Start with the sample size, which is denoted by the letter "N" in the formula. The sample size can be thought of as the number or the amount of signals for which a frequency spectrum needs to be found. A sample size is usually used in statistical calculations when calculating the entire population is either impractical or unnecessary. However, it is vital to ensure the sample size is neither too large or too small, while representing an adequate subset of the larger population to be tested.
Begin the calculation by using the synthesis and analysis equations for the DFT formula. DFT, or discrete Fourier transformation, involves two separate equations. These dual equations take both the sine and cosine waves of the frequency and separate them into their respective categories. In the calculation, both sets of waves have their individual frequencies spaced between zero and one-half of the sample's rate as N or the sample's number is increased to infinity.
Calculate the analysis equations of both the sine and cosine waves. The sample size will run from zero to N-1. The cosine formula is denoted as ReX(w) = sum of N to infinity times x [n] cos (wn), where w is between zero and pi, or 3.14. The sine formula is denoted as ImX(w) = sum of N to infinity times x[n] sin (wn). In the formulas above, x[n] represents the time domain signal and is considered to be finite.
Synthesize the two analysis equations by using the DTFT synthesis formula. In this formula, x[n] or the time domain signal equals one divided by pi times the fraction of the sampling rate. This fraction numerically runs from 0 to 0.5. It is also represented by w, which is the fraction of the sampling rate expressed in its natural frequency terms of 0 to pi. The results of the cosine equation is subtracted from the results of the sine equation and multiplied by the first half of the DTFT synthesis formula.