Fast Fourier Transform in R. What am I doing wrong? - r

Question

I am a non-expert in Fourier analysis and quite don't get what R's function fft() does. Even after crossreading a lot I couldnt figure it out.
I built an example.
require(ggplot2)
freq <- 200 #sample frequency in Hz
duration <- 3 # length of signal in seconds
#arbitrary sine wave
x <- seq(-4*pi,4*pi, length.out = freq*duration)
y <- sin(0.25*x) + sin(0.5*x) + sin(x)
which looks like:
fourier <- fft(y)
#frequency "amounts" and associated frequencies
amo <- Mod(fft(y))
freqvec <- 1:length(amo)
I ASSUME that fft expects a vector recorded over a timespan of 1 second, so I divide by the timespan
freqvec <- freqvec/duration
#and put this into a data.frame
df <- data.frame(freq = freqvec, ammount = amo)
Now I ASSUMABLY can/have to omit the second half of the data.frame since the frequency "amounts" are only significant to half of the sampling rate due to Nyquist.
df <- df[(1:as.integer(0.5*freq*duration)),]
For plotting I discretize a bit
df.disc <- data.frame(freq = 1:100)
cum.amo <- numeric(100)
for (i in 1:100){
cum.amo[i] <- sum(df$ammount[c(3*i-2,3*i-1,3*i)])
}
df.disc$ammount <- cum.amo
The plot function for the first 20 frequencies:
df.disc$freq <- as.factor(df.disc$freq)
ggplot(df.disc[1:20,], aes(x=freq, y=ammount)) + geom_bar(stat = "identity")
The result:
Is this really a correct spectrogram of the above function? Are my two assumptions correct? Where is my mistake? If there is no, what does this plot now tell me?
EDIT:
Here is a picture without discretization:
THANKS to all of you,
Micha.

Answer 1

Okay, okay. Due to the generally inferior nature of my mistake the solution is quite trivial.
I wrote freq = 200 and duration = 3. But the real duration is from -4pi to 4 pi, hence 8pi resulting in a "real" sample frequency of 1/ ((8*pi)/600) = 23.87324 which does not equal 200.
Replacing the respective lines in the example code by
freq <- 200 #sample frequency in Hz
duration <- 6 # length of signal in seconds
x <- seq(0,duration, length.out = freq*duration)
y <- sin(4*pi*x) + sin(6*pi*x) + sin(8*pi*x)
(with a more illustrative function) yields the correct frequencies as demonstrated by the following plot (restricted to the important part of the frequency domain):