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

2. Frequency Domain Representation of Discrete-Time Signals &
Systems
EEE328 Digital Signal Processing
Lecture 5
Ankara University Electrical and Electronics Eng. Dept. EEE328

3. Agenda
• Frequency Domain by Fourier Transform
• Eigenfunctions for LTI Systems
• Ideal Filters
Ankara University Electrical and Electronics Eng. Dept. EEE328

4. Discrete-Time Fourier Transform (D-T FT)
𝑋 𝑒𝑗𝜔 = ෍
𝑛=−∞
∞
𝑥[𝑛]𝑒−𝑗𝜔𝑛
𝑥 𝑛 =
1
2𝜋
න
2𝜋
𝑋(𝑒𝑗𝜔)𝑒𝑗𝜔𝑛𝑑𝜔
Ankara University Electrical and Electronics Eng. Dept. EEE328
D-T FT
D-T Inverse FT

5. Eigenfunction and Eigenvalue
Ankara University Electrical and Electronics Eng. Dept. EEE328
𝑥 𝑛 = 𝑒𝑗𝜔𝑛
𝑦 𝑛 = 𝐻(𝑒𝑗𝜔)𝑒𝑗𝜔𝑛
𝑦 𝑛 = ෍
𝑘=−∞
∞
ℎ[𝑘]𝑒−𝑗𝜔(𝑛−𝑘)
𝑦 𝑛 = 𝑒𝑗𝜔𝑛
෍
𝑘=−∞
∞
ℎ[𝑘]𝑒−𝑗𝜔𝑘
𝐻(𝑒𝑗𝜔
) = ෍
𝑘=−∞
∞
ℎ[𝑘]𝑒−𝑗𝜔𝑘
𝑦 𝑛 = 𝐻(𝑒𝑗𝜔)𝑒𝑗𝜔𝑛
𝐻 𝑒𝑗𝜔 = 𝐻𝑅 𝑒𝑗𝜔 + 𝑗𝐻𝐼(𝑒𝑗𝜔)
𝐻𝑅 𝑒𝑗𝜔 : Real Part
𝐻𝐼 𝑒𝑗𝜔 : Imaginary Part
𝐻 𝑒𝑗𝜔
= |𝐻 𝑒𝑗𝜔
|𝑒𝑗∢𝐻(𝑒𝑗𝜔)

6. Ideal Frequency Selective Filters
Ankara University Electrical and Electronics Eng. Dept. EEE328
-2π+ωc -2π -2π+ωc -π -ωc 0 ωc π 2π-ωc 2π 2π+ωc
HLP(ejω)
Ideal Low-Pass Filter
-2π -π +ωc -π -ωc 0 ωc π π+ωc 2π
HHP(ejω)
Ideal High-Pass Filter
ω
ω
…
…
…
…
1
1

7. Ideal Frequency Selective Filters
Ankara University Electrical and Electronics Eng. Dept. EEE328
-π -ωb -ωa 0 ωa ωb π
HBS(ejω)
Ideal Band-Stop Filter
HBP(ejω)
Ideal Band-Pass Filter
ω
…
…
ω
…
…
-π -ωb -ωa 0 ωa ωb π
1
1

8. 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