ENT 281 Signal and Systems Lecture 6 Fourier

ENT 281 Signal and Systems Lecture 6 Fourier Transform Dr. Abdul Halim Ismail 1 Semester 1, 2017/2018 Session

Contents 1) Fourier Transform (FT) 2) Magnitude and Phase Representation of FT 3) Dirichlet’s Condition for FT Existence 4) FT for Standard Signal 5) Properties of CTFT 6) Application of the FT: Ideal Filter School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 2

Fourier Transform (FT) ü Fourier transform is a transformation technique which transforms signals from the continuous time domain to the corresponding frequency domain or vice versa. ü Applies for both periodic and nonperiodic (aperiodic) signals. ü In general, there are FOUR categories of Fourier Transform. (See table at next page) ü The Fourier transform derived in this chapter is the Continuous-time Fourier transform (CTFS) ü Applications: Analysis of linear time-invariant (LTI) systems, cryptography, signal analysis, signal processing, astronomy, RADAR and etc. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 3

Fourier Transform (FT) Categories of Fourier Representation of a LTI system Time Property Periodic Nonperiodic Continuous (t) Fourier Series (FS) Fourier Transform (FT) Discrete [n] Discrete Time Fourier Series (DTFS) Discrete Time Fourier Transform (DTFT) School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 4

Fourier Transform (FT) Fourier Transform or the Fourier integral of x(t) is defined as: X(ω) represents the frequency spectrum of x(t) and is called the spectral density function. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 5

Fourier Transform (FT) The inverse Fourier Transform of X(ω): Sometime, this symbol also been used to represent Inverse Fourier Transform Fourier transform pair can be denoted as: or School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 6

Magnitude and Phase Representation of FT The magnitude and phase representation of FT is the tool used to analyze the transformed signal. In general X(ω) is a complex valued function of ω. Therefore, X(ω) can be written as: XR(ω): is the real part of X(ω). XL(ω): is the imaginary part of X(ω). School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 7

Magnitude and Phase Representation of FT The magnitude of X(ω): The phase of X(ω): The plot of spectrum. versus is known as amplitude The plot of spectrum. versus is known as phase The amplitude spectrum and phase spectrum together is called frequency spectrum. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 8

Dirichlet’s Condition for FT Existence The conditions for a function x(t) to have Fourier transform, called Dirichlet’s condition, are: 1. x(t) is absolutely integrable over the interval of -∞ to ∞, that is: 2. x(t) has a finite number of discontinuities in every finite time interval. Further, each of these discontinuities must be finite. 3. x(t) has a finite number of maxima and minima in every finite time interval School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 9

FT for Standard Signal We look at several standard signals in order to apply the Fourier Transform: 1. Impulse function 2. One-sided Real Exponential signal 3. Double-sided Real Exponential signal 4. Complex Exponential signal 5. Constant Amplitude Signal 6. Signum function 7. Unit Step function 8. Rectangular pulse 9. Sine signal 10. Cosine signal School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 10

FT for Standard Signal 1) Impulse Function, denoted by : Given, Then, or School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 11

FT for Standard Signal 1) Impulse Function Hence, the Fourier transform of a unit impulse function is unity. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 12

FT for Standard Signal 2) One-sided Real Exponential signal School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 13

FT for Standard Signal 2) One-sided Real Exponential signal Amplitude spectrum Phase spectrum School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 14

FT for Standard Signal 3) Double-sided Real Exponential signal School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 15

FT for Standard Signal 3) Double-sided Real Exponential signal Cont. …. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 16

FT for Standard Signal 3) Double-sided Real Exponential signal Cont. … or for all ω School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 17

FT for Standard Signal 4) Complex Exponential signal Complex Exponential Signal is commonly denoted by . To find the FT of complex exponential function, consider finding the inverse FT of. Let School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 18

FT for Standard Signal 5) Constant Amplitude, e. g. x(t) = 1 • Since x(t) = 1 is not absolutely integrable, we cannot find its FT directly. So the FT of x(t) = 1 is determined through inverse Fourier transform of • Consider School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 19

FT for Standard Signal 5) Constant Amplitude, e. g. x(t) = 1 School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 20

FT for Standard Signal 6) Signum function School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 21

FT for Standard Signal 6) Signum function Cont. … School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 22

FT for Standard Signal 6) Signum function Cont. … School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 23

FT for Standard Signal 7) Unit Step function School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 24

FT for Standard Signal 7) Unit Step function Cont. … School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 25

FT for Standard Signal 7) Unit Step function Cont. … School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 26

FT for Standard Signal 8) Rectangular Pulse A Rectangular pulse rect(t), is denoted by . School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 27

FT for Standard Signal 8) Rectangular Pulse X( ) School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 28

FT for Standard Signal 8) Rectangular Pulse School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 29

FT for Standard Signal 9) Sine wave School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 30

FT for Standard Signal 10) Cosine wave School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 31

Properties of Continuous-Time Fourier Transform (CTFT) The properties of CTFT can be delineated as follows; 1. Linearity 2. Symmetry 3. Time Shifting (Delay) 4. Time Scaling 5. Time-Differentiation 6. Time-Integration 7. Convolution 8. Duality 9. Modulation School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 32

Properties of CTFT 1) Linearity • The FT of two functions can be represented as; • Then, , where a and b are arbitrary constants. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 33

Properties of CTFT 1) Linearity (Example) Suppose we want to find the Fourier transform (FT) of cos ω0 t. The cosine signal can be written as a sum of two exponentials as follows: The linearity property of the FT is: Similarly the FT for sin ω0 t is: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 34

Properties of CTFT 2) Symmetry If x(t) is a real-valued time signal, then: , X* is complex conjugate. If X( ) is in polar form: i. e. , the magnitude spectrum is an even function of frequency and the phase spectrum is an odd function of frequency Taking the complex conjugate both sides Replacing each ω by - ω The left-hand sides of the last 2 equations are equal School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 35

Properties of CTFT 3) Time Shifting (Delay) If: Then: Similarly: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 36

Properties of CTFT 4) Time Scaling If: Then: , where is real constant School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 37

Properties of CTFT 4) Time Scaling (Example) Suppose we want to determine the Fourier transform of the pulse The Fourier transform of is: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 38

Properties of CTFT 5) Time Differentiation & 6) Time Integration Time Differentiation: If: Then: Time Integration: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 39

Properties of CTFT 5) Time Differentiation (Example) Consider the unit-step function. The function can be written: The first term has πδ(ω) The derivatives of discontinuous signals in terms of the delta function School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 40

Properties of CTFT 7) Convolution If: and: Then: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 41

Properties of CTFT 7) Convolution (Example) Consider an LTI system with impulse response h(t). Compute the convolution using the convolution property of the FT whose input is the unit step function u(t). Solution: Taking the inverse FT: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 42

Properties of CTFT 8) Duality If: Then: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 43

Properties of CTFT 8) Modulation If: Then: Convolution in the frequency domain is carried out exactly like convolution in the domain. That is, School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 44

Properties of CTFT 8) Modulation The importance of this property is that the spectrum of a signal such as x(t) cos ω0 t can be easily computed. Since, It follow that School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 45

Example: Modulation Consider the signal: Where p(t) is the periodic impulse train with equal-strength impulses. Analytically, p(t) can be written as: Solution: Using the sampling property of the delta function, we obtain: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP)

Cont. … The periodic of the periodic impulse train p(t) it itself a periodic impulse train; Specifically: By modulation property, School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP)

Complete Fourier Transform Pairs: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 48

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Commonly used FT pairs: School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 50

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Application of the FT: Ideal Filter Amplitude Modulation • The goal of communication system: to convey the information from one point to another. • The signal need to be converted into useful form before sending the information through the transmission channel which is known as modulation. • Reasons for conversion: 1) To transmit information efficiently, 2) to overcome hardware limitation. 3) to reduce noise and interference, 4) to utilize the EM spectrum efficiently. • Modulation is the process of merging two signals to form a third signal with desirable characteristics of both. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 52

Application of the FT: Ideal Filter Amplitude Modulation Consider the signal multiplier. x(t) X y(t) m(t) The output is the product of information-carrying signal x(t) and the signal m(t) as carrier signal. This scheme is known as amplitude modulation, which has many forms depending on m(t). We concentrate only on the case m(t) = cos ω0 t , which represent the practical form of modulation and is referred as double-sideband (DSB) amplitude modulation. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 53

Application of the FT: Ideal Filter Amplitude Modulation The output of the multiplier is. Use the property of convolution in the frequency domain to obtain its spectrum School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 54

Application of the FT: Ideal Filter Amplitude Modulation The magnitude spectra of x(t) and y(t) given by: The part of the spectrum of Y(ω) centered at + ω0 is the result of convolving X(ω) with δ(ω-ω0 ) , And the part centered at –ω0 is the result of convolving X(ω) with δ(ω+ω0 ). This process of shifting the spectrum of the signal by is necessary because low-frequency (baseband) information signal cannot be propagated easily by radio waves. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 55

Application of the FT: Ideal Filter Amplitude Modulation The process of extracting the information signal from the modulated signal is called demodulation. Synchronous demodulation =>used to perform amplitude demodulation. The output of the multiplier is, Hence, School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 56

Application of the FT: Ideal Filter Amplitude Modulation To extract the original information signal x(t), the signal z(t) is passed through the system with frequency response H(ω). Magnitude Spectrum of z(t) The low-pass-filter frequency spectrum School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 57

Application of the FT: Ideal Filter Amplitude Modulation Such a system is referred to as a low-pass filter, since it passes only low-frequency components of the input signal and filters out all frequency higher than ωB, the cutoff frequency of the filter. b)The low-pass-filter frequency spectrum, c) The extracted information • Note that if , and there were no transmission losses involved, then the energy of final signal is one-fourth that of the original signal because the total demodulated signal contains energy located at ω= 2ω0, that is eventually discarded by the receiver. School of Mechatronic Engineering Universiti Malaysia Perlis (Uni. MAP) 58