Exp7:Fourier Analysis of Signals Purposes 1.Learn the Fourier analysis methods. Learn how to measure frequency spectra of several common used signals. 3 Learn how to use the Cassy Leb,a computer aided data acquisition and signal processing system. Principle of the Experiment In various fields of science and technology,one may meet complex signals.No matter how complex the signals are,they can be decomposed into summation of sine waves with different frequency.Fourier analysis is just the mathematical tool to do the job,decomposing the original signal to different frequency parts.The Fourier transfer result,which is called the frequency spectrum,represents the basic characteristics of the signal,the frequency and amplitude. Spectrum of a periodic function A signal f(t),with period T,can be expanded into a Fourier series: )-a+acos(2xnv.t)+b,Sin(2xv.t)] -CO)+Clcos(2)+] (1) where, Co-va+b好， 0=g d Where,C(n)and o(n)are the amplitude and the phase of the n-th harmonic wave.C(n),as function of n,is called the amplitude spectrum.0(n),as function of n,called a phase spectrum. In normal circumstances,one only cares about the amplitude spectrum.Therefore,if someone says spectrum,he is talking the amplitude spectrum. The spectrum of the non-periodic function Non-periodic function can be treated as a periodic function with period of oo.Therefore, replacing the Fourier series shown above for the period function case with the Fourier integral, one can have Fourier transfer for non-period function. The Spectrum of RLC circuit Fig.1 RLC Circuit The RLC transient response can be described as,19 Exp7: Fourier Analysis of Signals Purposes 1. Learn the Fourier analysis methods. 2. Learn how to measure frequency spectra of several common used signals. 3. Learn how to use the Cassy Leb, a computer aided data acquisition and signal processing system. Principle of the Experiment In various fields of science and technology, one may meet complex signals. No matter how complex the signals are, they can be decomposed into summation of sine waves with different frequency. Fourier analysis is just the mathematical tool to do the job, decomposing the original signal to different frequency parts. The Fourier transfer result, which is called the frequency spectrum, represents the basic characteristics of the signal, the frequency and amplitude. ⚫ Spectrum of a periodic function A signal ƒ (t), with period T, can be expanded into a Fourier series: ƒ(t)=    = +  +  n 1 0 n 0 n 0 a a cos(2 nv t) b Sin (2 v t) =     = + + 1 ( ) 0 (0) cos(2 ) ( ) n C C n nv t  n (1) where, C(n)= 2 n 2 a n + b , d n tg   =  −1 ( ) Where, C(n) and φ(n) are the amplitude and the phase of the n-th harmonic wave. C(n), as function of n, is called the amplitude spectrum. θ(n), as function of n, called a phase spectrum. In normal circumstances, one only cares about the amplitude spectrum. Therefore, if someone says spectrum, he is talking the amplitude spectrum. ⚫ The spectrum of the non-periodic function Non-periodic function can be treated as a periodic function with period of ∞. Therefore, replacing the Fourier series shown above for the period function case with the Fourier integral, one can have Fourier transfer for non-period function. ⚫ The Spectrum of RLC circuit Fig.1 RLC Circuit The RLC transient response can be described as