In practice, it is the Fast Fourier Transform (FFT), a more efficient equivalent of the DFT given by the above equation, which is used. The FFT results can be used to calculate both a magnitude spectrum and a phase spectrum; both spectra are necessary in order to resynthesize the waveform using the Inverse Discrete Fourier Transform (IDFT). The magnitude spectrum gives the distribution of energy among the different frequency components making up the waveform, while the phase spectrum gives the starting phases of these same frequency components. As mentioned before, in speech processing phase contributions are largely ignored. What we usually see as the graphic output of a spectrogram program is a two-dimensional array, having time as the columnar axis, frequency as the row axis, and intensity as the value being plotted. This array may be written to a disk file and stored for later use; however, DFT data is about one order of magnitude greater than waveform data, so that in this course we will often generate the DFT data without saving it.
Figure 8 shows a waveform and spectrogram for the word ``adoring" with the above parameter settings. Figure 9 shows an expanded view of a small piece of the spectrogram in Figure 8. Figure 10 is a ``3-D" view of approximately the same section of the word as Figure 9.