Although the DFT is the major topic of this chapter, it's appropriate, now, to introduce the inverse discrete Fourier transform (IDFT). Typically we think of the DFT as transforming timedomain data into a frequencydomain representation. Well, we can reverse this process and obtain the original time domain signal by performing the IDFT on the X(m) frequencydomain values. The standard expressions for the IDFT are
and equally,
Equation 323'
Remember the statement we made in Section 3.1 that a discrete time domain signal can be considered the sum of various sinusoidal analytical frequencies and that the X(m) outputs of the DFT are a set of N complex values indicating the magnitude and phase of each analysis frequency comprising that sum. Equations (323) and (323') are the mathematical expressions of that statement. It's very important for the reader to understand this concept. If we perform the IDFT by plugging our results from DFT Example 1 into Eq. (323), we'll go from the frequencydomain back to the timedomain and get our original real Eq. (311') x(n) sample values of
Notice that Eq. (323)'s IDFT expression differs from the DFT's Eq. (32) only by a 1/N scale factor and a change in the sign of the exponent. Other than the magnitude of the results, every characteristic that we've covered thus far regarding the DFT also applies to the IDFT.
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