Test: Z Transform Properties - 1

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).

Explanation: Let us divide the given x(n) into x1(n)= 3(2ⁿ)u(n) and x2(n)= 4(3ⁿ)u(n)
and x(n)=x1(n)-x2(n)
From the definition of z-transform X1(z)= 3/(1-2z^-1) and X2(z)= 4/(1-3z^-1)
So, from the linearity property of z-transform
X(z)=X1(z)-X2(z)
=> X(z)= 3/(1-2z^-1)-4/(1-3z^-1).

Explanation: By Euler’s identity, the given signal x(n) can be written as

Explanation: According to the definition of Z-transform

Explanation:

Explanation: We know that from the definition of z-transform

Explanation: Given ROC of X(z) is r1<|z|<r2
Then ROC of X(a^-1z) will be given by r1<|a^-1z |<r2=|a|r1<|z|<|a|r2

Explanation:

Explanation: From the definition of z-transform, we have

Explanation:
From the definition of z-transform, we have