1. EEE 321
Signals and Systems
Ankara University
Faculty of Engineering
Electrical and Electronics Engineering Department
2. Properties of Discrete-Time
Fourier Transform
EEE321 Signals and Systems
Lecture 13
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
3. Agenda
โข Properties of Discrete Time Fourier Transform
โข Systems Characterized By Linear Constant-Coefficient Difference
Equations
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
4. Periodicity
โข ๐ ๐๐๐ = ๐ ๐๐ ๐+2๐
โข Periodic in ๐ with period 2๐
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
5. Linearity
โข ๐ฅ[๐] ๐ ๐๐๐ and ๐ฆ ๐ ๐ ๐๐๐
โข ๐๐ฅ[๐] + ๐๐ฆ[๐] ๐๐ ๐๐๐ + ๐๐(๐๐๐)
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
6. Time Shifting and Frequency Shifting
โข ๐ฅ[๐] ๐ ๐๐๐
โข ๐ฅ[๐ โ ๐0] ๐โ๐๐๐0๐(๐๐๐)
โข ๐๐๐0๐๐ฅ ๐ ๐(๐๐ ๐โ๐0 )
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
7. Conjugation and Conjugate Symmetry
โข ๐ฅ[๐] ๐ ๐๐๐
โข ๐ฅโ[๐] ๐โ(๐โ๐๐)
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
8. Differencing
โข ๐ฅ[๐] ๐ ๐๐๐
โข ๐ฅ ๐ โ ๐ฅ ๐ โ 1 1 โ ๐โ๐๐ ๐(๐๐๐)
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
9. Parsevalโs Relation
โข ๐=โโ
โ ๐ฅ[๐] 2 =
1
2๐ 0
2๐
๐(๐๐๐)
2
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
10. Convolution Property
โข ๐ฆ[๐] = โ[๐] โ ๐ฅ[๐] ๐ ๐๐๐ = ๐ป(๐๐๐)๐(๐๐๐)
โข Time domain: convolution
โข Frequency domain: multiplication
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
11. Multiplication Property
โข ๐[๐] = ๐ [๐]๐[๐] ๐ ๐๐๐ =
1
2๐ 0
2๐
๐ ๐๐๐ ๐ ๐๐(๐โ๐) ๐๐
โข Time domain: multiplication
โข Frequency domain: periodic convolution
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
12. Systems Characterized by Linear Constant
Coefficient Difference Equations
โข ๐ ๐๐๐ = ๐ป ๐๐๐ ๐(๐๐๐)
โข ๐ป ๐๐๐
= ๐=0
๐
๐๐๐โ๐๐๐
๐=0
๐ ๐๐๐โ๐๐๐
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems
13. References
โข Signals and Systems, 2nd Edition, Oppenheim, Willsky, Nawab
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems