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

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

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

Periodicity of Frequency Response 1 Frequency response is periodic with the period of 2π. Implication? Hossein Sameti, CE, SUT, Fall 1992

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

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

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

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

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

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

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

DTFT Pairs Hossein Sameti, CE, SUT, Fall 1992

DTFT Pairs Hossein Sameti, CE, SUT, Fall 1992

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

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

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

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

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

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

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

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

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

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

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

Example: Hossein Sameti, CE, SUT, Fall 1992

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