This set of Tough Signals & Systems Questions and Answers focuses on “Properties of Z-Transforms – 2”.
1.Find the Z-transform of the causal sequence x(n) = {1,0,-2,3,5,4}. (1 as the reference variable)
a) 1 – 2z-2 + 3z-3 + 5z-4 + 4z-5
b) 1 – 2z2 + 3z3 + 5z4 + 4z5
c) z-1 – 2z2 + 3z3 + 5z4 + 4z5
d) z – 2z3 + 3z4 + 5z5 + 4z6
View Answer
Explanation: Given sequence values are :
x(0)=1, x(1)=0, x(2)=-2, x(3)=3, x(4)=5, x(5)=4.
We know that
\(X(Z) = \sum\limits_{n=-∞}^{∞} x(n) z^{-n}\)
X(Z) = x(0) + x(1) z-1 + x(2) z-2 + x(3) z-3 + x(4) z-4 + x(5) z-5
X(Z) = 1 – 2z-2 + 3z-3 + 5z-4 + 4z-5.
2. Find the Z-transform of the anticausal sequence x(n) = {4,2,3,-1,-2,1}. (1 as the reference variable)
a) 4z5 + 2z4 + 3z3 – z2 – 2z + 1
b) 4z-5 + 2z-4 + 3z-3 -z-2 – 2z-1 + 1
c) -4z5 – 2z4 – 3z3 + z2 + 2z – 1
d) -4z-5 – 2z-4 – 3z-3 + z-2 + 2z-1 – 1
View Answer
Explanation: Given sequence values are :
x(-5)=4, x(-4)=2, x(-3)=3, x(-2)=-1, x(-1)=-2, x(0)=1
We know that
\(X(Z) = \sum\limits_{n=-∞}^{∞} x(n) z^{-n}\)
X(Z) = x(-5) z5 + x(-4) z4 + x(-3) z3 + x(-2) z2 + x(-1)z + x(0)
X(Z) = 4z5 + 2z4 + 3z3 – z2 – 2z + 1.
3. Find the Z-transform of x(n) = u(-n).
a) \(\frac{1}{z-1}\)
b) \(\frac{1}{z+1}\)
c) \(\frac{1}{1-z}\)
d) \(\frac{-1}{z+1}\)
View Answer
Explanation: Given x(n) = u(-n)
Time reversal property of Z-transform states that
If x(n) ↔ X(z), then x(-n) ↔ X(\(\frac{1}{z}\))
Z[u(-n)] = \(\left(\frac{1}{z-1}\right)_{z=(1/z)} = \frac{1/z}{(1/z)-1} = \frac{1}{1-z}\).
4. Find the Z-transform of x(n) = u(-n-2).
a) \(\frac{z^2}{z-1}\)
b) \(\frac{z^2}{1-z}\)
c) \(\frac{z^2}{1+z}\)
d) \(\frac{z^2}{2-z}\)
View Answer
Explanation: Given x(n) = u(-n-2)
Time shifting property of Z-transform states that
If x(n) ↔ X(z), then x(n-m) ↔ z-m X(z)
Z[u(-n-2)] = Z{u[-(n+2)]}=z2 Z[u(-n)] = \(\frac{z^2}{1-z}\).
5. Find the Z-transform of x(n) = n2 u(n).
a) \(\frac{z(z-1)}{(z-1)^3}\)
b) \(\frac{z(z+1)}{(z-1)^3}\)
c) \(\frac{z(z+1)}{(z+1)^3}\)
d) \(\frac{z(z-1)}{(z+1)^3}\)
View Answer
Explanation: Given x(n) = n2 u(n)
We know that X(z) = Z[x(n)] = Z[u(n)] = \(\frac{z}{1-z}\)
The multiplication of n or differentiation in z-domain property of Z-transform states that
If x(n) ↔ X(z), then nk x(n) ↔ (-1)k zk \(\frac{d^k X(z)}{dz^k}\)
Z[n2 u(n)] = z2 \(\frac{d^2 X(z)}{dz^2} = z^2 \frac{d^2}{dz^2}[\frac{z}{1-z}] = \frac{z(z+1)}{(z-1)^3}\).
6. Find the Z-transform of x(n) = 2n u(n-2).
a) \(\frac{z}{z-2}\)
b) \(\frac{z}{z+2}\)
c) \(\frac{z}{z(z-2)}\)
d) \(\frac{4}{z(z-2)}\)
View Answer
Explanation: Given x(n) = 2n u(n-2)
Time shifting property of Z-transform states that
If x(n) ↔ X(z), then x(n-m) ↔ z-m X(z)
Z[u(n-2)] = z-2 Z[u(n)] = \(z^{-2} \frac{z}{z-1} = \frac{1}{z(z-1)}\)
The multiplication by an exponential sequence property of Z-transform states that
If x(n) ↔ X(z), then an x(n) ↔ X(z/a)
Z[2n u(n-2)] = Z[u(n-2)]|z=(z/2) = \(\Big[\frac{1}{z(z-1)}\Big]_{z=(z/2)}\)
\( = \frac{1}{(z/2)[(z/2)-1]} = \frac{4}{z(z-2)}\).
7. Find the Z-transform of x(n) = n[an u(n)].
a) \(\frac{z}{z(z-a)}\)
b) \(\frac{az}{z(z-a)}\)
c) \(\frac{az}{z(z+a)}\)
d) \(\frac{a}{z(z-a)^2}\)
View Answer
Explanation: Given x(n) = n[an u(n)]
We know that an u(n) ↔ \(\frac{z}{z-a}\)
Time differentiation property states that
If x(n) ↔ X(z), then nx(n) ↔ -z \(\frac{dX(z)}{dz}\)
Z[x(n)] = Z{n[an u(n)]} = -z \(\frac{dX(z)}{dz} = -z \frac{d}{dz} \Big[\frac{z}{z-a}\Big] = \frac{a}{z(z-a)^2}\).
8. Find the Z-transform of x(n) = (\(\frac{1}{2}\))n u(n)*(\(\frac{1}{4}\))n u(n).
a) \(\frac{z}{z-(1/2)} \frac{z}{z-(1/4)}\)
b) \(\frac{z}{z-(1/2)} + \frac{z}{z-(1/4)}\)
c) \(\frac{z}{z+(1/2)} * \frac{z}{z-(1/4)}\)
d) \(\frac{z}{z-(1/2)} – \frac{z}{z+(1/4)}\)
View Answer
Explanation: We know that an u(n) ↔ \(\frac{z}{z-a}\)
Let x1 (n)=(\(\frac{1}{2}\))n u(n) and x2 (n) = (\(\frac{1}{4}\))n u(n)
∴X1 (z) = \(\frac{z}{z-(1/2)}\) and X2 (z) = \(\frac{z}{z-(1/4)}\)
Given x(n) = x1 (n) * x2 (n)
The convolution property of Z-transform states that
x1 (n) * x2 (n) ↔ X1 (z) X2 (z)
∴Z[x(n)] = X(z) = Z[x1 (n)*x2 (n)] = X1 (z) X2 (z) = \(\frac{z}{z-(1/2)} \frac{z}{z-(1/4)}\).
9. Find x(∞) if X(z) = \(\frac{Z+1}{(z-0.6)^2}\).
a) 1
b) 0
c) ∞
d) 0.6
View Answer
Explanation: Given X(z) = \(\frac{Z+1}{(z-0.6)^2}\)
The final value theorem of Z-transform states that
If x(n) ↔ X(z), then x(∞) = Ltz→1 (z-1)X(z)
x(∞) = Ltz→1 (z-1)X(z) = Ltz→1 (z-1) \(\frac{Z+1}{(z-0.6)^2}\) = 0.
10. Find x(∞) if X(z) = \(\frac{z+3}{(z+1)(z+2)}\).
a) ∞
b) -1
c) 1
d) 0
View Answer
Explanation: Given X(z) = \(\frac{z+3}{(z+1)(z+2)} = \frac{z[1+(3/z)]}{z^2 [1+(1/z)][1+(2/z)]} = \frac{1}{z} \frac{1+(3/z)}{[1+(1/z)][1+(2/z)]}\)
The initial value theorem of Z-transform states that
If x(n) ↔ X(z), then x(0) = Ltz→∞ X(z)
x(0) = Ltz→∞ X(z) = \(Lt_{z→∞} \frac{1}{z} \frac{[1+(3/z)]}{[1+(1/z)][1+(2/z)]}\) = 0.
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