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A sequence x(n) with the z-transform X(z) = z4 + z2 – 2z + 2 – 3z– 4 is applied as an input to a linear, time-invariant system with the impulse response h(n) = 2δ(n – 3) where
The output at n = 2 is—
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- – 6
- Zero
- 2
- – 4
Correct Option: D
Given that,
X(z) = z4 + z2 – 2z + 2 – 3z– 4
z{h(n)} = 2z– 3 = H(z)
we know that
Y(z) = H(z) X(z) = 2z– 3[z4 + z2 – 2z + 2 – 3z4]
= 2z + 2z– 1 – 2z– 2 + 2z– 3 – 2z– 7
or y(n)= 2δ(n + 1) + 2δ(n – 1) – 4δ(n – 2) + 4δ(n – 3) – 6δ(n – 7)
The output at n = 2
y(2) = 2δ(3) + 2δ(1) – 4δ(0) + 4δ(– 1) – 6δ(– 5)
or y(2) = 2.0 + 2.0 – 4.1 + 4.0 + 6.0
or y(2) = – 4
Hence alternative (D) is the correct choice.
