Section 3.14.  THE DFT FREQUENCY RESPONSE TO A COMPLEX INPUT

3.14. THE DFT FREQUENCY RESPONSE TO A COMPLEX INPUT

In this section, we'll determine the frequency response to an N -point DFT when its input is a discrete sequence representing a complex sinusoid expressed as x _c ( n ). By frequency response we mean the DFT output samples when a complex sinusoidal sequence is applied to the DFT. We begin by depicting an x _c ( n ) input sequence in Figure 3-40. This time sequence is of the form

where k is the number of complete cycles occurring in the N samples. Figure 3-40 shows x _c ( n ) if we happen to let k = 2. If we denote our DFT output sequence as X _c ( m ), and apply our x _c ( n ) input to the DFT expression in Eq. (3-2) we have

Equation 3-61

If we let N = K, n = p , and q = –2 p ( k – m )/ N , Eq. (3-61) becomes

Equation 3-62

Why did we make the substitutions in Eq. (3-61) to get Eq. (3-62)? Because, happily, we've already solved Eq. (3-62) when it was Eq. (3-39). That closed form solution was Eq. (3-41) that we repeat here as

Equation 3-63

Replacing our original variables from Eq. (3-61), we have our answer:

Equation 3-64

Like the Dirichlet kernel in Eq. (3-43), the X _c ( m ) in Eq. (3-64) is a complex expression where a ratio of sine terms is the amplitude of X _c ( m ) and the exponential term is the phase angle of X _c ( m ). At this point, we're interested only in the ratio of sines factor in Eq. (3-64). Its magnitude is shown in Figure 3-41. Notice that, because x _c ( n ) is complex, there are no negative frequency components in X _c ( m ). Let's think about the shaded curve in Figure 3-41 for a moment. That curve is the continuous Fourier transform of the complex x _c ( n ) and can be thought of as the continuous spectrum of the x _c ( n ) sequence. ^[] By continuous spectrum we mean a spectrum that's defined at all frequencies, not just at the periodic f _s / N analysis frequencies of an N -point DFT. The shape of this spectrum with its main lobe and sidelobes is a direct and unavoidable effect of analyzing any finite-length time sequence, such as our x _c ( n ) in Figure 3-40.

^[] Just as we used L'Hopital's rule to find the peak value of the Dirichlet kernel in Eq. (3-44), we could also evaluate Eq. (3-64) to show that the peak of X _c ( m ) is N when m = k.

We can conceive of obtaining this continuous spectrum analytically by taking the continuous Fourier transform of our discrete x _c ( n ) sequence, a process some authors call the discrete-time Fourier transform (DTFT), but we can't actually calculate the continuous spectrum on a computer. That's because the DTFT is defined only for infinitely long time sequences, and the DTFT's frequency variable is continuous with infinitely fine-grained resolution. What we can do, however, is use the DFT to calculate an approximation of x _c ( n )'s continuous spectrum. The DFT outputs represented by the dots in Figure 3-41 are a discrete sampled version of x _c ( n )'s continuous spectrum. We could have sampled that continuous spectrum more often, i.e., approximated it more closely, with a larger DFT by appending additional zeros to the original x _c ( n ) sequence. We actually did that in Figure 3-21.

Figure 3-41 shows us why, when an input sequence's frequency is exactly at the m = k bin center, the DFT output is zero for all bins except where m = k . If our input sequence frequency was k +0.25 cycles in the sample interval, the DFT will sample the continuous spectrum shown in Figure 3-42, where all of the DFT output bins would be nonzero. This effect is a demonstration of DFT leakage described in Section 3.8.

Again, just as there are several different expressions for the DFT of a rectangular function that we listed in Table 3-2, we can express the amplitude response of the DFT to a complex input sinusoid in different ways to arrive at Table 3-3.

At this point, the thoughtful reader may notice that the DFT's response to a complex input of k cycles per sample interval in Figure 3-41 looks suspiciously like the DFT's response to an all ones rectangular function in Figure 3-32(c). The reason the shapes of those two response curves look the same is because their shapes are the same. If our DFT input sequence was a complex sinusoid of k = 0 cycles, i.e., a sequence of identical constant values, the ratio of sines term in Eq. (3-64) becomes

which is identical to the all ones form of the Dirichlet kernel in Eq. (3-48). The shape of our X _c ( m ) DFT response is the sinc function of the Dirichlet kernel.

Table 3-3. Various Forms of the Amplitude Response of the DFT to a Complex Input Sinusoid Having k Cycles in the Sample Interval