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6.003: Signals and Systems
Fourier Representations
October 27, 2011 1
Fourier Representations
Fourier series represent signals in terms of sinusoids.
→ leads to a new representation for systems as filters.
2
Fourier Series
Representing signals by their harmonic components.
ωω0 2ω0 3ω0 4ω0 5ω0 6ω0
0 1 2 3 4 5 6 ← harmonic #
DC→
fun
da
men
tal→
seco
nd
ha
rmo
nic→
thir
dh
arm
on
ic→
fou
rth
ha
rmo
nic→
fift
hh
arm
on
ic→
sixth
ha
rmo
nic→
3
Musical Instruments
Harmonic content is natural way to describe some kinds of signals.
Ex: musical instruments (http://theremin.music.uiowa.edu/MIS )
piano
t
cello
t
bassoon
t
oboe
t
horn
t
altosax
t
violin
t
bassoon
t1252
seconds
4
.html
Musical Instruments
Harmonic content is natural way to describe some kinds of signals.
Ex: musical instruments (http://theremin.music.uiowa.edu/MIS )
piano
k
cello
k
bassoon
k
oboe
k
horn
k
altosax
k
violin
k5
.html
Musical Instruments
Harmonic content is natural way to describe some kinds of signals.
Ex: musical instruments (http://theremin.music.uiowa.edu/MIS )
piano
t
piano
k
violin
t
violin
k
bassoon
t
bassoon
k6
.html
Harmonics
Harmonic structure determines consonance and dissonance.
octave (D+D’) fifth (D+A) D+E
t ime(per iods of "D")
harmonics
0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112
–1
0
1
D
D'
D
A
D
E
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7
7
Harmonic Representations
What signals can be represented by sums of harmonic components?
ωω0 2ω0 3ω0 4ω0 5ω0 6ω0
T= 2πω0
t
T= 2πω0
t
Only periodic signals: all harmonics of ω0 are periodic in T = 2π/ω0. 8
Harmonic Representations
Is it possible to represent ALL periodic signals with harmonics? What about discontinuous signals?
2πω0
t
2πω0
t
Fourier claimed YES — even though all harmonics are continuous!
Lagrange ridiculed the idea that a discontinuous signal could be written as a sum of continuous signals.
We will assume the answer is YES and see if the answer makes sense. 9
Separating harmonic components
Underlying properties.
1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency:
jkω0t jlω0t j(k+l)ω0t e × e = e .
2. The integral of a harmonic over any time interval with length equal to a period T is zero unless the harmonic is at DC: t0+T 0, k = 0
jkω0tdt ≡ jkω0tdt =e e t0 T T, k = 0
= Tδ[k]
10
Separating harmonic components
Assume that x(t) is periodic in T and is composed of a weighted sum of harmonics of ω0 = 2π/T .
∞jω0kt x(t) = x(t + T ) = ake
Then ∞
−∞=k
f
f
x(t)e −jlω0tdt = jω0kt −jω0ltdteake T T k=−∞
∞fjω0(k−l)tdt= ak e
Tk=−∞ ∞
= akTδ[k − l] = Tal k=−∞
Therefore
f
1 1 −j 2π ktak = x(t)e −jω0ktdt = x(t)e T dt T TT T
11
∫
∫∫∫∫
∫
Fourier Series
Determining harmonic components of a periodic signal.
1 2π−j ktx(t)e (“analysis” equation) ak = dt TT T
∞f 2πj kt x(t)= x(t + T ) = (“synthesis” equation) Take k=−∞
12
∫
Let ak represent the Fourier series coefficients of the following square wave.
t
12
−12
0 1
Check Yourself
How many of the following statements are true?
1. ak = 0 if k is even 2. ak is real-valued 3. |ak| decreases with k2
4. there are an infinite number of non-zero ak 5. all of the above
13
� �
Check Yourself
Let ak represent the Fourier series coefficients of the following square wave.
t
12
−12
0 1
ak = x(t)e −j 2π T ktdt = −
1 2
0
− 1 2
e −j2πktdt + 1 2
1 2
0 e −j2πktdt
T
1 jπk − e −jπk = 2 − e j4πk ⎧ ⎨ 1 ; if k is odd = jπk ⎩ 0 ; otherwise
14
∫ ∫ ∫
Check Yourself
Let ak represent the Fourier series coefficients of the following square wave. 1 ; if k is odd
ak = jπk 0 ; otherwise
How many of the following statements are true? √ 1. ak = 0 if k is even 2. ak is real-valued X 3. |ak| decreases with k2 X √ 4. there are an infinite number of non-zero ak 5. all of the above X
15
Let ak represent the Fourier series coefficients of the following square wave.
t
12
−12
0 1
Check Yourself
How many of the following statements are true? 2
√ 1. ak = 0 if k is even 2. ak is real-valued X 3. |ak| decreases with k2 X √ 4. there are an infinite number of non-zero ak 5. all of the above X
16
Fourier Series Properties
If a signal is differentiated in time, its Fourier coefficients are multi
plied by j 2π k.T
Proof: Let ∞f 2π
x(t) = x(t + T ) = ake k=−∞
j kt T
then f∞ 2π 2π x(t) = x(t + T ) = j j kt k ak e T
T k=−∞
17
Let bk represent the Fourier series coefficients of the following triangle wave.
t
18
−18
0 1
Check Yourself
How many of the following statements are true?
1. bk = 0 if k is even 2. bk is real-valued 3. |bk| decreases with k2
4. there are an infinite number of non-zero bk 5. all of the above
18
Check Yourself
The triangle waveform is the integral of the square wave.
t
12
−12
0 1
t
18
−18
0 1
Therefore the Fourier coefficients of the triangle waveform are 1
j2πk times those of the square wave.
1 1 −1 = × = ; k odd bk jkπ j2πk 2k2π2
19
Check Yourself
Let bk represent the Fourier series coefficients of the following triangle wave.
−1 bk = ; k odd
2k2π2
How many of the following statements are true? √ 1. bk = 0 if k is even √ 2. bk is real-valued √ 3. |bk| decreases with k2
√ 4. there are an infinite number of non-zero bk√ 5. all of the above
20
Let bk represent the Fourier series coefficients of the following triangle wave.
t
18
−18
0 1
Check Yourself
How many of the following statements are true? 5
√ 1. bk = 0 if k is even √ 2. bk is real-valued √ 3. |bk| decreases with k2
√ 4. there are an infinite number of non-zero bk√ 5. all of the above
21
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
0∑k = −0k odd
−12k2π2 e
j2πkt
18
−18
0 1
22
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
1∑k = −1k odd
−12k2π2 e
j2πkt
18
−18
0 1
23
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
3∑k = −3k odd
−12k2π2 e
j2πkt
18
−18
0 1
24
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
5∑k = −5k odd
−12k2π2 e
j2πkt
18
−18
0 1
25
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
7∑k = −7k odd
−12k2π2 e
j2πkt
18
−18
0 1
26
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
9∑k = −9k odd
−12k2π2 e
j2πkt
18
−18
0 1
27
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
19∑k = −19k odd
−12k2π2 e
j2πkt
18
−18
0 1
28
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
29∑k = −29k odd
−12k2π2 e
j2πkt
18
−18
0 1
29
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: triangle waveform
t
39∑k = −39k odd
−12k2π2 e
j2πkt
18
−18
0 1
Fourier series representations of functions with discontinuous slopes converge toward functions with discontinuous slopes.
30
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
0∑k = −0k odd
1jkπ
e j2πkt
12
−12
0 1
31
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
1∑k = −1k odd
1jkπ
e j2πkt
12
−12
0 1
32
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
3∑k = −3k odd
1jkπ
e j2πkt
12
−12
0 1
33
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
5∑k = −5k odd
1jkπ
e j2πkt
12
−12
0 1
34
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
7∑k = −7k odd
1jkπ
e j2πkt
12
−12
0 1
35
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
9∑k = −9k odd
1jkπ
e j2πkt
12
−12
0 1
36
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
19∑k = −19k odd
1jkπ
e j2πkt
12
−12
0 1
37
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
29∑k = −29k odd
1jkπ
e j2πkt
12
−12
0 1
38
Fourier Series
One can visualize convergence of the Fourier Series by incrementally adding terms.
Example: square wave
t
39∑k = −39k odd
1jkπ
e j2πkt
12
−12
0 1
39
Fourier Series
Partial sums of Fourier series of discontinuous functions “ring” near discontinuities: Gibb’s phenomenon.
This ringing results because the magnitude of the Fourier coefficients
t
12
−12
0 1
9%
1is only decreasing as k 1 (while they decreased as
k2 for the triangle).
You can decrease (and even eliminate the ringing) by decreasing the magnitudes of the Fourier coefficients at higher frequencies.
40
Fourier Series: Summary
Fourier series represent periodic signals as sums of sinusoids.
• valid for an extremely large class of periodic signals
• valid even for discontinuous signals such as square wave
However, convergence as # harmonics increases can be complicated.
41
Filtering
The output of an LTI system is a “filtered” version of the input.
Input: Fourier series → sum of complex exponentials. ∞
x(t) = x(t + T ) = ake k=−∞
f j 2π kt T
Complex exponentials: eigenfunctions of LTI systems. 2πkt → H(j k)e j
2 2π πj kt T Te T
Output: same eigenfunctions, amplitudes/phases set by system. ∞ ∞ff 2π 2 2π πj kt → y(t) = akH(j k)e j kt x(t) = T Take
T k=−∞ k=−∞
42
Filtering
Notion of a filter.
LTI systems • cannot create new frequencies. • can scale magnitudes and shift phases of existing components.
Example: Low-Pass Filtering with an RC circuit
+−
vi
+
vo
−
R
C
43
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
Lowpass Filter
Calculate the frequency response of an RC circuit.
+−
vi
+
vo
−
R
C
KVL: vi(t) = Ri(t) + vo(t) C: i(t) = Cvo(t) Solving: vi(t) = RC vo(t) + vo(t)
Vi(s) = (1 + sRC)Vo(s) Vo(s) 1
H(s) = = Vi(s) 1 + sRC
44
Lowpass Filtering
x(t) =
Let the input be a square wave.
t
12
−12
0 T
T π2=0ω;kt 0jωe
jπk 1
odd k
f
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|X(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠X
(jω
)|
45
Lowpass Filtering
x(t) =
Low frequency square wave: ω0 << 1/RC.
t
12
−12
0 T
T π2=0ω;kt 0jωe
jπk 1
odd k
f
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
46
Lowpass Filtering
x(t) =
Higher frequency square wave: ω0 < 1/RC.
t
12
−12
0 T
T π2=0ω;kt 0jωe
jπk 1
odd k
f
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
47
Lowpass Filtering
x(t) =
Still higher frequency square wave: ω0 = 1/RC.
t
12
−12
0 T
T π2=0ω;kt 0jωe
jπk 1
odd k
f
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
48
Lowpass Filtering
x(t) =
High frequency square wave: ω0 > 1/RC.
t
12
−12
0 T
T π2=0ω;kt 0jωe
jπk 1
odd k
f
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
49
Fourier Series: Summary
Fourier series represent signals by their frequency content.
Representing a signal by its frequency content is useful for many signals, e.g., music.
Fourier series motivate a new representation of a system as a filter.
50
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6.003 Signals and SystemsFall 2011
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