Info
EE-2025
Spring-2000
Web-CT Check the Bulletin Board for msgs
Lecture 6 Fourier Series Coefficients 31-Jan-00
Old Quizzes & Problems are linked !
Quiz #1 on 4-Feb (Friday)
Prob Set #3 due This Week
Solution will be posted Thurs @ 6PM
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Lab Info
Lab #2 Report
ERRATA ? ALWAYS Check Bulletin Board Turn in during your lab time Write-up sections 4 and 5 Include INSTRUCTOR VERIFICATION
READING ASSIGNMENTS
This Lecture:
Notes on Fourier Series
17 pages, posted to WebCT Replace pp 62-66 in Chapter 3
Other Reading: Lab #3 was posted Friday
Learn some Music Notation
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Next Lecture: Chap. 4 on Sampling
LECTURE OBJECTIVES
Work with the Fourier Series Integral
HISTORY
Jean Baptiste Joseph Fourier
1807 thesis (memoir)
On the Propagation of Heat in Solid Bodies
ak =
1 T0
T0
0
x (t )e
j ( 2 k / T0 ) t
dt
ANALYSIS via Fourier Series
For PERIODIC signals: x(t+T) = x(t) SPECTRUM from the Fourier Series
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Heat ! Napoleonic era
http://www-groups.dcs.st-and.ac.uk/~history/ Mathematicians/Fourier.html
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6
SPECTRUM DIAGRAM
Recall Complex Amplitude vs. Freq
1 2
Xk *
4e j / 2
250
7e
j / 3
10 7e
j / 3
1 2
Xk
Xk = Ak e j k
100 0 100
4e j / 2
250 f (in Hz)
x ( t ) = X0 +
k =1
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N
{
1 2
* Xk e j 2 fk t + 1 Xk e j 2 fk t 2
}
8
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Harmonic Signal
x (t ) = A0 + Ak cos(2 kf0 t + k )
k =1 N * x (t ) = X0 + 1 Xk e j 2 kf0 t + 1 Xk e j 2 kf0 t 2 2 k =1 k =1 N N
Fourier Series Synthesis
x (t ) =
N
k =
ak e j 2 kf0 t
ak = 1 Xk = 1 Ak e j k 2 2
PERIOD/FREQUENCY of COMPLEX EXPONENTIAL:
0 =
2 k T0
= ( )k = 2 ( f0 )k
2 T0
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x (t ) = A0 + Ak cos(2 kf0 t + k )
k =1
Xk = Ak e
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j k
COMPLEX AMPLITUDE
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Harmonic Signal (3 Freqs)
a-1 a-5 a1 a5
SYNTHESIS vs. ANALYSIS
SYNTHESIS
Easy Given (k,Ak,k) create x(t)
ANALYSIS
Hard Given x(t), extract (k,Ak,k) How many? Need algorithm for computer
a-3
a3
Synthesis can be HARD
Synthesize Speech so that it sounds good
ak is the complex amplitude for kf0
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STRATEGY
ANALYSIS
Get representation from the signal Works for PERIODIC Signals
Fourier Series Integral
HOW do you determine ak from x(t) ?
Series
Fourier INTEGRAL over one period
ak =
1 T0
T0
x (t )e j 2 k t / T0 dt
0
1 ak = T0
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T0
x (t )e j 0 kt dt
0
14
1 a0 = T0
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T0
FUNDAMENTAL FREQ: f0=1/T0
x(t )dt
0
(DC Component)
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* a k = ak
when x (t ) is real
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ORTHOGONALITY of exp(j)
INTEGRATE over ONE PERIOD
ORTHOGONALITY of exp(j)
INTEGRATE over ONE PERIOD
T0
0
e
j 2 mt / T0
dt = =
T0 j 2 m T0 j 2 m
e
j 2 mt / T0 T0 0 j 2 m
(e
1)
T 1 0 j 2 l t / T0 j 2 k t / T0 0 k l e dt = e T0 0 1 k = l
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T0
0
e j m 0 t dt = 0
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2 0 = T0
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1 0 j 2 ( l k ) t / T0 dt e T0 0
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T
SQUARE WAVE EXAMPLE
FS for a SQUARE WAVE
1 x (t ) = 0 for T0 = 4:
0 t < T0 1 2 T0 t < T0
1 2
x(t) 1
1 ak = T0
2
T0
x (t )e j 2 k t / T0 dt
0
( k 0)
1 ak = 1e j 2 k t / 4 dt = 40
1 2 4 t
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1 4 ( j k / 2 )
e
j kt / 2 2 0
2
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0
=
1 ( 2 j k )
(e
j k
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1 ( 1)k 1) = j 2 k
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DC Coefficient, a0
Fourier Coefficients ak
1 ak = T0
T0
x ( t )e
0 T0
j 2 k t / T0
dt
( k = 0)
ak is a function of k
Complex Amplitude for k-th Harmonic Does NOT depend on the period, T0
1 a0 = T0
a0 =
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1 x(t )dt = T0 (AREA) 0
2 0
1 4
dt = 1 (2 0) = 4
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1 2
20
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1 j k j k 1 e ak = = 0 j 2 k 1 2
k = 1, 3,K k = 2, 4,K k=0
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