Digital Signal Processing

Properties of Z Transform

Question 1
Marks : +2 | -2
Pass Ratio : 100%
What is the signal x(n) whose z-transform X(z)=log(1+az-1);|z|>|a|?
\\((-1)^n.\\frac{a^n}{n}.u(n-1)\\)
\\((-1)^n.\\frac{a^n}{n}.u(n+1)\\)
\\((-1)^{n-1}.\\frac{a^n}{n}.u(n-1)\\)
\\((-1)^{n-1}.\\frac{a^n}{n}.u(n+1)\\)
Explanation:
Given X(z)=log(1+az-1)
Question 2
Marks : +2 | -2
Pass Ratio : 100%
If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then what is the z-transform of correlation between the two signals?
X1(z).X2(z-1)
X1(z).X2(z-1)
X1(z).X2(z)
X1(z).X2(-z)
Explanation:
We know that rx1x2(l)=x1(l)*x2(-l)
Question 3
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 4
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal x(n)=an(sinω0n)u(n)?
\\(\\frac{az^{-1} sin\\omega_0}{1+2 az^{-1} cos\\omega_0+a^2 z^{-2}}\\)
\\(\\frac{az^{-1} sin\\omega_0}{1-2 az^{-1} cos\\omega_0- a^2 z^{-2}}\\)
\\(\\frac{(az)^{-1} cos\\omega_0}{1-2 az^{-1} cos\\omega_0+a^2 z^{-2}}\\)
\\(\\frac{az^{-1} sin\\omega_0}{1-2 az^{-1} cos\\omega_0+a^2 z^{-2}}\\)
Explanation:
Question 5
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 6
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 7
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 8
Marks : +2 | -2
Pass Ratio : 100%
What is the convolution x(n) of the signals x1(n)={1,-2,1} and x2(n)={1,1,1,1,1,1}?
{1,1,0,0,0,0,1,1}
{-1,-1,0,0,0,0,-1,-1}
{-1,1,0,0,0,0,1,-1}
{1,-1,0,0,0,0,-1,1}
Explanation:
Question 9
Marks : +2 | -2
Pass Ratio : 100%
What is the signal whose z-transform is given as X(z)=\\(\\frac{1}{2Ï€j} \\oint X_1 (v) X_2 (\\frac{z}{v})v^{-1} dv\\)?
x1(n)*x2(n)
x1(n)*x2(-n)
x1(n).x2(n)
x1(n)*x2*(n)
Explanation:
From the convolution property in z-domain we have,
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