测量与信号基础英文PPT (13).pdf

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1、Chapter 3 Frequency Spectrum Analysis Fundamentals of Measurement and Signal Analysis 3.2 Fourier series representation of periodic signals A signal is said to be periodic if it satisfies the condition:x(t)=x(t+T)Where T is called the fundamental period,and f0=1/T is called the fundamental frequency

2、.In mathematics,any periodic signal can be represent as the sum of harmonically related sinusoid signals,which are called the Fourier series represntation.f0f20f30f40TT1)Orthogonal vectors a10 b1 For a X-Y table,it can move to any point with two motors.0 X Y y1x13.2 Fourier series representation of

3、periodic signals Two vectors are orthogonal if and only if their inner product is zero.In a orthogonal vector space any vector can be represented by orthogonal vectors.Fxy11XYa A11Fb BX Y Z z111Fx xy yzX Y0A B02121()()0,()(),f t ft dtijf t ft dtKijijttijttijij nnx tc f tc ftc ftttt()()().(),112212Wh

4、ere Ci is the projection coefficient in fi(t).2)Orthogonal function Definition of the orthogonal functions set：3.2 Fourier series representation of periodic signals In similar way,Two functions are orthogonal if their inner product is zero.In a orthogonal function space,any signal can be represented

5、 by orthogonal functions:3)Orthogonality of Sines and Cosines 3.2 Fourier series representation of periodic signals Sines and Cosines in harmonics are Orthogonal,they consist of an orthogonal basis.af t0()sin(2)bf t0()cos(2)cf t0()sin(2 2)df t0()cos(2 2)ef t0()sin(2 3)ff t0()cos(2 3)nf tmf t dt 000T

6、sin(2)sin(2)0,nf tmf t dt 000Tcos(2)cos(2)0,nf tmf t dt 000Tsin(2)cos(2)0,nm3.2 Fourier series representation of periodic signals Example：f tf t dt 000Tsin(2)*sin(2 2)0f tf t dt 00-cos(2)*cos(2 2)0f tf t dt 000Tsin(2)*cos(2 2)0f tf t dt 000Tsin(2 3)*cos(2 5)0 x=linspace(0,2*pi,1024);y1=sin(x);y2=sin

7、(2*x);y3=cos(x);y4=cos(2*x);y5=sin(3*x);y6=cos(5*x);z1=y1.*y2;subplot(4,1,1);plot(x,z1,linewidth,2);z2=y3.*y4;subplot(4,1,2);plot(x,z2,linewidth,2);z3=y1.*y4;subplot(4,1,3);plot(x,z3,linewidth,2);z4=y5.*y6;subplot(4,1,4);plot(x,z4,linewidth,2);aTx t dtTT2()0/2/2nTTaTx tnf t dt n0/2/22()cos(2),1,2,3,

8、.nTTbTx tnf t dt n0/2/22()sin(2),1,2,3,.F0 X Y y1x1(a)annnx tant bnf t02010()(cos 2 fsin 2)；n(1,2,3,.)4)Fourier series representation 3.2 Fourier series representation of periodic signals f0=1/T x(t)=x(t+T)Fourier coefficients:annnx tant bnf t02010()(cos 2 fsin 2)n(1,2,3,.)In the form of amplitude/p

9、hase:x tAntannn0()cos(2 f)201abnnnA22；arctgbannnAuxiliary angle formula of trigonometric function 22Ascossin()=arctgAinBABB3.2 Fourier series representation of periodic signals Amplitude C1 C2 C3 Ci is the Fourier coefficient 5)Spectrum of periodic signals:3.2 Fourier series representation of period

10、ic signals Spectrum annnx tant bnt02010()(cossin)Antannn0cos(2 f)201Power spectrum f 5 10 20 30 8 16 0 2Anf 5 10 20 30 0 2-2 4-4 anf 5 10 20 30 0 2-2 4-4 bnReal spectrum Imaginary spectrum Amplitude spectrum f 5 10 20 30 2 3 1 4 0 Anf 5 10 20 30 0 nPhase spectrum Types of spectrum:3.2 Fourier series

11、 representation of periodic signals abnnnA22；arctgbannnabnnnA22；()y tA tT0/2()y tATt/20Example：Fourier series of square wave aTx t dtTT0/2/21()0aTx tnt dtnTT0/2/22()cos(2 f)00/2/22()sin(2 f)4/nTTbTx tnt dtA n；n(1,3,5,7.)3.2 Fourier series representation of periodic signals Sine Cosine symmetrical an

12、tisymmetry;nx tAtAtAt1()4sin()43sin(3)45sin(5).3.2 Fourier series representation of periodic signals Real spectrum Imaginary spectrum A-A A-A bn3.2 Fourier series representation of periodic signals 6)Synthesis of periodic signals Any periodic signal can be represent as the sum of harmonically sines

13、and cosines.Therefore,a periodic signal can be synthesized by sines and cosines.Square wave:xxxsin()sin(3)/3sin(5)/5.Sawtoolth:xxxsin()sin(3)/2sin(3)/3.Triangle:xxxxsin()-sin(3)/9sin(5)/25-sin(7)/49.Fourier Series of periodic signal 3.2 Fourier series representation of periodic signals x=linspace(0,

14、6*pi,1000);y1=sin(x);y2=sin(3*x)/9;y3=sin(5*x)/25;y4=sin(7*x)/49;y=y1-y2+y3-y4;plot(x,y,linewidth,1);xlim(0,6*3.14);grid on Triangle:Square wave:x=linspace(0,6*pi,1000);z=zeros(1,1000);k=1;for i=1:15 y=(1/k)*sin(k*x);z=z+y;plot(x,z,linewidth,1);xlim(0,6*3.14);ylim(-1.2,1.2);grid(on);pause(1);k=k+2;e

15、nd 3.2 Fourier series representation of periodic signals Example：-Design one yourself 3.2 Fourier series representation of periodic signals 3.2 Fourier series representation of periodic signals Fs=44100;dt=1.0/Fs;T=1.5;N=T/dt;x=linspace(0,T,N);A=0.8;F=400;F1=5;y1=A*0.5*cos(2*3.14*(F+F1)*x);y2=-A*0.5

16、*cos(2*3.14*(F-F1)*x);y3=A*0.5/9*cos(2*3.14*(F+3*F1)*x);y4=-A*0.5/9*cos(2*3.14*(F-3*F1)*x);y5=A*0.5/25*cos(2*3.14*(F+5*F1)*x);y6=-A*0.5/25*cos(2*3.14*(F-5*F1)*x);y=y1+y2+y3+y4+y5+y6;plot(x,y,b,linewidth,1);xlim(0.0,0.4);ylim(-1,1);grid on;Example：Telephone ring tone ）y tt1(0.8*sin(2 400);）ytt2(trang

17、le(2 5);ty tytz()1()*2()Triangle:xxxsin()-sin(3)/9sin(5)/25-.ttz()0.8*sin(2 400)*sin(2 4t)-sin(2 12t)/9sin(2 20t)/25sin()*sin()12cos()cos()tttz()0.8*0.5*cos(2 405)0.8*0.5*cos(2 395)tt-0.8*0.5/9*cos(2 415)0.8*0.5/9*cos(2 385)tt0.8*0.5/25*cos(2 425)-0.8*0.5/25*cos(2 375)%Pure Tone%Fade in&out 3.2 Four

18、ier series representation of periodic signals 7)Walsh orthogonal function set aWalsht()(0,)b Walsht()(1,)cWalsht()(2,)dWalsht()(3,)0 t 1 1 3.2 Fourier series representation of periodic signals WalshHadamard transform:The physical meaning of Walsh spectrum is not so clear as the Fourier spectrum 3.2

19、Fourier series representation of periodic signals Homeworks 2.Synthetize a telephone ring tone with a fade in and fade out of sawtooth.1.Design a GUI program to synthetize periodic signals,such as square wave,sawtooth wave and triangular wave.3.2 Fourier series representation of periodic signals Fundamentals of Measurement and Signal Analysis