Fourier Series and Fourier Transform

E1 10 Fourier Series and Transforms (2014-5559) Fourier Transform - Parseval and Convolution: 7 – 2 / 10 Question: What is the Fourier transform of w(t)=u(t)v(t)? Let u(t)= R +∞ h=−∞ U(h)ei2πhtdh and v(t)= R g=−∞ V(g)ei2πgtdg [Note use of different dummy variables] w(t)=u(t)v(t) = R +∞ h=−∞ U(h)ei2πhtdh R +∞ g=−∞ V(g

4: Parseval’s Theorem and Convolution

E1 10 Fourier Series and Transforms (2014-5543) Parseval and Convolution: 4 – 3 / 9 The average power of a periodic signal is given by Pu, D u(t)2 E This is the average electrical power that would be dissipated if the signal represents the voltage across a 1Ω resistor

Ae2 Mathematics: Fourier Series & Parseval’s equality

Ae2 Mathematics: Fourier Series & Parseval’s equality 1 Fourier series Let’s begin with the Fourier series for a periodic function f(x), periodic on [−L,L] Its Fourier series representation is f(x) = 1 2 a 0 + X∞ n=1 ˆ a ncos nπx L + b nsin nπx ˙ (1) and the Fourier coefficientsa nand b nare given by a n= 1 L Z L −L f(x)cos nπx

Lecture 16 - Parseval’s Identity

12 1 GEOMETRIC INTERPRETATION OF PARSEVAL’S FORMULA For Fourier Sine Components: 2 L L 0 f(x) 2 dx = n=1 b2 n (12 10) Example 12 3 Consider f(x)=x2 −π

Chapter 1 Fourier Series - University of Minnesota

Parseval’s equality becomes ˇ2 8 = X1 j=1 1 (2j 1)2: 1 3 Fourier series on intervals of varying length, Fourier series for odd and even functions Although it is convenient to base Fourier series on an interval of length 2ˇ there is no necessity to do so Suppose we wish to look at functions f(x) in L2[ ; ] We simply make the change of

Lecture 8 Properties of the Fourier Transform

Parseval’s Theorem (Parseval proved for Fourier series, Rayleigh for Fourier transforms Also called Plancherel’s theorem) Recall signal energy of x(t) is E x = Z 1 1 jx(t)j2 dt Interpretation: energy dissipated in a one ohm resistor if x(t) is a voltage Can also be viewed as a measure of the size of a signal Theorem: E x = Z 1 1 jx(t)j2