Fourier Series. Fourier Series

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Ξένη Λιάπης
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1 ECE 37 Z. Aliyazicioglu Elecrical & Compuer Egieerig Dep. Cal Poly Pomoa Periodic sigal is a fucio ha repeas iself every secods. x() x( ± ) : period of a fucio, : ieger,,3, x() 3 x() x()

2 Periodic sigal ca be represeed as sum of siusoidals if he sigal is square-iegrable over a arbirary ierval (). + x () d < So, i ca be expressed as x() a + a cos( ω ) + b si( ω ) c + c cos( ω + θ ) Xe ω j where π ω ω fudameal frequecy of he periodic fucio i [rad/s]. are ω, harmoic for,3,4,... frequecies he parameers are called Fourier series expasio or coefficies ad give by + a + x() d a x()cos( ω ) d,, 3,... + ()si( ) ω,, 3,... b x d c a + b θ a + jω X xe () d b a,, 3,... c Where is arbirary. I ca be se or / a

3 Usig Euler s rule, ca be wrie as X + + X x ( )cos( ω d ) j x ( )si( ω d ) X a j b If x() is a real-valued periodic sigal, we have + + jω jω X x() e d x() e d X X * * X a + j b o obai ad a Re{ X } b Im{ X } jθ X ce,,,3 Remember ha + cos( ω ) d for all + si( ω ) d for all + cos( ω ) si( mω ) d for all ad m + cos( ω ) cos( mω ) d for all m for all m + si( ω ) si( mω ) d for all m for all m

4 Example: Vm v () V m Fid he Fourier series of he followig periodic sigal v() 3 + Vm Vm m a x() d d V Vm Vm ω ω a cos( ) d cos( ) d V m cos( ω ) + si( ω ) ω ω V π m cos( ) for all ω ω Vm Vm ω ω b si( ) d si( ) d V m si( ω ) cos( ω ) ω ω V π m Vm cos( ) for ω π he v () a b si( ω ) + Vm Vm v () si( ω ) π Vm Vm Vm Vm v ( ) si( ω) si( ω) si(3 ω)... π π 3π

5 Le s assume ha V m V ad ms π ω π rad/s >> Vm; >>.; >> w*pi/; >> :.:.; >> vvm/-vm/pi*si(w*); >> plo (,v) >> hold o; >> vvm/-vm/pi*si(w*)- Vm/(*pi)*si(*w*); >> plo (,v) >> v3vm/-vm/pi*si(w*)- Vm/(*pi)*si(*w*)- Vm/(3*pi)*si(3*w*); >> plo (,v3) >> v4vm/-vm/pi*si(w*)- Vm/(*pi)*si(*w*)- Vm/(3*pi)*si(3*w*)- Vm/(4*pi)*si(4*w*); >> plo (,v4) >> xlabel ('[s]') >> ile('v()') V b m for,,3,... π c a b V π + m b θ a 9 for,,3,... a ω Vm X a j b j π v () c + c cos( + θ ) X c jθ V m X ce e π j9

6 he Effec of symmery o he Fourier Coefficies Eve-fucio symmery Eve-fucio is defied as x() x( ) + a x() d / 4 a x()cos( ω ) d x() b forall he Effec of symmery o he Fourier Coefficies Odd-fucio symmery Odd-fucio is defied as x() x( ) + a x() d x() / 4 b x()si( ω ) d a for all -

7 Example: x() A - - Assume ha,a, ad Deermie Fourier series coefficies of i expoeial ad rigoomeric form. 4 Plo he discree specrum of x(). s s x() x( ) + a x() d / 4 a x()cos( ω ) d b forall Example: a d /4 /4 /4 /4 /4 /4 cos( ω ) si( ω ) ω 4 4 a d a 4 π π si( ) si π 4 π,,,,,,,... π 3π 5π 7π x( ) + cos( ω) cos(3 ω) + si(5 ω) cos(7 ω)... π 3π 5π 7π

8 >> :.:8; >> 4; >> w*pi/; >> v/+/pi*cos(w*)- /(3*pi)*cos(3*w*)+/(5*pi )*cos(5*w*)- /(7*pi)*cos(7*w*); >> plo (,v) >> xlabel ('[s]') >> ile('v()') Example : X 4 e d e e j4ω jω jω e e ω j si( ω ) si( ω ) ω ω si( π /4) si( π /) π /4 π / sic( ) jω jω jω where si( π x) sic( x) π x

9 Example.: (co) x () Xe sic( ) e jω jω Sice is real ad eve, a X sic( ) b X si( X ) a x( ) + sic( )cos( ω ) Sice ω π /4 π x () + sic( )cos( ) θ, π c sic( ) x(),3, 5, has odd umbers harmoics. he eve umbers harmoics are zero. X is always real, so ha he phase is eiher zero or π. he magiude of discree specrum is show i ex page. Example.: (co) X sigal as sic fucio >> -::; >> x.5*sic(/); >> sem (,x) >> ile('he /*sic(/) sigal'); >> xlabel('');

10 Example.: (co) Fourier series approximaio of sigal x() for.,,3,5, ad 7 >> -5:.:5; >> ; >> x.5; >> plo (,x) >> hold o >> ; >> a(sic(/)*cos(*pi**/4)); >> xx+a; >> plo (,x) >> 3; >> a(sic(/)*cos(*pi**/4)); >> xx+a; >> plo (,x) >> 5; >> a(sic(/)*cos(*pi**/4)); >> plo (,x) >> xx+a; >> plo (,x) >> 7; >> a(sic(/)*cos(*pi**/4)); >> xx+a; >> plo (,x,'r') >> ile(' approximaio for Differe values') >> Example. : x() - - /4 - / - a + x() d + jω X xe () d,, 3,... / jω jω X e d e d + /

11 Example. : (co) X e e jω jω / jω jω / X e e e e jω jω ( jω / jω ) ( jω jω / ) X e + e e π j π π π j j j X e + e e jπ jπ jπ jπ X e jπ jπ Example. : (co) X e e e jπ π π π j j j π j π X e si π π π si π j j X e e sic( ) π -j >> :; >> x(./(pi*)).*(si(pi/*)).*exp(-j*(pi*/)); ±, ±, ± 3, X -j j -j. +j -j. 7 +j -j. 99 +j -j. 77 +j

12 Example. : (co) X a j b b(-)*imag(x) b x ( ).73siω si3ω si5ω +.88si7 ω x( ) siω si3ω si5ω si7 ω... π Example. : (co) >> :.:.; >> b.73*si(*pi**); >> plo (,b) >> hold o >> b3.444*si(*3*pi**); >> bb+b3; >> plo (,b,'r') >> b5.546*si(*5*pi**); >> bb+b3+b5; >> plo (,b,'g') >> b7.889*si(*7*pi**); >> bb+b3+b5+b7; >> plo (,b,'y') >> b9.447*si(*9*pi**); >> plo (,b,'m') >> ile ('sum (b_ si( \omega )') >>

13 Example. : (co) >>:; >> x(./()).*(si(pi/*)).*exp(j *(pi*/)); xabs(x); hea(8*agle(x))/pi; subplo (,,); sem(,x) xlabel('\omega_'); ylabel('x_'); ile('x_(\omega_)'); subplo (,,) sem(,hea) xlabel('\omega_') ylabel('\hea') ile('\hea (\omega_)'); Problem. x() A - Assume ha,a, ad Deermie Fourier series coefficies of i expoeial ad rigoomeric form. 4 Plo he discree specrum of x(). Compare wih Example s s

14 Problem: Wrie he Fourier series for he followig periodic sigal ad plo he sum of firs harmoics x() - - /4 - /4 - x( ).73cosω.444cos3ω +.546cos5ω.88cos7 ω +...

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