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meiling chensignals & systems1 Lecture #04 Fourier representation for continuous-time signals

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meiling chensignals & systems2 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

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meiling chensignals & systems3 Orthogonal function ： A set of function Is called orthogonal in the interval if whereis the complex conjugate of if then inis orthonormal Continuous-time signals

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meiling chensignals & systems4 For any function We choose a orthogonal function set to be the basis Euler-Fourier formula The question is how to find C i

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meiling chensignals & systems5 Generalized Fourier series ： Fourier series of function f(t)

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meiling chensignals & systems6 example of orthogonal function : in the interval proof

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meiling chensignals & systems7 For any function f(t) in the interval

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meiling chensignals & systems8 If f(t) is real function let

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meiling chensignals & systems9 Fourier series ：

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meiling chensignals & systems10 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.

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meiling chensignals & systems11 MIT signals & systems

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meiling chensignals & systems12 Example:

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meiling chensignals & systems14 Frequency spectrum

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meiling chensignals & systems15 Fourier transform f(t) is not periodic function if T ∞

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meiling chensignals & systems16 Fourier transform of f(t) Inverse Fourier transform Comparing with Laplace transform

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meiling chensignals & systems17 The properties of Fourier transform (i) Linearity (ii) Reversal (iii) Scaling in time

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meiling chensignals & systems18 (iv) Delay (v) Frequency shifting modulation (vi) Frequency differentiation (vii) Convolution

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meiling chensignals & systems19

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meiling chensignals & systems20 (viii) multiplication (ix) Derivative (x) Integration

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meiling chensignals & systems example

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meiling chensignals & systems22

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meiling chensignals & systems23 example

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meiling chensignals & systems24 example

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meiling chensignals & systems25 Cardinal sine function

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meiling chensignals & systems26 Parseval’s theorem ( 時域頻域能量守恒 ) If f(t) is real function

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meiling chensignals & systems27 Example

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meiling chensignals & systems29 Example: Fourier series

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meiling chensignals & systems31 Example : Fourier transform

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