# Numerically how to approximate exponential decay in a discrete signal

• I
• cppIStough
cppIStough
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.

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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