By Simon Haykin

ISBN-10: 013322760X

ISBN-13: 9780133227604

Represents the main complete remedy to be had of neural networks from an engineering standpoint. Examines the entire very important elements of this rising know-how. DLC: Adaptive filters.

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**Extra resources for Adaptive Filter Theory (3rd Edition)**

**Example text**

If a signal is periodic, determine its fundamental period. ( ( a ) x(r) = cos 2r 1) +- ( 1 ("4") (2+ . ( y) ( g ) x[nl=cos - cos (h) x[n] = cos Am. 52. 53. ( a ) What is S(2t)? ( b ) What is S[2n]? Am. 54. See Prob. 17. 55. = -S1(t) Use Eqs. 99). Evaluate the following integrals: Ans. 56. =0 Consider a continuous-time system with the input-output relation Determine whether this system is ( a ) linear, ( b ) time-invariant, ( c ) causal. 57. Consider a continuous-time system with the input-output relation Determine whether this system is ( a ) linear, ( b ) time-invariant.

When the inputs to the system are xl[n], x,[n], and x,[n], the outputs of the system are yl[n], y,[n], and y , [ n ] as shown. Determine whether the system is linear. From Fig. 1-39 it is seen that x 3 [ n ]= x l [ n ] + x , [ n - 21 CHAP. 1) SIGNALS AND SYSTEMS Fig. 1-39 Thus, if T is linear, then T{xJnI) = T{x,[nl} + T(x,[n - 21) = y , [ n I + y , [ n - 21 which is shown in Fig. 1-40. From Figs. 1-39 and 1-40 we see that y 3 b I +y,[nI + y J n - 21 Hence, the system is not linear. - 2 - 1 0 1 2 3 4 n -2 - 1 Fig.

11 SIGNALS AND SYSTEMS ( d l x(t) = cos r + sin f i r =x,(r) +x2(r) where x,(t) = cos r = cos o , t is periodic with TI = 27r/01 = 27r and x2(t) = sin f i t = sin w2t is periodic with T2 = 27r/02 = fir. Since T,/T2 = fi is an irrational number, x(t) is nonperiodic. (e) Using the trigonometric identity sin2 0 = t(l - cos 201, we can write where x,(t) = $ is a dc signal with an arbitrary period and x2(t) = - $ cos2r = - I cos 0 2 t is periodic with T2 = 2n/w2 = 7. Thus, x(t) is periodic with fundamental period To = T.

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