Slide 1 ELT2035 Signals & Systems Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi Lesson 1 Introduction to signals ❑ Started to be in E[.]
ELT2035 Signals & Systems Lesson 1: Introduction to signals Hoang Gia Hung Faculty of Electronics and Telecommunications University of Engineering and Technology, VNU Hanoi Course overview ❑ Started to be in ECE curricula in the 1980s (the first textbook was Signals and Systems, by A V Oppenheim and A S Willsky, published in 1983) ❑ Concepts of signals and systems ❑ Mathematical descriptions of signals and systems ❑ Analysis of Linear Time Invariant Systems Provide the necessary background for follow-up courses at UET: ELT3051 – Control Engineering ELT3144E – Digital Signal Processing ELT3057 – Digital Communications and Coding Theory ELT3094 – Introduction to Signal Processing for Multimedia Systems ELT3281 – Microprocessor and embedded system What is signal? ❑ Roughly speaking, anything that carries information can be considered a signal: speech, ECG, VN index, … ❑ Plotted against time, which is called an independent variable ❑ A signal may have more independent variables: pictures, videos, … What is system? ❑ Roughly speaking, any physical device or computer program can be considered a system if the application of a signal to the device or program generates a new signal Input signal System Output signal Design/build a system to obtain desirable outputs from the input Classification of signals ❑ Continuous – Discrete time signals ❑ What is time? ❑ ❑ ❑ ❑ ❑ ❑ Periodic – Nonperiodic signals Causal – Anticausal – Noncausal signals Odd – Even signals Deterministic – Random signals Finite – infinite length signals Multichannel – multidimensional signals Conversion of a CT signal to a DT signal by sampling Energy and power of signals ❑ The total energy of a continuous time signal f(t) is ∞ 𝐸𝑓 = න 𝑓(𝑡) 𝑑𝑡 −∞ ❑ And its average power is 𝑇 Τ2 𝑃𝑓 = lim න 𝑓(𝑡) 𝑑𝑡 𝑇→∞ 𝑇 −𝑇 Τ ❑ Similarly, for a discrete time signal f(n) ∞ 𝐸𝑓 = 𝑓[𝑛] 𝑛=−∞ 𝑁 𝑃𝑓 = lim 𝑓[𝑛] 𝑁→∞ 2𝑁 + 𝑛=−𝑁 Energy and power signals ❑ A signal is referred to as an energy signal iff the total energy of the signal is bounded ❑ A signal is referred to as a power signal iff the average power of the signal is bounded ❑ Quiz: Find the energy and power of 𝑓 𝑡 = sin 𝑡 ❑ Solution: 𝑇 𝑇 𝑇 ➢ 𝐸𝑓 = lim −𝑇 sin 𝑡 𝑑𝑡 = lim −𝑇 sin2 𝑡 𝑑𝑡 = lim ቂ−𝑇 𝑑𝑡 − 𝑇→∞ 𝑇 𝑡 − 𝑇→∞ −𝑇 cos 2𝑡 𝑑𝑡ቃ = lim 𝑇 𝑇→∞ 2𝑇 −𝑇 ➢ 𝑃𝑓 = lim 𝑇→∞ sin 2𝑡 sin 𝑡 𝑑𝑡 = 𝑇 ቚ = 𝑇→∞ 𝑇−(−𝑇) sin 2𝑇−sin(−2𝑇) lim − 𝑇→∞ −𝑇 𝑇−(−𝑇) lim 2𝑇 𝑇→∞ − sin 2𝑇−sin(−2𝑇) = ∞ = ❑ The energy and power classifications of signals are mutually exclusive ➢ There are signals that are neither energy nor power signals Basic operations on signals ❑ Objective: design/built a system to manipulate signals How are signals be manipulated inside a system? ❑ Operations performed on dependent variables: amplitude scaling, addition, multiplication, differentiation, integration ❑ Operations on thedamped independent variable: The performed product is called an exponentially signal ➢ Time scaling ➢ Reflection ➢ Time shifting A system is usually built by combining multiple basic operations on input signals to obtain the desirable output signals Examples of signals multiplication ❑ Sketch the signal 𝑥 𝑡 = 4𝑒 −2𝑡 cos(6𝑡 − 60°) Examples of time shifting a signal 𝑒 −2𝑡 , 𝑡 ≥ ❑ Given 𝑓 𝑡 = ቊ , sketch the signals 𝑓 𝑡 − & 𝑓 𝑡 + 0, 𝑡 < Examples of time scaling a signal ❑ A signal 𝑓 𝑡 is depicted below Sketch 𝑓 2𝑡 & 𝑓 𝑡 Elementary signals 1, 𝑡 ≥ ❑ Unit step signal: 𝑢(𝑡) = ቊ 0, 𝑡 < ❑ Unit impulse signal (a.k.a Dirac delta function): 𝛿 𝑡 = ∀𝑡 ≠ ∞ and −∞ 𝛿 𝑡 𝑑𝑡 = Notice that 𝛿 𝑡 is undefined at 𝑡 = ❑ ❑ ❑ ❑ Unit ramp signal: 𝑡𝑢(𝑡) Sinusoidal signal: 𝐴 cos(𝜔𝑡 + 𝜑) (Real) exponential signal: 𝐵𝑒 𝛼𝑡 (Complex) exponential signal: 𝑒 𝑠𝑡 where 𝑠 = 𝜎 + 𝑗𝜔 Why we need elementary signals? Modelling natural signals Construct more complex signals System identification Impulse properties ❑ The important of the unit impulse is not its shape but the fact that its width approaches zero while its area remains unity ❑ Multiplication of a unit impulse 𝛿 𝑡 by a function 𝑥(𝑡) that is known to be continuous at 𝑡 = 0: ➢ 𝑥 𝑡 𝛿 𝑡 = 𝑥(0)𝛿(𝑡) ➢ Similarly, 𝑥 𝑡 𝛿 𝑡 − 𝑇 = 𝑥(𝑇)𝛿(𝑡 − 𝑇), provided 𝑥(𝑡) is continuous at 𝑡 = 𝑇 ❑ Sampling/sifting property: ∞ ∞ ∞ ➢ −∞ 𝑥 𝑡 𝛿 𝑡 𝑑𝑡 = −∞ 𝑥 𝛿 𝑡 𝑑𝑡 = 𝑥 −∞ 𝛿 𝑡 𝑑𝑡 = 𝑥 Similarly, ∞ −∞ 𝑥 𝑡 𝛿 𝑡 − 𝑇 𝑑𝑡 = 𝑥(𝑇) ➢ The area under the product of a function with an unit impulse equals the value of that function where the unit impulse is located 𝑎 ❑ Time-scaling property: 𝛿 𝑎𝑡 = 𝛿(𝑡) Proof: HW ❑ Unit impulse is not an ordinary function but rather a generalized function ➢ In this approach, 𝛿 𝑡 is defined by its effect on other functions at every instant of time (i.e the sampling property) Example of constructing a complex signal from elementary signals Practice ❑ Classification of signal: determining the type of a given signal ❑ Calculation of the total energy and power of a given signal ❑ Performing basic operations, especially a combination of time scaling and time shifting, on a given signal ❑ Construction of a complex signal from several elementary signals