Young Won Lim9/23/16
● CTFS: Continuous Fourier Series● CTFT: Continuous Time Fourier Transform● DTFT: Discrete Time Fourier Transform● DFT: Discrete Fourier Transform
Fourier Analysis Overview (0B)
Young Won Lim9/23/16
Copyright (c) 2009 - 2016 Young W. Lim.
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Fourier Analysis Overview (0B) 3 Young Won Lim
9/23/16
Fourier Analysis Methods
Discrete Frequency
CTFS CTFT
DTFS / DFT DTFT
C k X ( jω)
γk / X [k ] X ( j ω̂)
Continuous Frequency
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
Periodic in timeAperiodic in freq
Aperiodic in timeAperiodic in freq
Periodic in timePeriodic in freq
Aperiodic in timePeriodic in freq
Fourier Analysis Overview (0B) 4 Young Won Lim
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Time and Frequency Domains
Discrete Frequency
CTFS CTFT
DTFS / DFT DTFT
Continuous Frequency
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
+∞−∞+∞−∞
ω0 ω0
+π−π+π−π
ω̂0 ω̂0
+∞0+∞0
00
0 0
0 0
T s T s T s T s
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Time Domain
Discrete Frequency
CTFS CTFT
DTFS / DFT DTFT
Continuous Frequency
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
x (t )
x [n]
x (t )
x [n]
Singal Period
Sample Counts
Sample Period
T 0
N 0
T s
Singal Period
Sample Counts
Sample Period
T 0
N 0
T s
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Frequency Domain
Discrete Frequency
CTFS CTFT
DTFS / DFT DTFT
C k X ( jω)
γk / X [k ] X ( j ω̂)
Continuous Frequency
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
+∞−∞+∞−∞
ω0 ω0
ω̂0 ω̂0
Frequency Resolution
Frequency Resolution
No
rmal
ized
Fre
qu
ency
+π−π−π +π
Fourier Analysis Overview (0B) 7 Young Won Lim
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Discrete Time and Periodic Frequency
Discrete Frequency
DTFS / DFT DTFT
Continuous Frequency
Dis
cret
e T
ime
Periodic in discrete freq Periodic in continuous freq
Discrete Time Discrete Time
T s T sT s T s
ω̂sω̂s
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Periodic Time and Discrete Frequency
Discrete Frequency
CTFS
DTFS / DFT
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
+∞−∞
ω0 ω0
+π−π
ω̂0 ω̂0
Discrete Frequency
Discrete Frequency
Periodic in continuous time
Periodic in discrete time
T0
N0
Fourier Analysis Overview (0B) 9 Young Won Lim
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Discrete Time Resolution
Discrete Frequency
DTFS / DFT DTFT
Continuous Frequency
Dis
cret
e T
ime
Periodic in discrete freq Periodic in continuous freqDiscrete Time Discrete Time
ωs =2π
T sω̂s =
2π
1
T s T s T s T s
ωs : replication frequencyω̂s : normalized replication frequency
ωs =2π
T sω̂s =
2π
1
ωs : replication frequencyω̂s : normalized replication frequency
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Discrete Frequency Resolution
ω0 =2π
T 0
ω̂0 =2πN 0
period : T 0 seconds
period : N0 samples
Discrete Frequency
CTFS
DTFS / DFT
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
+∞−∞
ω0 ω0
+π−π
ω̂0 ω̂0
Discrete Frequency
Discrete Frequency
Periodic in continuous time
Periodic in discrete time
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Normalized Frequency
Discrete Frequency
DTFS / DFT DTFTγk / X [k ] X ( j ω̂)
Continuous Frequency
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
ω̂0 ω̂0
No
rmal
ized
Fre
qu
ency
+π−π−π +π
ω̂ = ω⋅T sω̂0 =2π
N0
ω̂s =2π
1= (2π
T s)T s
= (2π
T 0)T s
ω̂s =2π
1= (2π
T s)T s
continuous variable ω̂s
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Normalized by 1/Ts
ω0 =2π
N 0T s
ω0 =2π
T 0
ω0
1/T s
=2π
N0T s
T s
ω̂0 =2π
N0
Discrete Frequency
Co
nti
nu
ou
s T
ime
Dis
cret
e T
ime
normalized frequency resolution
frequency resolution
+∞−∞
+π−π
ω̂0 ω̂0
+∞0
0
0
0
T s T s
CTFS
DTFS / DFT
ω0 ω0
T 0 = N0T s
⋅T s ⋅ f s
Fourier Analysis Overview (0B) 13
CTFT pair of an impulse train
2π
T s
ωs 2ωs 3ωs−ωs−2ωs−3ωs
1
T s 2T s 3T s−T s−2T s−3T s
p (t) = ∑n=−∞
+∞
δ(t − nT s) P ( jω) = ∑k=−∞
∞
( 2π
T s ) δ(ω − kωs)
ωs =2π
T s
T s
2π
Ts
Sampling Replication
Fourier Analysis Overview (0B) 14
Sampling and Replicating
2πT s
ωs =2π
T s
T s
Sampling Replication
1
T 0
ω0 =2π
T 0
=2π
N1T 1
=2π
N 2T 2
Period Resolution
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Sampling Period and the Number of Samples
x (t )
T 0 period
N 1 samplesT 1
N 2 samplesT 2
(N 1 < N2)
(T 1 > T 2)sampling period : T 1
number of samples : N1
x [n]
x [n]
fundamental period T 0 = T 1 N1 = T 2 N2
sampling period : T 2
number of samples : N2 (N 1 < N2)
(T 1 > T 2)
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Periodic Relationship
sampling period T s :
no of samples N0 :
fundamental period T 0
T 1 > T 2
N 1 < N 2
T 0 = T 1N1 = T 2 N2
frequency resolution ω0
ω0 =2π
T 0
=2π
N1T 1
=2π
N 2T 2
replication period ω1 , ω2 :
ω1 =2π
T 1
ω2 =2π
T 2<
ω̂0 =2π
N1
ω̂0 =2π
N 2>
normalized frequency resolution ω̂0
ω̂0 = ω0T 1 ω̂0 = ω0T 2
Fourier Analysis Overview (0B) 17 Young Won Lim
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Sampling Period and Replication Period
ωs =2π
T s
ω1
2πT 1
2πT2
T 0 period
T 1
T 2
ω2
ω1
ω2
Fourier Analysis Overview (0B) 18 Young Won Lim
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Frequency Resolution
+ω2−ω2 0
2πT 1
2πT2
ω0 =2π
T 0
=2 π
N1T 1
ω0 =2π
T 0
=2π
N2T 2
the same signal period T0
ω1
ω2
+ω1−2ω10−ω1 +2ω1
the same frequency resolution ω0
the same signal period T0
the same frequency resolution ω0
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Replication Frequency
ω1 =2 π
T 1
+ω2−ω2 0
2πT 1
2πT2
ω2 =2π
T 2
ω1
ω2
+ω1−2ω10−ω1 +2ω1
large T1
small T2
small replication freq ω1
large replication freq ω2
small number of samples N 1
large number of samples N1
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Normalized Frequency for Comparison
2πT 1
2πT2
ω1
ω2
ω1 =2π
T 1
ω2 =2π
T 2
2π
2π
replication freq ω1
replication freq ω2
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Normalized Frequency Resolution
2πT 1
2πT2
ω0 =2π
N 1T 1
ω0 =2π
N 2T 2
ω1
ω2
ω̂0 =2π
N 1
ω̂0 =2π
N 2
ω̂0 = ω0T 1
ω̂0 = ω0T 2
coarse frequency resolution
fine frequency resolution
Fourier Analysis Overview (0B) 22 Young Won Lim
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Multiplication with an Impulse Train
1
x (t)
p (t)
x (t)⋅p(t) Multiplication with a dense impulse train
∑n=−∞
+∞
δ(t − nT s)
∑n=−∞
+∞
x(nT s) δ(t − nT s) x [n]
Fourier Analysis Overview (0B) 23 Young Won Lim
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Convolution with an Impulse Train
1
x (t)
p (t)
x (t)∗p(t) Multiplication with a sparse impulse train
1
Fourier Analysis Overview (0B) 24 Young Won Lim
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Convolution & Multiplication Properties
x (t )∗ y (t ) X ( jω)⋅Y ( jω)
x (t )⋅ y (t)12π
X ( jω)∗ Y ( jω)
x (t )∗ y (t ) X (f )⋅Y (f )
x (t )⋅ y (t) X (f )∗ Y ( f )
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Types of Fourier Transforms
Continuous Time Fourier Transform
Discrete Time Fourier Transform
Continuous Time
Discrete Time
Continuous Frequency
Continuous Frequency
CTFT
DTFT
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CTFT → DTFT
1
x (t)
p (t)
X ( jω)
P( jω)
x (t)⋅p(t) 12 π
X ( jω)∗ P( jω)Multiplication Convolution
A
AT s
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CTFT → CTFS
1
x (t)
p (t)
X ( jω)
P( jω)
x (t)⋅p(t) X ( jω)⋅P( jω)Convolution Multiplication
A
Young Won Lim9/23/16
References
[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] M.J. Roberts, Fundamentals of Signals and Systems[4] S.J. Orfanidis, Introduction to Signal Processing[5] K. Shin, et al., Fundamentals of Signal Processing for Sound and Vibration Engineerings