Fourier series-1

Fourier series-1 - ECE 307 Fourier Series Z. Aliyazicioglu...

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1 ECE 307 1-1 Fourier Series Z. Aliyazicioglu Electrical & Computer Engineering Dept. Cal Poly Pomona ECE 307 ECE 307 1-2 Signals • Consider a simple signal: A m the amplitude of the cosine waveform; f a the frequency of the signal in hertz; t the independent variable “ time”; and θ the phase angle, usually expressed in degrees, () cos (2 ) ma vt A ft π θ =+
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2 ECE 307 1-3 Signals We have seen a single-frequency signal. AC signal () 5cos (2 200 60) vt t π =+ D t=0:0.0001:0.02; x=5*cos(2*pi*200*t+pi/3); plot(t,x) xlabel('t') ylabel('x(t)') grid on 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -5 -4 -3 -2 -1 0 1 2 3 4 5 t x(t) A m the amplitude of the cosine waveform; f a the frequency of the signal in hertz; t the independent variable “ time”; and θ a the phase angle, usually expressed ECE 307 1-4 Signals we have seen a single-frequency signal with DC. AC signal t=0:0.0001:0.02; x=3+5*cos(2*pi*200*t-pi/4); plot(t,x) xlabel('t') ylabel('x(t)') grid on ( ) 3 5cos(2 200 45 ) t D 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -2 -1 0 1 2 3 4 5 6 7 8 t
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3 ECE 307 1-5 Signals We have seen Multi-frequency signal. t=0:0.00001:0.02; x=10*cos(2*pi*200*t)+6*cos(2*pi*400*t+pi/9) +2*cos(2*pi*600*t+50*pi/180) +0.5*cos(2*pi*2800*t+80*pi/180); plot(t,x) grid on xlabel('t') ylabel('x(t)') 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 -10 -5 0 5 10 15 20 t x(t) ( ) 10cos(2 200 ) 6cos(2 400 20 ) 2cos(2 600 50 ) 0.5cos(2 800 80 ) vt t t t t ππ π =+ + + + ++ DD D ECE 307 1-6 Signals •D r a w i n g a spectrum that represents the signal in the frequency domain. In simple terms, a spectrum is a graph of the magnitude or the phase angle of a signal plotted against frequency as the independent variable. ( ) 10cos(2 200 ) 6cos((2 400 20 ) 2cos((2 600 50 ) 0.5cos((2 800 80 ) t t t t + + + + + D 80 0.5 800 2 6 10 0 V 50 600 20 400 0 200 0 0 θ f 0 100 200 300 400 500 600 700 800 0 5 10 15 f V 0 100 200 300 400 500 600 700 800 0 20 40 60 80 f θ figure1.m
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4 ECE 307 1-7 Fourier Series Periodic signal is a function that repeats itself every T seconds. () ( ) x tx t n T T: period of a function, n: integer 1,2,3,… T 2T 3T t x(t) T2 T t x(t) T 2T t x(t) ECE 307 1-8 Fourier Series Periodic signal can be represented as sum of sinusoidal if the signal is square-integrable over an arbitrary interval (). 1 1 tT t xt d t + < ∞ So, it can be expressed as 0 00 0 11 1 cos ( ) s in ( ) cos( ) nn n jn t n n x ta a n t b n t ccn t Xe ω ωθ ∞∞ == = =−∞ =+ + + = ∑∑ where 0 0 2 T π = fundamental frequency of the periodic function in [rad/s]. are harmonic frequencies 0 ,2 , 3 , 4 , . . . nf o r n = 0 The exponential form
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5 ECE 307 1-9 Fourier Series The parameters are called Fourier series expansion or coefficients and given by 10 1 0 0 1 () tT t ax t d t T + = 1 0 0 2 ()cos ( ) n t t n t d t T ω + = 1, 2, 3, . .. n = 1 0 0 2 ()s in ( ) n t bx t n t d t T + = n = 22 nn n ca b =+ tan n n n b a θ =− 0 1 0 1 jn t n t X xte d t T + = 1, 2, 3,. .. n = ∓∓∓ Where t 1 is arbitrary. It can be set or 1 0 t = /2 = − 00 = ECE 307 1-10 Fourier Series Using Euler’s rule, can be written as n X 11 ( )cos( ) ( )sin( ) n tt X xt n td t j t TT ωω ++ ∫∫ If x(t) is a real-valued periodic signal, we have * * jn t jn t n Xx t e d t x t e d t XX == = n Xa j b To obtain and { } 2Re aX = { } 2Im bX 1 ,1 , 2 , 3 2 n j Xc e n n X ajb
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6 ECE 307 1-11 Fourier Series Remember that 10 1 0 cos( ) 0 for all n tT t ntd t ω + = 1 0 sin( ) 0 for all n t t + = 1 00 cos( ) sin( ) 0 for all n and m t nt
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Fourier series-1 - ECE 307 Fourier Series Z. Aliyazicioglu...

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