SIGNALS & SYSTEMS. Achmad Rizal Danisya/ /

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1 SIGNALS & SYSTEMS Achmad Rizal Danisya/ /

2 Lecture rules 14 Meetings including 2 Quizes, Mid & Final Exam Quizes = 20 % Team Points = 15 % Mid Exam = 25 % Final Exam = 25 % Assignments = 15 % NO REMEDY

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4 Assessment Mark Range Point Explaination A > 80,0 4 Very Good B 65,00-79,99 3 Good C 50,00-64,99 2 Enough D 30,00-49,99 1 Bad E 29,99 0 Very Bad

5 Syllabus Pengertian Sinyal Dasar sinyal berbasis waktu-kontinu. Dasar sinyal berbasis waktu-diskret. Beberapa sinyal penting. Sistem & Klasifikasi Sistem Realisasi sistem dalam bentuk blok diagram Kestabilan sistem linear Permasalahan dan penyelesaian persamaan differensial Persamaan diferensial sistem dan fungsi sistem H(s). Respon Impuls h(t). Permasalahan dan penyelesaian beberapa sinyal Operasi konvolusi sistem kontinu. Tanggapan pulsa h(n). Operasi konvolusi sistem diskret.

6 Syllabus (cont) Definisi dan contoh alih ragam Laplace. Teorema alih ragam Laplace Alih ragam Laplace balik. Penerapan alih ragam Laplace dalam berbagai study kasus Definisi dan evaluasi alih ragam - Z. Teorema alih ragam - Z. Penerapan alih ragam Z dalam berbagai study kasus Alih ragam -Z balik. Penerapan alih ragam -Z. UTS

7 Syllabus (cont) Deret Fourier untuk sinyal periodik. Karakteristik alih ragam Fourier Perbandingan alih ragam Laplace dengan alih ragam Fourier. Penerapan alih ragam Fourier dalam berbagai study kasus Deret Fourier untuk sinyal periodik diskret. Teorema pencuplikan dan masalah aliasing dan leakage. Alih ragam Fourier diskret (DFT/ =Discrete Fourier Transform). Karakteristik DFT Membahas Fast Fourier Transform. Membahas penerapan DFT dan FFT dalam penyelesaian masalah-masalah pengolahan sinyal digital. 14 Mei 2015 QUIZ 2 Konsep dari keadaan (state). 21 Mei 2015 Persamaan keadaan sistem kontinu. 28 Mei 2015 Persamaan keadaan sistem diskret. Penerapan metode state space

8 Books to read Oppenheim, A.V., A.S. Willsky dan I.T. Young., 1983, Signals and Systems, Prentice-Hall,, Englewoods Cliffs, new Jersey Hsu, Hwei P., 1995, Schaum's outline of theory and problems of signals and systems, McGraw-Hill Companies, Inc, United States of America

9 INTRODUCTION TO SIGNALS & SYSTEMS (LECTURE 01)

10 What is a Signal?

11 signals A Signal is the function of one or more independent variables that carries some information to represent a physical phenomenon. Example : in a RC circuit the signal may represent the voltage across the capacitor or the current flowing in the Resistor.

12 Signal Examples Acoustic signals Acoustic pressure (sound) over time Mechanical signals Velocity of a car over time Video signals Intensity level of a pixel (camera, video) over time

13 Continuous-time Signal A continuous-time signal, also called an analog signal, is defined along a continuum of time. x t dt t2 t1

14 Discrete time Signal A discrete-time signal is defined at discrete times. n2 n=n1 x[n]

15 Elementary Signals Sinusoidal & Exponential Signals Sinusoids and exponentials are important in signal and system analysis because they arise naturally in the solutions of the differential equations. Sinusoidal Signals can expressed in either of two ways : cyclic frequency form- A sin 2πf o t = A sin ( 2π T o )t radian frequency form- A sin ω o t ω o = 2 πf o = 2π/To T o = Time Period of the Sinusoidal Wave

16 Signal s Energy and Power For an instance, v(t) and i(t) are voltage and current to a resistance R, then the Power of a signal in a certain t time is : p t = v t i t = 1 R v2 (t) (1.1) So the total Energy for time interval t 1 t t 2 is t2 t1 p t dt = t2 t1 1 R v2 t dt (1.2)

17 Sinusoidal & Exponential Signals Contd. x t = A sin 2Пf o t + θ = A sin (ω o t + θ) x(t) = Ae = Aejω t Complex Exponential = A[cos (ω o t) + j sin (ω o t)] at Real Exponential Sinusoidal signal θ = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal f o = fundamental cyclic frequency of sinusoidal signal ω o = radian frequency

18 Signal s Energy and Power (with complex value) the total Energy for a Continuous-time Signal with Complex Properties ranging t 1 t t 2 is t2 t1 x t 2 dt (1.3) the total Energy for a Discrete-time Signal with Complex Properties ranging n 1 n n 2 is n2 n=n1 x n 2 (1.3)

19 Unit Step Function 1, t 0 u t 1/ 2, t 0 0, t 0 Precise Graph Commonly-Used Graph

20 Signum Function 1, t 0 sgn t 0, t 0 2u t 1 1, t 0 Precise Graph Commonly-Used Graph The signum function, is closely related to the unit-step function.

21 Unit Ramp Function t t, t 0 ramp u u 0, t 0 t d t t The unit ramp function is the integral of the unit step function. It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.

22 Rectangular Pulse or Gate Function Rectangular pulse, a t 1/ a, t a / 2 0, t a/ 2

23 t Unit Impulse Function t As a approaches zero, g approaches a unit step andg approaches a unit impulse Functions that approach unit step and unit impulse So unit impulse function is the derivative of the unit step function or unit step is the integral of the unit impulse function

24 Representation of Impulse Function The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.

25 Properties of the Impulse Function The Sampling Property g The Scaling Property t t dt g t a a t t0 t t0 The Replication Property g(t) δ(t) = g (t)

26 Unit Impulse Train The unit impulse train is a sum of infinitely uniformlyspaced impulses and is given by T t t nt, n an integer n

27 The Unit Rectangle Function The unit rectangle or gate signal can be represented as combination of two shifted unit step signals as shown

28 The Unit Triangle Function A triangular pulse whose height and area are both one but its base width is not, is called unit triangle function. The unit triangle is related to the unit rectangle through an operation called convolution.

29 Sinc Function sinc t sin t t

30 Discrete-Time Signals Sampling is the acquisition of the values of a continuous-time signal at discrete points in time x(t) is a continuous-time signal, x[n] is a discretetime signal x n x nt where T is the time between samples s s

31 Discrete Time Exponential and Sinusoidal Signals DT signals can be defined in a manner analogous to their continuous-time counter part x[n] = A sin (2πn/No + θ) = A sin (2πF o n + θ) Discrete Time Sinusoidal Signal x[n] = e n Discrete Time Exponential Signal n = the discrete time A = amplitude θ = phase shifting radians, N o = Discrete Period of the wave 1/N 0 = F o = Ω o /2π = Discrete Frequency

32 Discrete Time Sinusoidal Signals

33 Discrete Time Unit Step Function or Unit Sequence Function u n 1, n 0 0, n 0

34 Discrete Time Unit Ramp Function, 0 n n n ramp u 1 0, n 0 m n m

35 Discrete Time Unit Impulse Function or Unit Pulse Sequence n 1, n 0 0, n 0 n an for any non-zero, finite integer a.

36 Unit Pulse Sequence Contd. The discrete-time unit impulse is a function in the ordinary sense in contrast with the continuous-time unit impulse. It has a sampling property. It has no scaling property i.e. δ[n]= δ[an] for any non-zero finite integer a

37 Operations of Signals Sometime a given mathematical function may completely describe a signal. Different operations are required for different purposes of arbitrary signals. The operations on signals can be Time Shifting Time Scaling Time Inversion or Time Folding

38 Time Shifting The original signal x(t) is shifted by an amount tₒ. X(t) X(t-to) Signal Delayed Shift to the right

39 Time Shifting Contd. X(t) X(t+to) Signal Advanced Shift to the left

40 Time Scaling For the given function x(t), x(at) is the time scaled version of x(t) For a 1,period of function x(t) reduces and function speeds up. Graph of the function shrinks. For a 1, the period of the x(t) increases and the function slows down. Graph of the function expands.

41 Time scaling Contd. Example: Given x(t) and we are to find y(t) = x(2t). The period of x(t) is 2 and the period of y(t) is 1,

42 Time scaling Contd. Given y(t), find w(t) = y(3t) and v(t) = y(t/3).

43 Time Reversal Time reversal is also called time folding In Time reversal signal is reversed with respect to time i.e. y(t) = x(-t) is obtained for the given function

44 Time reversal Contd.

45 Operations of Discrete Time Functions Time shifting n n n0, n0 an integer

46 Operations of Discrete Functions Contd. Scaling; Signal Compression n Kn K an integer > 1

47 KULSUM KUis kecil SepulUh Menit cos ω 0 t δ t dt =? dengan T 5 0 = Gambarkan x t + 2 dan x 1 t untuk 2 x t = (t + 2)(u t u(t 4)) 3. Buktikan bahwa x t = cos (t) adalah sinyal genap dengan penjumlahan