1. CE 40763 Digital Signal Processing Fall 1992 Discrete-time Fourier Transform
Hossein Sameti
Department of Computer Engineering
Sharif University of Technology

2. Motivation: Eigen vector of matrix A:
In other words, once matrix A is multiplied by vector X, the direction of X is preserved.
Eigen function of a system:
αФ(n)
Ф(n)
System
- Subscript a denotes an analog signal.
Hossein Sameti, CE, SUT, Fall 1992

3. Motivation: x(n) y(n) LTI System Frequency response
- Subscript a denotes an analog signal.
magnifies the input
based on freq ω.
Clarification: Some textbooks use instead of
Hossein Sameti, CE, SUT, Fall 1992

4. Periodicity of Frequency Response1
Frequency response is periodic with the period of 2π. Implication?
Hossein Sameti, CE, SUT, Fall 1992

5. Low/high Frequencies in Discrete-time domain
Hossein Sameti, CE, SUT, Fall 1992

6. Convergence of the Frequency Response
The same condition as the stability condition
Hossein Sameti, CE, SUT, Fall 1992

7. Discrete-Time Fourier Transform
Same mathematical representation as the freq. response
Existence of DTFT:
x(n) is absolutely summable.
Inverse DTFT:
Fourier analysis considers signals to be constructed from a sum of complex exponentials with appropriate frequencies, amplitudes and phase.
Frequency components are the complex exponentials which, when added together, make up the signal.
Hossein Sameti, CE, SUT, Fall 1992

8. Example of calculating IDTFT
IDTFT of the ideal low-pass filter:
Hossein Sameti, CE, SUT, Fall 1992

9. Example of calculating IDTFT
sin⁡( 𝜔 𝑐 𝑛) 𝑛 ⟺𝑋 𝜔 = 1, |𝜔|≤ 𝜔 𝑐 0, 𝜔 𝑐 < 𝜔 ≤𝜋
Hossein Sameti, CE, SUT, Fall 1992

10. Real and Imaginary parts of DTFT
What happens if a>1?
Hossein Sameti, CE, SUT, Fall 1992

11. Magnitude and Angle of DTFT
Hossein Sameti, CE, SUT, Fall 1992

12. DTFT Pairs
Hossein Sameti, CE, SUT, Fall 1992

13. DTFT Pairs
Hossein Sameti, CE, SUT, Fall 1992

14. Properties of DTFT Linearity: Time-shifting: Time-reversal:
Convolution :
x(n)
y(n)
LTI System
h(n)
Hossein Sameti, CE, SUT, Fall 1992

15. Properties of DTFT Cross-correlation: Frequency Shifting:
Parseval’s Theorem:
Hossein Sameti, CE, SUT, Fall 1992

16. Properties of DTFT Modulation: Multiplication:
Differentiation in the freq. domain:
Conjugation:
Hossein Sameti, CE, SUT, Fall 1992

17. Symmetry Properties of DTFT
Conjugate Symmetric:
Conjugate Anti-Symmetric:
Why are these properties important?
Conjugate Symmetric
Conjugate Anti-symmetric
Hossein Sameti, CE, SUT, Fall 1992

18. Symmetry Properties of DTFT
Hossein Sameti, CE, SUT, Fall 1992

19. Symmetry Properties of DTFT
: real
If a sequence is real, then its DTFT is conjugate symmetric.
Hossein Sameti, CE, SUT, Fall 1992

20. Symmetry Properties of DTFT
: real
: real
: real
: real
: real
Hossein Sameti, CE, SUT, Fall 1992

21. Symmetry Properties of DTFT
Proakis, et.al
Hossein Sameti, CE, SUT, Fall 1992

22. Example: Determining an inverse fourier transform 22
Hossein Sameti, CE, SUT, Fall 1992

23. Example: Determining the Impulse response from the frequency response
Hossein Sameti, CE, SUT, Fall 1992 23

24. Example: Determining the Impulse response for a Difference Equation
To find the impulse response h[n], we set
Applying the DTFT to both sides of equation. We obtain 24
Hossein Sameti, CE, SUT, Fall 1992

25. Example:
Hossein Sameti, CE, SUT, Fall 1992

26. Summary
Reviewed Discrete-time Fourier Transform, some of its properties and FT pairs
Next: the Z-transform
Hossein Sameti, CE, SUT, Fall 1992