3-2-1 From Fourier Series to Fourier Transform 3-2-1 From Fourier Series to Fourier Transfor t)=∑ Ce CTh f(t)=∑cnen af(t) e indt Bandwidth X pulse width=const. o=2/T t 2π/r= BW Bandwidth Non-periodic signals equal to the period of periodic a→do Most energy of periodic signals is distributed non→ between zero to a certain frequency. $3-2-2 Fourier Transform 3-2-1 From Fourier series to Fourier transform M f(t)=∑ f(t)e In wot f(t)= T/2 f(t)edt limTcn=limf(t)e limTcn=limf(t)e lf(t)e u dt=F(o)[ Image Cf(t)e at dt=F(o) Inction Fourier Transform f)=∑cem-∑ee-oe f(t=f(t+D f(t),Too Origi( f(t)=2m"F( o)-elotdo Diserete Continuou spectrum Cn (ng Mr Laplace and Mr. Fourier Pierre-Simon, marquis de Laplace (23 April 1749-5 March 1827)was a uwm时 statistics. He summarized and ulus, opening up a broader range ms. In statistics, the so-called Bayesian interpretation of probability was on, and invented the Laplace transform which In Baptiste Joseph Fourier(March 21, 1768- May 16, 1830)was amed in his honour. Fourier is also generally credited with the di greenhouse effect北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 a 2 − τ 2 τ τ - T T t f(t) jnω t n n 0 n e 0 f(t) c ) 2 nω Sa( T a c = ⋅ = ∑ ∞ =−∞ τ τ C n ω T a τ 2π/τ = BW Bandwidth Bandwidth × pulse width = const. ω0 =2π/T §3-2-1 From Fourier Series to Fourier Transform Most energy of periodic signals is distributed between zero to a certain frequency. 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 ‰周期信号的∑ 傅氏级数: ∞ =−∞ = n jnω t ne 0 f(t) c ∫ + = T/2 -T/2 -jnω t n f(t)e dt T 1 c 0 T→∞ lim Non-periodic signals equal to the period of periodic signals TÆ∝ 2 − τ 2 τ τ -T T t T 2 ω0 π = ω0 = 2π/T → 0 nω ω ω dω 0 0 → → §3-2-1 From Fourier Series to Fourier Transform 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 ∑ ∞ = −∞ = n jn ω t n e 0 f(t) c ∫ + = T/2 -T/2 - jn ω t n f(t) e dt T 1 c 0 f(t) e dt F(ω) limTc lim f(t) e dt - -jωt T/2 -T/2 -jnω t T n T 0 = ⋅ = = ⋅ ∫ ∫ +∞ ∞ + →∞ →∞ ∑ ∑ ∞ =−∞ ∞ =−∞ = = n 0 jnω t n n jnω t f(t) cne 0 Tc e 0 ω 2π 1 F(ω) e dω 2 1 f(t) jωt = ⋅ ∫ +∞ π −∞ Image function Original function §3-2-2 Fourier Transform *** 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 ∑ ∞ = −∞ = n jn ω t n e 0 f(t) c ∫ + = T/2 -T/2 - jn ω t n f(t) e dt T 1 c 0 f(t) e dt F(ω) limTc lim f(t) e dt - -jωt T/2 -T/2 -jnω t T n T 0 = ⋅ = = ⋅ ∫ ∫ +∞ ∞ + →∞ →∞ §3-2-1 From Fourier Series to Fourier Transform *** Fourier series 傅氏级数 Fourier Transform 傅氏变换 f(t)=f(t+T) 0 0 nω ω Discrete spectrum Cn(nω0) spectrum analysis of periodic signals Cn(nω0) spectrum analysis of any signals f(t), TÆ∞ ω dω F(ω) Æ∞ ω F(ω) Continuous spectrum 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 北京大学 wwhu 北京大学 wwhu 北京大学 wwhu 北京大学 Mr. Laplace and Mr. Fourier Pierre-Simon, marquis de Laplace (23 April 1749 – 5 March 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799-1825). This seminal work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation of probability was mainly developed by Laplace. He formulated Laplace's equation, and invented the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in applied mathematics, is also named after him. Jean Baptiste Joseph Fourier (March 21, 1768 – May 16, 1830) was a French mathematician and physicist best known for initiating the investigation of Fourier series and their application to problems of heat flow. The Fourier transform is also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect