Now that we know what the DFT frequency response is to a complex sinusoidal input, it's easy to determine the DFT frequency response to a real input sequence. Say we want the DFT's response to a real cosine sequence, like that shown in Figure 340(a), expressed as
where k is the integral number of complete cycles occurring in the N samples. Remembering Euler's relationship cos(ø) = (ejø + e–jø)/2, we can show the desired DFT as Xr(m) where
Equation 372
Fortunately, in the previous section we just finished determining the closed form of a summation expression like those in Eq. (372), so we can write the closed form for Xr(m) as
Equation 373
We show the magnitude of those two ratio of sines terms as the sinc functions in Figure 343. Here again, the DFT is sampling the input cosine sequence's continuous spectrum and, because k = m, only one DFT bin is nonzero. Because the DFT's input sequence is real, Xr(m) has both positive and negative frequency components. The positive frequency portion of Figure 343 corresponds to the first ratio of sines term in Eq. (373) and the second ratio of sines term in Eq. (373) produces the negative frequency components of Xr(m).
Figure 343. Npoint DFT frequency magnitude response to a real cosine having integral k cycles in the Npoint time sequence xr(n) = cos(2pnk/N).
DFT leakage is again demonstrated if our input sequence frequency were shifted from the center of the kth bin to k+0.25 as shown in Figure 344. (We used this concept of real input DFT amplitude response to introduce the effects of DFT leakage in Section 3.8.)
Figure 344. Npoint DFT frequency magnitude response showing spectral leakage of a real cosine having k+0.25 cycles in the Npoint time sequence xr(n).
In Table 34, the various mathematical expressions for the (positive frequency) amplitude response of the DFT to a real cosine input sequence are simply those expressions in Table 33 reduced in amplitude by a factor of 2.
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