it discards the imaginary part real-valued signals.

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0 ' x(n). 1

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One could state, however, that this system is invertible over the set of

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Consider the cascade of two systems. S I and S2.

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(a) If both SI and S2 are linear, shift-invariant, stable, and causal, will the cascade also be linear,

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shift-invariant, stable, and causal

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(b) If both S I and S2 are nonlinear, will the cascade be nonlinear

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(c) If both SI and S2are shift-varying, will the cascade be shift-varying

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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.

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CHAP. 11

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SIGNALS AND SYSTEMS

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.

( 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

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

(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.

and the second is a demodulator,

y ( n ) = w ( n ) . e-Inq

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

and S2 is a down-sampler

y(n) = w(2n)

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.

Convolution 1.24

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

Because we are convolving two finite-length sequences, the index of the first nonzero value in the convolution is equal to the sum of the indices of the first nonzero values of the two sequences that are being convolved. In this case, the index is n = - 12, and the value is y(-12) = x2(-6) = 9 Similarly, the index of the last nonzero value is at n = 48 and the value is

The convolution of two finite-length sequences will be finite in length. 1s it true that the convolution of a finite-length sequence with an infinite-length sequence will be infinite in length