4.1
Contents
Continuous Time Fourier Transform Pair for Aperiodic and Periodic Signals
- $ x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j \omega) e^{j\omega t} \, d\omega $ (4.8)
- $ X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} \, dt $ (4.9)
The Fourier transform exists if the signal is absolutely integrable or if the signal has a finite number of discontinuities within any finite interval. (See Page 290)
Fourier Transform from the Fourier Series
- This is useful for signals that fail to satisfy the previous properties of a signal that is guaranteed a Fourier Transform.
A signal represented as the sum of complex exponentials:
- $ x(t) = \sum_{k = -\infty}^{+\infty} a_ke^{jk\omega_0t} $
with ak's:
- $ a_k = \frac{1}{T}\int_{T} x(t)e^{-jk\omega_0 t} \, dt $
$ \xrightarrow{\mathcal{F}} $
- $ X(j\omega) = \sum_{k = -\infty}^{+\infty} 2\pi a_k \delta(\omega - k\omega_0) $
Properties of CT Fourier Transforms
Linearity
Time Shifting
Conjugation and Conjugate Symmetry
Differentiation and Integration
Time and Frequency Scaling
Quiz?
Duality
Parseval's Relation
Lecture 15: Sections 4.2-4.7. Lecture 16: Lecture 17: Lecture 18: October break-no classes Lecture 11: Sections 4.0-4.1 Lecture 12: Sections 4.2-4.4. 4.7 Lecture 13: Sections 4.5-4.7. Lecture 14: Sections 5.0-5.3. Lecture 15: Sections 5.4-5.5, 5.8
