Signals and Systems Fall 2003 Lecture #10 7 October 2003 Examples of the dt fourier Transform 2. Properties of the dt Fourier transform 3. The Convolution Property and its Implications and Uses
Signals and Systems Fall 2003 Lecture #10 7 October 2003 1. Examples of the DT Fourier Transform 2. Properties of the DT Fourier Transform 3. The Convolution Property and its Implications and Uses
DT Fourier transform pair x]→X(1) X(-)=∑mln Analysis equation FT 2丌 X( jEjUn Synthesis equation Inverse ft
DT Fourier Trans form Pair – Analysis Equation – F T – Synthesis Equation – I n v e r s e F T
Convergence Issues Synthesis Equation: None, since integrating over a finite interval Analysis equation: Need conditions analogous to CTFT, e.g ∑ Finite energy or ∑aml<∞- Absolutely summable
Convergence Issues Synthesis Equation: None, since integrating over a finite interval Analysis Equation: Need conditions analogous to CTFT, e.g. — Absolutely summable — Finite energy
Examples Parallel with the ct examples in lecture #8 X()=∑6mle-ln 2)rn=dn-no -shifted unit sample X()=∑6n 0e4 Same amplitude(1) as above, but with a linear phase -wno
Examples Parallel with the CT examples in Lecture #8
More examples 3)m=anan,a< 1- Exponentially decaying function ∑(a Infinite sum formula <1 ae (1-a w)+jasin e X(eu) 2a cos+a =0:X( a<0 2a+ X(eu) √1+2a+a 0
More Examples Infinite sum formula