Fourier Transform FLB 23053 PDF

SIGNALS AND SYSTEMS FLB 23053 CHAPTER 3 FOURIER TRANSFORM 1  Representation Of Signals in Time and Frequency Domain A signal can be represented in the time domain or in the frequency domain. All electr...

SIGNALS AND SYSTEMS FLB 23053 CHAPTER 3 FOURIER TRANSFORM 1  Representation Of Signals in Time and Frequency Domain A signal can be represented in the time domain or in the frequency domain. All electronic signals can be visualized using two basic methods - the time domain and the frequency domain. Time Domain Frequency Domain The time domain is the form of visualization that The frequency domain is very important in the field of Signal shows variations of a signal with time. Processing. It shows the variation of the signal with respect to The most common time domain instrument is the frequency. The most common instrument for displaying the oscilloscope, which has graduations of volts on the frequency domain is the spectrum analyzer. Y-axis and graduations of time on the X-axis. The frequency domain representation is also called the spectrum of the signal. Signals in Time and Frequency Domains SignalsinTimeandFrequency Domains Time Domain Frequency Domain AMPLITUDE TimeDomain FrequencyDomain T1=1/f1 AMPLITUDE AMPLITUDE T=1/f 1 1 AMPLITUDE TIME f1 FREQUENCY TIME f1 FREQUENCY T2=1/f2 AMPLITUDE AMPLITUDE T2=1/f2 AMPLITUDE AMPLITUDE TIME f2 FREQUENCY TIME f2 FREQUENCY T = period f = frequency T=period f =frequency LECTURE 2 2-1 LECTURE2 2-1 2  Frequency Domain Representation of Non Periodic Signals The continuous time Fourier transform (CTFT) is a generalization of the Fourier series. It can be also called Fourier series representation for continuous time aperiodic signals as it only applies to continuous time aperiodic signals. The Fourier transform is used to transform a signal from the time domain to the frequency domain. For certain signals, Fourier transform can be performed analytically with calculus.   j t The Fourier transform of x(t) is defined as: X ( ) = x ( t )e − dt − The Fourier transform allows you to compute the frequency domain representation of a signal from the time domain signal.  1 There is also the inverse Fourier transform: x(t ) = 2  X ( )e jt d − The inverse Fourier transform allows you to compute the time domain representation from the frequency domain signal. 3 FLB23053 SIGNAL AND SYSTEM - BUSTANI  THE CONTINUOUS TIME FOURIER TRANSFORM FOR APERIODIC SIGNALS The Fourier transform and the inverse Fourier transform are known as the Fourier pair. The following is used for shorthand notation of the forward and inverse Fourier transforms:  F {x ( t )} = X ( ) =  x ( t )e − jt dt −  1 F −1 { X ( )} = x ( t ) = 2  X ( )e  d j t − Also, The Fourier transform can be defined in terms of frequency of Hertz as and corresponding inverse Fourier transform is 4  A summary of the Fourier series and Fourier transform expressions for both continuous-time and discrete-time signals are given. C o n t i n u o u s time Di s crete time Time domain Frequency domain Time domain Frequency domain + 1 jk  0 t = x(t)e −  ck e jk ( 2  / N ) n 1 x (t ) =  ck e jk  0 t c dt x[n ] = ck =  x [ n ]e − jk ( 2  / N ) n Fourier k = − k T T k= N N k= N Series cont i nuous time discrete f r equency discrete time discrete frequency per i odic in time aper iodic in frequency per i odic in time per i odic in frequency x (t ) = X ( j ) = x[n ] = X (e j ) = 1 + j t + − j t X ( j ) e d (t )e dt 1 j jn d + Fourier 2 − x  X (e )e  x [ n ]e − j n − 2 2 n = − T r ansform cont i nuous time cont inuous f r equency discrete time cont inuous f r e quency aper iodic in time aper iodic in frequency aper iodic in time per i odic in frequency 5 Fourier transform of Basic Signals A. Fourier transform of δ Functions B. Fourier transform of Constant Signal C. Fourier transform of Exponential Signal D. Fourier transform of the Sinusoidal Signal E. Fourier transform of Rectangular Signal 6 A. Fourier transform of δ Functions i) δ functions in time 7 ii) Shifted δ functions in time 8 B. Fourier transform of Constant Signal 9 Shifted δ functions in frequency 10  C. Fourier transform of Exponential Signal Consider the (non-periodic) signal (solution to first order ODE) x(t) = e − at u(t) a0 x(t) Then the Fourier transform is: ∞ 𝑋 j𝜔 = ∫ 𝑒−𝑎𝑡𝑢 𝑡 𝑒 −j𝜔𝑡 𝑑𝑡 −∞ ∞ = ∫ 𝑒 −(𝑎+j𝜔)𝑡 𝑑𝑡 0 ∞ 1 = 𝑒 −(𝑎+j𝜔)𝑡 | −(𝑎 + j𝜔) 0 1 = a=1 (𝑎 + j𝜔) 1 (𝑎 − j𝜔) = Showing the (𝑎2 + 𝜔2) (𝑎2 + 𝜔2) magnitude and 1 If x(t) is a real signal 1) X(0) is real, 2) Im(X(-j)) = -Im(X(-j)) = 𝑒 j tan−1(−𝜔/𝑎) phase shift (𝑎2 + 𝜔2) 11 EXAMPLE 1 Consider the odd two sided exponential function fα(t) defined as Determine the Fourier transforms of the following signals: where α > 0 By adapting slightly our earlier calculation for the even two sided exponential function we find The parameter α controls how rapidly the exponential function varies: 12 13  D. Fourier transform of the Sinusoidal Signal i) Fourier transform of the cosine The cosine signal x(t) = cos(ω0t) does not have the Fourier transform in the ordinary sense. It does, however, have a generalized Fourier transform: 14 ii) Fourier transform of the Sine 15 E. Fourier transform of Rectangular Signal 16 17 EXAMPLE 1 Determine the Fourier transforms of the following signals and sketch IX(ω)I: 18  Properties of the Fourier transform The Fourier transform has many useful properties that make calculations easier and also help thinking about the structure of signals and the action of systems on signals. Linearity 19 Time shift Given that the Fourier transform of the signal x(t) is X (ω) , i.e. x(t) ⇔ X(ω) , prove from first principle that 20  Time Differentiation Taking the derivative of the inverse Fourier transform 21 Convolution in time domain 22 Table 1 23 Table 2 24  CT Fourier Transform Pair Euler’s Formula 25 END ASSIGNMENT 10% 28 29 30  Fourier Transform for Solving Differential Equations What are Differential Equations? A differential equation is an equation that contains that involves the derivatives (a rate of change) of the dependent variables with respect to the independent variables. The differential equation helps us present a relationship between the changing quantity with respect to the change in another quantity. Ordinary differential equations applications in real life are used - to calculate the movement or flow of electricity (Electrical) - to calculate motion of an object to and fro like a pendulum (Mechanical) - to check the growth of diseases in graphical representation (Medical) Example: (d3y/dt3) + (d2x/dt2) = x Notes: y and x are dependent variables and t is an independent variable : commonly used notations for derivatives. (dy/dx) = y', (d2y/dx2) = y'', (d3y/dx3) = y''' Solving a differential equation is referred to as integrating a differential equation and a solution of a differential equation is an expression for the dependent variable in terms of the independent variable which satisfies the differential equation. 31  Linear time-invariant (LTI) system A linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (x ∗ h)(t) where h(t) is called the system's impulse response A good example of an LTI system is any electrical circuit consisting of resistors, capacitors and inductors Analysis of LTI CT System using Fourier Transform Fourier Fourier Inverse Transform Transform Fourier Transform Frequency Response, H(ω) = 32 The impulse response and frequency response are two attributes that are useful for characterizing linear time-invariant (LTI) systems. Impulse Response of LTI System A system's impulse response (often annotated as h(t) for continuous-time systems) is defined as the output signal that results when an impulse is applied to the system input. The transfer function of LTI system (in frequency domain) The transfer function of a continuous-time LTI system may be defined using Laplace transform or Fourier transform. Also, the transfer function of the LTI system can only be defined under zero initial conditions. The block diagram of a continuous-time LTI system is shown in the following figure. The transfer function 𝐻(𝜔) of an LTI system can be defined in one of the following ways − - The transfer function of an LTI system is defined as the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal provided that the initial conditions are zero. - The transfer function is defined as the ratio of output to input in frequency domain when the initial conditions are neglected. - The transfer function of the LTI system is the Fourier transform of its impulse response. 33  General Procedure for Solving Differential Equations The general procedure for solving ODE using Fourier transform: 1. Use Fourier transform to convert the ODE into algebraic equation. 2. Solve the algebraic equation for the unknown function 3. Use Partial fraction expansion to express the unknown function as the sum of first and second order terms 4. Use inverse Fourier transform to obtain the solution to the original problem 3 FLB23053 SIGNAL AND SYSTEM - BUSTANI 2 35 36 37