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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 $t<t_0$ must satisfy $y(t)=0$ for all $t<t_0$.

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!

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If you are given an input $x(t)$ with $x(t)=0$ for $t<t_0$, 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 $y(0^-)=0$.

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    $\begingroup$ 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. $\endgroup$ – Adi Oct 10 at 23:25
  • $\begingroup$ Also, is the system time-invariant because we define it by its impulse / unit response? $\endgroup$ – Adi Oct 12 at 11:11

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