Digital Signal Processing

Properties of Z Transform

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal x(n)=[3(2n)-4(3n)]u(n)?
\\(\\frac{3}{1-2z^{-1}}-\\frac{4}{1-3z^{-1}}\\)
\\(\\frac{3}{1-2z^{-1}}-\\frac{4}{1+3z^{-1}}\\)
\\(\\frac{3}{1-2z}-\\frac{4}{1-3z}\\)
None of the mentioned
Explanation:
Let us divide the given x(n) into x1(n)=3(2n)u(n) and x2(n)= 4(3n)u(n)
Question 2
Marks : +2 | -2
Pass Ratio : 100%
If Z{x(n)}=X(z) and the poles of X(z) are all inside the unit circle, then the final value of x(n) as \\(n\\rightarrow\\infty\\) is given by i.e., \\(\\lim_{n\\rightarrow\\infty}\\)x(n)=?
\\(\\lim_{z \\rightarrow 1} [(z-1) ⁡ X(z)] \\)
\\(\\lim_{z \\rightarrow 0} [(z-1) ⁡ X(z)] \\)
\\(\\lim_{z \\rightarrow -1} [(z-1) X(z)] \\)
\\(\\lim_{z \\rightarrow 1} [(z+1) ⁡ X(z)] \\)
Explanation:
According to the Final Value theorem of z-transform we have,
Question 3
Marks : +2 | -2
Pass Ratio : 100%
If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then Z{x1(n)*x2(n)}=?
X1(z).X2(z)
X1(z)+X2(z)
X1(z)*X2(z)
None of the mentioned
Explanation:
According to the convolution property of z-transform, the z-transform of convolution of two sequences is the product of their respective z-transforms.
Question 4
Marks : +2 | -2
Pass Ratio : 100%
If x(n) is causal, then \\(\\lim_{z\\rightarrow\\infty}\\) X(z)=?
x(-1)
x(1)
x(0)
Cannot be determined
Explanation:
According to the initial value theorem, X(z)=x(0)+x(1)z-1+x(2)z-2+….
Question 5
Marks : +2 | -2
Pass Ratio : 100%
Which of the following justifies the linearity property of z-transform?[x(n)↔X(z)].
x(n)+y(n) ↔ X(z)Y(z)
x(n)+y(n) ↔ X(z)+Y(z)
x(n)y(n) ↔ X(z)+Y(z)
x(n)y(n) ↔ X(z)Y(z)
Explanation:
According to the linearity property of z-transform, if X(z) and Y(z) are the z-transforms of x(n) and y(n) respectively then, the z-transform of x(n)+y(n) is X(z)+Y(z).
Question 6
Marks : +2 | -2
Pass Ratio : 100%
If X(z) is the z-transform of the signal x(n), then what is the z-transform of x*(n)?
X(z*)
X*(z)
X*(-z)
X*(z*)
Explanation:
According to the conjugation property of z-transform, we have
Question 7
Marks : +2 | -2
Pass Ratio : 100%
If the ROC of X(z) is r1<|z|<r2, then what is the ROC of X(a-1z)?
|a|r1<|z|<|a|r2
|a|r1>|z|>|a|r2
|a|r1<|z|>|a|r2
|a|r1>|z|<|a|r2
Explanation:
Given ROC of X(z) is r1<|z|<r2
Question 8
Marks : +2 | -2
Pass Ratio : 100%
If X(z) is the z-transform of the signal x(n) then what is the z-transform of anx(n)?
X(az)
X(az-1)
X(a-1z)
None of the mentioned
Explanation:
We know that from the definition of z-transform
Question 9
Marks : +2 | -2
Pass Ratio : 100%
X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal nx(n)?
\\(-z\\frac{dX(z)}{dz}\\)
\\(z\\frac{dX(z)}{dz}\\)
\\(-z^{-1}\\frac{dX(z)}{dz}\\)
\\(z^{-1}\\frac{dX(z)}{dz}\\)
Explanation:
Question 10
Marks : +2 | -2
Pass Ratio : 100%
If X(z) is the z-transform of the signal x(n), then what is the z-transform of the signal x(-n)?
X(-z)
X(z-1)
X-1(z)
None of the mentioned
Explanation:
From the definition of z-transform, we have