1
GATE ECE 2014 Set 3
Numerical
+2
-0
The z-transform of the sequence x$$\left[ n \right]$$ is given by x(z)= $${1 \over {{{(1 - 2{z^{ - 1}})}^2}}}$$ , with the region of convergence $$\left| z \right| > 2$$. Then, $$x\left[ 2 \right]$$ is ____________________.
2
GATE ECE 2014 Set 3
Numerical
+2
-0
Let $${H_1}(z) = {(1 - p{z^{ - 1}})^{ - 1}},{H_2}(z) = {(1 - q{z^{^{ - 1}}})^{ - 1}}$$ , H(z) =$${H_1}(z)$$ +r $${H_2}$$. The quantities p, q, r are real numbers. Consider , p=$${1 \over 2}$$, q=-$${1 \over 4}$$ $$\left| r \right|$$ <1. If the zero H(z) lies on the unit circle, the r = ____________________________.
3
GATE ECE 2014 Set 2
+2
-0.6
The input-output relationship of a causal stable LTI system is given as
𝑦[𝑛] = 𝛼 𝑦[𝑛 − 1] + $$\beta$$ x[n].
If the impulse response h[n] of this system satisfies the condition $$\sum\limits_{n = 0}^\infty h$$[n] = 2, the relationship between α and is $$\alpha$$ and $$\beta$$ is
A
𝛼 = 1 − $${\beta \over 2}$$
B
𝛼 = 1 + $${\beta \over 2}$$
C
𝛼 = 2𝛽
D
𝛼 = −2𝛽
4
GATE ECE 2014 Set 1
+2
-0.6
Let x $$\left[ n\right]$$= $${\left( { - {1 \over 9}} \right)^n}\,u(n) - {\left( { - {1 \over 3}} \right)^n}u( - n - 1).$$ The region of Convergence (ROC) of the z-tansform of x$$\left[ n \right]$$
A
is $$\left| z \right| > {1 \over 9}$$
B
is $$\left| z \right| < {1 \over 3}$$
C
is $${1 \over 3} > \left| z \right| > {1 \over 9}$$
D
does not exist.
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