Question

Consider a discrete time signal x[n] and its z-transform, X(z) given as

$X\left(z\right)=\frac{z^2+5z}{z^2-2z-3}$

If ROC of $X\left(z\right)is|z|<1$, then signal x[n] would be

• A. $\left[2\right(3{)}^n-(-1{)}^n]u\left[n\right]$
• B. $[-2(3{)}^n+(-1{)}^n]u[-n-1]$
  
• C. $-2\left(3{)}^{n}u\right[-n-1]-(-1{)}^{n}u\left[n\right]$
• D. $\left[2\right(3{)}^n+1]+u[n]$

Answer 1

The correct option is B $[-2(3{)}^n+(-1{)}^n]u[-n-1]$


$X\left(z\right)=\frac{z^2+5z}{z^2-2z-3}=\frac{z(z+5)}{(z-3)(z+1)}$

$\frac{X\left(z\right)}{z}=\frac{z+5}{(z-3)(z+1)}=\frac{2}{z-3}-\frac{1}{z+1}$

Thus, $X\left(z\right)=\frac{2z}{z-3}-\frac{z}{z+1}$
ROC : |z| < 1, which is not exterior of circle outside the outermost pole = 3

So, x[n] is anti-causal given as,

$x\left[n\right]=[-2(3{)}^n+(-1{)}^n]u[-n-1]$