Fourier SeriesKEEE343 Communication Theory

Lecture #4, March 15, 2011Prof. Young-Chai Kokoyc@korea.ac.kr

2011년 3월 15일 화요일

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

Review

·System classification·Continuous-time, linear, time-invariant systems

·Convolution of two continuous-time signals

·Impulse response to the Linear Time-Invariant (LTI) Systems

y(t) = x(t) ∗ h(t) =� ∞

−∞x(τ)h(t− τ) dτ

LTISystem

δ(t) h(t)

x(t) y(t) = x(t) ∗ h(t)

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Summary

·Properties of the Convolution Integral·Fourier series·Periodic signal

·Complex exponential Fourier Series

·Trigonometric Fourier Series

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Properties of the Convolution Integral

·Associativity

·Commutativity

·Distributive with respect to addition

·Convolution with the unit impulse

·Convolution with the shifted unit impulse

·Derivative property

x(t) ∗ (v(t) ∗ w(t)) = (x(t) ∗ v(t)) ∗ w(t)

x(t) ∗ v(t) = v(t) ∗ x(t)

x(t) ∗ (v(t) + w(t)) = x(t) ∗ v(t) + x(t) ∗ w(t)

x(t) ∗ δ(t) = x(t)

x(t) ∗ δ(t− t0) = x(t− t0)

d

dt[x(t) ∗ v(t)] = dx(t)

dt∗ v(t) d

2

dt2[x(t) ∗ v(t)] = dx(t)

dt∗ dv(t)

dt

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Example: Convolution Integral

·Suppose that , where is the rectangular pulse.

·Find the convolution integral:

x(t) = h(t) = p(t) p(t)

p( t )

0 T t

y(t) =

� ∞

−∞x(τ)h(t− τ) dτ

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• There are four cases to be considered:

• Case 1:

• Case 2:

t ≤ 0

0 T t

t-T t

⇒ y(t) = 0

0 ≤ t ≤ T

0 T t t - T t

y(t) =

� t

0dτ = t

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• Case 3:

• Case 4:

0 ≤ t− T ≤ T → T ≤ t ≤ 2T

0 T t

t - T t

y(t) =

� T

t−Tdτ = T − (t− T ) = 2T − t

T ≤ t− T −→ 2T ≤ t

0 T t

t - T t

y(t) = 0

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y(t) = x(t) ∗ h(t)

0 T t

2T

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Fourier Series

·Motivation·Representation of continuous-time, periodic signals in the frequency domain.

·Periodic signals occur frequently in the communication signals.

·Outline·Biography of J. Fourier

·Fourier series of periodic functions

·Examples of Fourier series

·Fourier transforms of periodic functions - relations to Fourier series

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J. Fourier

·Joseph Fourier

• was born in Auxerre, France on March 21, 1768 and died in Paris on May 4, 1830.

• was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems to heat transfer and vibration.

• The Fourier transform and Fourier’s Law are also named in his honor.

• Fourier is also generally credited with the discovery of the greenhouse effect.

• Detailed biography can be found at http://en.wikipedia.org/wiki/Joseph_Fourier.

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http://en.wikipediahttp://en.wikipediahttp://en.wikipediahttp://en.wikipedia

Periodic Signals

·Periodic signals if

·Fundamental period of is the smallest positive value of

·Fundamental frequency

·Two basic examples of periodic signals·Real sinusoidal signal

·Imaginary exponential signal

x(t+ T ) = x(t), for all t

T0 x(t) T

f0 =1

T0

x(t) = cos(ω0t+ φ)

x(t) = ejω0t

where is called the fundamental angular frequency.ω0 = 2π/T0 = 2πf0

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Fourier’s Insight

·Fourier’s insight was that (under certain circumstances), one can write a series expansion for a - periodic function in terms of sines and cosines.

·Then it was proved that any periodic signal can be converged to the sum of orthogonal sines and cosines (or exponential) functions.

2π

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Complex Exponential Fourier Series

·Periodic time function· is a periodic time function with

·Such a periodic function can be expanded in an infinite series of exponential time functions called the Fourier series.

x(t) T

x(t)

T 0 t

x(t) =∞�

k=−∞cke

jkω0t =∞�

k=−∞cke

j2πkt/T0

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Coefficients of Complex Exponential Fourier Series

·Exponential Fourier series representation

·Note: are known as complex Fourier coefficients and are given as

·where denotes the integral over any one period of and 0 to or to are commonly used.

·Setting k=0, we have

ck

ck =1

T0

�

T0

x(t)e−jkω0t dt

x(t)�

T0 T0−T0/2 +T0/2

x(t) =∞�

k=−∞cke

jkω0t =∞�

k=−∞cke

j2πkt/T0

c0 =1

T0

�

T0

x(t) dt

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• equals the average value of over a period.

• When is real, then it follows

• Therefore,

c0 x(t)

x(t)c−k = c

∗k

|c−k| = |c∗k|

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