Fourier Series and TransformKEEE343 Communication Theory

Lecture #5, March 17, 2011Prof. Young-Chai Kokoyc@korea.ac.kr

2011년 3월 17일 목요일

mailto:koyc@korea.ac.krmailto:koyc@korea.ac.kr

Summary

2011년 3월 17일 목요일

Summary

·Generalized Fourier series·Fourier transform·

2011년 3월 17일 목요일

Generalized Fourier Series

φ1(t),φ2(t), · · · ,φN (t)x(t) x(t)

xa(t) =N�

n=1

Xnφn(t), t0 ≤ t ≤ t0 + T

x(t) T

{φn(t)}Nn=1� t=t0+T

t0

φm(t)φ∗n(t) dt = cnδnm =

�cn, n = m0, n �= m (all m and n)

cn = 1 n φn(t)�s

where δnm is called Kronecker delta function

2011년 3월 17일 목요일

Generalized Fourier Series

·Represent the time function or signal on a second interval as a weighted sum of linearly independent orthogonal basis functions.

·Consider a set of time function, which are specified independently of , and seek a series expansion form which approximates

·Orthogonal function,

·where if for all , the are said to be normalized (or orthonormal functions).

φ1(t),φ2(t), · · · ,φN (t)x(t) x(t)

xa(t) =N�

n=1

Xnφn(t), t0 ≤ t ≤ t0 + T

x(t) T

{φn(t)}Nn=1� t=t0+T

t0

φm(t)φ∗n(t) dt = cnδnm =

�cn, n = m0, n �= m (all m and n)

cn = 1 n φn(t)�s

where δnm is called Kronecker delta function

2011년 3월 17일 목요일

·Consider the orthonormal basis set

·Error in the approximation of : Integral-Square Error (ISE) defined as

·Using the orthnormality of , that is,

·we can show the ISE as

{φn(t)}Nn=1

x(t)

Error = �N =

�

T|x(t)− xa(t)|2 dt

�N =

�

T|x(t)|2 dt−

N�

n=1

�X∗n

�

Tx(t)φ∗n(t) dt+Xn

�

Tx∗(t)φn(t) dt

�

+N�

n=1

|Xn|2

{φn(t)}Nn=1�

Tφn(t)φ

∗m(t) dt = 1

2011년 3월 17일 목요일

• To find the that minimize we add and subtract the quantity:

• which yields

• We have the minimum ISE only when we choose

• The resulting minimum-error coefficient => Fourier coefficients

X �ns �NN�

n=1

�����

Tx(t)φ∗n(t) dt

����2

�N =

�

T|x(t)|2 dt−

N�

n=1

�����

Tx(t)φ∗n(t) dt

����2

+N�

n=1

����Xn −�

Tx(t)φ∗(t) dt

����2

Xn =

�

Tx(t)φ∗(t) dt

2011년 3월 17일 목요일

• Minimum value for

• If we can find an infinite set of orthonormal functions such that

for any signal that is integrable square,

we say that are complete.

�N

(�N )min =

�

T|x(t)|2 dt−

N�

n=1

�����

Tx(t)φ∗(t) dt

����2

=

�

T|x(t)|2 dt−

N�

n=1

|Xn|2

limN→∞

(�N )min = 0,

�

T|x(t)|2 dt < ∞

{φn(t)}Nn=1

2011년 3월 17일 목요일

• In the sense that ISE is zero, we can writex(t) =

N�

n=1

Xnφn(t) (ISE=0)

2011년 3월 17일 목요일

Complex Exponential Fourier Series

·Given a signal defined over the interval with the definition

we define the complex exponential Fourier series as

·where

·It can be shown to represent the signal exactly in the interval , except at a point of jump discontinuity where it converges to the arithmetic mean of the left-hand and right-hand limits.

x(t)

ω0 = 2πf0

x(t) =∞�

n=−∞Xne

jnω0t, t0 ≤ t ≤ t0 + T

Xn =1

T0

� t0+T0

t0

x(t)e−jnω0t dt

(t0, t0 + T0)

x(t) (t0, t0 + T0)

2011년 3월 17일 목요일

Example of Fourier Series

·Consider the signal

·where Using Euler’s theorem and trigonometric identity, we can rewrite

Hence, we have

x(t) = cos(ω0t) + sin2(2ω0t)

ω0 = 2π/T

x(t) = cos(ω0t) +1

2− 1

2cos(4ω0t)

=1

2ejω0t +

1

2e−jω0t +

1

2− 1

4ej4ω0t − 1

4e−j4ω0t

X0 =1

2

X1 =1

2= X−1

X4 = −1

4= X−4

2011년 3월 17일 목요일

Symmetry Properties of the Fourier Coefficients

·Note that the Fourier coefficients are given as

·Assuming is real, we have the symmetry property such as

·Writing , we have the symmetry property such as

·For real signal, the magnitude of the Fourier coefficients is an even function of n, and the argument is odd.

·Assuming is is even, that is,

Xn =1

T0

� t0+T0

t0

x(t)e−jnω0t dt

x(t)

X∗n = X−n

Xn = |Xn|ej∠Xn

|Xn| = |X−n| ∠Xn = −∠X−n

x(t) x(t) = x(−t)

Xn =1

T0

� T0/2

−T0/2x(t) cos(nω0t) dt−

j

T0

� T0/2

−T0/2x(t) cos(nω0t) dt

=1

T0

� T0/2

−T0/2x(t) cos(nω0t) dt

purely real value

2011년 3월 17일 목요일

Trigonometric Form of the Fourier Coefficients

·We can regroup the complex exponential Fourier series by pairs of terms of the form:

·Hence, can be rewritten as

·Using , we can rewrite as

where

Xnejnω0t +X−ne

−jnω0t = |Xn|ej(nω0t+∠Xn) + |Xn|e−j(nω0t+∠Xn)

= 2|Xn| cos(nω0t+ ∠Xn)x(t)

x(t) = X0 +N�

n=1

2|Xn| cos(nω0t+ ∠Xn)

cos(x+ y) = cosx cos y − sinx sin y x(t)

x(t) = X0 +N�

n=1

An cos(nω0t) +N�

n=1

Bn sin(nω0t)

An = 2|Xn| cos∠Xn =2

T0

� t0+T0

t0

x(t) cos(nω0t) dt

Bn = −2|Xn| sin∠Xn =2

T0

� t0+T0

t0

x(t) sin(nω0t) dt

2011년 3월 17일 목요일

Example: Periodic Pulse Train

• Find the complex Fourier coefficients Xnx(t)

T/2 0 t

A

-T/2 T 0 -T 0

x(t) =

�A, −T2 ≤ t ≤

T2

0, for the remainder of the period

2011년 3월 17일 목요일

• Complex Fourier coefficients

where we define sinc function as

Xn

Xn =

� T/2

−T/2A exp(−j2πnf0t)dt

=A

−j2πnf0exp(−j2πnf0t)

����t=T/2

t=−T/2

= A[exp(−jπnf0T )− exp(jπnf0T )]

−j2πnf0

=A

πnf0

[exp(jπnf0T )− exp(−jπnf0T )]j2

=A

πnf0sin (πnf0T ) = AT sinc(nf0T )

sinc(x) =sin(πx)

πx

2011년 3월 17일 목요일

• Fourier series of a periodic signal

• Recall for the real signal

• for the real signal

Real signal x(t)

x(t)

x(t) = X0 +N�

n=1

An cos(nω0t) +N�

n=1

Bn sin(nω0t)

X−n = X∗n

An andBn x(t)

An = Xn +X−n = Xn +X∗n = 2�[Xn]

Bn = j(Xn −X−n) = j(Xn −X∗n) = −2�[Xn]

2011년 3월 17일 목요일

Even and Odd signals

• If is even, then from

Then,

• If is odd, then in a similar way. Then,

x(t) Bn = 0

Bn =1

T0

�

T0

x(t) sinnω0t� �� �even×odd

dt = 0

x(t) = X0 +∞�

n=1

An cosnω0t, ω0 =2π

T0

x(t) An = 0

x(t) =∞�

n=1

Bn sinnω0t, ω0 =2π

T0

2011년 3월 17일 목요일

Amplitude and Phase Spectra

• Recall , forwhere

• Generally, is complex value and can be represented as

• Amplitude spectrum: Plot of versus • Phase spectrum: Plot of versus• For a real periodic signal , we have . Thus,

x(t) =∞�

n=−∞Xne

jnω0t ω0 = 2π/T0

Xn =1

T0

�

T0

x(t)e−jnω0t dt

Xn

Xn = |Xn|ejφk

|Xn| ω

φk ω

x(t) X−n = X∗n

|X−n| = |Xn|� �� �Even fucntion

, φ−n = −φn� �� �Odd fucntion

2011년 3월 17일 목요일