# Initial Rest Condition for LCCDE causal LTI systems

I am self studying Alan Opennheim's course Signals and Systems. I am a math major and have no background in EE.

I understand that for a LCCDE system to be linear its auxiliary conditions must be 0.

I also understand that for the system to be causal every output $$y(t)$$ corresponding to input $$x(t)$$ s.t. $$x(t) = 0$$ for $$t must satisfy $$y(t)=0$$ for all $$t.

What i am having trouble understanding is why this imposes the initial condition $$y(t_0)=0$$ for the response to the unit step function. Couldn't we have set the initial condition to be $$y(t_1)=0$$ for $$t_1>t_0$$ and kept the system causal and LTI?

In the 6th lecture in the video series Alan Oppenheim finds the unit step function response to the causal LTI system described by $$y'(t) +ay(t) = x(t)$$. He imposes the initial condition $$y(0)=0$$, as i mentioned above i can't understand why this is imposed by the properties of the system. He then goes on to find the unit impulse response, and finds that it is $$e^{-at}u(t)$$ where $$u(t)$$ is the unit step. Clearly this function does not satisfy the initial condition which were imposed on the unit step response, I am having trouble understanding why this is valid.

Any help is appreciated, Thanks!

If you are given an input $$x(t)$$ with $$x(t)=0$$ for $$t, and you specify an initial condition $$y(t_1)=0$$ for $$t_1>t_0$$, then the resulting system is generally non-causal, because we already know the system's response at $$t_1>t_0$$, regardless of the input signal in the interval $$[t_0,t_1]$$.
For a system to be linear (in the sense used by Oppenheim) and causal, we need to specify the initial condition at the point $$t_0^-$$, i.e., we require that the left-sided limit $$y(t_0^-)$$ satisfies $$y(t_0^-)=0$$.
For the given example, if an impulse is applied at $$t=0$$, we require $$y(0^-)=0$$ for the system to be linear and causal. Note that there is no requirement on the right-sided limit $$y(0^+)$$.
Clearly, the response $$y(t)=e^{-at}u(t)$$ satisfies the initial condition $$y(0^-)=0$$.
• Thanks for the reply. With an initial condition at $t_1 > t_0$, wouldn't the system still be causal because the response at $t_1$ depends only past values of $t$. Also can you recommend a reference, Oppenheim makes no mention of one sided limits. – Adi Oct 10 '19 at 23:25