Signals & Systems The Continuous-Time Fourier Transform Adapted - PowerPoint PPT Presentation

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Signals & Systems The Continuous-Time Fourier Transform Adapted From: Lecture Notes From MIT Dr. Hamid R. Rabiee Fall 2013

Lecture 8 (Chapter 4) Content • Derivation of the CT Fourier Transform pair • Examples of Fourier Transform • Fourier Transforms of Periodic Signals • Properties of the CT Fourier Transform Sharif University of Technology, Department of Computer Engineering, Signals & Systems 2

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Fourier’s Derivation of the CTFT • x ( t ) - an aperiodic signal - view it as the limit of a periodic signal as T → ∞ • For a periodic signal, the harmonic components are spaced ω 0 = 2 π /T apart ... As T → ∞ , ω 0 → 0, and harmonic components are spaced • closer and closer in frequency Fourier Fourier series integral Sharif University of Technology, Department of Computer Engineering, Signals & Systems 3

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Motivating Example: Square wave T 1 kept fixed and T increases 4 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Motivating Example: Square wave T 1 kept fixed and T increases  2sin( ) k T 0 1  a  k k T 0 Discrete frequency points  become denser in 2sin( ) T 1  ω as T increases Ta  k 5 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) So, on with the Derivation ...  T T    ( ), x t t   For simplicity, assume 2 2   ( ) x t x ( t ) has a finite duration. T   , | | periodic t   2   , ( ) ( ) AsT x t x t for allt 6 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Derivation (Continued…)   2      jk t ( ) 0 ( ) x t a e 0 k T  k T T 2 2 1 1         jk t jk t ( ) ( ) a x t e dt x t e dt 0 0 k T T T T   2 2  ( ) ( ) x t x t in this interval  1     jk t ( ) 0 x t e dt T       j t ( ) ( ) X jw x t e dt If we defined  7 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Derivation (Continued…)  1   T T    jk t    0 ( ) ( ) ( ) x t x t X jk e Thus, for t 0 2 2 T  k  1      jk t 0 ( ) X jk e 0 0  2  k      , As T w dw 0 0 We can get the CT fourier transform pairs 8 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) For What Kinds of Signals Can We Do This? (1) It works also even if x ( t ) is infinite duration, but satisfies:     2 | ( ) | a) Finite energy x t dt  In this case, there is zero energy in the error   1        j t 2 ( ) ( ) ( ) e t x t X j e dt | ( ) | 0 e t dt Then  2   b) Dirichlet conditions c) By allowing impulses in x(t )or in X(j ω ), we can represent even more signals E.g. It allows us to consider FT for periodic signals 9 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #1   ( ) ( ) ( ) a x t t         j t ( ) ( ) 1 X j t e dt    1       j t ( ) t e d Synthesis equation for δ( t )  2     ( ) ( ) ( ) b x t t t 0            j t j t ( ) ( ) X j t t e dt e 0 0  10 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #1   ( ) ( ) ( ) a x t t         j t ( ) ( ) 1 X j t e dt    1       j t ( ) t e d Synthesis equation for δ( t )  2     ( ) ( ) ( ) b x t t t 0            j t j t ( ) ( ) X j t t e dt e 0 0  11 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #2: Exponential Function    at ( ) ( ), 0 x t e u t a             j t at j t ( ) ( ) X j x t e dt e e dt  0 ∞ 1 1       ( ) a j t ( ) e     a j a j 0 12 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #2: Exponential Function    at ( ) ( ), 0 x t e u t a             j t at j t ( ) ( ) X j x t e dt e e dt  0 ∞ 1 1       ( ) a j t ( ) e     a j a j 0 13 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Example #2 (continued) Even symmetry Odd symmetry Sharif University of Technology, Department of Computer Engineering, Signals & Systems 14

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #3: A Square Pulse in the Time-domain 15 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #3: A Square Pulse in the Time-domain  T 1 2sin T       j t 1 ( ) X j e dt   T 1     ( ) X j d  Note the inverse relation between the two widths ⇒ Uncertainty principle 16 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) e  2  at ( ) x t Exampl1e #4: 17 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) e  2  at ( ) x t Exampl1e #4:      2   at j t ( ) X j e e dt      j j  2   2  2 [ ( ) ] ( )  a t j t a  2 2 a a a e dt     2 j   2  ( ) a t   [ 2 ]. 4 e a dt e a   2   (Pulse width in t ) • (Pulse width in ω )  4 a e ⇒ ∆ t •∆ ω ~ (1/a 1/2 ) • (a 1/2 ) = 1 a Uncertainty Principle! Cannot make both ∆ t and ∆ ω arbitrarily small. Also a Gaussian! 18 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) CT Fourier Transforms of Periodic Signals       Suppose ( ) ( ) X j 0   — periodic in t with 1 1   e        frequency ω o j t j t ( ) ( ) x t e dw 0   0 2 2  That is — All the energy is concentrated in one frequency — ω o More generally 19 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #5: 𝑦 𝑢 = cos 𝜕 0 𝑢 20 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Example #5: 1 1        j t j t ( ) cos x t t e e 0 0 0 2 2        ( ) ) ) X jw   “Line spectrum” 21 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)      ( ) ( ) x t t nT (Sampling function) Example #6:  n x(t) 22 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)      ( ) ( ) x t t nT (Sampling function) Example #6:  n x(t) Same function in the frequency-domain! (period in t ) T ⇔ (period in ω ) 2 π / T Inverse relationship again! 23 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Properties of the CTFT      ( ) ( ) ( ) ( ) 1) Linearity ax t by t aX j bY j      2) Time Shifting j t ( ) ( ) x t t e X j 0 0 Proof: 3)FT magnitude unchanged 24 Sharif University of Technology, Department of Computer Engineering, Signals & Systems

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4) Properties (Continued) 4)Linear change in FT phase 5) Conjugate Symmetry Even Odd Even Odd 25 Sharif University of Technology, Department of Computer Engineering, Signals & Systems