This document summarizes properties of the discrete-time Fourier transform covered in a signals and systems lecture, including: periodicity with period 2π; linearity of the transform; time and frequency shifting properties; conjugation and conjugate symmetry; differencing; Parseval's relation; convolution and multiplication properties in the time and frequency domains; and that systems characterized by linear constant-coefficient difference equations have a frequency response equal to the ratio of polynomials in e-jw.

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
• Properties of Discrete Time Fourier Transform
• Systems Characterized By Linear Constant-Coefficient Difference
Equations
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

4. Periodicity
• 𝑋 𝑒𝑗𝜔 = 𝑋 𝑒𝑗 𝜔+2𝜋
• Periodic in 𝜔 with period 2𝜋
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

5. Linearity
• 𝑥[𝑛] 𝑋 𝑒𝑗𝜔 and 𝑦 𝑛 𝑌 𝑒𝑗𝜔
• 𝑎𝑥[𝑛] + 𝑏𝑦[𝑛] 𝑎𝑋 𝑒𝑗𝜔 + 𝑏𝑌(𝑒𝑗𝜔)
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

6. Time Shifting and Frequency Shifting
• 𝑥[𝑛] 𝑋 𝑒𝑗𝜔
• 𝑥[𝑛 − 𝑛0] 𝑒−𝑗𝜔𝑛0𝑋(𝑒𝑗𝜔)
• 𝑒𝑗𝜔0𝑛𝑥 𝑛 𝑋(𝑒𝑗 𝜔−𝜔0 )
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

7. Conjugation and Conjugate Symmetry
• 𝑥[𝑛] 𝑋 𝑒𝑗𝜔
• 𝑥∗[𝑛] 𝑋∗(𝑒−𝑗𝜔)
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

8. Differencing
• 𝑥[𝑛] 𝑋 𝑒𝑗𝜔
• 𝑥 𝑛 − 𝑥 𝑛 − 1 1 − 𝑒−𝑗𝜔 𝑋(𝑒𝑗𝜔)
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

9. Parseval’s Relation
• 𝑛=−∞
∞ 𝑥[𝑛] 2 =
1
2𝜋 0
2𝜋
𝑋(𝑒𝑗𝜔)
2
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

10. Convolution Property
• 𝑦[𝑛] = ℎ[𝑛] ∗ 𝑥[𝑛] 𝑌 𝑒𝑗𝜔 = 𝐻(𝑒𝑗𝜔)𝑋(𝑒𝑗𝜔)
• Time domain: convolution
• Frequency domain: multiplication
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

11. Multiplication Property
• 𝑟[𝑛] = 𝑠[𝑛]𝑝[𝑛] 𝑅 𝑒𝑗𝜔 =
1
2𝜋 0
2𝜋
𝑆 𝑒𝑗𝜃 𝑃 𝑒𝑗(𝜔−𝜃) 𝑑𝜃
• Time domain: multiplication
• Frequency domain: periodic convolution
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

12. Systems Characterized by Linear Constant
Coefficient Difference Equations
• 𝑌 𝑒𝑗𝜔 = 𝐻 𝑒𝑗𝜔 𝑋(𝑒𝑗𝜔)
• 𝐻 𝑒𝑗𝜔
= 𝑘=0
𝑀
𝑏𝑘𝑒−𝑗𝑘𝜔
𝑘=0
𝑁 𝑎𝑘𝑒−𝑗𝑘𝜔
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems

13. References
• Signals and Systems, 2nd Edition, Oppenheim, Willsky, Nawab
Ankara Univeristy
Electrical and Electronics Engineering Department, EEE321
Signals and Systems