83.1 Discrete-Time Fourier Transform Convergence Condition-An infinite series of the form Y(e0)=∑xnle Jon 1=-0 may or may not converge ·Let K X￥k(e)=∑ xine0n n=-K
§3.1 Discrete-Time Fourier Transform • Convergence Condition - An infinite series of the form = =− − n j j n X (e ) x[n]e = =− − K n K j j n XK (e ) x[n]e may or may not converge • Let
83.1 Discrete-Time Fourier Transform Then for uniform convergence of X(ejo lim X(e)-Xx(e)=0 K→∞ NoW, ifx is an absolutely summable sequence,I.e, ∑xn]<∞ 1=-0
§3.1 Discrete-Time Fourier Transform • Then for uniform convergence of X(ej) lim ( ) − ( ) = 0 → j K j K X e X e n=− x[n] • Now, if x[n] is an absolutely summable sequence, i.e., if
83.1 Discrete-Time Fourier Transform ·The en X(e0)=∑ x[n]e Jon< 2x[n]<o n=-∞0 for all values of o Thus, the absolute summability of xn is a sufficient condition for the existence of the dtFT X(ejo
§3.1 Discrete-Time Fourier Transform • Then = =− =− − n n j j n X (e ) x[n]e x[n] • Thus, the absolute summability of x[n] is a sufficient condition for the existence of the DTFT X(ej) for all values of
83.1 Discrete-Time Fourier Transform Example- The sequence xn=aun for ak I is absolutely summable as ∑on]=∑a/1 <OO 0 and its DTFT X(ejo) therefore converges to 1/(1-ae Jo )uniformly
§3.1 Discrete-Time Fourier Transform • Example - The sequence x[n] = n[n] for ||< 1 is absolutely summable as − = = = =− 1 1 [ ] n 0 n n n n and its DTFT X(ej) therefore converges to 1/(1- e -j) uniformly
83.1 Discrete-Time Fourier Transform Since 2 xn2≤∑ 1=-00 1=-0 an absolutely summa ble sequence has always a finite energy However, a finite-energy sequence is not necessarily absolutely summable
§3.1 Discrete-Time Fourier Transform • Since [ ] [ ] , 2 2 =− n=− n x n x n • However, a finite-energy sequence is not necessarily absolutely summable an absolutely summable sequence has always a finite energy