# Impulse Response for an Anti Causal Linear Shift Invariant System

I have a system given by $$y[n] - \frac{1}{4} y[n-1] - \frac{1}{8} y[n-2] =3x[n]$$

I'm asked to determine the impulse response of both an anti-causal and a causal Linear Shift Invariant System characterized by this difference equation.

The impulse response to the causal system is $$2\left(\frac{1}{2}\right)^{n}u[n]+\left(-\frac{1}{4}\right)^{n}u[n]$$

I found this without using Z transform or Fourier Transform but by solving the difference equation using the initial condition $$h[n]=0$$ for $$n<0$$.

How can I find the anti-causal impulse response without using Z-transform or Fourier Transform. What boundary conditions do I use now?

• Hint: a causal system has an impulse response which is a right-handed signal of the form $h[n<0]=0$. Hence, an anti-causal system will have a left-handed impulse response of the form $h[n>0]=0$. Jan 26 '21 at 7:17

You've already found the solutions of the characteristic equation as $$\lambda_1=\frac12$$ and $$\lambda_2=-\frac14$$, so you know that the solution must have the form

$$y[n]=c_1\left(\frac12\right)^n+c_2\left(-\frac14\right)^n\tag{1}$$

For the anti-causal solution we know that $$y[n]=0$$ for $$n>0$$. The constants $$c_1$$ and $$c_2$$ are simply determined by making sure that the difference equation is satisfied for $$x[n]=\delta[n]$$, given that we require $$y[n]=0$$ for $$n>0$$:

\begin{align}n=2:\quad&&&-\frac18y[0]&=0\\n=1:\quad&&-\frac14y[0]&-\frac18y[-1]&=0\\n=0:\quad&y[0]&-\frac14 y[-1]&-\frac18y[-2]&=3\end{align}\tag{2}

From the first equation in $$(2)$$ we see that the impulse response actually starts at $$n=-1$$, not at $$n=0$$. The constants $$c_1$$ and $$c_2$$ can be determined from the second and third equation, which simplify to

$$y[-1]=0\quad\textrm{and}\quad -\frac18 y[-2]=3\tag{3}$$

Plugging $$(3)$$ into $$(1)$$ finally gives the actual values of $$c_1$$ and $$c_2$$, which I'm sure you can figure out yourself. The final anti-causal solution has the form

$$y[n]=\left[c_1\left(\frac12\right)^n+c_2\left(-\frac14\right)^n\right]u[-n-1]\tag{4}$$

• I got it....thanks a lot Matt L....I confused anti causal signal with non causal signal! Jan 26 '21 at 12:49