I Numerically how to approximate exponential decay in a discrete signal

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To approximate exponential decay in a discrete signal using Laplace or Z-transform methods, the discussion emphasizes the need for a model that reflects the known system behavior, such as f(t) = A exp(-αt)cos(2πft + φ). While regression techniques are acknowledged, the focus is on finding a pseudo-analytic approach that aligns with the exponential decay requirement. The conversation highlights the importance of specifying prior knowledge about the system, as this can significantly influence the choice of modeling techniques. Curve fitting is suggested as a potential method, but it must adhere to the constraints of continuity and the specific functional form of the data. The discussion ultimately points to the complexity of the data and the necessity for a tailored approach to accurately capture its characteristics.
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Given a vector of numbers, say [exp(-a t) ] for t - [1, 2, 3, 4, 5] and choose maybe a = -2.4, how can I approximate -2.4 from using Laplace transform methods?

I know you can use regression for this, but I'd like to know the Laplace transform (or Z-transform since it is discrete) approach.
 
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Say given number sequence is f(t), plot t - log f(t) and find the approxmate line to connect the points and its tan. It is my idea, though Laplace transform plays no role here.
 
anuttarasammyak said:
Say given number sequence is f(t), plot t - log f(t) and find the approxmate line to connect the points and its tan. It is my idea, though Laplace transform plays no role here.
yea this is regression. was looking for laplace transform or some psuedo-analytic manner
 
This would be about statistics and curve fitting, I think. You'll have some basic assumptions as constraints for your model, things like continuity, that you haven't told us. Then I would just use a polynomial fit. It 's not that that's the correct answer, it will be just as likely to be wrong as other models. But since you haven't specified any prior knowledge of the nature of the system producing the data, I don't see a better approach.

Or, maybe I misunderstood and you KNOW that the system is ##e^{-at}##, in which case the answer is almost trivial.
 
DaveE said:
This would be about statistics and curve fitting, I think. You'll have some basic assumptions as constraints for your model, things like continuity, that you haven't told us. Then I would just use a polynomial fit. It 's not that that's the correct answer, it will be just as likely to be wrong as other models. But since you haven't specified any prior knowledge of the nature of the system producing the data, I don't see a better approach.

Or, maybe I misunderstood and you KNOW that the system is ##e^{-at}##, in which case the answer is almost trivial.
The data can be chaotic. Even curve fitting assumes a functional form (polynomial, which I cannot use, must be exponential decay and sinusoidal, so I think ##f(t) = A \exp(-\alpha t)\cos(2\pi f t + \phi)##.

I saw this post and thought there would be a nice implementation for extracting both the sinusoidal frequency and exponential decay:
https://dsp.stackexchange.com/quest...a-signal-into-exponentally-decaying-sinusoids
 
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