Now that we understand the DFT's overall Npoint (or Nbin) frequency response to a real cosine of k cycles per sample interval, we conclude this chapter by determining the frequency response of a single DFT bin. We can think of a single DFT bin as a kind of bandpass filter, and this useful notion is used, for example, to describe DFT scalloping loss (Section 3.10), employed in the design of frequencydomain filter banks, and applied in a telephone frequency multiplexing technique known as transmultiplexing[15]. To determine a DFT singlebin's frequency response, consider applying a real xr(n) cosine sequence to a DFT and monitoring the output magnitude of just the m = k bin. Let's say the initial frequency of the input cosine sequence starts at a frequency of k<m cycles and increases up to a frequency of k>m cycles in the sample interval. If we measure the DFT's m = k bin during that frequency sweep, we'll see that its output magnitude must track the input cosine sequence's continuous spectrum, shown as the shaded curves in Figure 345.
Figure 345. Determining the output magnitude of the mth bin of an Npoint DFT: (a) when the real xr(n) has k = m–2.5 cycles in the time sequence; (b) when the real xr(n) has k = m–1.5 cycles in the time sequence; (c) when the real xr(n) has k = m cycles in the time sequence; (d) the DFT singlebin frequency magnitude response of the m = k bin.
Table 34. Various Forms of the Positive Frequency Amplitude Response of the DFT to a Real Cosine Input Having k Cycles in the Sample Interval
Description 
Expression 

Real input DFT amplitude response in terms of the integral frequency variable m [From Eq. (373)] 

Alternate form of the real input DFT amplitude response in terms of the integral frequency variable m [based on Eq. (349)] 
Equation 375 
Amplitude normalized real input DFT response in terms of the integral frequency variable m 
Equation 376 
Real input DFT response in terms of the sample rate fs in Hz 
Equation 377 
Amplitude normalized real input DFT response in terms of the sample rate fs in Hz 
Equation 378 
Amplitude normalized real input DFT response in terms of the sample rate ws in radians/s 
Equation 379 
Figure 345(a) shows the m = k bin's output when the input xr(n)'s frequency is k = m–2.5 cycles per sample interval. Increasing xr(n)'s frequency to k = m–1.5 cycles per sample interval results in the m = k bin's output, shown in Figure 345(b). Continuing to sweep xr(n)'s frequency higher, Figure 345(c) shows the m = k bin's output when the input frequency is k = m. Throughout our input frequency sweeping exercise, we can see that the m = k bin's output magnitude must trace out the cosine sequence's continuous spectrum, shown by the solid curve in Figure 345(d). This means that a DFT's singlebin frequency magnitude response, to a real input sinusoid, is that solid sinc function curve defined by Eqs. (374) through (379).
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