06ec44-signals and system –session4-2009• · 06ec44-signals and system –chapter 5.2-2009•...
TRANSCRIPT
![Page 1: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/1.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 1
Chapter 5.2 Fourier Transform for Discrete time Signal
DTFT
5.2.1 Fourier Transform for Discrete Time Signal Definition
Development of the Discrete-Time Fourier Transform
Fig 1: is a Finite duration signal x[n]; Fig 2 periodic signal
FT of an Aperiodic Signal • Define
![Page 2: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/2.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 2
DTFT
It is convenient to think of ak as being defined for all
integers K. So:
1) ak+N = ak – Special property of DT Fourier coefficients
2) We only use N consecutive values of ak in the synthesis
equation
3) Since N is periodic, it is specified by a sequence of N
numbers, either in the
Time or in the Frequency domain.
4) It is convenient to think of ak as being defined for all
integers K. So:
a) ak+N = ak – Special property of DT Fourier coefficients
![Page 3: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/3.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 3
b) We only use N consecutive values of ak in the synthesis
equation
c) Since N is periodic, it is specified by a sequence of N
numbers, either in the Time or in the Frequency domain
CT Verses FT an OverView
Periodicity of a DT Fourier Transform
![Page 4: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/4.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 4
High Frequency Low Frequency Concept
Example 1 :
![Page 5: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/5.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 5
Magnitude and Phase Spectra of the FT of the FT of above Example:
x[n], for a>1 and a<1
Example 2:
![Page 6: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/6.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 6
Example 3:
![Page 7: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/7.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 7
Convergence of DT Fourier Transform
![Page 8: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/8.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 8
The Analysis Equation will converge: Either if the x[n] is absolutely summable, or
has finite Energy, that is,
DTFT Summarize
It is convenient to think of ak as being defined for all integers K.
So:
1) ak+N = ak – Special property of DT Fourier coefficients
2) We only use N consecutive values of ak in the synthesis
equation
3) Since N is periodic, it is specified by a sequence of N
numbers, either in the Time or in the Frequency domain
5.2.1 Properties of Discrete-Time Fourier Transform
• Linearity
• Time Shifting and Frequency Shifting
• Conjugation and Conjugate Symmetry
• Differencing and Accumulation
• Time Reversal
• Time Expansion
• Differentiation in frequency
• Parseval’s Relation
Discrete-Time Fourier Transform Pair
• Notation
• Synthesis Equation
![Page 9: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/9.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 9
• Analysis Equation
• Also if
• Periodicity of DTFT
![Page 10: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/10.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 10
![Page 11: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/11.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 11
![Page 12: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/12.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 12
![Page 13: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/13.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 13
![Page 14: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/14.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 14
![Page 15: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/15.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 15
![Page 16: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/16.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 16
![Page 17: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/17.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 17
![Page 18: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/18.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 18
![Page 19: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/19.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 19
![Page 20: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/20.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 20
References • Figures and images used in these lecture notes are adopted
from “Signals & Systems” by Alan V. Oppenheim and
Alan S. Willsky, 1997
• Feng-Li Lian, NTU-EE and Mark Fowler Signals and
Systems.
• Text and Reference Books have been referred during the
notes preparation.
![Page 21: 06EC44-Signals and System –Session4-2009• · 06EC44-Signals and System –Chapter 5.2-2009• Krupa Rasane(KLE) Page 3 b) We only use N consecutive values of a k in the synthesis](https://reader033.vdocuments.us/reader033/viewer/2022060707/60734ea090b0e65dd720ed2d/html5/thumbnails/21.jpg)
06EC44-Signals and System –Chapter 5.2-2009•
Krupa Rasane(KLE) Page 21