Hello,

In a signal processing exercise I have a system described by the following input output difference equation

\( y(n) = x(n) + y(n-1) + y(n-2) \)

and is it required to determine the initial conditions \(y(n-1)\) and \(y(n-2)\) such that \( y(0) = y(1) = 1\) when \( x(n) = \delta(n) \).

At first glance, I would say that if

\( x(n) = \delta(n) = 1 \qquad \text{if}\, n = 0\)

then

\(

\begin{aligned}

y(0) & = x(0) + y(-1) + y(-2) \\

1 & = 1 + y(-1) + y(-2)

\end{aligned}

\)

so the sum of both initial conditions should be zero. Then, solving for \( y(1) \):

\(\begin{aligned}

y(1) & = x(1) + y(0) + y(-1)\\

1 &= 0 + 1 + y(-1)

\end{aligned}\)

seems that both initial conditions must be zero.

But I am not sure about the correctness of my solution.

Initial conditions (IC) are usually assumed to be zero in filtering problems, but unfortunately DSP textbooks usually show a lack of interest for IC, in most of the cases even without mentioning them at all...

Could someone confirm or give an hint (not the solution) about the correct way to solve it in case of wrong solution ?

Thanks in advance.

s.

In a signal processing exercise I have a system described by the following input output difference equation

\( y(n) = x(n) + y(n-1) + y(n-2) \)

and is it required to determine the initial conditions \(y(n-1)\) and \(y(n-2)\) such that \( y(0) = y(1) = 1\) when \( x(n) = \delta(n) \).

At first glance, I would say that if

\( x(n) = \delta(n) = 1 \qquad \text{if}\, n = 0\)

then

\(

\begin{aligned}

y(0) & = x(0) + y(-1) + y(-2) \\

1 & = 1 + y(-1) + y(-2)

\end{aligned}

\)

so the sum of both initial conditions should be zero. Then, solving for \( y(1) \):

\(\begin{aligned}

y(1) & = x(1) + y(0) + y(-1)\\

1 &= 0 + 1 + y(-1)

\end{aligned}\)

seems that both initial conditions must be zero.

But I am not sure about the correctness of my solution.

Initial conditions (IC) are usually assumed to be zero in filtering problems, but unfortunately DSP textbooks usually show a lack of interest for IC, in most of the cases even without mentioning them at all...

Could someone confirm or give an hint (not the solution) about the correct way to solve it in case of wrong solution ?

Thanks in advance.

s.

Last edited: