Note 15. Exploring DFT Leakage
This note uses the model of signal truncation developed in Note 14 and applies this model to sinusoidal signals to demonstrate how leakage can corrupt a DFT result—often to the point where the DFT results can be misleading when used to estimate signal frequency. The insights provided by this demonstration are the foundation for the introduction and analysis of windowing techniques in Notes 23 through 27.
The effects of truncation are absent in cases where the input signal is a sinusoid having a frequency that exactly matches one of the DFT bin frequencies. For example:
• Consider a DFT with N = 32 and T = 1.
• The signal of interest is x(t) = cos(2πfct), where fc = 4F.
• Figure 15.1 shows the DFT magnitude spectrum for this signal; all of the response is concentrated in the bins corresponding to ±4F, as expected.
Figure 15.1. DFT magnitude spectrum of a rectangularly windowed, 32-sample segment of a sinusoid having a frequency of fc = 4F.
However, in general, the effects of truncation are very noticeable. Again, consider a DFT with N = 32 and T = 1:
• The signal of interest is changed to x(t) = cos(2πfct), where fc = 4.5F.
• Intuition suggests that the postive-frequency response would be split equally between bins 4 and 5, and that the negative-frequency response would be split equally between bins 27 and 28.
• Figure 15.2 shows the DFT magnitude spectrum for this signal. The responses in bins 4 and 5 are not equal. Furthermore, there is a non-zero response in each of the 32 bins.
Figure 15.2. DFT magnitude spectrum of a rectangularly windowed, 32-point segment of a sinusoid having a frequency of fc = 4.5F
How can a small change in input frequency result in Figures 15.1 and 15.2 being so different? To explore leakage in a quantitative fashion, we need to draw upon the results of Note 14, as described in Recipe 15.1 and demonstrated in Example 15.1.
Example 15.1. DTFT for Sinusoidal Pulses
Consider a DFT with N = 32 and T = 1. The signal of interest is x(t) = cos 2 π fct. Apply Recipe 15.1 for the cases of fc = 4F and fc = 4.5F.
• The effects of truncation are hidden in cases where the frequency of the input sinusoid exactly matches one of the DFT bin frequencies. Specifically, in the case of fc = 4F = 4/(NT), the magnitude spectrum has a null at every bin frequency except for the bins corresponding to ±4F, as shown in Figure 15.3. (Note: Figures 15.3 and 15.4 are unusual in that they each contain both a DFT result having a discrete-frequency abscissa and a DTFT result having a continuous-frequency abscissa. See Math Box 15.1 for a discussion of how to set up the equations so that the DFT and DTFT results plot correctly on the same set of axes.)
Figure 15.3. Fourier spectra of a rectangularly windowed segment of a sinusoid having a frequency of fc = 4F. The solid trace represents the magnitude of the DTFT, and the small circles indicate the magnitudes produced by a 32-point DFT.
Figure 15.4. Fourier spectra of a rectangularly windowed segment of a sinusoid having a frequency of fc = 4.5F. The solid trace represents the magnitude of the DTFT, and the small circles indicate the magnitudes produced by a 32-point DFT.
• When the sinusoid’s frequency does not exactly equal one of the DFT bin frequencies, there is generally some nonzero magnitude response in every bin. This is illustrated in Figure 15.4 for fc = 4.5F. In this case, the main lobes of X(f) are wide enough to each span two DFT bins, thus making it difficult to discern the frequency of the sinusoid.
• Intuition might suggest that a sinusoid at fc = 4.5F should have equal magnitude responses in the bins for f = 4F and f = 5F, but examination of Figure 15.4 reveals that this is not the case. The responses in bins 4 and 5 are unequal because of asymmetries in the side lobe that take place in the interval going from bin 5 to bin 27 and in the interval going from bin 28 wrapping around to bin 4.
Math Box 15.1. Plotting DFT and DTFT Results Together in the Same Graph
Figures 15.3 and 15.4 are unusual in that they each contain both a DFT result and a DTFT result. These two types of transform results are usually plotted on diffrently scaled axes.
• The result of a DTFT is typically plotted against a real-valued frequency axis. Because the DTFT result is periodic with a period of T –1, only a single period of the result is usually plotted. For a two-sided plot format, the period plotted is either for cyclic frequencies in the interval [–½T –1, ½T –1] or for radian frequencies in the interval [–πT –1, πT –1]. For a single-sided plot format, the period plotted is either for cyclic frequencies in the interval [0, T –1] or for radian frequencies in the interval [0, 2πT –1]. When the conventional normalization of T = 1 is followed, the plotted frequency interval becomes either –½ ≤ f ≤ ½ or –π ≤ ω ≤ π, for two-sided plots, and either 0 ≤ f ≤ 1 or 0 ≤ ω ≤ 2π, for one-sided plots.
• On the other hand, DFT results are usually plotted against a sequence of integer-valued frequency indices or “bin numbers” m = 0, 1, 2, . . ., N –1. The cyclic frequency corresponding to DFT bin m is given by
To cover the same [0, T–1] frequency interval as the DTFT, the DFT frequency indices would need to run for N + 1 values from m = 0 through m = N.
• Because this particular note is concerned with leakage in the DFT, and the DTFT is playing only a supporting role, it seems logical to keep the plotting format consistent with typical DFT plots. Therefore, for Figures 15.3 and 15.4, the values of this normalized frequency are numerically equal to the corresponding DFT bin numbers.
Recipe 15.1. Computing the DTFT for Sinusoidal Pulses
• An ideal sinusoid of cyclic frequency fc has a continuous-time Fourier transform whose magnitude consists of two Dirac delta functions, one located at f = fc and the other located at f = –fc. The phasing of these two delta functions depends upon the phase of the sinusoid.
• For x(t) = cos 2 π fct, the delta functions are both real and positive.
• For x(t) = sin 2 π fct, the delta functions are both purely imaginary; the delta function at f = fc is negative and the other is positive.
• If x(t) is multiplied by a time domain window, w(t), the DTFT for the resulting product is equal to the convolution of X(f) and W(f).
• The result of convolving W(f) with a delta function, δ(fc), shifts W(f) so that its peak is centered at fc. The Fourier spectrum for a rectangularly windowed sinusoid is the sum of two shifted instances of W(f), with one instance centered at f = fc and one instance centered at f = –fc:
15.1
where
15.2
• The DTFT for a rectangularly windowed, weighted sum of M sinusoids is given by
