Tune

1. EEE328
Digital Signal Processing
Ankara University
Faculty of Engineering
Electrical and Electronics Engineering Department
Ankara University Electrical and Electronics Eng. Dept. EEE328

2. Fourier Transform and Its Properties
EEE328 Digital Signal Processing
Lecture 6
Ankara University Electrical and Electronics Eng. Dept. EEE328

3. Agenda
• Fourier Transform Representation of Sequences
• Properties of Fourier Transform
• Fourier Transform Theorems
Ankara University Electrical and Electronics Eng. Dept. EEE328

4. D-T FT
𝑋 𝑒𝑗𝜔 = ෍
𝑛=−∞
∞
𝑥[𝑛]𝑒−𝑗𝜔𝑛
𝑥 𝑛 =
1
2𝜋
න
2𝜋
𝑋(𝑒𝑗𝜔
)𝑒𝑗𝜔𝑛
𝑑𝜔
Ankara University Electrical and Electronics Eng. Dept. EEE328
𝑋 𝑒𝑗𝜔
= 𝑋𝑅 𝑒𝑗𝜔
+ 𝑗𝑋𝐼(𝑒𝑗𝜔
)
𝑋 𝑒𝑗𝜔 = |𝑋 𝑒𝑗𝜔 |𝑒𝑗∢𝑋(𝑒𝑗𝜔)

5. D-T FT
ℎ 𝑛 =
1
2𝜋
න
2𝜋
𝐻(𝑒𝑗𝜔
)𝑒𝑗𝜔𝑛
𝑑𝜔
Ankara University Electrical and Electronics Eng. Dept. EEE328
Impulse Response by Inverse D-T FT
Convergence
𝑋 𝑒𝑗𝜔 = ෍
𝑛=−∞
∞
𝑥[𝑛]𝑒−𝑗𝜔𝑛 ≤ ෍
𝑛=−∞
∞
𝑥 𝑛 𝑒−𝑗𝜔|
≤ ෍
𝑛=−∞
∞
𝑥 𝑛 < ∞

6. Symmetry Property of D-T FT
Ankara University Electrical and Electronics Eng. Dept. EEE328
𝑥 𝑛 = 𝑥𝑒 𝑛 + 𝑥𝑜[𝑛]
𝑥𝑒 𝑛 =
1
2
𝑥 𝑛 + 𝑥∗
−𝑛 = 𝑥𝑒
∗
[−𝑛]
𝑥𝑜 𝑛 =
1
2
𝑥 𝑛 − 𝑥∗ −𝑛 = −𝑥𝑜
∗[−𝑛]
𝑋 𝑒𝑗𝜔 = 𝑋𝑒 𝑒𝑗𝜔 + 𝑋𝑜 𝑒𝑗𝜔
𝑋𝑒 𝑒𝑗𝜔 =
1
2
[𝑋 𝑒𝑗𝜔 + 𝑋∗ 𝑒−𝑗𝜔 ]
𝑋𝑜 𝑒𝑗𝜔 =
1
2
[𝑋 𝑒𝑗𝜔 − 𝑋∗ 𝑒−𝑗𝜔 ]
𝑋𝑒 𝑒𝑗𝜔 = 𝑋𝑒
∗ 𝑒𝑗𝜔
𝑋𝑜 𝑒𝑗𝜔 = −𝑋𝑜
∗ 𝑒𝑗𝜔

7. Fourier Transform Theorems
Ankara University Electrical and Electronics Eng. Dept. EEE328
𝑋 𝑒𝑗𝜔 = ℱ{𝑥 𝑛 }
𝑥 𝑛 = ℱ−1{𝑋 𝑒𝑗𝜔 }
ℱ
𝑥 𝑛 ⟷ 𝑋 𝑒𝑗𝜔
ℱ
𝑦 𝑛 ⟷ 𝑌 𝑒𝑗𝜔
𝑎𝑥 𝑛 + 𝑏𝑦 𝑛 ՞
ℱ
𝑋 𝑒𝑗𝜔
+ 𝑌 𝑒𝑗𝜔
𝑥 𝑛 − 𝑛𝑑 ՞
ℱ
𝑒−𝑗𝜔𝑛𝑑𝑋 𝑒𝑗𝜔
𝑥 −𝑛 ՞
ℱ
𝑋 𝑒−𝑗𝜔
𝑥 𝑛 ∗ 𝑦[𝑛] ՞
ℱ
𝑋 𝑒𝑗𝜔
𝑌 𝑒𝑗𝜔
𝑥 𝑛 𝑦 𝑛 ՞
ℱ 1
2𝜋
න
2𝜋
𝑋 𝑒𝑗𝜃 𝑌 𝑒𝑗 𝜔−𝜃 𝑑𝜃

8. • Parseval’s Theorem
𝑥[𝑛] ՞
ℱ
𝑋(𝑒𝑗𝜔)
𝐸 = ෍
𝑛=−∞
∞
|𝑥 𝑛 |2 =
1
2𝜋
න
2𝜋
|𝑋 𝑒𝑗𝜔 |2 𝑑𝜔
Ankara University Electrical and Electronics Eng. Dept. EEE328

9. References
• Signals & Systems, Second Edition, A. V. Oppenheim, A. S. Willsky with S. H.
Nawab, Prentice Hall, 1997
• Discrete-Time Signal Processing, Second Edition, A. V. Oppenheim, R. W. Schafer
with J. R. Buck, Prentice Hall, 1999
Ankara University Electrical and Electronics Eng. Dept. EEE328