DTFS, DTFT, DFS, DFT, FFT -...
Transcript of DTFS, DTFT, DFS, DFT, FFT -...
![Page 1: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/1.jpg)
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
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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
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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
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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
![Page 5: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/5.jpg)
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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
![Page 6: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/6.jpg)
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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
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Negative frequency?
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Negative Frequency?
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Negative Frequency?
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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
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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~
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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~
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Concept of DTFS
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The DTFS
Note: we could divide x[n] or X[k] by N
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The DTFS
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The DTFS
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The DTFS
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The DTFS
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The DTFS
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Example: periodic square
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Example: periodic square
- We know that
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Example: periodic square
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Example: periodic square
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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−∑
![Page 25: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/25.jpg)
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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~
![Page 26: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/26.jpg)
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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
− →← →←
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Properties of DTFS
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Summary of Properties
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Symmetry Properties
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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
![Page 31: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/31.jpg)
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Periodic Convolution
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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
![Page 33: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/33.jpg)
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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
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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
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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
![Page 36: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/36.jpg)
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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~
![Page 37: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/37.jpg)
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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
![Page 38: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/38.jpg)
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~
![Page 39: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/39.jpg)
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
=
=
![Page 40: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/40.jpg)
40
Finite-length x[n] & Periodic Signals
![Page 41: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/41.jpg)
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−
![Page 42: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/42.jpg)
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
![Page 43: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/43.jpg)
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
![Page 44: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/44.jpg)
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
![Page 45: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/45.jpg)
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
![Page 46: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/46.jpg)
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 ==
=
![Page 47: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/47.jpg)
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][~
![Page 48: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/48.jpg)
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]
![Page 49: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/49.jpg)
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~
![Page 50: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/50.jpg)
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
![Page 51: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/51.jpg)
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 ~
![Page 52: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/52.jpg)
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 →←
![Page 53: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/53.jpg)
53
DFT: x[n] finite duration
![Page 54: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/54.jpg)
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
![Page 55: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/55.jpg)
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
![Page 56: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/56.jpg)
56
DFT: Example 1summary
The larger the DFT size K, the more details of the INVERSE DFT, i.e., x[n ] can be seen
![Page 57: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/57.jpg)
57
DFT: example 2
NLwhereLk >=−≤≤ ,10
![Page 58: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/58.jpg)
58
DFT: example 3
NLwhereLk >=−≤≤ ,10
![Page 59: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/59.jpg)
59
DFT: example 3
![Page 60: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/60.jpg)
60
Properties of DFT (very similar to that of DTFS)
Linearity
Duality
[ ] [ ][ ] [ ]
[ ] [ ] [ ] [ ]kbXkaXnbxnaxkXnxkXnx
DFT
DFT
DFT
2121
22
11
+ →←+ →← →←
[ ] [ ][ ] ( )( )[ ]N
DFT
DFT
kNxnXkXnx
− →← →←
![Page 61: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/61.jpg)
61
Example: Duality
![Page 62: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/62.jpg)
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 -
>
=−=≤≤−
∞≤≤∞−==>≤≤
=>=<=
≤≤
![Page 63: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/63.jpg)
63
Circular Shift property[ ] [ ]
( )( )[ ] [ ] ( )mNkjDFTN
DFT
ekXmnxkXnx
/21-Nn0 π− →←≤≤− →←
![Page 64: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/64.jpg)
64
![Page 65: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/65.jpg)
65
Circular Shift property
![Page 66: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/66.jpg)
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 =
![Page 67: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/67.jpg)
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
![Page 68: DTFS, DTFT, DFS, DFT, FFT - users.encs.concordia.causers.encs.concordia.ca/.../teach/elec442/notes/Ch8_DTFT_DFT_FS.pdf · o Introduction o DTFS & Properties o FT of periodic signals](https://reader033.fdocuments.net/reader033/viewer/2022052117/5ab74add7f8b9a684c8b5988/html5/thumbnails/68.jpg)
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
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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
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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
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Circular convolution example
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Symmetry Property
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Symmetry Properties
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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
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Discrete-time signal transforms
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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
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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
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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
82
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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
87
|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:
89
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Matlab examples: z-Transform
90
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Matlab examples: z-Transform
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Matlab examples: z-Transform
92
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