Berkeley Notes
  • Introduction
  • EE120
    • Introduction to Signals and Systems
    • The Fourier Series
    • The Fourier Transform
    • Generalized transforms
    • Linear Time-Invariant Systems
    • Feedback Control
    • Sampling
    • Appendix
  • EE123
    • The DFT
    • Spectral Analysis
    • Sampling
    • Filtering
  • EECS126
    • Introduction to Probability
    • Random Variables and their Distributions
    • Concentration
    • Information Theory
    • Random Processes
    • Random Graphs
    • Statistical Inference
    • Estimation
  • EECS127
    • Linear Algebra
    • Fundamentals of Optimization
    • Linear Algebraic Optimization
    • Convex Optimization
    • Duality
  • EE128
    • Introduction to Control
    • Modeling Systems
    • System Performance
    • Design Tools
    • Cascade Compensation
    • State-Space Control
    • Digital Control Systems
    • Cayley-Hamilton
  • EECS225A
    • Hilbert Space Theory
    • Linear Estimation
    • Discrete Time Random Processes
    • Filtering
  • EE222
    • Real Analysis
    • Differential Geometry
    • Nonlinear System Dynamics
    • Stability of Nonlinear Systems
    • Nonlinear Feedback Control
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  1. EE120

Appendix

Theory of Distributions

The Theory of Distributions is the mathematical framework which underlies the generalized Fourier Transforms.

Definition 41

Given a test function xxx, a distribution TTT operates on xxx to produce a number <T,x><T,x><T,x>.

Definition 42

The distrbution induced by a function ggg is defined as

<Tg,x>=∫−∞∞g(t)∗x(t)dt<T_g, x> = \int_{-\infty}^{\infty}{g(t)^*x(t)dt}<Tg​,x>=∫−∞∞​g(t)∗x(t)dt

Notice two things:

  • <Tg,x><T_g, x><Tg​,x> is linear

  • <αTg,x>=α∗<Tg,x><\alpha T_g, x> = \alpha^*<T_g, x><αTg​,x>=α∗<Tg​,x>

With these definitions, we can now define the Dirac delta in terms of distrubions. Let ggg be any function such that

∫∞∞g(t)dt=1\int_{\infty}^{\infty}{g(t)dt} = 1∫∞∞​g(t)dt=1

Define gϵg_\epsilongϵ​ to be

gϵ=1ϵg(tϵ)g_\epsilon = \frac{1}{\epsilon}g(\frac{t}{\epsilon})gϵ​=ϵ1​g(ϵt​)

Now we can defined δ(t)=lim⁡ϵ→0Tgϵ\delta(t) = \lim_{\epsilon \rightarrow 0}{T_{g_\epsilon}}δ(t)=limϵ→0​Tgϵ​​

<δ,x>=∫−∞∞δ(t)x(t)dt=x(0)<\delta, x> = \int_{-\infty}^{\infty}{\delta(t)x(t)dt} = x(0)<δ,x>=∫−∞∞​δ(t)x(t)dt=x(0)

which is the property of the Dirac Delta we want. Now we can define the generalized Fourier Transform in terms of distributions.

Definition 43

The generalized Continuous Time Fourier Transform of a distrubtion TTT is

<FT,X>=2π<T,x><FT, X> = 2\pi<T, x><FT,X>=2π<T,x>

for test function xxx whose Fourier Transform is XXX

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