Signals and Systems Fall 2003 Lecture#8 30 September 2003 1. Derivation of the Ct Fourier Transform pair Examples of Fourier Transforms 3. Fourier Transforms of Periodic signals 4. Properties of the CT Fourier transform
Signals and Systems Fall 2003 Lecture #8 30 September 2003 1. Derivation of the CT Fourier Transform pair 2. Examples of Fourier Transforms 3. Fourier Transforms of Periodic Signals 4. Properties of the CT Fourier Transform
Fouriers derivation of the ct Fourier transform 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 Oo=2π/ T apart ·AST→∞,n→0, and harmonic components are spaced closer and closer in frequency Fourier series - Fourier integral
Fourier’s Derivation of the CT Fourier Transform • 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 ... • A s T → ∞, ω0 → 0, and harmonic components are spaced closer and closer in frequency ⇓ Fourier series ⎯ ⎯ → Fourier integral
Motivating Example: Square wave Increases T T=4T1 ept fixed 2 sin(hwoT1 200 kwoN frequency T=8T1 pc become 40 4 enser in 2 sintI ”0 oas T k Increases w=kwo -8 800 mmN0V←mwum
Discrete frequency points become denser in ω as T increases Motivating Example: Square wave increases kept fixed
So on with the derivation x(t) For simplicity, assume x(t has a finite duration here 2<t X periodic -2T 0 T1 T 2T T/2 T/2 as T c(t=a(t) for all t
So, on with the derivation ... For simplicity, assume x ( t) has a finite duration
Derivation(continued) ∑ ake T k swot 已 clte swot (t)=a(t) in this inte kwo t If we define X(w) lte jut dt then Eq (1) Hwo)
Derivation (continued)
Derivation(continued) Thus. for T <t< T X(jkwo)e kwot k= k ∑nX(ko) AsT→∞,∑0→∫du, we get the Ct Fourier Transform pair weSt dw Synthesis equation T X(j)= co e(t)e-jut dt Analysis equation
Derivation (continued)
For what kinds of signals can we do this? (1) It works also even if x()is infinite duration, but satisfies a) Finite energy a(t) dt In this case there is zero energy in the error (t)=(t) lewT dw Then b) dirichlet conditions 1 a(t) at points of continuity (i)2 oo X(w)ejt dw= midpoint at discontinuity (iii) Gibb's phe c) By allowing impulses in x(t)or in Xgo), we can represent even more signals E. g. It allows us to consider Ft for periodic signals
a) Finite energy In this case, there is zero energy in the error For what kinds of signals can we do this? (1) It works also even if x(t) is infinite duration, but satisfies: E.g. It allows us to consider FT for periodic signals c) By allowing impulses in x(t) or in X(jω), we can represent even more signals b) Dirichlet conditions
Example #1 (a)m(t)=6( X(j)=/6(te-ot=1 6(t) 2丌J-∞ Synthesis equation for d(t) (b)x(t)=6(t-to) (t-toe Judt
Example #1 (a) (b)
Example #2: Exponential function e u(t),a> 1/a X(w c(te cte- jwt dt e-(a+ju)t )t at3 atW X(jo)=1/(a2+o) ∠Ⅹ(j0)=tan(oa) π/2 1/a 兀/4 1/ay2 a a a a /2 Even symmetry Odd symmetry
Example #2: Exponential function Even symmetry Odd symmetry
Example #3: a square pulse in the time-domain SIn w X(w) X(o) 2T xO /1TT1 Note the inverse relation between the two widths Uncertainty principle Useful facts about ctft's a(tdt Example above:/ a(t)dt=2T1=X( XGw) Ex. above: 2(0 X(ju) (Area of the triangle)
Example #3: A square pulse in the time-domain Useful facts about CTFT’s Note the inverse relation between the two widths ⇒ Uncertainty principle
