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( a ) Linearity, shift-invariance, stability, and causality are easily shown to be preserved in a cascade. For example, the response of S I to the input nxl ( n ) h x z ( n ) will be a w l ( n ) b w 2 ( n )due to the linearity of S,. With this as the input to S2, the response will be, again by linearity, a y , ( n ) hy7(n). Therefore, if both S I and S2 are linear, the cascade will be linear.

1 2(18)

Similarly, for shift-invariance, if x ( n - no) is input to S , , the response will be w ( n - no). In addition, because S2 is shift-invariant, the response to w ( n - n o ) will be y(n - n o ) . Therefore, the response of the cascade to x(n - no) is y ( n - no), and the cascade is shift-invariant. To establish stability, note that with SI being stable, if x ( n ) is bounded, the output w ( n ) will be bounded. With w ( n ) a bounded input to the stable system S2, the response y ( n ) will also be bounded. Therefore, the cascade is stable. Finally, causality of the cascade follows by noting that if S2 is causal, y ( n ) at time n = no depends only on w ( n ) for n 5 no. With S I being causal, w ( n ) for n 5 no will depend only on the input x ( n ) for n 5 no, and it follows that the cascade is causal.

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1 2(18) 23

( b ) If SI and S2 are nonlinear, it is not necessarily true that the cascade will be nonlinear because the second system may undo the nonlinearity of the first. For example, with

lim f (x) = lim [f (x) f (a) + f (a)]

7.40 Applying principles 5 and 6 (a) Find the length of the leg of an isosceles right triangle whose hypotenuse has length 28 [Fig. 7-44(a)].

although both SI and Sz are nonlinear, the cascade is the identity system and, therefore, is linear.

Fig. 7-44

(c) As in ( b ) , if S I and S2 are shift-varying, it is not necessarily true that the cascade will be shift-varying. For example. if the first system is a modulator.

(b) An isosceles trapezoid has base angles measuring 45 . If the upper base has length 12 and the altitude has length 3, find the lengths of the lower base and each leg [Fig. 7-44(b)].

and the second is a demodulator,

= lim [f (x) f (a)] + lim f (a)

the cascade is shift-invariant, even though a modulator and a demodulator are shift-varying. Another example is when S l is an up-sampler

1 2 (28) 22

y(n) = w(2n)

7.1. Express each of the following ratios in lowest terms: (a) 20 cents to 5 cents (b) 5 dimes to 15 dimes (c) 30 lb to 25 lb (d) 20 to 14 (e) 27 min to 21 min (f) 50% to 25% (g) 15 to 75 (h) 33% to 77% (i) $2.20 to $3.30 (j) $.84 to $.96 (k) (l)

In this case, the cascade is shift-invariant, and y ( n ) = x ( n ) . However, if the order of the systems is reversed, the cascade will no longer be shift-invariant. Also, if a linear shift-invariant system, such as a unit delay, is inserted between the up-sampler and the down-sampler, the cascade of the three systems will, in general, be shift-varying.

(m) 5 ft to 1ft 4 (n)

The first nonzero value of a finite-length sequence x(n) occurs at index n = -6 and has a valuex(-6) = 3, and the last nonzero value occurs at index n = 24 and has a value x(24) = -4. What is the index of the first nonzero value in the convolution y(n) = x(n) * x(n) and what is its value What about the last nonzero value

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