Signal and Systems: Verify the impulse response of this system

  • #1
SumDood_
30
6
Homework Statement
Verify that the impulse response of this system is h(t) = e^(-2t)u(t)
Relevant Equations
Impulse response => x(t) = impulse
Verify that the impulse response of this system is ##h(t) = e^{-2t}u(t)## for the following system
$$\frac{dy(t)}{dt} + 2y(t) = x(t)$$

So this is what I did first,
$$
\begin{align}
\frac{dh(t)}{dt} + 2h(t) &= \delta (t) \\
\frac{d}{dt}e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\
-2e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\
0 &= \delta (t)
\end{align}
$$
Of course, my solution is wrong. Honestly, at the beginning, I didn't know what I am supposed to get that would verify that the impulse response of the system is ##h(t) = e^{-2t}u(t)##.
So, first, what mistake did I make when simplifying after substituting ##h(t)## and ##x(t)##?
Second, what form does the final statement need to take to actually verify the impulse response? Should I end up with a true statement?
 
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  • #2
SumDood_ said:
Homework Statement: Verify that the impulse response of this system is h(t) = e^(-2t)u(t)
Relevant Equations: Impulse response => x(t) = impulse

Verify that the impulse response of this system is ##h(t) = e^{-2t}u(t)## for the following system
$$\frac{dy(t)}{dt} + 2y(t) = x(t)$$

So this is what I did first,
$$
\begin{align}
\frac{dh(t)}{dt} + 2h(t) &= \delta (t) \\
\frac{d}{dt}e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\
-2e^{-2t}u(t) + 2e^{-2t}u(t) &= \delta (t) \\
0 &= \delta (t)
\end{align}
$$
Of course, my solution is wrong. Honestly, at the beginning, I didn't know what I am supposed to get that would verify that the impulse response of the system is ##h(t) = e^{-2t}u(t)##.
So, first, what mistake did I make when simplifying after substituting ##h(t)## and ##x(t)##?
Second, what form does the final statement need to take to actually verify the impulse response? Should I end up with a true statement?
## \frac{d}{dt}[e^{-2t}u(t)] \neq -2e^{-2t}u(t) ##. Use the product rule, ##u(t)## is also a differentiable function of ##t##.
 
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  • #3
DaveE said:
## \frac{d}{dt}[e^{-2t}u(t)] \neq -2e^{-2t}u(t) ##. Use the product rule, ##u(t)## is also a differentiable function of ##t##.
Got the right solution, thanks!
 
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Likes berkeman and DaveE

Related to Signal and Systems: Verify the impulse response of this system

What is an impulse response in the context of signals and systems?

An impulse response is the output of a system when it is subjected to an impulse input. It characterizes the system's behavior and response to any arbitrary input through the principle of superposition and convolution.

How do you mathematically verify the impulse response of a system?

Mathematically, to verify the impulse response of a system, you apply an impulse function (often denoted as δ(t) in continuous time or δ[n] in discrete time) to the system and analyze the output. This output is the impulse response, typically denoted as h(t) for continuous systems or h[n] for discrete systems.

What are the steps to experimentally determine the impulse response of a system?

To experimentally determine the impulse response:1. Generate an impulse signal.2. Apply this impulse signal to the input of the system.3. Measure and record the output of the system.4. The recorded output is the impulse response of the system.

Why is the impulse response important in system analysis?

The impulse response is crucial because it completely characterizes a linear time-invariant (LTI) system. Once the impulse response is known, the output for any arbitrary input can be determined using the convolution operation. It helps in understanding the system's dynamics and stability.

Can the impulse response be determined for non-linear or time-variant systems?

No, the impulse response is a concept that applies specifically to linear time-invariant (LTI) systems. For non-linear or time-variant systems, the impulse response does not fully characterize the system, and other methods are required to analyze such systems.

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