ii. fourier series - naval postgraduate...

06/09/10 2003rws/jMc-modif SuFY10 (MPF)-Textbook Sections II & III 1

[p. 3] Fourier series definition[p. 3] Complex exponential expansion[p. 6] Fourier analysis equation[p. 8] Examples

[p. 16] Fourier series properties[p. 16] Gain[p. 17] Derivative[p. 18] Time delay[p. 19] Properties summary

[p. 28] Evaluating FS coefficients using MATLAB[p. 29] Fourier series synthesis operation[p. 34] Gibb’s phenomenon

II. Fourier Series


Fourier Series

Most signals may be represented as Sinusoids with DIFFERENT Frequencies

Periodic signals may be represented by

01

( ) cos(2 )N

k k kk

x t A f t Aπ ϕ=

= + +∑

020

0 01

( ) e

Real Periodic signals may be represented as

( ) cos(2 )

j kf tk

k

k kk

x t a A

x t A kf t A

π

π ϕ

∞

=−∞

∞

=

= +

= + +

∑

∑


Relationship between ak and Ak in the FS expression for real signals

tfkj

kk eatx 02)( π∑

∞

−∞=

=


SYNTHESIS vs. ANALYSIS operations

SYNTHESISEasyGiven (ωk,Ak,φk) create x(t)

Synthesis can be HARD

Synthesize speech so that it sounds good

ANALYSISHardGiven x(t), extract (ωk,Ak,φk) How many harmonics?Need algorithm for computer

0( ) cos(2 )k k kk

x t A f t Aπ ϕ= + +∑


STRATEGY: x(t) ak

Fourier Analysis equationGet representation from the signalWorks for PERIODICPERIODIC Signals

Fourier SeriesAnswer is: an INTEGRAL over one period

∫ −=0

0

0

0)(1

T

dtetxa tkjTk

ω

[p. 48-50, textbook]

tfkj

kkeatx 02)( π∑

∞

−∞=

=

02 fπ


0 0Fundamental Frequency 1 /f T=

Summary: Fourier Series Integral

HOW to determine ak from x(t)?

0

0

0

0

0

(2 / )1

0

10

0

( )

( ) (DC component)

Tj T kt

k T

T

T

a x t e dt

a x t dt

π−=

=

∫

∫

*Property: when ( ) is realk ka a x t− =


Example: Compute the FS coefficients of x(t) without integration

x1 (t)=cos(2t)x2 (t)=3sin(3t+π/3)x3 (t)=cos(5t)-2sin(15t)x4 (t)=cos(2t)+2


Example: Compute the FS coefficients of x(t)

2( ) sin (3 )x t tπ=


Example: Compute the FS coefficients of x(t)

)3(sin)( 3 ttx π=


Example: Compute the FS coefficients of x(t)A periodic signal x(t) is described over one period [0, T0 ], by the equation

(a) Sketch the periodic function x(t) for the specific case tc =T0 /2(b) Determine the D.C. coefficient of the Fourier Series, a0 .

0

, 0( )

0, ,c

c

t t tx t

t t T≤ ≤⎧

= ⎨ ≤ ≤⎩


Example: A periodic signal is represented by the Fourier Series formula:

(a) Sketch the two-sided spectrum of this signal. Label all complex amplitudes in polar form.

(b) Determine the fundamental frequency (in Hz) and the fundamental period (in secs.) of this signal.

30( )

1/(4 2 ) 3, 2, 1,0

0 | | 3

j ktk

k

k

x t a e

jk for ka

for k

π∞

=−∞

=

+ = ± ± ±⎧= ⎨ >⎩

∑


Fourier Series Coefficients Properties

Note: It is possible to relate Fourier series coefficients of related signals without starting from scratch!

Example 1: gain propertyAssume we know the FS coefficients ak ’s for x(t)

02( ) jk f tk

k

x t a e π∞

=−∞

= ∑Assume we want to compute the Fourier series coefficients of y(t)=Kx(t)

02( ) jk f tk

ky t b e π

∞

=−∞

= ∑


Example 2: derivative propertyAssume we know the FS coefficients ak ’s for x(t)

02( ) jk f tk

k

x t a e π∞

=−∞

= ∑Assume we want to compute the Fourier series coefficients of y(t)=dx(t)/dt

12( ) jk f tk

k

y t b e π∞

=−∞

= ∑

Question 1: How is the period (fundamental frequency) of x(t) related to the period (fundamental frequency) of y(t)?

Question 2: How do we compute bk ’s ?


Example 3: time-delay propertyAssume we know the FS coefficients ak ’s for x(t)

02( ) jk f tk

kx t a e π

∞

=−∞

= ∑

Assume we want to compute the Fourier series coefficients of y(t)=x(t-D)

12( ) jk f tk

ky t b e π

∞

=−∞

= ∑

Question 1: How is the period of x(t) related to the period of y(t)?

Question 2: How do we compute bk ’s ?


x(t) ak y(t) bk

x(t) and y(t) with same period T

Summary Basic FS Coefficient Properties

x(t) real ==> ak =a-k*

x(t) real + even ⇒

ak real

x(t) real + odd ⇒

ak imaginary


Property Signal Coefficients

Linearity Ax(t) +By(t) Aak +Bbk

Time-shift

Time reversal

Conjugation

( ) 2 21x k

kT

P x t dt aT

= =∑∫

( )x t−( )0x t t−

( )*x t

0 0jk tka e ω−

ka−*

ka−

Parseval’s relation:

x(t) ak y(t) bk

x(t) and y(t) with same period T

Summary Basic FS Coefficient Properties


t-1

1

Example

06/09/10 2003rws/jMc-modif SuFY10 (MPF)-Textbook Sections II & III 22[Reference: Oppenheim & Wilsky, Signals & Systems, Prentice Hall]


Using FS coefficients properties, match the Frequency representation to the correct signal


More Fourier Series Coefficients Properties

0

0

0 0

0

0 0

0 0

0 0 0

1

10 0

10 0

( )

( )[cos( ) sin( )]

{ ( ) cos( ) ( )sin( ) }

T

T

T T

jk tk T

T

T

a x t e dt

x t k t j k t dt

x t k t dt j x t k t dt

ω

ω ω

ω ω

−=

= +

= +

∫

∫

∫ ∫

02 fπFS coefficient ak may be expressed as:

0

0 0

0 0

2 1( ) ; ( )T

j kf kt j ktk k T

kx t a e a x t e dtπ ω

∞−

=−∞

= =∑ ∫

Real part: R Imaginary part: I


Signal odd/even properties and integration

1) Signal real and even:

2) Signal real and odd:

Signal is real & odd: FS coefficients contribution is due to sine terms contribution only x(t) can be written in terms of sine functions only & ak are purely imaginary

Ex: sine function

Signal is real and even: FS coefficients contribution is due to cosine terms contribution only x(t) can be written in terms of cosine terms only & ak are purely real

Ex: cosine function


Signal is real & has half wave symmetry x(t)=-x(t+T/2):FS coefficients contribution is due to odd terms contribution only ak with odd k terms only are non zero


Evaluating Fourier Series Coefficients using MATLAB

Use the MATLAB function int.m

Y= int(S,a,b) returns the definite integral from a to b of each element of S with respect to each element's default symbolic variable. a and b are symbolic or double scalars.

Example: compute the first 3 harmonics of the signalx(t)=cos(2πf0 t), f0 =0.2Hz.

MATLAB codesyms t % defines the symbolic variable f0=1; T=1 % defines the signal frequenfor k=0:3

a(k)=(1/T)*INT(cos(2*pi*f0*t)*exp(j*2*pi*k*t*f0),0,1);end

>a=0, ½,0,0


Fourier Series Synthesis Operation

HOW do you APPROXIMATE x(t) based on FS coefficients information?

Use a FINITE number of coefficients

∫ −=0

0

00

)/2(1 )(T

tkTjTk dtetxa π

*Recall: w hen ( ) is realk ka a x t− =

tfkjN

Nkk eatx 02)( π∑

−=

=


Fourier Series Synthesis Overall process

Question: What happens when the FS decomposition is truncated?


Partial reconstruction using Synthesis: use 1st & 3rd Harmonics only for the square wave decomposition, what happens?

))75(2cos(32))25(2cos(2

21)( 22

ππ ππ

ππ

−+−+= ttty


Partial reconstruction using Synthesis: Repeat using up to 7th Harmonic for the square wave decomposition

21 2 2( ) cos(50 ) sin(150 ) ...2 3

2 2sin(250 ) sin(350 )5 7

y t t t

t t

ππ ππ π

π ππ π

= + − + +

+ +


Partial Fourier Synthesis & the square wave

…+++= )3sin(32)sin(2

21)( 00 tttxN ω

πω

π


Gibbs’ Phenomenon

Convergence at DISCONTINUITY of x(t)There is always an overshoot9% for the Square Wave case

ii. fourier series - naval postgraduate...

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