Digital Signal Processing

Properties of Z Transform

Question 1
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 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%
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 4
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 5
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 6
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 7
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 8
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal x(n)=δ(n-n0)?
zn0
z-n0
zn-n0
zn+n0
Explanation:
From the definition of z-transform,
Question 9
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 10
Marks : +2 | -2
Pass Ratio : 100%
What is the z-transform of the signal defined as x(n)=u(n)-u(n-N)?
\\(\\frac{1+z^N}{1+z^{-1}}\\)
\\(\\frac{1-z^N}{1+z^{-1}}\\)
\\(\\frac{1+z^{-N}}{1+z^{-1}}\\)
\\(\\frac{1-z^{-N}}{1-z^{-1}}\\)
Explanation: