By Monson Hayes
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Additional resources for Schaum's Outline of Digital Signal Processing
13 A linear system is one that is both homogeneous and additive. (a) Give an example of a system that is homogeneous but not additive. (b) Give an example of a system that is additive but not homogeneous. There are many different systems that are either homogeneous or additive but not both. One example of a system that is homogeneous but not additive is the following: ~ ( n= ) x(n - I)x(n) x(n + I) Specifically, note that if x(n) is multiplied by a complex constant c, the output will be ~ ( n= ) cx(n-l)cx(n) cx(n + I) =c x(n-I)x(n) x(n + 1) which is c times the response to x(n).
By definition, x ( n ) = x , ( n ) x,(n). Therefore, + Note that x,(n)x,(n) is the product of an even sequence and an odd sequence and, therefore, the product is odd. Because the sum for all n of an odd sequence is equal to zero, Thus, the power in x ( n ) is m m which says that the power in x ( n ) is equal to the sum of the powers in its even and odd parts. , evaluate the sum) This is a direct application of the geometric series With the substitution of -n for n we have Therefore, it follows from the geometric series that SIGNALS AND SYSTEMS [CHAP.
27 Derive a closed-form expression for the convolution of x ( n ) and h ( n ) where I N-6 x(n) = (6) u(n) h ( n ) = ( f ) " u ( n - 3) Because both sequences are infinite In length. it is easier to evaluate the convolution sum directly: Note that because x ( n ) = 0 for n < 0 and h ( n ) = 0 for n < 3 , y ( n ) will be equal to zero for n < 3. Substituting x ( n ) and h ( n ) into the convolution sum, we have Due to the step u ( k ) , the lower limit on the sum may be changed to k = 0, and because u ( n - k - 3) is zero for k > n - 3 , the upper limit may be changed to k = n - 3 .
Schaum's Outline of Digital Signal Processing by Monson Hayes