## The DFT

• In the first week of the semester, the following formula for the Discrete Fourier Transform (DFT) was given:
 (11)

• From the previous discussion, you should now have some intuition on how the DFT works.

• In effect, an inner product is computed of a given discrete-time signal x[n] and N complex exponential signals representing sine and cosine terms at discrete frequencies . Each of these inner product computations produces a single complex valued weight'' that indicates the relative strength of a specific sinusoidal frequency in the signal.

• Figure 4 shows the first 5 cosine and sine terms of the complex exponential signals used in a length N=32 DFT.

• The frequencies evaluated by the DFT are directly related to the size N of the signal x[n]. These frequencies are evenly distributed from 0 to , so that the larger the value of N, the more precise the estimate of frequency content in the signal. However, from the sampling theorem, only those frequency components less than or equal to fs/2 are unique.

• As mentioned earlier in this section, sinusoids are periodic functions. Thus, for the DFT to work perfectly, the signal x[n] would have to be periodic with a frequency that is an integral multiple of .

• Remembering that periodic signals must have harmonic spectra, we see that the DFT computes the amplitudes and phases of a harmonic set of sinusoidal components, each having an integer number of periods in N samples.

• If the signal x[n] is not periodic in N, the DFT will only provide an approximation of its actual frequency content.