Practice Question on "Digital Signal Processing"
Topic: Computing a z-transform
Contents
Question
Compute the compute the z-transform (including the ROC) of the following DT signal:
$ x[n]= n^2 \left( u[n+3]- u[n-1] \right) $
(Write enough intermediate steps to fully justify your answer.)
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Answer 1
Andrei Henrique Patriota Campos x[n] = n2(u[n + 2] − u[n − 1]).
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
$ = \sum_{n=-3}^{0} n^2 z^{-n} $
= 9z3 + 4z2 + z
= z3(9 + 4z − 1 + z − 2)
= X(z) = (9 + 4z − 1 + z − 2) / (z − 3), for all z in complex plane.
- TA's comment: z can not be $ \infty $ for the z transform to converge
Answer 2
x[n] = n2(u[n + 3] − u[n − 1])
x[n] = n2(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n))
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
$ X(z) = \sum_{n=-\infty}^{+\infty} n^2(\delta(n+3)+\delta(n+2)+\delta(n+1)+\delta(n)) z^{-n} $
X(z) = 9z3 + 4z2 + z + 1 for all z in complex plane
- TA's comment: When n=0,x[n]=0. So the constant term is 0. ROC is everywhere except z=infinity
Answer 3
Write it here.
Answer 4
Write it here.
Answer 5
Tony Mlinarich
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
X(z) = n2(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n) + δ(n − 1))z − n
X(z) = 9z3 + 4z2 + z + 1/z<\span>
- TA's comment: u[n+3]-u[n-1] is non-zero only when n=-3,-2,-1,0. So x[n]= n2(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n)). ROC is everywhere except z=infinity
Answer 7
Yixiang Liu
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
$ X(z) = \sum_{n=-\infty}^{+\infty} n^{2}[{u[n+3]-u[n-1]}]z^{-n} $
This expression equals to zero except n = -3, -2, -1
so X(z) = x[ − 3]z3 + x[ − 2]z2 + x[ − 1]z1
= 9z^{3} + 4z^{2} + z
- TA's comment: ROC is everywhere except z=infinity.
Answer 8
Xi Wang
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
= X(z) = (9z + 3 + 4z + 2 + z). The range of the value of z is from negative infinity to positive infinity
- TA's comment: Show your derivation
Answer 9
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
$ X(z) = \sum_{n=-3}^{+1} x[n] z^{-n} $
= X(z) = 9z + 3 + 4z +2 + z + 1 for all z in complex plane
- TA's comment: In your second step, the summation should be from -3 to 0 . There should be no constant termsince x[0]=0. ROC is everywhere except z=infinity
Answer 10
Cary Wood
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n} $
$ X(z) = \sum_{n=-3}^{0} x[n] z^{-n} $
= X(z) = 9z + 3 + 4z + 2 + z, for all z in complex plane
- TA's comment: ROC is everywhere except z=infinity.
Answer 11
Shiyu Wang
x[n] = n2(u[n + 3] − u[n − 1])
x[n] = n2 (-3=< n < 1)
$ X(z) = \sum_{n=-3}^{0} n^2 z^{-n} $
x(z)=9z3+4z2+z, for all z in complex plane except z=infinity
- TA's comment: Simple and straightforward.
Answer 12
Matt Miller
x[n] = n2(u[n+3]-u[n-1])
x[n] = n2u[n+3] - n2u[n-1]
x[n] = n2|0-3
$ X(z) = \sum_{n=-3}^{0} n^2 z^{-n} $
X(z) = (-3)2z3 + (-2)2z2 + (-1)2z1 + (0)2z0
X(z) = 9z3 + 4z2 + z
lim z->inf X(1/2) = 0, lim z->0 X(1/2) = inf --> valid for all Z in complex plane.
- TA's comment: In the third step, it's better write it as a summation.
