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Lecture #04. Fourier representation for continuous-time signals. Fourier representations. Fourier Series (FS) : for periodic signals Fourier-Transform (FT) : for nonperiodic signals Discrete-time Fourier series (DTFS): for discrete-time periodic signals
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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
