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 time-domain data into a frequency-domain representation. Well, we can reverse this process and obtain the original time- domain signal by performing the IDFT on the X(m) frequency-domain values. The standard expressions for the IDFT are
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 (3-23) and (3-23') 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. (3-23), we'll go from the frequency-domain back to the time-domain and get our original real Eq. (3-11') x(n) sample values of
Notice that Eq. (3-23)'s IDFT expression differs from the DFT's Eq. (3-2) 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.