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
​

Sampling

EE123

Last modified 9mo ago

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