This is not quite a homework question, but I hope close enough that it can be posted here. I'm going through a signal processing book on my free time, doing all the problems and so on, and I've come across a problem which I'm not too sure about how to solve. Here it is--(adsbygoogle = window.adsbygoogle || []).push({});

Consider the sequence:

h: Z -> C (integers to complex)

h(n) = [C1*(p1^n) + C2*(p2^n) + . . . + CM*(pM^n)]u(n) (0),

where u(n) = 1 for n >= 0, 0 else.

without using Z transforms, show that h(n) satisfies the difference equation for n >= M:

h(n) + a1*h(n-1) + a2*h(n-2) + . . . + aM*h(n-M) = 0 (1),

where {1,a1,a2,...,aM} are the coefficients of the polynomial whose roots are (complex) numbers {p1,p2,...,pM}, that is,

1 + a1*(z^-1) + a2*(z^-2) + ... + aM*(z^-M) =

(1-p1*(z^-1))*(1-p2*(z^-2))*...*(1-pM*(z^-M))

Note: the Ci are arbitrary and the restriction n >= M is necessary.

By the substitution of (0) into (1) one gets an equation in ai,Ci,pi, but I don't quite see how to show that the coefficients ai are what the problem says they are. I'm thinking about matrix multiplication, determinants and characteristic polynomials but my memory of that stuff is fading and I sold back all my college books which talk specifically about those. Do any of you have any ideas, or know how to do this type of problem? Help would be appreciated. Thanks!

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# Difference equation

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