2 Content Introduction More on Impulse Function Fourier Transform Related to Impulse FunctionFourier Transform of Some Special FunctionsFourier Transform vs. Fourier Series
3 IntroductionSufficient condition for the existence of a Fourier transformThat is, f(t) is absolutely integrable.However, the above condition is not the necessary one.
4 Some Unabsolutely Integrable Functions Sinusoidal Functions: cos t, sin t,…Unit Step Function: u(t).Generalized Functions:Impulse Function (t); andImpulse Train.
5 Fourier Transforms of Special Functions More onImpulse Function
6 Dirac Delta FunctionandtAlso called unit impulse function.
7 Generalized Function (t): Test Function The value of delta function can also be defined in the sense of generalized function:(t): Test FunctionWe shall never talk about the value of (t).Instead, we talk about the values of integrals involving (t).
8 Properties of Unit Impulse Function Pf)Write t as t + t0
9 Properties of Unit Impulse Function Pf)Write t as t/aConsider a>0Consider a<0
13 Generalized Derivatives The derivative f’(t) of an arbitrary generalized function f(t) is defined by:Show that this definition is consistent to the ordinary definition for the first derivative of a continuous function.=0
35 Fourier Transform of Unit Step Function ()|F(j)|t1f(t)F
36 Fourier Transforms of Special Functions Fourier Transform vs. Fourier Series
37 Find the FT of a Periodic Function Sufficient condition --- existence of FTAny periodic function does not satisfy this condition.How to find its FT (in the sense of general function)?
38 Find the FT of a Periodic Function We can express a periodic function f(t) as:
39 Find the FT of a Periodic Function We can express a periodic function f(t) as:The FT of a periodic function consists of a sequence of equidistant impulses located at the harmonic frequencies of the function.
40 Example: Impulse Train tT2T3TT2T3TFind the FT of the impulse train.
41 Example: Impulse Train tT2T3TT2T3TFind the FT of the impulse train.cn
42 Example: Impulse Train 0tT2T3TT2T3TFind the FT of the impulse train.cn
44 Find Fourier Series Using Fourier Transform f(t)tT/2T/2fo(t)t
45 Find Fourier Series Using Fourier Transform Sampling the Fourier Transform of fo(t) with period 2/T, we can find the Fourier Series of f (t).Find Fourier Series Using Fourier TransformT/2T/2f(t)tfo(t)
46 Example: The Fourier Series of a Rectangular Wave f(t)d1ttfo(t)1
47 Example: The Fourier Transform of a Rectangular Wave f(t)d1tF [f(t)]=?