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This lecture covers Fourier representations for continuous-time signals and systems, including Fourier Series and Transform, as well as Discrete-time Fourier Series and Transform. It delves into orthogonality of functions, Euler-Fourier formula, Generalized Fourier Series, and Frequency Spectrum. Learn how functions can be extended into infinite sums of sine and cosine functions and explore the properties of Fourier Transform. Examples and applications are also discussed.
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Lecture #04 Fourier representation for continuous-time signals signals & systems
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 signals & systems
A set of function Is called orthogonal in the interval if where is the complex conjugate of then if in is orthonormal Continuous-time signals Orthogonal function： signals & systems
Euler-Fourier formula The question is how to find Ci For any function We choose a orthogonal function set to be the basis signals & systems
Generalized Fourier series： Fourier series of function f(t) signals & systems
example of orthogonal function : in the interval proof signals & systems
For any function f(t) in the interval signals & systems
If f(t) is real function let let signals & systems
Fourier series： signals & systems
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. signals & systems
signals & systems MIT signals & systems
Example: signals & systems
Frequency spectrum signals & systems
Fourier transform f(t) is not periodic function if T∞ signals & systems
Fourier transform of f(t) Inverse Fourier transform Comparing with Laplace transform signals & systems
The properties of Fourier transform (i) Linearity (ii) Reversal (iii) Scaling in time signals & systems
(iv) Delay (v) Frequency shifting modulation (vi) Frequency differentiation (vii) Convolution signals & systems
(viii) multiplication (ix) Derivative (x) Integration signals & systems
example signals & systems
example signals & systems
example signals & systems
Cardinal sine function signals & systems
Parseval’s theorem (時域頻域能量守恒) If f(t) is real function signals & systems
Example signals & systems
Example: Fourier series signals & systems
Example : Fourier transform signals & systems
