Harmonik Fonksiyonların Toplamı Periyodik Fonksiyon:

Örnek 4.1: )65.0t9cos(2.3)1.2t6cos(4)7.1t3cos(2)t(x

2T3 1

2T6 2

2T9 3

093.2T1

047.1T2

70.0T3

093.2ts 035.020

70.0t

4. Periyodik sinyaller, fft

t

1fs

: Örnekleme frekansı

clc;clear;w1=3;t1=2*pi/w1;dt=t1/20/3;t=0:dt:t1-dt;x1=2*cos(3*t-1.7);x2=-4*cos(6*t+2.1);x3=3.2*cos(9*t-0.65);subplot(2,2,1);plot(t,x1)subplot(2,2,2);plot(t,x2)subplot(2,2,3);plot(t,x3)x=x1+x2+x3;subplot(2,2,4);plot(t,x)pause;close(figure(1))y=[x,x,x];plot (y)

)65.0t9cos(2.3)1.2t6cos(4)7.1t3cos(2)t(x

fx=fft(x);ampl=2*abs(fx)/length(x);fi=angle(fx);

MatLab’da x: Periyodik sinyalin örnekleri (Örnekleme frekansı: fs=1/dt)

1n

nn0 )tncos(A2

A)t(x 2T

1nA)n(ampl

w1=3;t1=2*pi/w1;dt=t1/20/3;t=0:dt:t1-dt;x1=2*cos(3*t-1.7);x2=-4*cos(6*t+2.1);x3=3.2*cos(9*t-0.65);

fx=fft(x);ampl=2*abs(fx)/length(x);fi=angle(fx);

)21.2t6cos(4)1.2t6cos(4x2

Örnek 4.2:)s(H)t(x )t(y

30s6s

)1s2(60)s(H 2

x(t) periyodik (T=6.4 s). İlk 4 harmoniğini dikkate alınız.

Sistemin öz değerleri: -3±4.6i → Δt=0.0574 s

Girdiden Δt= 6.4/20/4 = 0.08 Δt=0.0574 s, fs=17.42 Hz

MatLAB’da x: x(t) nin 128 örneği (Δt=0.05).

fx=fft(x) , abs(fx): 230.4, 102.4, 51.2, 76.8, 58.88, 3.2, ……

x(t)=?, y(t)=?

angle(fx): 0, -1.86, 2.1, -2.4, 1.6, 0.89, ….

)1.2t9634.1cos(8.0)86.1t9817.0cos(6.18.1)t(x

)6.1t9270.3cos(92.0)4.2t9452.2cos(2.1

Yanıt:

43210 yyyyy)t(y

30

60)8.1(y0

)4.2t9451.2cos(A)2.1(y 333

s=2.9451i; h=60*(s+1)/(s^2+6*s+30);a3=abs(h),fi3=angle(h)

33 3fi,A3a