6.003: Signals and Systems
DT Fourier Representations
November 10, 2011
1
Mid-term Examination #3
Wednesday, November 16, 7:30-9:30pm,
No recitations on the day of the exam.
Coverage:
Lectures 1–18
Recitations 1–16
Homeworks 1–10
Homework 10 will not be collected or graded.
Solutions will be posted.
Closed book: 3 pages of notes (8 12
Conflict? Contact before Friday, Nov. 11, 5pm.
2
Review: DT Frequency Response
The frequency response of a DT LTI system is the value of the
system function evaluated on the unit circle.
cos(Ωn)
H(z)
|H(ejΩ )| cos Ωn + ∠H(ejΩ )
H(e jΩ ) = H(z)|z=e jΩ
3
Comparision of CT and DT Frequency Responses
CT frequency response: H(s) on the imaginary axis, i.e., s = jω.
DT frequency response: H(z) on the unit circle, i.e., z = e jΩ .
ω
s-plane
z -plane
σ
H(ejΩ )
|H(jω)|
1
0
ω
−π
4
0
πΩ
Check Yourself
A system H(z) =
1 − az
has the following pole-zero diagram.
z−a
z -plane
Classify this system as one of the following filter types.
1. high pass
3. band pass
5. band stop
2. low pass
4. all pass
0. none of the above
5
Check Yourself
Classify the system ...
1 − az
H(z) =
z−a
Find the frequency response:
1 − ae jΩ
e−jΩ − a
= e jΩ jΩ
H(e jΩ ) = jΩ
e −a
e −a
← complex
← conjugates
jΩ ) = 1.
Because complex conjugates have equal magnitudes, H(e
→ all-pass filter
6
Check Yourself
A system H(z) =
1 − az
has the following pole-zero diagram.
z−a
z -plane
Classify this system as one of the following filter types.
1. high pass
3. band pass
5. band stop
2. low pass
4. all pass
0. none of the above
7
4
Effects of Phase
8
Effects of Phase
9
Effects of Phase
http://public.research.att.com/~ttsweb/tts/demo.php
10
Effects of Phase
artificial speech synthesized by Robert Donovan
11
Effects of Phase
x[n]
y[n] = x[−n]
???
artificial speech synthesized by Robert Donovan
12
Effects of Phase
x[n]
y[n] = x[−n]
???
How are the phases of X and Y related?
13
Effects of Phase
How are the phases of X and Y related?
ak =
x[n]e
−jkΩ0 n
n
x[−n]e−jkΩ0 n =
bk =
n
x[m]e jkΩ0 m = a−k
m
Flipping x[n] about n = 0 flips ak about k = 0.
Because x[n] is real-valued, ak is conjugate symmetric: a−k = a∗k .
bk = a−k = a∗k = |ak |e−j∠ak
The angles are negated at all frequencies.
14
Review: Periodicity
DT frequency responses are periodic functions of Ω, with period 2π.
If Ω2 = Ω1 + 2πk where k is an integer then
H(e jΩ2 ) = H(e j(Ω1 +2πk) ) = H(e jΩ1 e j2πk ) = H(e jΩ1 )
The periodicity of H(e jΩ ) results because H(e jΩ ) is a function of e jΩ ,
which is itself periodic in Ω. Thus DT complex exponentials have
many “aliases.”
e jΩ2 = e j(Ω1 +2πk) = e jΩ1 e j2πk = e jΩ1
Because of this aliasing, there is a “highest” DT frequency: Ω = π.
15
Review: Periodic Sinusoids
There are (only) N distinct complex exponentials with period N .
(There were an infinite number in CT!)
If y[n] = e jΩn is periodic in N then
y[n] = e jΩn = y[n + N ] = e jΩ(n+N ) = e jΩn e jΩN
and e jΩN must be 1, and ejΩ must be one of the N th roots of 1.
Example: N = 8
z -plane
16
Review: DT Fourier Series
DT Fourier series represent DT signals in terms of the amplitudes
and phases of harmonic components.
DT Fourier Series
ak = ak+N =
1
N
x[n]= x[n + N ] =
X
x[n]e−jkΩ0 n ; Ω0 =
n=<N >
X
ak ejkΩ0 n
2π
N
(“analysis” equation)
(“synthesis” equation)
k=<N >
17
DT Fourier Series
DT Fourier series have simple matrix interpretations.
x[n] = x[n + 4] =
X
ak ejkΩ0 n =
k=<4>
⎡
⎤
⎡
x[0]
1
⎢ x[1]
⎥ ⎢ 1
⎢
⎥ ⎢
⎢
⎥ =
⎢
⎣
x[2]
⎦
⎣
1
x[3]
1
1
ak = ak+4 =
4
⎡
⎤
⎡
a0
1
⎢ a ⎥ 1
⎢ 1
⎢
⎢ 1⎥
⎢ ⎥= ⎢
⎣
a2 ⎦
4 ⎣
1
1
a3
X
ak ejk
k=<4>
⎤⎡
2π n
4
=
X
ak j kn
k=<4>
⎤
1
1
1
a0
⎥
⎢
j
−1 −j ⎥ ⎢ a1 ⎥
⎥
⎥⎢ ⎥
−1
1
−1 ⎦ ⎣ a2 ⎦
a3
−j −1
j
X
1
X −jk 2π n 1
X
x[n]e−
e
N =
x[n]j −kn
jkΩ0 n =
4
4
n=<4>
1
−j
−1
j
n=<4>
1
−1
1
−1
⎤⎡
⎤
1
x[0]
⎢
⎥
j
⎥
⎥ ⎢ x[1]
⎥
⎥⎢
⎥
−1
⎦ ⎣ x[2]
⎦
−j
x[3]
These matrices are inverses of each other.
18
n=<4>
Scaling
DT Fourier series are important computational tools.
However, the DT Fourier series do not scale well with the length N.
ak = ak+2 =
1 X
1 X −jk 2π n 1 X
2
x[n]e−jkΩ0 n =
e
=
x[n](−1)−kn
2
2
2
n=<2>
a0
1
1
=
2 1
a1
1
−1
n=<2>
x[0]
x[1]
n=<2>
1 X
1 X −jk 2π n 1 X
4
ak = ak+4 =
x[n]e−jkΩ0 n =
e
=
x[n]j −kn
4
4
4
n=<4>
⎡
⎤
⎡
1
a0
⎢ a ⎥ 1
⎢ 1
⎢ 1⎥
⎢
⎢ ⎥= ⎢
⎣
a2 ⎦
4 ⎣
1
a3
1
1
−j
−1
j
n=<4>
1
−1
1
−1
⎤⎡
⎤
1
x[0]
⎢
⎥
j
⎥
⎥ ⎢ x[1]
⎥
⎥⎢
⎥
−1
⎦ ⎣ x[2]
⎦
−j
x[3]
Number of multiples increases as N 2 .
19
n=<4>
Fast Fourier “Transform”
Exploit structure of Fourier series to simplify its calculation.
Divide FS of length 2N into two of length N (divide and conquer).
Matrix formulation of 8-point FS:
⎡
⎤
⎡
0
c0
W
8
⎢ c1 ⎥ ⎢ W
0
⎢ ⎥ ⎢ 8
⎢ c ⎥ ⎢ W
0
⎢ 2⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ c3 ⎥ ⎢ W
8
⎢ ⎥ =
⎢
⎢ c ⎥ ⎢ W
0
⎢ 4⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ c5 ⎥ ⎢ W
8
⎢ ⎥ ⎢
⎣
c6 ⎦
⎣
W
0
8
c7
W
80
W
80
W
8
1
W
8
2
W
8
3
W
8
4
W
8
5
W
8
6
W
8
7
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
3
W
8
6
W
8
1
W
8
4
W
8
7
W
8
2
W
8
5
W
80
W
8
4
W
80
W
8
4
W
80
W
8
4
W
80
W
8
4
2π
where WN = e−j N
8 × 8 = 64 multiplications
20
W
80
W
8
5
W
8
2
W
8
7
W
8
4
W
8
1
W
8
6
W
8
3
W
80
W
8
6
W
8
4
W
8
2
W
80
W
8
6
W
8
4
W
8
2
⎤⎡
⎤
W
80
x[0]
⎢
⎥
W
8
7 ⎥
⎥ ⎢ x[1]
⎥
⎥
⎢
6
W
8
⎥ ⎢ x[2]
⎥
⎥
⎥⎢
⎥
⎢ x[3]
⎥
W
8
5 ⎥
⎥⎢
⎥
⎢
⎥
W
8
4 ⎥
⎥ ⎢ x[4]
⎥
⎥⎢
⎥
W
8
3 ⎥ ⎢ x[5]
⎥
⎥⎢
⎥
W
2 ⎦ ⎣ x[6]
⎦
8
W
8
1
x[7]
FFT
Divide into two 4-point series (divide and conquer).
Even-numbered entries
⎡ ⎤ ⎡ 0
W4 W
40
a0
⎢ a ⎥ ⎢ W
0 W
1
⎢ 1⎥ ⎢ 4
4
⎢ ⎥ =
⎢ 0
⎣
a2 ⎦
⎣
W
4 W
42
a3
W
40 W
43
in x[n]:
Odd-numbered entries
⎡ ⎤ ⎡ 0
b0
W4 W
40
⎢ b ⎥ ⎢ W
0 W
1
⎢ 1⎥ ⎢ 4
4
⎢ ⎥ =
⎢ 0
⎣
b2 ⎦
⎣
W
4 W
42
b3
W40 W
43
in x[n]:
W
40
W
42
W
40
W
42
W
40
W
42
W
40
W
42
⎤⎡
⎤
W
40
x[0]
⎢
⎥
W
43 ⎥
⎥ ⎢ x[2]
⎥
⎥
⎢
⎥
W
42 ⎦ ⎣ x[4]
⎦
W
41
x[6]
⎤⎡
⎤
W
40
x[1]
⎢
⎥
W
43 ⎥
⎥ ⎢ x[3]
⎥
⎥⎢
⎥
W
42 ⎦ ⎣ x[5]
⎦
W
41
x[7]
Sum of multiplications = 2 × (4 × 4) = 32: fewer than the previous 64.
21
FFT
Break the original 8-point DTFS coefficients ck into two parts:
ck = dk + ek
where dk comes from the even-numbered x[n] (e.g., ak ) and ek comes
from the odd-numbered x[n] (e.g., bk )
22
FFT
The 4-point DTFS coefficients ak of the even-numbered
⎡ ⎤ ⎡ 0
⎤⎡
⎤ ⎡ 0
a0
W
4 W
40 W
40 W
40
W
8 W
80 W
80
x[0]
⎢ a ⎥ ⎢ W
0 W
1 W
2 W
3 ⎥⎢ x[2]
⎥ ⎢ W
0 W
2 W
4
⎥ ⎢ 8
⎢ 1⎥ ⎢ 4
4
4
4
⎥⎢
8
8
⎥=⎢
⎢ ⎥=⎢ 0
⎥⎢
⎣
a2 ⎦
⎣
W
4 W
4
2 W
40 W
4
2 ⎦⎣ x[4]
⎦ ⎣ W
80 W
8
4 W
80
a3
W
40 W
4
3 W
4
2 W
4
1
x[6]
W
80 W
8
6 W
8
4
x[n]
⎤⎡
⎤
W
80
x[0]
⎢
⎥
W
8
6 ⎥
⎥⎢ x[2]
⎥
⎥⎢
⎥
W
8
4 ⎦⎣ x[4]
⎦
W
8
2
x[6]
contribute to the 8-point DTFS coefficients dk :
⎡
⎤
⎡
0
d0
W
8
⎢ d1 ⎥ ⎢ W
0
⎢ ⎥ ⎢ 8
⎢ d ⎥ ⎢ W
0
⎢ 2⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ d3 ⎥ ⎢ W
8
⎢ ⎥ =
⎢
⎢ d ⎥ ⎢ W
0
⎢ 4⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ d5 ⎥ ⎢ W
8
⎢ ⎥ ⎢
⎣
d6 ⎦
⎣
W
0
8
d7
W80
W
80
W
8
1
W
8
2
W
8
3
W
8
4
W
8
5
W
8
6
W
8
7
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
3
W
8
6
W
8
1
W
8
4
W
8
7
W
8
2
W
8
5
W
80
W
8
4
W
80
W
8
4
W
80
W
8
4
W
80
W
8
4
23
W
80
W
8
5
W
8
2
W
8
7
W
8
4
W
8
1
W
8
6
W
8
3
W
80
W
8
6
W
8
4
W
8
2
W
80
W
8
6
W
8
4
W
8
2
⎤
⎤
⎡
x[0]
W
80
⎢
⎥
W
8
7 ⎥
⎥ ⎢ x[1]
⎥
⎥
⎢
6
W
8
⎥ ⎢ x[2]
⎥
⎥
⎥⎢
⎥
⎢ x[3]
⎥
W
8
5 ⎥
⎥⎢
⎥
⎢ x[4]
⎥
W
8
4 ⎥
⎥⎢
⎥
⎥⎢
⎥
W
8
3 ⎥ ⎢ x[5]
⎥
⎥⎢
⎥
W
8
2 ⎦
⎣
x[6]
⎦
x[7]
W
81
FFT
The 4-point DTFS coefficients ak of the even-numbered
⎡ ⎤ ⎡ 0
⎤⎡
⎤ ⎡ 0
a0
W
4 W
40 W
40 W
40
W
8 W
80 W
80
x[0]
⎢ a ⎥ ⎢ W
0 W
1 W
2 W
3 ⎥⎢ x[2]
⎥ ⎢ W
0 W
2 W
4
⎥ ⎢ 8
⎢ 1⎥ ⎢ 4
4
4
4 ⎥⎢
8
8
⎥=⎢
⎢ ⎥=⎢ 0
⎥⎢
⎣
a2 ⎦
⎣
W
4 W
42 W
40 W
42 ⎦⎣ x[4]
⎦ ⎣ W
80 W
84 W
80
a3
W
40 W
43 W
42 W
41
x[6]
W
80 W
86 W
84
x[n]
⎤⎡
⎤
W
80
x[0]
⎢
⎥
W
86 ⎥
⎥⎢ x[2]
⎥
⎥⎢
⎥
W
84 ⎦⎣ x[4]
⎦
W
82
x[6]
contribute to the 8-point DTFS coefficients dk :
⎡
⎤
⎡
0
d0
W
8
⎢ d1 ⎥ ⎢ W
0
⎢ ⎥ ⎢ 8
⎢ d ⎥ ⎢ W
0
⎢ 2⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ d3 ⎥ ⎢ W
8
⎢ ⎥ =
⎢
⎢ d ⎥ ⎢ W
0
⎢ 4⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ d5 ⎥ ⎢ W
8
⎢ ⎥ ⎢
⎣
d6 ⎦
⎣
W
0
8
W80
d7
W
80
W
82
W
84
W
86
W
80
W
82
W
84
W
86
W
80
W
84
W
80
W
84
W
80
W
84
W
80
W
84
24
W
80
W
86
W
84
W
82
W
80
W
86
W
84
W
82
⎤
⎤
⎡
x[0]
⎥
⎥⎢
⎥
⎥⎢
⎥ ⎢ x[2]
⎥
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥ ⎢ x[4]
⎥
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎦
⎣
x[6]
⎦
FFT
The 4-point DTFS coefficients ak of the even-numbered
⎡ ⎤ ⎡ 0
⎤⎡
⎤ ⎡ 0
a0
W
4 W
40 W
40 W
40
W
8 W
80 W
80
x[0]
⎢ a ⎥ ⎢ W
0 W
1 W
2 W
3 ⎥⎢ x[2]
⎥ ⎢ W
0 W
2 W
4
⎥ ⎢ 8
⎢ 1⎥ ⎢ 4
4
4
4 ⎥⎢
8
8
⎥=⎢
⎢ ⎥=⎢ 0
⎥⎢
⎣
a2 ⎦
⎣
W
4 W
42 W
40 W
42 ⎦⎣ x[4]
⎦ ⎣ W
80 W
84 W
80
a3
W
40 W
43 W
42 W
41
x[6]
W
80 W
86 W
84
x[n]
⎤⎡
⎤
W
80
x[0]
⎢
⎥
W
86 ⎥
⎥⎢ x[2]
⎥
⎥⎢
⎥
W
84 ⎦⎣ x[4]
⎦
W
82
x[6]
contribute to the 8-point DTFS coefficients dk :
⎡
⎤
⎡
⎤
⎡
0
d0
a0
W
8
⎢ d1 ⎥ ⎢ a1 ⎥ ⎢ W
0
⎢ ⎥ ⎢ ⎥ ⎢ 8
⎢ d ⎥ ⎢ a ⎥ ⎢ W
0
⎢ 2⎥ ⎢ 2⎥ ⎢ 8
⎢ ⎥ ⎢ ⎥ ⎢ 0
⎢ d3 ⎥ ⎢ a3 ⎥ ⎢ W
8
⎢ ⎥ =
⎢ ⎥ =
⎢
⎢ d ⎥ ⎢ a ⎥ ⎢ W
0
⎢ 4⎥ ⎢ 0⎥ ⎢ 8
⎢ ⎥ ⎢ ⎥ ⎢ 0
⎢ d5 ⎥ ⎢ a1 ⎥ ⎢ W
8
⎢ ⎥ ⎢ ⎥ ⎢
⎣
d6 ⎦
⎣
a2 ⎦
⎣
W
0
8
W80
a3
d7
W
80
W
82
W
84
W
86
W
80
W
82
W
84
W
86
W
80
W
84
W
80
W
84
W
80
W
84
W
80
W
84
25
W
80
W
86
W
84
W
82
W
80
W
86
W
84
W
82
⎤
⎤
⎡
x[0]
⎥
⎥⎢
⎥
⎥⎢
⎥ ⎢ x[2]
⎥
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥ ⎢ x[4]
⎥
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎦
⎣
x[6]
⎦
FFT
The 4-point DTFS coefficients ak of the even-numbered
⎡ ⎤ ⎡ 0
⎤⎡
⎤ ⎡ 0
a0
W
4 W
40 W
40 W
40
W
8 W
80 W
80
x[0]
⎢ a ⎥ ⎢ W
0 W
1 W
2 W
3 ⎥⎢ x[2]
⎥ ⎢ W
0 W
2 W
4
⎥ ⎢ 8
⎢ 1⎥ ⎢ 4
4
4
4 ⎥⎢
8
8
⎥=⎢
⎢ ⎥=⎢ 0
⎥⎢
⎣
a2 ⎦
⎣
W
4 W
42 W
40 W
42 ⎦⎣ x[4]
⎦ ⎣ W
80 W
84 W
80
a3
W
40 W
43 W
42 W
41
x[6]
W
80 W
86 W
84
x[n]
⎤⎡
⎤
W
80
x[0]
⎢
⎥
W
86 ⎥
⎥⎢ x[2]
⎥
⎥⎢
⎥
W
84 ⎦⎣ x[4]
⎦
W
82
x[6]
contribute to the 8-point DTFS coefficients dk :
⎤ ⎡ ⎤ ⎡
0
W
8
a0
d0
⎢ d1 ⎥ ⎢ a1 ⎥ ⎢ W
0
⎢ ⎥ ⎢ ⎥ ⎢ 8
⎢ d ⎥ ⎢ a ⎥ ⎢ W
0
⎢ 2⎥ ⎢ 2⎥ ⎢ 8
⎢ ⎥ ⎢ ⎥ ⎢ 0
⎢ d3 ⎥ ⎢ a3 ⎥ ⎢ W
8
⎢ ⎥ = ⎢ ⎥ =
⎢
⎢ d ⎥ ⎢ a ⎥ ⎢ W
0
⎢ 4⎥ ⎢ 0⎥ ⎢ 8
⎢ ⎥ ⎢ ⎥ ⎢ 0
⎢ d5 ⎥ ⎢ a1 ⎥ ⎢ W
8
⎢ ⎥ ⎢ ⎥ ⎢
⎣ d6 ⎦ ⎣ a2 ⎦ ⎣
W
0
8
W
80
a3
d7
⎡
W
80
W
82
W
84
W
86
W
80
W
82
W
84
W
86
W
80
W
84
W
80
W
84
W
80
W
84
W
80
W
84
26
W
80
W
86
W
84
W
82
W
80
W
86
W
84
W
82
⎤
⎤
⎡
x[0]
⎥
⎥⎢
⎥
⎥⎢
⎥ ⎢ x[2]
⎥
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥ ⎢ x[4]
⎥
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎥
⎥⎢
⎦
⎣
x[6]
⎦
FFT
The ek components result
⎡ ⎤ ⎡ 0
b0
W4 W
40 W
40
⎢ b ⎥ ⎢ W
0 W
1 W
2
⎢ 1⎥ ⎢ 4
4
4
⎢ ⎥=⎢ 0
⎣
b2 ⎦ ⎣ W
4 W
4
2 W
40
b3
W40 W
4
3 W
4
2
⎡
⎤
⎡
0
e0
W
8
⎢ e1 ⎥ ⎢ W
0
⎢ ⎥ ⎢ 8
⎢ e ⎥ ⎢ W
0
⎢ 2⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ e3 ⎥ ⎢ W
8
⎢ ⎥ =
⎢
⎢ e ⎥ ⎢ W
0
⎢ 4⎥ ⎢ 8
⎢ ⎥ ⎢ 0
⎢ e5 ⎥ ⎢ W
8
⎢ ⎥ ⎢
⎣
e6 ⎦
⎣
W
0
8
W
80
e7
W
80
W
8
1
W
8
2
W
8
3
W
8
4
W
8
5
W
8
6
W
8
7
from the odd-number entries in x[n].
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
2
W
8
4
W
8
6
⎤⎡
⎤ ⎡ 0
x[1]
W
40
W
8
⎢
⎥
⎥
⎢
3
W
4
⎥⎢ x[3]
⎥ ⎢ W
80
⎥=⎢
⎥⎢
W
4
2 ⎦⎣ x[5]
⎦ ⎣ W
80
W
4
1
x[7]
W
80
W
80
W
8
3
W
8
6
W
8
1
W
8
4
W
8
7
W
8
2
W
8
5
W
80
W
8
4
W
80
W
8
4
W
80
W
8
4
W
80
W
8
4
27
W
80
W
8
5
W
8
2
W
8
7
W
8
4
W
8
1
W
8
6
W
8
3
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
6
W
8
4
W
8
2
W
80
W
8
6
W
8
4
W
8
2
W
80
W
8
4
W
80
W
8
4
⎤⎡
⎤
x[1]
W
80
⎢
⎥
W
8
6 ⎥
⎥⎢ x[3]
⎥
⎥⎢
⎥
W
8
4 ⎦⎣ x[5]
⎦
W
8
2
x[7]
⎤
⎡
⎤
W
80
x[0]
⎢
⎥
W
8
7 ⎥
⎥ ⎢ x[1]
⎥
⎢
⎥
W
8
6 ⎥
⎥ ⎢ x[2]
⎥
⎥
⎢
⎥
⎢
⎥
W
8
5 ⎥
⎥ ⎢ x[3]
⎥
⎥
⎢
4
W
8
⎥ ⎢ x[4]
⎥
⎥
⎥⎢
⎥
W
8
3 ⎥ ⎢ x[5]
⎥
⎥⎢
⎥
W
8
2 ⎦
⎣
x[6]
⎦
W
81
x[7]
FFT
The ek components result
⎡ ⎤ ⎡ 0
b0
W4 W
40 W
40
⎢ b ⎥ ⎢ W
0 W
1 W
2
⎢ 1⎥ ⎢ 4
4
4
⎢ ⎥=⎢ 0
⎣
b2 ⎦ ⎣ W
4 W
4
2 W
40
b3
W40 W
4
3 W
4
2
⎡
⎤
⎡
e0
⎢ e1 ⎥ ⎢
⎢ ⎥ ⎢
⎢e ⎥ ⎢
⎢ 2⎥ ⎢
⎢ ⎥ ⎢
⎢ e3 ⎥ ⎢
⎢ ⎥ =
⎢
⎢e ⎥ ⎢
⎢ 4⎥ ⎢
⎢ ⎥ ⎢
⎢ e5 ⎥ ⎢
⎢ ⎥ ⎢
⎣
e6 ⎦
⎣
e7
W
80
W
8
1
W
8
2
W
8
3
W
8
4
W
8
5
W
8
6
W
8
7
from the odd-number entries in x[n].
⎤⎡
⎤ ⎡ 0
x[1]
W
40
W
8
⎢
⎥
⎥
⎢
3
W
4
⎥⎢ x[3]
⎥ ⎢ W
80
⎥=⎢
⎥⎢
W
4
2 ⎦⎣ x[5]
⎦ ⎣ W
80
W
4
1
x[7]
W
80
W
80
W
8
3
W
8
6
W
8
1
W
8
4
W
8
7
W
8
2
W
8
5
W
80
W
8
5
W
8
2
W
8
7
W
8
4
W
8
1
W
8
6
W
8
3
28
W
80
W
8
2
W
8
4
W
8
6
W
80
W
8
4
W
80
W
8
4
⎤⎡
⎤
x[1]
W
80
⎢
⎥
W
8
6 ⎥
⎥⎢ x[3]
⎥
⎥⎢
⎥
W
8
4 ⎦⎣ x[5]
⎦
W
8
2
x[7]
⎤⎡
⎤
W
80
⎢
⎥
W
8
7 ⎥
⎥ ⎢ x[1]
⎥
⎢
⎥
W
8
6 ⎥
⎥
⎥⎢
⎥
⎥
⎢
5
⎢
⎥
W
8
⎥
⎥ ⎢ x[3]
⎥
⎥
⎥
⎢
4
W
8
⎥ ⎢
⎥
⎥
⎢
⎥
W
8
3 ⎥ ⎢ x[5]
⎥
⎥
⎥⎢
⎦
W
8
2 ⎦ ⎣
W
81
x[7]
FFT
The ek components result
⎡ ⎤ ⎡ 0
b0
W4 W
40 W
40
⎢ b ⎥ ⎢ W
0 W
1 W
2
⎢ 1⎥ ⎢ 4
4
4
⎢ ⎥=⎢ 0
⎣
b2 ⎦ ⎣ W
4 W
42 W
40
b3
W40 W
43 W
42
⎡
⎤
⎡
0 ⎤
⎡
W
8 b0
e0
⎢ e1 ⎥ ⎢ W
1 b1 ⎥ ⎢
⎢ ⎥ ⎢ 8 ⎥ ⎢
⎢ e ⎥ ⎢ W
2 b ⎥ ⎢
⎢ 2⎥ ⎢ 8 2⎥ ⎢
⎢ ⎥ ⎢ 3 ⎥ ⎢
⎢ e3 ⎥ ⎢ W
8 b3 ⎥ ⎢
⎢ ⎥ =
⎢
⎥ ⎢
⎢ e ⎥ ⎢ W
4 b ⎥ =
⎢
4
0
⎢ ⎥ ⎢ 8 ⎥ ⎢
⎢ ⎥ ⎢ 5 ⎥ ⎢
⎢ e5 ⎥ ⎢ W
8 b1 ⎥ ⎢
⎢ ⎥ ⎢
⎥ ⎢
⎣
e6 ⎦
⎣
W
6 b2 ⎦
⎣
8
e7
W
87 b3
from the odd-number entries in x[n].
⎤⎡
⎤ ⎡ 0
x[1]
W
40
W
8
⎢
⎥
⎥
⎢
3
W
4 ⎥⎢ x[3]
⎥ ⎢ W
80
⎥=⎢
⎥⎢
W
42 ⎦⎣ x[5]
⎦ ⎣ W
80
W
41
x[7]
W
80
W
80
W
81
W
82
W
83
W
84
W
85
W
86
W87
W
80
W
83
W
86
W
81
W
84
W
87
W
82
W
85
29
W
80
W
82
W
84
W
86
W
80
W
85
W
82
W
87
W
84
W
81
W
86
W
83
W
80
W
84
W
80
W
84
⎤⎡
⎤
x[1]
W
80
⎢
⎥
W
86 ⎥
⎥⎢ x[3]
⎥
⎥⎢
⎥
W
84 ⎦⎣ x[5]
⎦
W
82
x[7]
⎤⎡
⎤
W
80
⎥
⎢
W
87 ⎥
⎥ ⎢ x[1]
⎥
⎥
⎢
W
86 ⎥
⎥
⎥⎢
⎥
⎥
⎢
5
⎥
⎢
W
8 ⎥
⎥ ⎢ x[3]
⎥
⎥
⎥
⎢
4
W
8 ⎥ ⎢
⎥
⎥
⎥
⎢
W
83 ⎥ ⎢ x[5]
⎥
⎥
⎥⎢
⎦
W
82 ⎦ ⎣
W
81
x[7]
FFT
The ek components result
⎡ ⎤ ⎡ 0
b0
W4 W
40 W
40
⎢ b ⎥ ⎢ W
0 W
1 W
2
⎢ 1⎥ ⎢ 4
4
4
⎢ ⎥=⎢ 0
⎣
b2 ⎦ ⎣ W
4 W
42 W
40
b3
W40 W
43 W
42
⎤ ⎡ 0 ⎤ ⎡
W8 b0
e0
⎢ e1 ⎥ ⎢ W 1 b1 ⎥ ⎢
⎢ ⎥ ⎢ 8 ⎥ ⎢
⎢e ⎥ ⎢W2 b ⎥ ⎢
⎢ 2⎥ ⎢ 8 2⎥ ⎢
⎢ ⎥ ⎢ 3 ⎥ ⎢
⎢ e3 ⎥ ⎢ W8 b3 ⎥ ⎢
⎢ ⎥=⎢
⎥ ⎢
⎢ e ⎥ ⎢ W 4 b ⎥ =
⎢
4
0
⎢ ⎥ ⎢ 8 ⎥ ⎢
⎢ ⎥ ⎢ 5 ⎥ ⎢
⎢ e5 ⎥ ⎢ W8 b1 ⎥ ⎢
⎢ ⎥ ⎢
⎥ ⎢
⎣ e6 ⎦ ⎣ W 6 b2 ⎦ ⎣
8
e7
W87 b3
⎡
from the odd-number entries in x[n].
⎤⎡
⎤ ⎡ 0
x[1]
W
40
W
8
⎢
⎥
⎥
⎢
3
W
4 ⎥⎢ x[3]
⎥ ⎢ W
80
⎥=⎢
⎥⎢
W
42 ⎦⎣ x[5]
⎦ ⎣ W
80
W
41
x[7]
W
80
W
80
W
81
W
82
W
83
W
84
W
85
W
86
W
87
W
80
W
83
W
86
W
81
W
84
W
87
W
82
W
85
30
W
80
W
82
W
84
W
86
W
80
W
85
W
82
W
87
W
84
W
81
W
86
W
83
W
80
W
84
W
80
W
84
⎤⎡
⎤
x[1]
W
80
⎢
⎥
W
86 ⎥
⎥⎢ x[3]
⎥
⎥⎢
⎥
W
84 ⎦⎣ x[5]
⎦
W
82
x[7]
⎤⎡
⎤
W
80
⎥
⎢
W
87 ⎥
⎥ ⎢ x[1]
⎥
⎥
⎢
W
86 ⎥
⎥
⎥⎢
⎥
⎥
⎢
5
⎥
⎢
W
8 ⎥
⎥ ⎢ x[3]
⎥
⎥
⎥
⎢
4
W
8 ⎥ ⎢
⎥
⎥
⎥
⎢
W
83 ⎥ ⎢ x[5]
⎥
⎥
⎥⎢
⎦
W
82 ⎦ ⎣
W
81
x[7]
FFT
Combine ak and bk to get ck .
⎡
⎤
⎡
⎤
⎡
⎤
⎡
0 ⎤
a0
d0 + e0
c0
W8 b0
⎢ c1 ⎥ ⎢ d1 + e1 ⎥ ⎢ a1 ⎥ ⎢ W 1 b1 ⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢ 8 ⎥
⎢c ⎥ ⎢d + e ⎥ ⎢a ⎥ ⎢W2 b ⎥
⎢ 2⎥ ⎢ 2
⎢ 2⎥ ⎢
2⎥
2⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢ 8 ⎥
⎢ c3 ⎥ ⎢ d3 + e3 ⎥ ⎢ a3 ⎥ ⎢ W83 b3 ⎥
⎢ ⎥ =
⎢
⎥ ⎢ ⎥ ⎢
⎥
⎢ c ⎥ ⎢ d + e ⎥ =
⎢ a ⎥ +
⎢ W 4 b ⎥
4⎥
⎢ 4⎥ ⎢ 4
⎢ 0⎥ ⎢ 8 0⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎥
⎢ c5 ⎥ ⎢ d5 + e5 ⎥ ⎢ a1 ⎥ ⎢ W85 b1 ⎥
⎢ ⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎥
⎣
c6 ⎦
⎣
d6 + e6 ⎦
⎣
a2 ⎦
⎣
W 6 b2 ⎦
8
c7
d7 + e7
a3
W87 b3
FFT procedure:
• compute ak and bk : 2 × (4 × 4) = 32 multiplies
• combine ck = ak + W8k bk : 8 multiples
• total 40 multiplies: fewer than the orginal 8 × 8 = 64 multiplies
31
Scaling of FFT algorithm
How does the new algorithm scale?
Let M (N ) = number of multiplies to perform an N point FFT.
M (1) = 0
M (2) = 2M (1) + 2 = 2
M (4) = 2M (2) + 4 = 2 × 4
M (8) = 2M (4) + 8 = 3 × 8
M (16) = 2M (8) + 16 = 4 × 16
M (32) = 2M (16) + 32 = 5 × 32
M (64) = 2M (32) + 64 = 6 × 64
M (128) = 2M (64) + 128 = 7 × 128
. . .
M (N ) = (log2 N ) × N
Significantly smaller than N 2 for N large.
32
Fourier Transform: Generalize to Aperiodic Signals
An aperiodic signal can be thought of as periodic with infinite period.
Let x[n] represent an aperiodic signal DT signal.
x[n]
1
n
0
“Periodic extension”: xN [n] =
∞
X
x[n + kN ]
k=−∞
xN [n]
1
n
N
Then x[n] = lim xN [n].
N →∞
33
Fourier Transform
Represent xN [n] by its Fourier series.
xN [n]
1
n
−N1 N1
ak =
N
2π
1 X
1 X1
xN [n]e−j N kn =
N
N
N
N
2π
e−j N kn =
n=−N1
1 sin N1 + 12 Ω
N
sin 12 Ω
sin 23 Ω
N ak
sin 21 Ω
k
Ω
Ω0 =
34
2π
N
Ω = kΩ0 = k
2π
N
Fourier Transform
Doubling period doubles # of harmonics in given frequency interval.
xN [n]
1
n
−N1 N1
2π
1 X
1
ak =
xN [n]e−j N kn =
N
N
N
N
N1
X
e
n=−N1
−j 2π
N kn
1 sin N1 + 12 Ω
=
N
sin 12 Ω
sin 23 Ω
N ak
sin 21 Ω
k
Ω
Ω0 =
35
2π
N
Ω = kΩ0 = k
2π
N
Fourier Transform
As N → ∞, discrete harmonic amplitudes → a continuum E(Ω).
xN [n]
1
n
−N1 N1
2π
1 X
1
ak =
xN [n]e−j N kn =
N
N
N
N
N1
X
e
n=−N1
−j 2π
N kn
1 sin N1 + 12 Ω
=
N
sin 12 Ω
sin 23 Ω
N ak
sin 21 Ω
k
Ω
Ω0 =
N ak =
X
n=<N >
2π
x[n]e−j N kn =
2π
N
X
n=<N >
36
Ω = kΩ0 = k
x[n]e−jΩn = E(Ω)
2π
N
Fourier Transform
As N → ∞, synthesis sum → integral.
xN [n]
1
n
−N1 N1
N
sin 23 Ω
N ak
sin 21 Ω
k
Ω
Ω0 =
N ak =
2π
x[n]e−j N kn =
X
n=<N >
x[n] =
X
k=<N >
2π
N
X
n=<N >
2π
N
x[n]e−jΩn = E(Ω)
1
Ω0
jΩn
E(Ω)e
→
E(Ω)e
jΩn dΩ
2π
2π
2π
k=<N >
X
2π
1
E(Ω) e
j N kn =
�N
ak
Ω = kΩ0 = k
37
Fourier Transform
Replacing E(Ω) by X(e jΩ ) yields the DT Fourier transform relations.
X(e jΩ )=
∞
X
x[n]e−jΩn
(“analysis” equation)
n=−∞
x[n]=
Z
1
X(e jΩ )e jΩn dΩ
2π 2π
(“synthesis” equation)
38
Relation between Fourier and Z Transforms
If the Z transform of a signal exists and if the ROC includes the
unit circle, then the Fourier transform is equal to the Z transform
evaluated on the unit circle.
Z transform:
∞
X
X(z) =
x[n]z −n
n=−∞
DT Fourier transform:
∞
X
X(e jΩ ) =
x[n]e−jΩn = H(z) z=e jΩ
n=−∞
39
Relation between Fourier and Z Transforms
Fourier transform “inherits” properties of Z transform.
Property
x[n]
X(z)
X(e jΩ )
Linearity
ax1 [n] + bx2 [n]
aX1 (s) + bX2 (s)
aX1 (e jΩ ) + bX2 (e jΩ )
Time shift
x[n − n0 ]
z −n0 X(z)
e−jΩn0 X(e jΩ )
Multiply by n
nx[n]
Convolution
(x1 ∗ x2 )[n]
−z
d
X(z)
dz
X1 (z) × X2 (z)
40
j
d
X(e jΩ )
dΩ
X1 (e jΩ ) × X2 (e jΩ )
DT Fourier Series of Images
Magnitude
Angle
41
DT Fourier Series of Images
Magnitude
Uniform Angle
42
DT Fourier Series of Images
Uniform
Magnitude
Angle
43
DT Fourier Series of Images
Different
Magnitude
Angle
44
DT Fourier Series of Images
Magnitude
Angle
45
DT Fourier Series of Images
Magnitude
Angle
46
DT Fourier Series of Images
Different
Magnitude
Angle
47
Fourier Representations: Summary
Thinking about signals by their frequency content and systems as
filters has a large number of practical applications.
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MIT OpenCourseWare
http://ocw.mit.edu
6.003 Signals and Systems
Fall 2011
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