Matlab-frequency bin of the positive and negative frequency - matlab

Question

I am using MATLAB tutorial Frequency-Domain Linear Regression. There is one part of code, where is necessary to determine "frequency bin of the positive and negative frequency", for fft and ifft, this part of code:
freqbin = 72/12;
freqbins = [freqbin 72-freqbin]+1;
tsfit = zeros(72,1);
tsfit(freqbins) = tsdft(freqbins);
tsfit = ifft(tsfit);
mu = mean(ts);
tsfit = mu+tsfit;
Length of time series is 72, and there is 12 months is one cycle. How there can be only one frequency bin for positive frequency, how do we know that is exactly 72/12+1 (+1 is because first bin is for zero frequency), is that some formula or what?

Answer 1

Let's start with some general notes. If the sample rate is Fs, and the number of samples in the FFT is N, then the frequency of the bin with MATLAB index k is:
f_k = (k-1) * Fs / N ;
Or, conversely,
k = f_k * N / Fs + 1;
Hence, the elements of the FFT vector correspond to frequencies 0, Fs/N, 2*Fs/N, ..., (N-1)*Fs/N.
Note that the frequencies are invariant to shifts by Fs. In other words, f and f+Fs and f-Fs are all equivalent. So, you could consider the first half of the FFT vector to correspond to positive frequencies (index: 1 to N/2; frequency: 0 to (N/2-1)/N*Fs), and the second half to correspond to negative frequencies (index: N/2+1 to N; frequency: -Fs/2 to -Fs/N). Here I include 0 in the set of positive frequencies for brevity.
Now, coming to your specific example:
N = 72
Fs = 12 ; % samples per year
So, the positive frequency range is from 0 to Fs/2 = 6 cycles per year, corresponding to indices 1 to 36, respectively.
You are interested in finding the FFT sample corresponding to the frequency 1 cycle per year. That corresponds to index 1 * 72 / 12 + 1 = 7.
Since the FFT is symmetric for real signals, you are also interested in finding the FFT sample corresponding to the frequency -1 cycle per year, which, using the shift-invariance property, is also equivalent to -1 + Fs = 11 cycles per year. The corresponding index is (12-1) * 72 / 12 + 1 = 67.
In summary,
There are 36 positive frequencies (including 0), not 1.
The algorithm is interested in only one positive frequency with the highest magnitude.
That frequency happens to be 1 cycle per year. The corresponding MATLAB index is 7.
The mirror image of the frequency is -1 cycle per year, which corresponds to MATLAB index 67.