(t) sin(2t)(t) Laplace transforms 5 s 2 + s Poles ±j, ±j5, ±j2 2 s Sinusoid and cosinusoid sin(t)(t) cos(t)(t) Laplace transforms s s 2 + s 2 + Poles ±j MAE43A Signals & Systems 24 Winter 24 Complex exponentials Blue Red e 2t sin(2t)(t) ±e 2t (t) Laplace transforms 2 (s + 2) = 2 s 2 +4s + 44 s +2 Poles -2±j2, -2 The (upper) red curve is called the envelope of the blue curve 2

13 7//4 MAE43A Signals & Systems 24 Winter 25 More complex exponentials Blue Red e 2t sin(2t)(t) e 2t (t) Laplace transform 2 (s 2) = 2 s 2 4s + 44 s 2 Poles 2±j2, 2 in the right half of the complex plane that is, the real part is positive These signals are unbounded MAE43A Signals & Systems 24 Winter 26 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 3

14 7//4 MAE43A Signals & Systems 24 Winter 27 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 MAE43A Signals & Systems 23 Winter 28 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 4

15 7//4 MAE43A Signals & Systems 24 Winter 29 Macknade rotary bulk sugar dryer - experiments MAE43A Signals & Systems 24 Winter 3 Input sugar temperature signal 5

16 7//4 MAE43A Signals & Systems 24 Winter 3 Input air temperature signal MAE43A Signals & Systems 24 Winter 32 Input air humidity signal 6

17 7//4 MAE43A Signals & Systems 24 Winter 33 Output sugar temperature signal A quantized signal MAE43A Signals & Systems 24 Winter 34 Macknade sugar dryer is a 3-input -output system 7

18 7//4 MAE43A Signals & Systems 24 Winter 35 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 ) T i a ( k τ ) = T i a ([ k - ] τ ) + ha τ + C pv E i ( k τ ) s a [ ][ T i ([ k - ] τ )- T i ([ k - ] τ )] C pa M i a ( k τ ) + C pv M i v ( k τ ) s a T s i ( k τ ) = T s L H 2 O E i ( k τ ) + ha τ 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 ) + α M 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 MAE43A Signals & Systems 24 Winter 36 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 8

19 7//4 MAE43A Signals & Systems 24 Winter 37 Signal norms a measure of signal size applez 4 General L p signal norms kxk p = x(t) p dt p applez 4 L 2 Euclidean signal norm kxk 2 = x(t) 2 dt often related to signal energy voltage, current, velocity signals 2 L norm Z 4 kxk = x(t) dt L norm or sup norm kxk 4 = sup x(t) t2(,) MAE43A Signals & Systems 24 Winter 38 Signal transforms Laplace and Fourier Expression of the signal in a different domain Laplace transform for signals defined on domain [ -, ] x(t) = X(s)e st ds, for t 2 j c j Fourier transform for signals defined on the domain [-, ] X(!) 4 = X(s) Z 4 = Z x(t)e st dt Z c+j x(t)e j!t dt, x(t) = 2 Z Since the transforms are invertible no information is lost in using them instead of the original time-domain description X(!)e j!t d!, for all t 9

MAE143A Signals & Systems

MAE143A Signals & Systems MAE43A Signals & Systems Winter 206 MAE43A Signals & Systems http://numbat.ucsd.edu/~bob/signals Tu, Th: 09:30-0:50 Pepper Canyon 06 We 09:00-09:50 Pepper Canyon 09 Professor Bob Bitmead rbitmead@ucsd.edu