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

Fourier Analysis Overview (0B) 5 Young Won Lim

9/23/16

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

Fourier Analysis Overview (0B) 6 Young Won Lim

9/23/16

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

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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

Fourier Analysis Overview (0B) 8 Young Won Lim

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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

9/23/16

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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9/23/16

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

Fourier Analysis Overview (0B) 11 Young Won Lim

9/23/16

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

Fourier Analysis Overview (0B) 12 Young Won Lim

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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

Fourier Analysis Overview (0B) 15 Young Won Lim

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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)

Fourier Analysis Overview (0B) 16 Young Won Lim

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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

9/23/16

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

9/23/16

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

Fourier Analysis Overview (0B) 19 Young Won Lim

9/23/16

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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9/23/16

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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9/23/16

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

9/23/16

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 )

Fourier Analysis Overview (0B) 25 Young Won Lim

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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