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This robust learn consultant offers 571 difficulties in indications and structures, absolutely solved step by step! From SchaumOs, the unique examine advisor, and studentsO favourite with over 30 million publications soldNthis solution-packed timesaver is helping you grasp all types of challenge you are going to face in your checks, from uncomplicated questions about linear time-invariant platforms to complicated Fourier research of discrete-time platforms and country house research. cross on to the solutions you would like with a whole index. suitable with any lecture room textual content, SchaumOs define of signs and platforms is so entire itOs the suitable instrument for graduate or specialist examination prep!

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**Extra resources for Schaum's outline of theory and problems of signal and systems**

**Example text**

Let T represent a continuous-time LTI system. Then show that T{es'} = ks' where s is a complex variable and h is a complex constant. Let y ( t ) be the output of the system with input x ( t ) = e". Then T { e S t )= y ( t ) Since the system is time-invariant, we have T(es('+'(l))= y ( I +to) for arbitrary real t o . Since the system is linear, we have T { ~ ~ ( ' + '= I IT{eS' ) ) e s ' ~ } = e " ~ T { e S ' )= e S ' ~ y ( t ) Hence, y(r +I,) = e H 0 y ( t ) CHAP. 11 SIGNALS AND SYSTEMS Setting t = 0, we obtain to) =y(0)eS1" Since to is arbitrary, by changing to to t,' we can rewrite Eq.

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. 42. 67). SIGNALS AND SYSTEMS [CHAP. 116) where x * [ n ] is the complex conjugate of x [ n ] . Then T { x l [ n I + x 2 [ n I } = { x ~ [ n+I ~ 2 [ n I } =* x T [ n I + x ? [ n I = ~ l [ n +I ~ 2 [ n I Next, if a is any arbitrary complex-valued constant, then T { a x [ n ] } = { a x [ n ] )* = cu*x*[n] = a * y [ n ] # a y [ n ] Thus, the system is additive but not homogeneous.

1-23(c) we obtain (d Fig. 5. Sketch and label the even and odd components of the signals shown in Fig. 1-24. Using Eqs. 6), the even and odd components of the signals shown in Fig. 1-24 are sketched in Fig. 1-25. [CHAP. 1 SIGNALS AND SYSTEMS (4 Fig. 6. Find the even and odd components of x ( r ) Let x,(r) and x,(I) = eJ'. be the even and odd components of ei', respectively. eJ' = x , ( I ) + x , ( I ) From Eqs. ( 1 . 6) and using Euler's formula, we obtain x,( I ) = $ ( e J r+ e - J ' ) = cos I x,,(I) = f(ei'-e-j') =jsint Show that the product of two even signals or of two odd signals is an even signal and that the product of an even and an odd signaI is an odd signal.