- #1
I know that, but for t<0 we have δΔ(t)=0, which implies that for t<0, δΔ'(t)=0. That's why I think your function is the same as -δ'(t).the_emi_guy said:The doublet is the derivative of the Dirac delta function.
asmani said:A clarification: This thread is moved from electrical engineering forum. Maybe it is a homework or coursework-type question, but actually it's not a homework or a coursework question, and I believe cannot be.
Anyway, here is my attempt:
$$\mathcal{F}\left \{ x(t) \right \}=sinc^2(f)\; ;\: \mathcal{F}\left \{ y(t) \right \}=2sinc(2f)$$
$$\mathcal{F}\left \{ h(t) \right \}=\frac{\mathcal{F}\left \{ y(t) \right \}}{\mathcal{F}\left \{ x(t) \right \}}=\frac{2sinc(2f)}{sinc^2(f)}=4\pi f\cot(2\pi f)$$
I guess that this function has no inverse Fourier transform, and thus there is no LTI system with x(t) as input and y(t) as output. Is this correct? If yes, how to prove?
rude man said:What reason do you have for assuming your H(f) has no inverse? Have you tried doing the integration?
Ref: G.A. Campbell and R. M. Foster, "Fourier Integrals for Practical Applications", D. Van Nostrand 1958.
PS - I did not check to see that you did X(f) and Y(f) correctly ...
the_emi_guy said:Think of it this way...
1 - What kind of circuit would convert your x(t) to your y(t)?
2 - Next, what would be the impulse response of that circuit? This would be your h(t).
Actually this argument was not correct!asmani said:The input to the system S' is y(t), and the output is x(t). We can observe that S' is the same S series with another system S'' which gives the output z(2t) for the input z(t). Obviously S'' is not a LTI system, so S' isn't.
rude man said:You're not integrating the function you sketched, which I presume is your H(f).
You're trying to integrate ∫H(f)ejωtdf over f = -∞ to f = +∞ with ω = 2πf.
asmani said:I thought these are known results:
http://en.wikipedia.org/wiki/Rectangular_function#Fourier_transform_of_the_rectangular_function
http://en.wikipedia.org/wiki/Triangular_function
An impulse response is a function that describes the behavior of a system in response to an impulse input. It represents the output of the system as a function of time, and can be used to understand the characteristics and properties of the system.
The impulse response of a system can be found by applying an impulse input to the system and measuring the resulting output. This can be done in a controlled laboratory setting or through mathematical modeling and simulation.
The impulse response is important because it provides valuable information about the behavior and properties of a system. It can be used to analyze and predict the system's response to different inputs, and to design and optimize the system for specific purposes.
Yes, the impulse response of a system can change over time due to various factors such as aging, wear and tear, and external influences. It is important to regularly monitor and update the impulse response to ensure accurate and reliable results.
Yes, there are different types of impulse responses depending on the type of system and the input used. Some common types include linear and non-linear impulse responses, as well as discrete and continuous impulse responses.