Chapter 1 Signal Representations and Mathematical Operations Fairouz Yousof 1 Chapter overview CHAPTER 1 – SIGNAL REPRESENTATIONS AND MATHEMATICAL OPERATIONS • 1.1 Mathematical representation of signals: sinusoidal, step, exponential and Impulse • 1.2 Terminology and units (frequency, amplitude, wavelength, period, power) • 1.3 Visualization of signals (graph plotting) • 1.4 Mathematical operations on signals 2 1.1 Mathematical representation of signals: sinusoidal, step, exponential and impulse Fairouz Yousof 3 Mathematial representation of signals : Piecewise function • A Function Can be in Pieces • We can create functions that behave differently based on the input (x) value. 4 Mathematial representation of signals Sinusoidal 5 Mathematial representation of signals : Sinusoidal 6 Mathematial representation of signals: Sinusoidal Fairouz Yousof 7 Mathematial representation of signals : Step • Definition: The unit step function, u(t), is defined as • That is, u is a function of time t, and u has value zero when time is negative (before we flip the switch); and value one when time is positive (from when we flip the switch). 8 Mathematial representation of signals : Step The Unit Step Function (Heaviside Function) • In engineering applications, we frequently encounter functions whose values change abruptly at specified values of time t. One common example is when a voltage is switched on or off in an electrical circuit at a specified value of time t. • The value of t = 0 is usually taken as a convenient time to switch on or off the given voltage. • The switching process can be described mathematically by the function called the Unit Step Function 9 Mathematial representation of signals : Step • Shifted Unit Step Function • In many circuits, waveforms are applied at specified intervals other than t=0. Such a function may be described using the shifted (aka delayed) unit step function. • Definition of Shifted Unit Step Function • A function which has value 0 up to the time t=a and thereafter has value 1, is written: 10 Mathematial representation of signals Step • Example 1 - Shifted Unit Step Function • f(t)=u(t−3) • The equation means f(t) has value of 0 when t<3 and 1 when t>3. • The sketch of the waveform is as follows: 11 Mathematial representation of signals Step • Rectangular Pulse • A common situation in a circuit is for a voltage to be applied at a particular time (say t = a) and removed later, at t = b (say). We write such a situation using unit step functions as: • V(t)=u(t−a)−u(t−b) • This voltage has strength 1, dura on (b−a). Fairouz Yousof 12 Mathematial representation of signals Step • Example 2 - Rectangular Pulse • The graph of V(t)=u(t−1.2)−u(t−3.8) is as follows. • Here, the dura on is 3.8−1.2=2.6. 13 Mathematial representation of signals Ramp Fairouz Yousof 14 Mathematial representation of signals : Exponential Fairouz Yousof 15 Mathematial representation of signals Exponential Fairouz Yousof 16 Mathematial representation of signals Exponential 17 Mathematial representation of signals Impulse Fairouz Yousof 18 1.2 Terminology and units (frequency, amplitude, wavelength, period, power) 19 Terminology and units Frequency, amplitude, wavelength, period, power • the rate at which a vibration occurs that constitutes a wave, either in a material (as in sound waves), or in an electromagnetic field (as in radio waves and light), usually measured per second. Fairouz Yousof 20 Terminology and units Frequency, amplitude, wavelength, period, power Fairouz Yousof 21 1.3 Visualization of signals (graph plotting) Fairouz Yousof 22 1.4 Mathematical operations on signals Fairouz Yousof 23 Mathematical operations on signals Fairouz Yousof 24 Mathematical operations on signals Fairouz Yousof 25 Mathematical operations on signals Fairouz Yousof 26 Mathematical operations on signals Fairouz Yousof 27 Mathematical operations on signals • X1(t) and X2(t) are two time dependent signals, performing the additional operation on them we get Fairouz Yousof 28 Mathematical operations on signals • Multiplication of signals is illustrated in the diagram below, where X1(t) and X2(t) are two time dependent signals, on whom after performing the multiplication operation we get Fairouz Yousof 29 Mathematical operations on signals • integration of signals is also applicable to only continuous time signals. The limits of integra on will be from − ∞ to present instance of time t. It is mathematically expressed as Fairouz Yousof 30 End of Chapter Fairouz Yousof 31
