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

- 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*f*_{s}/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.

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