Fourier Transform PDF

Signals and Systems Software Engineering & IT, 2nd year Dr. Eng. Iyad M. Abuhadrous Associate professor in Computer & Control (Robotics) [email protected]...

Signals and Systems Software Engineering & IT, 2nd year Dr. Eng. Iyad M. Abuhadrous Associate professor in Computer & Control (Robotics) [email protected] Egyptian Chinese University Faculty of Engineering & Technology  The Fourier Transform Signals Part 2 Dr. Iyad Abuhadrous 2 Fourier Transform Lecture Objective: Understand the role of Fourier Transform in signal analysis and its applications in engineering. Key Concepts: Signal representation in time and frequency domains. Fourier Transform as a tool to analyze periodic and non- periodic signals. Describe non-periodic signals for which no Fourier series Some typical aperiodic signals, u(t), exp[-t]u(t), rect(t/T). Applications: Frequency analysis in audio processing. Image compression (e.g., JPEG). Dr. Iyad Ayoub  The Continuous Fourier Transform The Fourier Transform decomposes a signal into its frequency components (spectral contents), allowing analysis of signals in the frequency domain. This is widely used in signal processing, communications, and image analysis Used to convert a time-domain waveform into its frequency- domain equivalent. Mathematical formula for analyzing the frequency content of a signal. ∞ 𝑿(𝝎) = න 𝒙(𝒕) 𝒆−𝒋𝝎𝒕 𝒅𝒕 X(w) is called the −∞ Fourier transform of x(t)  = 2 f 𝟏 ∞ 𝒙(𝒕) = න 𝑿(𝒋𝝎) 𝒆𝒋𝝎𝒕 𝒅𝝎 Inverse Fourier transform 𝟐𝝅 −∞ Signals Part 2 Dr. Iyad Abuhadrous 4 Signals Part 2 Dr. Iyad Abuhadrous 5 Signals Part 2 Dr. Iyad Abuhadrous 6 Signals Part 2 Dr. Iyad Abuhadrous 7  Continuous Fourier Transform Example - FT of the rectangular pulse  t  − jt   / 2 − j t X ( ) =  rect e dt =  e dt = − 1 − j  / 2 e − e j  / 2 ( ) −   − / 2 j       x(t ) = rect (t /  ) 2 sin  sin  𝜔𝜏 𝜔𝜏 =  2  =  2  = 𝜏. 𝑆𝑎 = 𝜏 sinc     2 2𝜋    2  x(t) X()  Time-domain 1 F representation Frequency-  domain t -/2 0 /2 − − − 0    result       Signals Part 2 Fourier Transform pairs Dr. Iyad Abuhadrous 9  Continuous Fourier Transform Example - FT of the rectangular pulse A rectangular pulse in the time domain produces a sinc function in the frequency domain. The width of the pulse is inversely proportional to the spread of its frequency components. This is widely used in signal transmission and filtering. Signals Part 2 Dr. Iyad Abuhadrous 10  Continuous Fourier Transform Example: Fourier Transform pairs (Impulse & DC) An impulse in the time domain corresponds to a constant in the frequency domain. – Time domain: δ(t) as a sharp spike. – Frequency domain: Flat constant. A constant signal in the time domain results in an impulse in the frequency domain. – Time domain: constant (for ex. f(t) = 1). – Frequency domain: δ(w). Impulse: "Contains all frequencies equally, useful for sampling and system analysis. DC Signal: "Represents zero-frequency content." Signals Part 2 Dr. Iyad Abuhadrous 11  Continuous Fourier Transform Example: Fourier Transform pairs (Impulse & DC) Ex. 1: An Impulse in the Time Domain Corresponds to a Constant in the Frequency Domain Time Domain: δ(t) (impulse function).  From the sampling F  (t ) =   (t ) e − jt dt = e − j 0 =1 property of the impulse, − δ(t) in the time domain corresponds to a flat constant (1) in the frequency domain. Signals Part 2 Dr. Iyad Abuhadrous 12  Continuous Fourier Transform Example: Fourier Transform pairs (Impulse & DC) Ex. 2: A Constant Signal in the Time Domain Results in an Impulse in the Frequency Domain Time Domain: f(t)=1 (constant signal). (2) means that the area under the spike is (2) This integral does not converge unless ω=0. However, in the context of generalized functions, the result is a scaled Dirac delta function. The factor 2π​ ensures that the inverse operation recovers the original time-domain signal exactly. A constant signal in the time domain corresponds to an impulse at ω=0 in the frequency domain. Signals Part 2 Dr. Iyad Abuhadrous 13  Continuous Fourier Transform Fourier Transform pairs (impulse & dc) From the inverse formula  F  ( ) = −  ( ) e d = 2 e = 2 −1 1 j t 1 j 0t 1 2 f(t) = 1 F() = 2  () 1 F (2) t  0 0 −1 1 ∞ The same: 𝐹 (1) = ‫׬‬ (1) 𝑒 𝑗 𝜔 𝑡 𝑑 𝜔= 𝛿 𝑡 2𝜋 −∞ Signals Part 2 Dr. Iyad Abuhadrous 14  Continuous Fourier Transform Fourier Transform pairs (Exp. & impulse) Inverse Fourier Transform of δ(ω−ω0) The shifted delta function δ(ω−ω0) in the frequency domain corresponds to a complex exponential in the time domain.  F  ( − 0 ) = −  ( − 0 ) e d = 2 e −1 1 jt 1 j0t 2 e   ( − 0 ) or 1 j0t e j0t  2. ( − 0 ) 2 Signals Part 2 Dr. Iyad Abuhadrous 15  Continuous Fourier Transform Fourier Transform pairs (Exp. & impulse (cosine)) For cos(w0t) the Fourier Transform represents it as a continuous signal composed of two impulses in the frequency domain. ( Since cos(0t ) = e + e − j0t 1 j0t 2 ) cos(0t )    ( + 0 ) +  ( − 0 ) f(t) F() F () () t 0  −0 0 0 Signals Part 2 Dr. Iyad Abuhadrous 16 Signals Part 2 Dr. Iyad Abuhadrous 18  Fourier Transform Properties Fourier Transform properties simplify the analysis of signals and systems by providing intuitive relationships between the time and frequency domains. Linearity x1 (t )  X 1 ( ) x2 (t )  X 2 ( ) ax1 (t ) + bx2 (t )  aX 1 ( ) + bX 2 ( ) Example: If f(t) and g(t) are sinusoids, their Fourier Transform can be computed by linear combination. Dr. Iyad Ayoub  Fourier Transform Properties Symmetry: The symmetry property in the Fourier Transform states that if the signal 𝑥(𝑡) is real-valued, its Fourier Transform 𝑋(𝜔) satisfies the conjugate symmetry: x(t )  X ( ) X (− )  X ( ) * * Denotes the complex conjugate Signals Part 2 Dr. Iyad Abuhadrous 20  FT: Time Scaling 1   f (t )  F ( ) f (at )  F   a a |a| > 1: compress time axis, expand frequency axis |a| < 1: expand time axis, compress frequency axis Effective extent in the time domain is inversely proportional to extent in the frequency domain (bandwidth). f(t) is wider  spectrum is narrower f(t) is narrower  spectrum is wider Signals Part 2 Dr. Iyad Abuhadrous 21  FT: Time Scaling F Signals Part 2 Dr. Iyad Abuhadrous 22  FT: Time-shifting Property Time shifting introduces a phase shift in the frequency domain. Does not change magnitude of the Fourier transform Shift the phase of the Fourier transform by -t0 If f (t )  F ( ) then ( ) f t − t0  e − jt F ( ) 0 Similarly f (t ).e j 0 t  F ( − 0 ) Frequency Shift (modulation) Dr. Iyad Ayoub Signals Part 2 Dr. Iyad Abuhadrous 23  FT: Frequency-shifting Property e j 0 t f (t )  F ( − 0 ) ( Since cos(0t ) = e + e − j0t 1 j 0 t 2 ) e − j 0 t f (t )  F ( + 0 ) sin(0t ) = ( 1 j 0 t e − e − j 0 t ) 2j cos(0t ) f (t )  (F ( + 0 ) + F ( − 0 )) 1 2 sin(0t ) f (t )  (F ( + 0 ) − F ( − 0 )) 1 2j Signals Part 2 Dr. Iyad Abuhadrous 25  FT: Convolution Property Convolution in time domain is equivalent to multiplication in the frequency domain. x(t )  X ( ) h(t )  H ( ) x(t )* h(t )  X ( )H ( ) Multiplication in time domain is equivalent to convolution in the frequency domain. x(t )  X ( ) m(t )  m( ) modulation x(t )m(t )  X ( )* M ( ) 1 2 Signals Part 2 Dr. Iyad Abuhadrous 26  FT: Time Differentiation Property From the chain rule, recall that Conditions  u dv = u v -  v du f(t) → 0, when |t| →  Let u = e − j  t and dv = df (t ), f(t) is differentiable so du = − j e − j  t Derivation of property: f (t )d (e − j  t )  B ( ) = e f (t )  Given f(t)  F(w) − jt − t = − −  d  = j  f (t )e − j  t dt = j F ( ) Let B ( ) = F  f (t )  −  dt  df (t ) B ( ) =   df (t ) e − j  t dt  j F ( ) −  dt dt  =  e − j  t df (t ) df (t ) n  ( j ) F ( ) n − n Signals Part 2 dt Dr. Iyad Abuhadrous 27  FT: Time Integration Property Find  f ( x )dx  ? t - From the property of time convolution :   f (x )dx =  f (x )u(t − x )dx t - - = f (t )  u (t )  1  F ( ) = F ( )   ( ) +  =  F (0)  ( ) +  j  j therefore, F ( ) f (x )dx  t  - t f (x )dx   F (0)  ( ) + F ( ) j if F (0) = 0  - j Signals Part 2 Dr. Iyad Abuhadrous 28  FT: Duality Property Forward/inverse transforms are similar 1  f (t ) = ( )  F ( ) =  f (t ) e − j t dt − F  e jt d − 2 f (t )  F ( ) F (t )  2 f (−  ) Example: rect(t/)   sinc(  / 2) – Apply duality  sinc(t /2)  2  rect(-/) – rect(·) is even  sinc(t /2)  2  rect(/) f(t)  F() 1  t -/2 0 /2 − − − 0          Signals Part 2 Dr. Iyad Abuhadrous 29  FT: Modulation Property y (t ) = f (t ) cos(0t ) f (t )m(t )  1 F ( )* M ( ) 2 Multiplication in time domain is equ. to convolution in the frequency domain Y ( ) = F ( )  ( ( + 0 ) +  ( − 0 )) 1 2 Recall that  x(t )   (t ) =   ( )x(t −  )d = x(t ) −  x(t )   (t − t0 ) =   ( − t0 )x(t −  )d = x(t − t0 ) − So, Y ( ) = F ( + 0 ) + F ( − 0 ) 1 1 2 2 Signals Part 2 Dr. Iyad Abuhadrous 30  Fourier Transform Applications Amplitude Modulation: How Fourier Transform helps in understanding AM signal spectra. Communication: Multiplexing in data transmission. Image Processing: Role in edge detection and image filtering. Signals Part 2 Dr. Iyad Abuhadrous 31  FT Applications - Amplitude Modulation Definition: Amplitude Modulation (AM) is a technique where the amplitude of the carrier wave is varied based on the message signal. AM is used in analog broadcasting, such as AM radio, analog TV signals, aircraft communication. Signals Part 2 Dr. Iyad Abuhadrous 32  FT Applications Amplitude Modulation Example: y(t) = f(t).cos(w0 t) F() 1 f(t) is an ideal lowpass signal Assume 1