This set of Signals & Systems Multiple Choice Questions & Answers (MCQs) focuses on “The Z-Transform”.
1. When do DTFT and ZT are equal?
a) When σ = 0
b) When r = 1
c) When σ = 1
d) When r = 0
View Answer
Explanation: Discrete Time Fourier Transform, X(e-jω) = \(\sum_{n=-∞}^∞ x(n) e^{-jωn}\)
Z-Transform, X(Z) = \(∑_{n=-∞}^∞ x(n) z^{-n}\), z = r ejω
When r=1, z = ejω and hence DTFT and ZT are equal.
2. Find the Z-transform of δ(n+3).
a) z
b) z2
c) 1
d) z3
View Answer
Explanation: Given x(n) = δ(n+3)
We know that δ(n+3) = \( \begin{cases}
1 &\text{\(n=-3\)} \\
0 &\text{otherwise} \\
\end{cases}\)
X(Z) = \(\sum\limits_{n=-\infty}^{\infty} x(n) z^{-n} = \sum\limits_{n=-\infty}^{\infty} δ(n+3) z^{-n}\) = z3.
3. Find the Z-transform of an u(n);a>0.
a) \(\frac{z}{z-a}\)
b) \(\frac{z}{z+a}\)
c) \(\frac{1}{1-az}\)
d) \(\frac{1}{1+az}\)
View Answer
Explanation: Given x(n) = an u(n)
We know that \( u(n)=\begin{cases}
1 &\text{\(n≥0\)} \\
0 &\text{\(n<0\)} \\
\end{cases}\)
X(Z) = \(\sum\limits_{n=-\infty}^{\infty} x(n) z^{-n} = \sum\limits_{n=-\infty}^{\infty} a^n u(n) z^{-n}\)
= \(\sum\limits_{n=0}^{∞} a^n (1) z^{-n} = \sum\limits_{n=0}^{∞} (az^{-1})^n = (1-az^{-1})^{-1}\)
= \(\frac{1}{1-az^{-1}} = \frac{z}{z-a}\).
4. Find the Z-transform of cosωn u(n).
a) \(\frac{z(z+cosω)}{z^2-2z cosω+1}\)
b) \(\frac{z(z-cosω)}{z^2-2z cosω+1}\)
c) \(\frac{z(z-cosω)}{z^2+2z cosω+1}\)
d) \(\frac{z(z+cosω)}{z^2+2z cosω+1}\)
View Answer
Explanation: Given x(n) = cosωn u(n)
We know that \( u(n)=\begin{cases}
1 &\text{\(n≥0\)} \\
0 &\text{\(n<0\)} \\
\end{cases}\)
Z[cosωn u(n)] = \(Z\Big[\frac{e^jωn+e^{-jωn}}{2} u(n)\Big] = \frac{1}{2} Z[e^{jωn} u(n)] + \frac{1}{2} Z[e^{-jωn} u(n)]\)
\(= \frac{1}{2} \left(\frac{z}{z-e^{jω}} + \frac{z}{z-e^{-jω}}\right) = \frac{1}{2} \Big[\frac{z(z-e^{-jω}) + z(z-e^{jω})}{(z-e^{jω})(z-e^{-jω})}\Big]\)
\(= \frac{1}{2} \Big\{\frac{z[2z-(e^{jω}+e^{-jω})]}{z^2-z(e^{jω}+e^{-jω})+1}\Big\} = \frac{z(z-cosω)}{z^2-2z cosω+1}\).
5. For causal sequences, the ROC is the exterior of a circle of radius r.
a) True
b) False
View Answer
Explanation: Consider a causal sequence, x(n) = rn u(n)
X(Z) = \(\sum\limits_{n=-∞}^{∞} x(n) z^{-n} = \sum\limits_{n=-∞}^{∞} r^n u(n) z^{-n} = \sum\limits_{n=0}^{∞} r^n (1) z^{-n} = \sum\limits_{n=0}^{∞} (rz^{-1})^n\)
The above summation converges for |rz-1|<1, i.e. for |z|>r
Hence, for the causal sequences, the ROC is the exterior of a circle of radius r.
6. x(n) = an u(n) and x(n) = -an u(-n-1) have the same X(Z) and ROC.
a) True
b) False
View Answer
Explanation: an u(n) ↔ \(\frac{1}{1-az^{-1}} = \frac{z}{z-a}\); ROC:|z|>|a|
-an u(-n-1) ↔ \(\frac{1}{1-az^{-1}} = \frac{z}{z-a}\); ROC:|z|<|a|
Hence, x(n) = an u(n) and x(n) = -an u(-n-1) have the same X(Z) and differ only in ROC.
7. Find the Z-transform of y(n) = x(n+2)u(n).
a) z2 X(Z) – z2 x(0) – zx(1)
b) z2 X(Z) + z2 x(0) – zx(1)
c) z2 X(Z) – z2 x(0) + zx(1)
d) z2 X(Z) + z2 x(0) + zx(1)
View Answer
Explanation: Given y(n) = x(n+2)u(n)
Y(z) = Z[y(n)] = Z[x(n+2)u(n)] = \(\sum\limits_{n=0}^{∞} x(n+2)u(n) z^{-n} = \sum\limits_{n=0}^{∞} x(n+2)z^{-n}\)
Let n + 2 = p,i.e.n = p – 2
Y(z) = \(∑_{p=2}^∞ x(p)z^{-(p-2)} = z^2 ∑_{p=2}^∞ x(p)z^{-p} = z^2 ∑_{p=0}^∞ x(p)z^{-p} – x(0) – x(1) z^{-1}\)
=z2 X(Z) – z2 x(0) – zx(1).
8. Find the Z-transform of x(n) = a|n|; |a|<1.
a) \(\frac{z}{z-a} – \frac{z}{z-(1/a)}\)
b) \(\frac{z}{z-(1/a)} – \frac{z}{z-a}\)
c) \(\frac{z}{z-a} + \frac{z}{z-(1/a)}\)
d) \(\frac{1}{z-a} – \frac{1}{z-(1/a)}\)
View Answer
Explanation: a^|n| = a^n u(n) + a-n u(-n-1) = an u(n) + \((\frac{1}{a})^n\) u(-n-1)
Z[a|n|] = Z[an u(n)] + Z[\((\frac{1}{a})^n\) u(-n-1)] = \(\frac{z}{z-a} – \frac{z}{z-(1/a)}\).
9. Find the Z-transform of u(-n).
a) \(\frac{1}{1-z}\)
b) \(\frac{1}{1+z}\)
c) \(\frac{z}{1-z}\)
d) \(\frac{z}{1+z}\)
View Answer
Explanation: Given x(n) = u(-n)
Z[x(n)] = X(Z) = \(∑_{n=-∞}^∞ x(n) z^{-n} = ∑_{n=-∞}^∞ u(-n) z^{-n} = ∑_{n=-∞}^0 (1) z^{-n}\)
=\(∑_{n=0}^∞ z^n = \frac{1}{1-z}\).
10. For a right hand sequence, the ROC is entire z-plane.
a) True
b) False
View Answer
Explanation: If x(n) is finite duration causal sequence (right-sided sequence),the X(z) converges for all values of z except at z = 0. Hence, the ROC is entire z-plane except at z = 0.
Sanfoundry Global Education & Learning Series – Signals & Systems.
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