Digital Signal Processing

One Sided Z Transform

Question 1
Marks : +2 | -2
Pass Ratio : 100%
If X+(z) is the one sided z-transform of x(n), then what is the one sided z-transform of x(n-k)?
z-k X+(z)
zk X+(z-1)
z-k \\([X^+(z)+\\sum_{n=1}^k x(-n)z^n]\\); k>0
z-k \\([X^+(z)+\\sum_{n=0}^k x(-n)z^n]\\); k>0
Explanation:
From the definition of one sided z-transform we have,
Question 2
Marks : +2 | -2
Pass Ratio : 100%
What is the one sided z-transform of x(n)=δ(n-k)?
z-k
zk
0
1
Explanation:
Since the signal x(n)= δ(n-k) is a causal signal i.e., it is defined for n>0 and x(n)=1 at z=k
Question 3
Marks : +2 | -2
Pass Ratio : 100%
If x(n)=an, then what is one sided z-transform of x(n-2)?
\\(\\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} + a^{-2}\\)
\\(\\frac{z^{-2}}{1-az^{-1}} – a^{-1}z^{-1} + a^{-2}\\)
\\(\\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} – a^{-2}\\)
\\(\\frac{z^{-2}}{1+az^{-1}} + a^{-1}z^{-1} + a^{-2}\\)
Explanation:
Question 4
Marks : +2 | -2
Pass Ratio : 100%
What is the one sided z-transform of x(n)=δ(n+k)?
z-k
0
zk
1
Explanation:
Since the signal x(n)=δ(n+k) is an anti causal signal i.e., it is defined for n<0 and x(n)=1 at z=-k. Since the one sided z-transform is defined only for causal signal, in this case X+(z)=0.
Question 5
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Pass Ratio : 100%
The z-transform of a signal x(n) whose definition is given by \\(X(z)=\\sum_{n=0}^{\\infty} x(n)z^{-n}\\) is known as _____________
Unilateral z-transform
Bilateral z-transform
Rational z-transform
None of the mentioned
Explanation:
The z-transform of the x(n) whose definition exists in the range n=-∞ to +∞ is known as bilateral or two sided z-transform. But in the given question the value of n=0 to +∞. So, such a z-transform is known as Unilateral or one sided z-transform.
Question 6
Marks : +2 | -2
Pass Ratio : 100%
If X+(z) is the one sided z-transform of the signal x(n), then \\(\\lim_{n \\rightarrow \\infty} x(n)=\\lim_{z\\rightarrow 1}(z-1) X^+(z)\\) is called Final value theorem.
True
False
Explanation:
In the above theorem, we are calculating the value of x(n) at infinity, so it is called as final value theorem.
Question 7
Marks : +2 | -2
Pass Ratio : 100%
For what kind of signals one sided z-transform is unique?
All signals
Anti-causal signal
Causal signal
None of the mentioned
Explanation:
One sided z-transform is unique only for causal signals, because only these signals are zero for n<0.
Question 8
Marks : +2 | -2
Pass Ratio : 100%
The impulse response of a relaxed LTI system is h(n)=anu(n), |a|<1. What is the value of the step response of the system as n→∞?
\\(\\frac{1}{1+a}\\)
\\(\\frac{1}{1-a}\\)
\\(\\frac{a}{1+a}\\)
\\(\\frac{a}{1-a}\\)
Explanation:
The step response of the system is y(n)=x(n)*h(n) where x(n)=u(n)
Question 9
Marks : +2 | -2
Pass Ratio : 100%
If x(n)=an, then what is one sided z-transform of x(n+2)?
\\(\\frac{z^{-2}}{1-az^{-1}}\\) + a-1z-1 + a-2
\\(\\frac{z^{-2}}{1-az^{-1}}\\) – a-1z-1 + a-2
\\(\\frac{z^2}{1-az^{-1}}\\) + a z + z2
\\(\\frac{z^2}{1+az^{-1}}\\) – z2 – az
Explanation:
We will apply the time advance theorem with the value of k=2.We obtain,
Question 10
Marks : +2 | -2
Pass Ratio : 100%
What is the one sided z-transform X+(z) of the signal x(n)={1,2,5,7,0,1}?
z2+2z+5+7z-1+z-3
5+7z+z3
z-2+2z-1+5+7z+z3
5+7z-1+z-3
Explanation:
Since the one sided z-transform is valid only for n>=0, the z-transform of the given signal will be X+(z)= 5+7z-1+z-3.