Signals Lecture 1 PDF

Lecture 1 Dr. Adel M. Soliman Course Specifications  Course aims Assessment Schedule and Weighting Course CLOs  Prerequisite: Electrical Circuits 1 (EPE 112)  Reference Book: Lecture Notes Signals and Systems ECE 271  Textbooks: Alan V. Oppenheim, Ala...

Lecture 1 Dr. Adel M. Soliman Course Specifications  Course aims Assessment Schedule and Weighting Course CLOs  Prerequisite: Electrical Circuits 1 (EPE 112)  Reference Book: Lecture Notes Signals and Systems ECE 271  Textbooks: Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, "Signals and Systems," Prentice-Hall signal processing series, 4th. Ed., 2004. HWEI P. HSU, Theory and Problems of Signals and Systems, Pearson, 2nd. Ed..  Course Description Signals and Systems. Linear Time-Invariant Systems. Laplace Transform and Continuous-Time LTI Systems. Fourier Analysis of Continuous-Time Signals and Systems. Fourier Analysis of Discrete-Time Signals and Systems. Z-Transform. Fast Fouries Transform Signals and Systems  Signals may describe a wide variety of physical phenomena. signals can be represented in many ways. Signals are represented mathematically as functions of one or more independent variables. The information in a signal is contained in a pattern of variations of some form. different sounds correspond to different patterns in the variations of acoustic pressure recording of speech. (The signal represents acoustic pressure variations as a function of time) a microphone senses variations in acoustic pressure, which are then converted into an electrical signal.  CLASSIFICATION OF SIGNALS SIGNAL Analog signal Digital signal 01011… Continuous time signal Discrete time signal 1. Continuous-Time and Discrete-Time Signals 𝑥𝑥 𝑡𝑡 Sampler 𝑥𝑥 𝑛𝑛 A continuous-time signal x(t) A discrete-time signal x[n] The independent variable is continuous. defined only at discrete times. The signals are defined for a continuum The independent variable takes of values of the independent variable. on only a discrete set of values. (Ts is the sampling interval) Example (1) For a continuous-time signal x ( t ) shown in figure. Sketch and label each of the following signals: a) 𝒙𝒙(𝒕𝒕 − 𝟐𝟐) b) 𝒙𝒙(−𝒕𝒕) c) 𝒙𝒙 𝟐𝟐𝟐𝟐 d) 𝒙𝒙(𝒕𝒕⁄𝟐𝟐) Solution (a) (b) c) 𝒙𝒙 𝟐𝟐𝟐𝟐 d) 𝒙𝒙(𝒕𝒕⁄𝟐𝟐) Solution (c) (d) compression expanding Example (2) For a discrete-time signal x [ n ] shown in figure. Sketch and label each of the following signals: a) 𝒙𝒙(𝒏𝒏 − 𝟐𝟐) b) 𝒙𝒙(−𝒏𝒏) c) 𝒙𝒙 𝟐𝟐𝟐𝟐 d) 𝒙𝒙(−𝒏𝒏 + 𝟐𝟐) Solution (a) Shifting (b) Folding (Flipping)  c) 𝒙𝒙(−𝒏𝒏 + 𝟐𝟐) d) 𝒙𝒙 𝟐𝟐𝟐𝟐 Solution (c) (d) Shifting and Folding Down Sampling 2. Real and Complex Signals A signal x(t) is a real signal if its value is a real number. A signal x(t) is a complex signal if its value is a complex number. where x1( t ) and x2(t ) are real signals and 𝑗𝑗 = −1 t represents either a continuous or a discrete variable. 3. Deterministic and Random Signals Deterministic signals: are those signals whose values are completely specified for any given time. Random signals are those signals that take random values at any given time and must be characterized statistically.  4. Even and Odd Signals even signal: odd signal: 4. Even and Odd Signals  4. Even and Odd Signals even signal Χ even signal = even signal odd signal Χ odd signal = even signal even signal Χ odd signal = odd signal Example (3) Find the even and odd components of : 𝑥𝑥 𝑡𝑡 = 𝑒𝑒 𝑗𝑗𝑗𝑗 Solution ⸪ 𝑥𝑥 𝑡𝑡 = 𝑥𝑥𝑒𝑒 𝑡𝑡 + 𝑥𝑥𝑜𝑜 (𝑡𝑡) 1 ⸪ 𝑥𝑥𝑒𝑒 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 + 𝑥𝑥(−𝑡𝑡) 2 1 𝑗𝑗𝑗𝑗 −𝑗𝑗𝑗𝑗 1 𝑥𝑥 ⸫ 𝑒𝑒 𝑡𝑡 = 𝑒𝑒 + 𝑒𝑒 = [(cos 𝑡𝑡 + 𝑗𝑗 sin 𝑡𝑡) + (cos 𝑡𝑡 − 𝑗𝑗 sin 𝑡𝑡)] 2 2 1 ⸫ 𝑥𝑥𝑒𝑒 𝑡𝑡 = 2 cos 𝑡𝑡 = cos 𝑡𝑡 2 1 ⸪ 𝑥𝑥𝑜𝑜 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 − 𝑥𝑥(−𝑡𝑡) 2 1 1 ⸫ 𝑥𝑥𝑜𝑜 𝑡𝑡 = 𝑒𝑒 𝑗𝑗𝑗𝑗 − 𝑒𝑒 −𝑗𝑗𝑗𝑗 = [(cos 𝑡𝑡 + 𝑗𝑗 sin 𝑡𝑡) − (cos 𝑡𝑡 − 𝑗𝑗 sin 𝑡𝑡)] 2 2 1 ⸫ 𝑥𝑥𝑜𝑜 𝑡𝑡 = 2 𝑗𝑗 sin 𝑡𝑡 = 𝑗𝑗 sin 𝑡𝑡 2 Example (4) Sketch and label the even and odd components of the signals shown in Fig : (a) (b) Solution 1 1 (a) 𝑥𝑥𝑒𝑒 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 + 𝑥𝑥(−𝑡𝑡) & 𝑥𝑥𝑜𝑜 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 − 𝑥𝑥(−𝑡𝑡) 2 2 ⸪ 𝟏𝟏 𝟏𝟏 𝒙𝒙 −𝒕𝒕 𝒙𝒙 𝒕𝒕 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝒙𝒙 𝒕𝒕 𝟐𝟐 −𝟏𝟏 𝒙𝒙 −𝒕𝒕 𝟐𝟐  Solution (b) 1 1 (b) 𝑥𝑥𝑒𝑒 𝑡𝑡 = 𝑥𝑥 𝑛𝑛 + 𝑥𝑥(−𝑛𝑛) & 𝑥𝑥𝑜𝑜 𝑡𝑡 = 𝑥𝑥 𝑛𝑛 − 𝑥𝑥(−𝑛𝑛) 2 2 𝟏𝟏 ⸪ 𝟏𝟏 𝒙𝒙 − 𝒏𝒏 𝒙𝒙 𝒏𝒏 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝒙𝒙 𝒏𝒏 𝟐𝟐 −𝟏𝟏 𝒙𝒙 − 𝒏𝒏 𝟐𝟐 5. Periodic and Nonperiodic Signals periodic continuous-time signals: Periodic discrete-time signals : Example (5) Determine whether or not each of the following signals is periodic. If a signal is periodic, specify its fundamental period.: (a) 𝑥𝑥1 𝑡𝑡 = 𝑒𝑒 𝑗𝑗𝑗𝑗 𝑡𝑡 (b) 𝑥𝑥2 𝑡𝑡 = 𝑗𝑗 𝑒𝑒 𝑗𝑗10𝑡𝑡 (c) 𝑥𝑥3 𝑡𝑡 = 3 𝑒𝑒 𝑗𝑗𝑗𝜋𝜋 𝑛𝑛+1⁄2 ⁄5 Solution (a) 𝑥𝑥1 𝑡𝑡 = 𝑒𝑒 𝑗𝑗𝑗𝑗 𝑡𝑡 = cos 20𝑡𝑡 + 𝑗𝑗 sin 20𝑡𝑡 ⸪ sinusoidal signals are periodic ⸫ x1(t) is a periodic complex exponential ⸪ 𝜔𝜔0 = 20 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 2𝜋𝜋 2𝜋𝜋 𝜋𝜋 ⸫ the fundamental period is : 𝑇𝑇0 = 𝜔𝜔0 = 20 = 10 (b) 𝑥𝑥2 𝑡𝑡 = 𝑗𝑗 𝑒𝑒 𝑗𝑗10𝑡𝑡 10𝑡𝑡+𝜋𝜋⁄2 𝑥𝑥2 𝑡𝑡 = 𝑗𝑗 𝑒𝑒 𝑗𝑗𝑗𝑗 𝑡𝑡 = 𝑒𝑒 𝑗𝑗 = cos 10𝑡𝑡 + 𝜋𝜋⁄2 + 𝑗𝑗 sin 10𝑡𝑡 + 𝜋𝜋⁄2 ⸪ sinusoidal signals are periodic ⸫ x2(t) is a periodic complex exponential ⸪ 𝜔𝜔0 = 10 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 2𝜋𝜋 2𝜋𝜋 𝜋𝜋 ⸫ the fundamental period is : 𝑇𝑇0 = 𝜔𝜔0 = 10 = 5 (c) 𝑥𝑥3 𝑛𝑛 = 3 𝑒𝑒 𝑗𝑗𝑗𝜋𝜋 𝑛𝑛+1⁄2 ⁄5 𝑛𝑛+1⁄2 ⁄5 𝑥𝑥3 𝑛𝑛 = 3 𝑒𝑒 𝑗𝑗3𝜋𝜋 = 3 cos 3𝜋𝜋 𝑛𝑛 + 1⁄2 ⁄5 + 𝑗𝑗 3 sin 3𝜋𝜋 𝑛𝑛 + 1⁄2 ⁄5 ⸪ sinusoidal signals are periodic ⸫ x3(n) is a periodic complex exponential 3𝜋𝜋 ⸪ 𝜔𝜔0 = 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 5 2𝜋𝜋 ⸫ the fundamental period is : 𝑁𝑁 = 𝑚𝑚 𝜔𝜔0 2𝜋𝜋 10 𝑁𝑁 = 𝑚𝑚 = 𝑚𝑚 3𝜋𝜋⁄5 3