Lecture 04 Fourier representation for continuoustime signals meiling

Lecture #04 Fourier representation for continuous-time signals meiling chen signals & systems 1

Fourier representations • Fourier Series (FS) : for periodic signals • Fourier-Transform (FT) : for nonperiodic signals • Discrete-time Fourier series (DTFS): for discrete-time periodic signals • Discrete-time Fourier transform : for discrete-time nonperiodic signals meiling chen signals & systems 2

Continuous-time signals Orthogonal function： A set of function Is called orthogonal in the interval if where if meiling chen is the complex conjugate of then in signals & systems is orthonormal 3

For any function The question is how to find Ci We choose a orthogonal function set to be the basis Euler-Fourier formula meiling chen signals & systems 4

Generalized Fourier series： Fourier series of function f(t) meiling chen signals & systems 5

example of orthogonal function : in the interval proof meiling chen signals & systems 6

For any function f(t) in the interval meiling chen signals & systems 7

If f(t) is real function let meiling chen let signals & systems 8

Fourier series： meiling chen signals & systems 9

A periodic signal satisfying he following conditions can be extended into an infinite sum of sine and cosine functions. 1. The single-valued function f(t) is bounded, and hence absolutely integrable over the finite period T; that is 2. The function has a finite number of maxima and minima over the period T. 3. The function has a finite number of discountinuity points over the period T. meiling chen signals & systems 10

meiling chen signals & systems MIT signals & systems 11

Example: meiling chen signals & systems 12

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Frequency spectrum meiling chen signals & systems 14

Fourier transform f(t) is not periodic function if T ∞ meiling chen signals & systems 15

Fourier transform of f(t) Inverse Fourier transform Comparing with Laplace transform meiling chen signals & systems 16

The properties of Fourier transform (i) Linearity (ii) Reversal (iii) Scaling in time meiling chen signals & systems 17

(iv) Delay (v) Frequency shifting modulation (vi) Frequency differentiation (vii) Convolution meiling chen signals & systems 18

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(viii) multiplication (ix) Derivative (x) Integration meiling chen signals & systems 20

example meiling chen signals & systems

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example meiling chen signals & systems 23

example meiling chen signals & systems 24

Cardinal sine function meiling chen signals & systems 25

Parseval’s theorem (時域頻域能量守恒) If f(t) is real function meiling chen signals & systems 26

Example meiling chen signals & systems 27

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Example: Fourier series meiling chen signals & systems 29

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Example : Fourier transform meiling chen signals & systems 31

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