Using R, (Please provide source code) Function - x(t) = exp(-a t) for t>0 (a >0)

Question

Using R, (Please provide source code)

Function - x(t) = exp(-a t) for t>0 (a >0) x(t) = 0 for t

Consider the above function x(t). Assume you have a finite (T=512 seconds) and sampled version of x(t), starting at time t=0. Please start with Dt=1 second and compute the complex absolute value squared discrete Fourier transform |X(f)| 2 of the finite, sampled time series xk. Please plot and label the results.

Do the same for Dt=0.5 seconds and Dt=0.25 seconds.

Compare your results with those you derived analytically above for the continueous transform X(f).

What are the similiarities and discrepancies?

Explanation / Answer

Hi Thanks for the question.Please find the code below

R-Code:

require(stats)
#Domain setup
T <- 512
dt <- 1 #s
n <- T/dt
F <-1/dt
df <- 1/T

#CREATE OUR TIME SERIES DATA
y <- exp(-t)

#CREATE OUR FREQUENCY ARRAY
f <- 1:length(t)/T

#FOURIER TRANSFORM WORK
Y <- fft(y)
mag <- sqrt(Re(Y)^2+Im(Y)^2)*2/n
phase <- atan(Im(Y)/Re(Y))
Yr <- Re(Y)
Yi <- Im(Y)

#PLOTTING
layout(matrix(c(1,2), 2, 1, byrow = TRUE))
plot(t,y,type=”l”,xlim=c(0,T))
plot(f[1:length(f)/2],mag[1:length(f)/2],type=”l”)

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