Connection between impulse response and LTI system

Question

I am always confused by this, please make it clearer for me here:
1)If I have a system which has Impulse response, which means \$H(j\omega)\$, so my system is LTI.
Am I right?
2) Same with transfer function \$H(s)\$, which is basically stronger.

If it is false, I have to find if it is LTI by usual stuff? with \$y(t)\$

For example: $$y'\left(t\right)+\frac{1}{RC}y\left(t\right)=\frac{1}{RC}x\left(t\right)$$ $$H\left(j\omega \right)=\frac{1}{RC}\left(\frac{1}{j\omega +\frac{1}{RC}}\right)$$ and the impulse signal: $$h\left(t\right)=\frac{1}{RC}\cdot e^{-\frac{t}{RC}}$$

can I know if it LTI based on the impulse response, or I must find \$y(t)\$ and then like the usual long way.

Answer 1

It is perfectly reasonable to talk about a linear time-varying system having an impulse response. It's just than instead of the impulse response being \$h(\tau)\$, it is \$h(t, \tau)\$, with \$t\$ being the time and \$\tau\$ being the time lag since the impulse.

When this is the case, \$H(j\omega)\$ (and \$H(s)\$) does not exist.

Howevever, \$H(j\omega)\$ and \$H(s)\$ do imply that the system in question has a time-invariant impulse response and is, therefor, time-invariant itself.

In your example:

$$y'\left(t\right)+\frac{1}{RC}y\left(t\right)=\frac{1}{RC}x\left(t\right)$$

Each one of the operations in this example are LTI:

• I see one addition (two if you bring everything over to one side or another). Addition is linear and time invariant.
• I see a time derivative. The time derivative is linear and time invariant.
• I see multiplication by constants. Multiplication by a constant is both linear and time invariant.

If you have an assemblage of systems that are each LTI, and the assembly is done using operations that don't violate linearity, then the result is linear and time invariant.

So you can stop right there and demonstrate that the system is LTI -- then you can go on to use Fourier* or Laplace analysis, knowing that you're "legal".

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* I really hesitate to add this because at your stage this will probably be confusing. However, honesty compels me: you can use Fourier analysis to usefully analyze some linear time varying systems. It's not as easy, and it's mostly useful for systems that are or resemble heterodyne radio systems (i.e., the only time-varying part is multiplication by a sine or other periodic wave someplace). But it can be done.

Answer 2

Not quite. You have it backwards. Every system will have an impulse response. If a system happens to be an LTI system, then that impulse response completely describes the system, and if you know the impulse response of that LTI system, you can calculate the output for any given system.