Practice Question on "Digital Signal Processing"

Topic: Computing a z-transform


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))zn

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.

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva