Because we estimated the Fourier transform by making the x-space function periodic, it also means that the reciprocal space amplitude function will be discrete, and harmonically related. Using this information, we can determine that the values for should be be separated by
where N is the number of points in , and is chosen to be odd to simplify matters. Furthermore, the maximum (most positive) and minimum (most negative) values of should be
Note, to take into account larger values of q (``higher frequencies''), we should decrease (``increase the sampling frequency''). To decrease (without increasing we should make our maximum value of x larger, and keep N the same.
In 6.011 you will learn more about the numerical issues concerning the Discrete Fourier Transform. You will also learn how to accelerate its computation by a method known as the Fast Fourier Transform (FFT).