o Introduction o DTFS & Properties o FT of periodic signalso DFT & Properties: Sampling of the DTFTo DTFT, DTFS, DFT, DFS, FFT, ZT: numericalo Summary

ELEC442: DSP

DTFS, DTFT, DFS, DFT, FFT

M.A. AmerConcordia UniversityElectrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997

•A.V. Oppenheim, R.W. Schafer and J.R. Buck, Discrete-Time Signal Processing

•M.J. Roberts, Signals and Systems, McGraw Hill, 2004

•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003

•Slides 2-22 are from http://metalab.uniten.edu.my/~zainul/images/Signals&Systems

2

Periodic DT Signals

A DT signal is periodic with period where is a positive integer if

The fundamental period of is the smallest positive value of for which the equation holds

Example:

is periodic with fundamental period

3

Fourier representation of signals

The study of signals and systems using sinusoidal representations is termed Fourier analysis, after Joseph Fourier (1768-1830)

The development of Fourier analysis has a long history involving a great many individuals and the investigation of many different physical phenomena, such as the motion of a vibrating string, the phenomenon of heat propagation and diffusion

Fourier methods have widespread application beyond signals and systems, being used in every branch of engineering and science

The theory of integration, point-set topology, and eigenfunction expansions are just a few examples of topics in mathematics that have their roots in the analysis of Fourier series and integrals

4

Approximation of Signals by Sinusoids

A signal can be approximated by a sum of many sinusoids at harmonic frequencies of the signal f0with appropriate amplitude and phase

The more harmonic components are added, the more accurate the approximation becomes

Instead of using sinusoidal signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies

A Fourier representation is unique, i.e., no two same signals in time domain give the same function in frequency domain

5

Overview of Fourier Analysis MethodsPeriodic in TimeDiscrete in Frequency

Aperiodic in TimeContinuous in Frequency

Continuous in Time

Aperiodic in Frequency

Discrete in Time

Periodic in Frequency

∑

∫

∞

−∞=

−

=

⇒⊗

=

⇒⊗

k

tjkk

Ttjk

k

eatx

dtetxT

a

0

0

)(

P-CTDT :SeriesFourier Inverse CT

)(1

DTP-CT :SeriesFourier CT

T

0

T

ω

ω

∫

∑

=

⇒+⊗

=

+⇒⊗∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

∑

∑

−

=

−

=

−

=

⇒⊗

=

⇒⊗

1

0

NN

1

0

NN

0

0

][1][

P-DTP-DT SeriesFourier DT Inverse

][][

P-DTP-DT SeriesFourier DT

N

k

knj

N

n

knj

ekXN

nx

enxkX

ω

ω

∫

∫

∞

∞−

∞

∞−

−

=

⇒⊗

=

⇒⊗

ωωπ

ω

ω

ω

dejXtx

dtetxjX

tj

tj

)(21)(

CTCT :TransformFourier CT Inverse

)()(

CTCT :TransformFourier CT

6

Overview of Fourier Analysis Methods

Variable Period Continuous Frequency

Discrete Frequency

DT x[n] n N k

CT x(t) t T k

Nkk /2πω =

Tkk /2πω =

ω

Ω

• DT-FS: Discrete in time; Periodic in time; Discrete in Frequency; Periodic in Frequency• DT-FT: Discrete in time; Aperiodic in time; Continous in Frequency; Periodic in Frequency• CT-FS: Continuous in time; Periodic in time; Discrete in Frequency; Aperiodic in Frequency• CT-FT: Continuous in time; Aperiodic in time; Continous in Frequency; Aperiodic in Frequency

7

Negative frequency?

8

Negative Frequency?

9

Negative Frequency?

10

Outline

o Introduction o DTFS & properties o DTFT of periodic signalso DFT: Sampling of the DTFTo DTFT, DTFS, DFT, DFS, FFT, ZT: numerical (Matlab)o Summary

11

Discrete-Time Fourier Series (DTFS)

Given a periodic sequence with period N so that

The FS can be written as

(Recall: the FS of continuous-time periodic signals require infinite many complex exponentials)

Not that for DT periodic signals we have

Due to the periodicity of the complex exponential we only need N exponentials for DT FS

The FS coefficients can be obtained via

]n[x~ ]rNn[x~]n[x~ +=

[ ] ( )∑ π=k

knN/2jekX~N1]n[x~

( )( ) ( ) ( ) ( )knN/2jmn2jknN/2jnmNkN/2j eeee πππ+π ==

[ ] ( )∑−

=

π=1N

0k

knN/2jekX~N1]n[x~

[ ] ( )∑−

=

π−=1N

0n

knN/2je]n[x~kX~

12

DTFS Pair

For convenience we sometimes use Analysis equation

Synthesis equation

( )N/2jN eW π−=

[ ]∑−

=

−=1N

0k

knNWkX~

N1]n[x~

[ ] ∑−

=

=1N

0n

knNW]n[x~kX~

13

Concept of DTFS

14

The DTFS

Note: we could divide x[n] or X[k] by N

15

The DTFS

16

The DTFS

17

The DTFS

18

The DTFS

19

The DTFS

20

Example: periodic square

21

Example: periodic square

- We know that

22

Example: periodic square

23

Example: periodic square

24

Example: periodic square

DTFS of an periodic rectangular pulse train The DTFS coefficients

[ ] ( )( )

( )( ) ( )

( )10/ksin2/ksine

e1e1ekX~ 10/k4j

k10/2j

5k10/2j4

0n

kn10/2j

ππ

=−−

== π−π−

π−

=

π−∑

25

Example: periodic impulse train

DFS of a periodic impulse train

Since the period of the signal is N

We can represent the signal with the DTFS coefficients as

[ ] =

=−δ= ∑∞

−∞= else0rNn1

rNn]n[x~r

[ ] ( ) ( ) ( ) 1ee]n[e]n[x~kX~ 0kN/2j1N

0n

knN/2j1N

0n

knN/2j ==δ== π−−

=

π−−

=

π− ∑∑

[ ] ( )∑∑−

=

π∞

−∞=

=−δ=1N

0k

knN/2j

re

N1rNn]n[x~

26

Properties of DTFS

Linearity

Shifting

Duality

[ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ]kX~bkX~anx~bnx~akX~nx~kX~nx~

21DFS

21

2DFS

2

1DFS

1

+ →←+ →← →←

[ ] [ ][ ] [ ]

[ ] [ ]mkX~nx~ekX~emnx~

kX~nx~

DFSN/nm2j

N/km2jDFS

DFS

− →← →←− →←

π

π−

[ ] [ ][ ] [ ]kx~NnX~

kX~nx~DFS

DFS

− →← →←

27

Properties of DTFS

28

Summary of Properties

29

Symmetry Properties

30

Periodic Convolution Take two periodic sequences

Form the product

The periodic sequence with given DTFS can be written as

Periodic convolution is commutative

[ ] [ ][ ] [ ]kXnx

kXnxDFS

DFS

22

11 ~~

~~

→← →←

[ ] [ ] [ ]kXkXkX 213~~~ =

[ ] [ ] [ ]∑−

=

−=1

0213

~~~ N

mmnxmxnx

[ ] [ ] [ ]∑−

=

−=1

0123

~~~ N

mmnxmxnx

31

Periodic Convolution

32

Outline

o Introduction o DTFS & properties o DTFT of periodic signalso DFT: Sampling of the DTFTo DTFT, DTFS, DFT, DFS, FFT, ZT: numerical (Matlab)o Summary

33

The DTFT

∫

∑

=

⇒+⊗

=

+⇒⊗

∞

−∞=

−

π

ωω

π

ωω

π

ωπ 2

2

2

)(21][

DT PCT :TransformFourier DT Inverse

][)(

PCTDT :TransformFourier DT

deeXnx

enxeX

njj

n

njj

• DTFT represents a DT aperiodic signal as a sum of infinitely many complex exponentials, with the frequency varying continuously in (-π, π)• DTFT is periodic

only need to determine it for

DTFT is continuous in frequency

34

The DTFT

From the numerical computation viewpoint, the computation of DTFT by computer has several problems: The summation over n is infinite The independent variable is continuous

DTFT and z-transform are not numerically computable transforms

nj

n

j enxeX ωω −∞

−∞=∑= ][)(

ω

35

FS versus FT

Aperiodic signals can be viewed as a periodic signal with an infinite period

FS: a representation of periodic signals as a linear combination of complex exponentials The FS cannot represent an aperiodic signal for all times

FT: apply to signals that are not periodic The FT can represent an aperiodic signal for all time

NN

1

0

NN

1

0

P-DTP-DF ][1][

P-DFP-DT ][][

:DTFS

0

0

⇒=

⇒=

∑

∑−

=

−

=

−

N

k

knj

N

n

knj

ekXN

nx

enxkX

ω

ω

DT PCT )(21][

PCTDT ][)(

:DTFT

22

2

⇒+=

+⇒=

∫

∑∞

−∞=

−

ππ

ωω

πωω

ωπ

deeXnx

enxeX

njj

n

njj

36

The FT of Periodic Signals

Periodic sequences are not absolute or square summable: no DTFT exist We can represent them as sums of complex exponentials: DTFS

We can combine DTFS and DTFT Periodic impulse train with values proportional to DTFS coefficients

This is periodic with 2π since DTFS is periodic

( ) [ ]∑∞

−∞=

−=

k

j

NkkX

NeX πωδπω 2~2~

37

The FT of Periodic Signals The inverse transform can be written as

FT Pair:

Example:

( ) [ ]∑∞

−∞=

−=

k

j

NkkX

NeX πωδπω 2~2~

( ) [ ]

[ ] [ ]∑∫∑

∫ ∑∫−

=

−

−

∞

−∞=

−

−

∞

−∞=

−

−

=

−

−=

1

0

22

0

2

0

2

0

~12~1

2~221~

21

N

k

nN

kjnj

k

nj

k

njj

ekXN

deN

kkXN

deN

kkXN

deeX

πωεπ

ε

ωεπ

ε

ωεπ

ε

ω

ωπωδ

ωπωδππ

ωπ

[ ]∑−

=

=1

0

2~ ~1][

N

k

nN

kjX ekX

Nn

π

38

Example Consider the periodic impulse train

The DTFS was calculated previously to be

Therefore the FT is

[ ]∑∞

−∞=

−δ=r

rNn]n[p~

[ ] k allfor 1kP~ =

( ) ∑∞

−∞=

ω

π

−ωδπ

=k

j

Nk2

N2eP~

39

Finite-length x[n] & Periodic Signals

Convolve with periodic impulse train

The FT of the periodic sequence is

This implies that

DFS coefficients of a periodic signal = equally spaced samples of the FT of one period

[ ] [ ]∑∑∞

−∞=

∞

−∞=

−=−δ∗=∗=rr

rNnxrNn]n[x]n[p~]n[x]n[x~

( ) ( ) ( ) ( )

( )

π

−ωδ

π=

π

−ωδπ

==

∑

∑∞

−∞=

πω

∞

−∞=

ωωωω

Nk2eX

N2eX~

Nk2

N2eXeP~eXeX~

k

Nk2jj

k

jjjj

[ ] ( )Nk2

jNk2j

eXeXkX~ π=ω

ωπ

=

=

40

Finite-length x[n] & Periodic Signals

41

Example

Consider

The FT is

The DFS coefficients

≤≤

=else0

4n01]n[x

( ) ( )( )2/sin

2/5sineeX 2jj

ωω

= ω−ω

[ ] ( ) ( )( )10/ksin

2/ksinekX~ 10/k4j

ππ

= π−

42

Outline

o Introduction to frequency analysiso DTFS & properties o DTFT of periodic signalso DFT: Sampling of the DTFT o DTFT, DTFS, DFT, DFS, FFT, ZT: numerical (matlab)o Summary

43

Sampling the DTFT:Sampling in frequency domain

In the DTFT

The summation over n is infinite The independent variable is continuous

DTFT is not numerically computable transform

To numerically represent the continuous frequency DTFT, we must take samples of it DFT

nj

n

j enxeX ωω −∞

−∞=∑= ][)(

ω

44

Sampling the DTFT:Review to sampling

Sampling is converting x(t) to x[n] T : sampling period in second; fs = 1/T : sampling frequency in Hz Ωs=2πfs : Sampling frequency in radian-per-second

In frequency domain: convolution of X(jw) with an impulse train

Creates replica of the FT of x(t); Replica are periodic with Ωs

If Ωs< ΩN sampling maybe irreversible due to aliasing of images

[ ] ( ) ∞<<∞−= nnTxnx c

( ) ( )( )∑∞

−∞=

Ω−Ω=Ωk

scs kjXT

jX 1

( )ΩjXc

( )ΩjXs

( )ΩjXs

ΩN-ΩN

ΩN-ΩN Ωs 2Ωs 3Ωs

-2Ωs Ωs3Ωs

ΩN-ΩN Ωs 2Ωs 3Ωs

-2Ωs

Ωs

3Ωs

Ωs<2ΩN

Ωs>2ΩN

45

Sampling the DTFT:Sampling in frequency domain

Consider an aperiodic x[n] with a DTFT Assume a sequence is obtained by sampling the DTFT

Since the DTFT is periodic, the resulting sequence is also periodic

could be the DFS of a sequence The corresponding sequence is

[ ] ( )( )

( )( ) 10 ;~ /2

/2−≤≤==

=LkeXeXkX kNj

kN

j π

πω

ω

( )ωjDTFT eXnx →←][

[ ]kX~

[ ] ( ) 10 and 10 ;~1][~ 1

0

/2 −≤≤−≤≤= ∑−

=

LkNnekXN

nxN

k

knNj π

46

Sampling the DTFT

We can also write it in terms of the z-transform

The sampling points are shown in figure

[ ] ( )( )

( )( )kNj

kN

j eXeXkX /2

/2

~ π

πω

ω ===

[ ] ( ) ( )( )( )kNj

ezeXzXkX kN

/2/2

~ ππ ==

=

47

Sampling the DTFT

The only assumption made on x[n]: its DTFT exist

Combine the equations gives

Term in the parenthesis [] is

( ) [ ]∑∞

−∞=

−=m

mjj emxeX ωω [ ] ( )∑−

=

=1

0

/2~1][~ N

k

knNjekXN

nx π[ ] ( )( )kNjeXkX /2~ π=

[ ] ( ) ( )

[ ] ( ) ( ) [ ] [ ]∑∑ ∑

∑ ∑∞

−∞=

∞

−∞=

−

=

−

−

=

∞

−∞=

−

−=

=

=

mm

N

k

mnkNj

N

k

knNj

m

kmNj

mnpmxeN

mx

eemxN

nx

~1

1][~

1

0

/2

1

0

/2/2

π

ππ

[ ] ( ) ( ) [ ]∑∑∞

−∞=

−

=

− −−==−r

N

k

mnkNj rNmneN

mnp δπ1

0

/21~

[ ] [ ] [ ]∑∑∞

−∞=

∞

−∞=

−=−∗=rr

rNnxrNnnxnx δ][~

48

Sampling the DTFT

FS are samples of the FT of one period

FS are still samples of the FT; But, one period is no longer identical to x[n]

49

Sampling the DTFT

DFS coefficients of a periodic sequence obtained through summing periodic replicas of aperiodic original sequence x[n]

If x[n] is of finite length & we take sufficient number of samples of its DTFT, x[n] can be recovered by

No need to know the DTFT at all frequencies, to recover x[n]

DFT: Representing a finite length sequence by samples of DTFT

[ ] [ ] −≤≤

=else

Nnnxnx

010~

50

Sampling in the frequency domain

The relationship between and one period of in the under-sampled case is considered a form of time domain aliasing

Time domain aliasing can be avoided only if has finite length just as frequency domain aliasing can be avoided only for

signals being band-limited If has finite length N and we take a sufficient number L

of equally spaced samples of its FT, then the FT is recoverable from these samples equivalently is recoverable from

Sufficient number L means: L>=NWe must have at least as many frequency samples as the

signal’s length

][nx ][~ nx

][nx

][nx

][nx ][~ nx

51

The DFT

Consider a finite length sequence x[n] of length N

For x[n] associate a periodic sequence The DFS coefficients of the periodic sequence are samples of the

DTFT of x[n]

Since x[n] is of length N there is no overlap between terms of x[n-rN] and we can write the periodic sequence as

To maintain duality between time and frequency We choose one period of as the DFT of x[n]

[ ] 10 of outside 0 −≤≤= Nnnx

[ ] [ ]∑∞

−∞=

−=r

rNnxnx~

[ ] ( )[ ] ( )( )[ ]NkXkXkX == N mod ~

[ ]kX~

[ ] [ ] −≤≤

=else

NkkXkX0

10~

[ ] ( )[ ] ( )( )[ ]Nnxnxnx == N mod ~

52

The DFT

Consider the DFS pair

The equations involve only one period so we can write

The DFT pair

[ ] ( )∑−

=

=1

0

/2~1][~ N

k

knNjekXN

nx π

[ ] ( )∑−

=

−=1

0

/2][~~ N

n

knNjenxkX π

[ ]( )

−≤≤= ∑−

=

−

else

NkenxkXN

n

knNj

0

10][~1

0

/2π [ ] ( )

−≤≤= ∑−

=

else

NkekXNnx

N

k

knNj

0

10~1][

1

0

/2π

[ ] ( )

N, LLk

enxkXN

n

knNj

>=−≤≤

= ∑−

=

−

10

][1

0

/2π [ ] ( )

NLwhereLk

ekXN

nxN

k

knNj

>=−≤≤

= ∑−

=

,10

1][1

0

/2π

[ ] ][nxkX DFT →←

53

DFT: x[n] finite duration

54

DFT: Example 1

DFT of a rect. pulse x[n], N=5 Consider x[n] of any length L>5 Let L=N=5 Calculate the DFS of the

periodic form of x[n]

[ ] ( )

( )

±±=

=

−−

=

=

π−

π−

=

π−∑

else0,...10,5,0k5

e1e1

ekX~

5/k2j

k2j

4

0n

n5/k2j

55

DFT: Example 1

Let L=2N=10 We get a different set

of DFT coefficients Still samples of the

DTFT but in different places

x[n] = Inverse X[k] depends on relation L & N

56

DFT: Example 1summary

The larger the DFT size K, the more details of the INVERSE DFT, i.e., x[n ] can be seen

57

DFT: example 2

NLwhereLk >=−≤≤ ,10

58

DFT: example 3

NLwhereLk >=−≤≤ ,10

59

DFT: example 3

60

Properties of DFT (very similar to that of DTFS)

Linearity

Duality

[ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ]kbXkaXnbxnaxkXnxkXnx

DFT

DFT

DFT

2121

22

11

+ →←+ →← →←

[ ] [ ][ ] ( )( )[ ]N

DFT

DFT

kNxnXkXnx

− →← →←

61

Example: Duality

62

Circular Shift property

( )( )[ ] ( )

( )N

NN

mN

Nnnmnxny

by shift circular toequivalent is mshift circular A -m)-(Nby shift circular left a toequivalent is mby shift Circular -

modulo where, ][ 1-Nn0 range in the be always bemust y[n] :shift"Circular "

m]-n[ :shiftlinear apply cannot We1-Nn0 range over the defined belonger nomay

m],-x[ny[n] shifted them,arbitrary an For -Nn and 0nfor 0x[n]-

1-Nn0for defined ]length x[n-NConsider -

>

=−=≤≤−

∞≤≤∞−==>≤≤

=>=<=

≤≤

63

Circular Shift property[ ] [ ]

( )( )[ ] [ ] ( )mNkjDFTN

DFT

ekXmnxkXnx

/21-Nn0 π− →←≤≤− →←

64

65

Circular Shift property

66

Circular Convolution Property

10 −≤≤ Nn

][~ nx

• Linear convolution: one sequence is multiplied by a time–reversed and linearly-shifted version of the other

•Circular convolution: the second sequence is circularly time-reversed and circularly-shifted it is called an N-point circular convolution

][ N ][][ 213 nxnxnx =

67

Circular Convolution Property

( ) ( ) ( )

( ) ( )( )

aliasing en with timConvolutioLinear n ConvolutioCircular

0

10][][

thenN, period of sequence periodic a ][~y form :y[n] of DFTget To)12 oflength max. has BUT length of

then,length of and If

)()()( so, is from DFT -

so, :nconvolutioCircular - so h[n],* x[n] y[n] :nconvolutioLinear

/2

===>

−≤≤−=

−

−

==

====−

∑∞

−∞=

else

NnrNnynw

nxN-(y[n]Nw[n]

Nh[n]x[n]

kHkXkYeYY(k) eY

X(k)H(k)W(k) n] x[n] N h[w[n] eHeXeY

r

kNjj

jjj

πω

ωωω

68

Circular Convolution: example 1

Circular convolution of two finite length sequences

][][ 01 nnnx −= δ

[ ] [ ] ( )( )[ ]∑−

=

−=1

0213

N

mNmnxmxnx

[ ] [ ] ( )( )[ ]∑−

=

−=1

0123

N

mNmnxmxnx

][][

][

23

1

0

0

kXWkX

WkXkn

N

knN

=

=

][][ 01 nnnx −= δ

69

Example 2: L=N

Two rect. X[n]: L=N=6

DFT of each sequence

Multiplication of DFTs

Inverse DFT

[ ] [ ] −≤≤

==else

Knnxnx

0101

21

[ ] [ ] =

=== ∑−

=

−

elsekN

ekXkXN

n

knN

j

001

0

2

21

π

[ ] [ ] [ ] =

==elsekN

kXkXkX0

02

213

[ ] −≤≤

=else

NnNnx

010

3

70

Example 2: L=2N

Augment zeros to each sequence L=2N=12

The DFT of each sequence

Multiplication of DFTs

[ ] [ ]Nk2j

NLk2j

21

e1

e1kXkX π−

π−

−

−==

[ ]2

Nk2j

NLk2j

3

e1

e1kX

−

−= π

−

π−

x[n] = Inverse DFT X[k] is not unique; depends on L and N

71

Circular convolution example

72

73

Symmetry Property

74

Symmetry Properties

75

Outline

o Introduction to frequency analysiso DTFS & properties o DTFT of periodic signalso DFT: sampling of the DTFTo DTFT, DTFS, DFT, DFS, FFT, ZT: numerical (matlab)o Summary

76

Discrete-time signal transforms

76

77

Numerical Calculation of FT

1. The original signal is digitized2. A Fast Fourier Transform (FFT) algorithm

is applied, which yields samples of the FT at equally spaced intervals

For a signal that is very long, e.g., a speech signal or a music piece, spectrogram is used FT over successive overlapping short

intervals

78

Matlab examples: DTFT

Suppose that:

Analytically, the DTFT is X(ejω): continuous function of ω X(ejω): periodic with period 2π

Plot it using

78

79

Matlab examples: DTFT

Signal x[n] DTFT

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Matlab examples: DFT

Close form X(ejω) not always easy To plot |X(ejω)|, we sampled from 0 to 2π

In code: w and X are vectors Small step size 0.001 to simulate continuous frequency

Workaround: DFT Uniform L-samples from DTFT from 0 to 2π Takes discrete values and returns discrete values No need to find |X(ejω)| analytically Fast implementation using the fast Fourier transform (FFT) Matlab: fft(x,L)

• L: number of samples to take• More L more resolution • Default L is N=length(x)

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Matlab examples: DFT

Calculating the DFT

Plotting the DFT against k

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Matlab examples: DFT

Notes: Default L=32 gives bad

resolution information lost x-axis not usefulCannot find fundamental

frequency 3π/8

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Matlab examples: DFT

Effect of increasing L (better resolution)

• L=64

• L-128

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Matlab examples: DFT Obtaining the frequency (x-axis)

Spike at 3π/8=1.17 Spike at 2π-3π/8 = 5.11 FFT calculates from 0 to 2π More familiar to shift using

fftshift

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Matlab examples: DFT

Spikes at 3π/8 and -3π/8

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Matlab examples: DFT

Sometimes we want frequency in Hz

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Matlab examples: DFT

|X[k]| vs. ωk

Discrete DFT

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|X(ejω)| vs. ω Continuous By interpolating DFT

|X(f)| vs. f Continuous f = (ω/ 2π) fs fs : sampling frequency fft values divided by N Peak at 0.5 (half our

amplitude of 1)

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Matlab examples: DFS No special function Same as DFT Provided signal corresponds to 1 period

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Matlab examples: z-Transform

Suppose that:

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Matlab examples: z-Transform

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Matlab examples: z-Transform

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Matlab examples: z-Transform

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Evaluate H2(ejω) directly from z-Transform

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Matlab examples: z-Transform

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Finding z-Transform analytically

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Outline

o Introduction to frequency analysiso DTFS & properties o DTFT & properties o FT of periodic signals o DTFT, DTFS, DFT, DFS, FFT, ZT: numerical (matlab)o FTTo Summary

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FFT: Fast Fourier transform

FFT is a direct computation of the DFT FFT is a set of algorithms for the efficient

and digital computation of the N-point DFT, rather than a new transform

Use the number of arithmetic multiplications and additions as a measure of computational complexity

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FFT

The DFT pair was given as Baseline for computational complexity:

Each DFT coefficient requires• N complex multiplications• N-1 complex additions

All N DFT coefficients require• N2 complex multiplications• N(N-1) complex additions

Complexity in terms of real operations• 4N2 real multiplications• 2N(N-1) real additions

[ ] ( )∑−

=

π=1N

0k

knN/2jekXN1]n[x

[ ] ( )∑−

=

π−=1N

0n

knN/2je]n[xkX

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FFT

Most fast methods are based on symmetry properties Conjugate symmetry

Periodicity in n and k

The Second Order Goertzel Filter• Approximately N2 real multiplications and 2N2 real additions• Do not need to evaluate all N DFT coefficients

Decimation-In-Time FFT Algorithms (N/2)log2N complex multiplications and additions

( ) ( ) ( ) ( ) ( ) ( )knN/2jnkN/2jkNN/2jnNkN/2j eeee π−π−π−−π− ==

( ) ( ) ( ) ( )( )nNkN/2jNnkN/2jknN/2j eee +π+π−π− ==

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Symmetry and periodicity of complex exponential

Complex conjugate symmetry

Periodicity in n and k

For example

The number of multiplications is reduced by a factor of 2

ImRe)( *][ knN

knN

knN

knN

nNkN WjWWWW −=== −−

nNkN

NnkN

knN WWW )()( ++ ==

Re])[Re][(Re

Re][ReRe][Re ][

knN

nNkN

knN

WnNxnxWnNxWnx

−+=

−+ −

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Outline

o Introduction to frequency analysiso DTFS & properties o DTFT & properties o FT of periodic signals o DTFT, DTFS, DFT, DFS, FFT, ZT: numerical (matlab)o FTTo Summary

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Overview of signal transformsVariable Period Continuous

FrequencyDiscrete Frequency

DT x[n] n N k

CT x(t) t T k

Nkk /2πω =

Tkk /2πω =

ω

Ω• DT-FS: Discrete in time; Periodic in time; Discrete in Frequency; Periodic in Frequency• DT-FT: Discrete in time; Aperiodic in time; Continous in Frequency; Periodic in Frequency• CT-FS: Continuous in time; Periodic in time; Discrete in Frequency; Aperiodic in Frequency• CT-FT: Continuous in time; Aperiodic in time; Continous in Frequency; Aperiodic in Frequency• DFT: Discrete in time; Aperiodic in time; Discrete in Frequency; Periodic in Frequency;

finite-duration x[n]• DFS: Discrete in time; Periodic in time (make finite-duration x[n] periodic);

Discrete in Frequency; Periodic in Frequency;

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Relationships between signal transforms

Ω= j ez

Continuous-time analog signal

x(t)

Discrete-time analog sequence

x [n]

Sample in timeSampling period = Ts

ω=2πfΩ = ω Ts,scale amplitude by 1/Ts

Sample in frequency,Ω = 2πn/N,N = Length

of sequence

ContinuousFourier Transform

X(f)

∞≤≤∞

∫∞

∞−

f-

dt e x(t) ft2 j- π

Discrete Fourier Transform

X(k)

10

e [n]x 1

0 =n

Nnk2j-

−≤≤

∑−

Nk

N π

Discrete-Time Fourier Transform

X(Ω)

π20

e [n]x - =n

j-

≤Ω≤

∑∞

∞

Ωn

LaplaceTransform

X(s)s = σ+jω

∞≤≤∞

∫∞

∞−

−

s-

dt e x(t) st

z-TransformX(z)

∞≤≤∞−

∑∞

∞−

z =n

n- z [n]x

s = jωω=2πf

C CC

C

C D

D

DC Continuous-variable Discrete-variable

Ω= j ez r

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Fourier versus Cosine Transform

Recall: the cosine wave starts out 1/4th later in its period

It has an offset Common to measure this offset in degree or radians One complete period equals 360° or 2π radian

The cosine wave thus has an offset of 90° or π/2 This offset is called the phase of a sinusoid We cannot restrict a signal x(t) to start out at zero

phase or 90° phase all the time Must determine its frequency, amplitude, and phase to

uniquely describe it at any one time instant With the sine or cosine transform, we are restricted to

zero phase or 90° phase

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DCT: One Dimensional

∑

+=

−

=

1

0 2)12(cos

21 n

t nftxCX tffπ

>

==

0,1

0,2

1

f

fCf

where

n = size

x = signal

X = transform coefficients