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)