LetConsider the behavior of the error X(ej)- Xk(ej) as K goes to .3.2.4 Convergence Conditionis said to exist if it converges in some sense.3 Discrete-Time Signals in the Frequency Domain1/16nUniform convergence of X(ej ) xn is absolute summable.for all values of as K goes to nA sufficient condition for uniform convergence:3 Discrete-Time Signals in the Frequency Domain2/16If xn is an absolutely summable sequence, i.e., ifThenfor all values of 3 Discrete-Time Signals in the Frequency Domain3/162. xn is a finite-energy sequence.mean-square convergence of X(ej): the total energy of the error X(ej)- Xk(ej) must approach zero at each value of as K goes to 3 Discrete-Time Signals in the Frequency Domain4/16(3.30) So,Examining 3 Discrete-Time Signals in the Frequency DomainnExample 3.8 Consider the DTFT5/16K=10K=20K=30K=40Gibbs phenomenon (discussed in Section 10.2.3)3 Discrete-Time Signals in the Frequency Domain6/16Shown below(3.27) 3 Discrete-Time Signals in the Frequency Domain7/16So, the DTFT of hLPndoes not converge uniformly to HLP(ej) for all values of , but converge to HLP(ej) in the mean-square sense.(3.30) 3 Discrete-Time Signals in the Frequency Domain8/16The type of sequences that a DTFT representation is possible using the Dirac Delta function () nneither absolutely summable nor square summablen,cos(0n+)3. The DTFT of xn is Dirac Delta function 3 Discrete-Time Signals in the Frequency Domain9/16w Dirac Delta function ( ) is the limiting form of a unit area pulse function p ( ) as goes to zero satisfyingThe sampling property of the Dirac delta function3 Discrete-Time Signals in the Frequency Domain10/16nExample 3.9 Consider the complex exponential sequencewhere ( ) is an impulse function of andIts DTFT is given by3 Discrete-Time Signals in the Frequency Domain11/16is a periodic function of w with a period 2 and is called a periodic impulse train. nTo verify that X(ej ) given above is indeed the DTFT of xn=ej 0n we compute the inverse DTFT of X(ej ) The function3 Discrete-Time Signals in the Frequency Domain12/16Where we have used the sampling property of the Dirac delta function3 Discrete-Time Signals in the Frequency Domain13/16 Sequence DTFTTable 3.3 Commonly Used DTFT Pairs3 Discrete-Time Signals in the Frequency Domain14/163.2.5 Strength of a DTFTp is a positive integer.In practice, the value of p used is typically 1 or 2 or .(3.35)The strength of a DT FT is given by its norm. of X(ej) is defied by 3 Discrete-Time Signals in the Frequency Domain15/16(3.36)Peak absolute value is root-mean-squared(rms) value of X(ej) is the mean absolute value of X(ej) 3 Discrete-Time Signals in the Frequency Domain16/16