fourier analysis overview (0a)...2016/10/06 · fourier analysis overview (0a) 3 young won lim...

Young Won Lim10/6/16

● CTFS: Continuous Time Fourier Series● CTFT: Continuous Time Fourier Transform● DTFS: Discrete Time Fourier Series● DTFT: Discrete Time Fourier Transform● DFT: Discrete Fourier Transform

Fourier Analysis Overview (0A)


Copyright (c) 2011 -2016 Young W. Lim.

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Fourier Analysis Overview (0A) 3 Young Won Lim

10/6/16

CTFS Correlation Process

1 cycle

2 cycles

3 cycles

4 cycles

Measure the degree of correlation with these cosine and sine waves whose frequencies are the integer multiples of the fundamental frequency

x (t )

Cn =1T∫0

Tx t e− j n0 t dt x(t) = ∑

n=−∞

+ ∞

C n e+ j nω0 t


10/6/16

DTFS Correlation Process

1 cycle

2 cycles

3 cycles

4 cycles

Measure the degree of correlation with these cosine and sine waves whose frequencies are the integer multiples of the fundamental frequency

x [n]

γk =1N∑n=0

N−1

x [n] e− j (2π

N )k n

x [n] = ∑k=−M

+M

γk e+ j( 2π

N )k n

Fourier Analysis Overview (0A) 5

CTFS → CTFT

0ω0 1ω0 2ω0 3ω0 4 ω0 5ω0 6ω0 7ω0

ω0

k ω0

ω

T 0→∞

ω0→0

xT 0(t ) = ∑

k=−∞

+∞

C k e+ j ω0k t⋅1

= ∑k=−∞

+∞

Ck e+ jω0k t⋅( T 0

2 π )⋅( 2π

T 0)

=12π

∑k=−∞

+∞

C k T 0 e+ jω0k t⋅( 2π

T 0)

xT 0(t ) =

12π

∑k=−∞

+∞

C k T 0 e+ jω0k t⋅ω0

x (t) =1

2 π∫−∞

+∞

X ( jω) e+ jω t dω

dω

Fourier Analysis Overview (0A) 6

DTFS → DTFT

0 ω̂0 1 ω̂0 2 ω̂0 3 ω̂0 4 ω̂0 5 ω̂0 6 ω̂0 7 ω̂0

ω0

dω

k ω̂0

ω̂

N 0→∞

ω̂0→0

0 2π

xN 0[n] = ∑

k=0

N 0

γk e+ j( 2π

N 0)k n

⋅1

= ∑k=0

N 0

γk e+ j (2 π

N 0)k n

⋅( N0

2π )⋅( 2π

N 0)

=12π

∑k=0

N 0

γk N 0 e+ j( 2π

N 0)k n

⋅( 2π

N 0)

xN 0[n] =

12 π

∑k=0

N 0

γk N0 e+ j (2 π

N 0)k n

⋅(2π

N 0)

x [n] =12π

∫−π

+π

X (e j ω̂) e+ j ω̂n d ω̂

0 2π


10/6/16

CTFS & DTFS Correlation Processes

R RI

T

R RI

x(t )e

+ j(2π

T )k t

x [n] e+ j(2π

N )k n 1N∑n=0

N −1

x [n] e− j (2π

N )nk

= γk

1T∫0

Tx (t ) e− jk ω0 t dt = Ck

N samples


10/6/16

DTFS and DFT

Discrete Time Fourier Series DTFS

Discrete Fourier Transform

X [k ] = ∑n = 0

N − 1

x [n] e− j2 /N k n x [n] =1N ∑

k = 0

N − 1

X [k ] e+ j (2π /N )k n

γ[k ] =1N

∑n = 0

N − 1

x [n] e− j(2π/N )k n x [n] = ∑k = 0

N − 1

γ[k ] e+ j(2π/N )k n

DTFS (x [n]) =1N

DFT (x [n])

γ[n] =1N

X [n ]

ω̂0 = ( 2π

N )


10/6/16

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1

γ[k ] e+ j k ω̂0n

Fourier Transform Types

Continuous Time Fourier Series

C k =1T∫0

Tx (t ) e− j k ω0 t dt

Discrete Time Fourier Series

x (t ) =12π

∫−∞

+∞


Continuous Time Fourier Transform

x [n] =12π

∫−π

+π

X ( j ω̂) e+ j ω̂nd ω̂

Discrete Time Fourier Transform

X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n

x (t ) = ∑k=−∞

+∞

Ck e+ j kω0 t

X ( jω) = ∫−∞

+∞

x (t) e− jω t dt


10/6/16

Frequency Resolution


C k =1T∫0

Tx (t ) e− j k ω0 t dt x (t ) = ∑

k=−∞

+∞

Ck e+ j kω0 t

ω0 = ( 2π

T 0)

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1



ω̂0 = ( 2π

N 0)

Signal Period

Sample Counts

T 0

N 0




10/6/16

Frequency Variable Notations


x (t ) =12π

∫−∞

+∞


x [n] =12π

∫−π

+π



X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n

X ( jω) = ∫−∞

+∞

x (t) e− jω t dt

jω → e jω→ X (e j ω

) jω always appears as e jω

j ω̂ → e j ω̂→ X (e j ω̂

) j ω̂ always appears as e j ω̂


10/6/16

A. CTFS


Periodic Continuous Time Signal

Aperiodic Discrete Frequency Spectrum

C k =1T∫0

Tx (t ) e− j k ω0 t dt x (t ) = ∑

k=−∞

+∞

Ck e+ j kω0 t


10/6/16

B. DTFS


Periodic Discrete Time Signal

Periodic Discrete Frequency Spectrum

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1



10/6/16

C. CTFT


Aperiodic Continuous Time Signal

Aperiodic Discrete Frequency Spectrum

x (t ) =12π

∫−∞

+∞

X ( jω) e+ jω t dωX ( jω) = ∫−∞

+∞

x (t) e− jω t dt


10/6/16

D. DTFT


Aperiodic Discrete Time Signal

Periodic Continuous Frequency Spectrum

x [n] =12π

∫−π

+π

X ( j ω̂) e+ j ω̂nd ω̂X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n


10/6/16

CTFS & CTFT


C k =1T∫0


x (t ) =12π

∫−∞

+∞



x (t ) = ∑k=−∞

+∞

Ck e+ j kω0 t

X ( jω) = ∫−∞

+∞

x (t) e− jω t dt

Continuous Frequency

Discrete Frequency


10/6/16

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1


DTFS & DTFT


x [n] =12π

∫−π

+π



X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n




10/6/16

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1


CTFS & DTFS


C k =1T∫0



x (t ) = ∑k=−∞

+∞

Ck e+ j kω0 t

Continuous Time

Discrete Time


10/6/16

CTFT & DTFT

x (t ) =12π

∫−∞

+∞



x [n] =12π

∫−π

+π



X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n

X ( jω) = ∫−∞

+∞

x (t) e− jω t dt

Continuous Time

Discrete Time


10/6/16

Fourier Analysis Methods

Discrete Frequency

CTFS CTFT

DTFS / DFT DTFT

C k X ( jω)

γk / X [k ] X ( j ω̂)


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


10/6/16

Types of Fourier Transforms


Continuous Time Discrete Frequency




Continuous Time

Discrete Time

Discrete Time



Discrete Frequency

CTFS

CTFT

DTFT

DTFS / DFT


10/6/16

1. CTFS → CTFT




Continuous Time

Continuous Time Continuous Frequency

CTFS

CTFT

CTFS CTFTDTFS DTFT

Time Aperiodic Continuous Frequency


10/6/16

2. DTFS → DTFT


Discrete Time Continuous Frequency

DTFT


Discrete Time Discrete Frequency

DTFS / DFT

CTFS CTFTDTFS DTFT


T=∞


10/6/16

3. CTFS ← CTFT




Continuous Time


CTFS

CTFT

CTFS CTFTDTFS DTFT

T=∞

Frequency Sampling Time Periodic


10/6/16

4. DTFS ← DTFT



DTFT



DTFS / DFT

CTFS CTFTDTFS DTFT

T=∞



10/6/16

5. CTFT → DTFT



CTFT



DTFT

Time Sampling

CTFS CTFTDTFS DTFT

Frequency Periodic


10/6/16

6. CTFS → DTFS

CTFS CTFTDTFS DTFT


Continuous Time Discrete FrequencyContinuous Time

CTFS



DTFS / DFT

Time Sampling Frequency Periodic


10/6/16

7. CTFT ← DTFT



CTFT



DTFT

CTFS CTFTDTFS DTFT

Continuous Time Frequency Aperiodic


10/6/16

8. CTFS ← DTFS

CTFS CTFTDTFS DTFT



CTFS



DTFS / DFT



10/6/16

****

CTFS CTFTDTFS DTFT


CTFS CTFTDTFS DTFT


CTFS CTFTDTFS DTFT

CTFS CTFTDTFS DTFT




10/6/16

*****

Frequency SamplingTime Aperiodic

CTFS

CTFT

DTFT

DTFS / DFT

Frequency SamplingTime Aperiodic


10/6/16

*****

Frequency AperiodicTime Sampling

CTFS

CTFT

DTFT

DTFS / DFT

Making AperiodicTime Sampling Frequency Aperiodic


10/6/16

9 CTFS → DTFT



CTFS



DTFT

CTFS CTFTDTFS DTFT




10/6/16

10 CTFS ← DTFT



CTFS



DTFT

CTFS CTFTDTFS DTFT Continuous Time Frequency Aperiodic



10/6/16

11. CTFT → DTFS



DTFS / DFT



CTFT

CTFS CTFTDTFS DTFT

Frequency SamplingTime Periodic



10/6/16

12. CTFT ← DTFS



DTFS / DFT



CTFT

CTFS CTFTDTFS DTFT Continuous Time Frequency Aperiodic



10/6/16

CTFS & DTFT



CTFS



DTFT

Sampled in frequency

Sampled in time

Periodic in time

Periodic in frequency

T

CTFS CTFTDTFS DTFT


10/6/16

CTFT & DTFS



DTFS / DFT



CTFT

CTFS CTFTDTFS DTFT

Sampling in time Periodic in frequency

Sampled in frequencyPeriodic in time


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

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