Essential Facts

• The modified periodogram is only slightly more costly to implement than the unmodified periodogram discussed in Note 26. The additional cost is associated with applying a window function to the data prior to performing the DFT.

• The response to a distinct sinusoidal component has a wider main lobe response than for the case of the unmodified technique.

• Side lobe leakage is reduced relative to the unmodified periodogram. The amount of reduction depends upon the particular window function that is used.

• The absence of any averaging still leaves two major disadvantages:

• No mitigation of additive noise that may be present in the signal

• High variability in the result when the input signal is inherently random (such as a communications signal that has been modulated by a random data sequence)

Example 29.1. Side Lobe Reduction in the Modified Periodogram

The MATLAB code in Computer Listing 29.1 generates a signal consisting of two cosine waves—one strong and one weak—and then computes the modified periodogram for this signal.

This code was run for several different window types: (1) rectangular window (equivalent to the unmodified periodogram), (2) Hann window, and (3) Hamming window. The amplitude of the stronger cosine was ten times larger than the amplitude of the weaker cosine. The results are shown in Figures 29.1 through 29.3. Of the three cases, the Hamming window clearly offers the best resolution of the weaker component.

Figure 29.1. Periodogram result for Example 29.1 using a rectangular window

Figure 29.2. Periodogram result for Example 29.1 using a Hann window

Figure 29.3. Periodogram result for Example 29.1 using the Hamming window

Example 29.2. Spectral Nulls in the Modified Periodogram

A PracSim simulation [4] was used to generate a 256-sample segment of a binary CPFSK signal with T_symb = 8, T_samp = 1, and f_d = 0.04375. The modified periodogram for this signal was then computed for the cases of: (1) rectangular window (equivalent to unmodified periodogram), (2) Hann window, and (3) Hamming window. The results are shown in Figures 29.4, 29.5, and 29.6, respectively. The use of a window improves the periodogram’s fidelity in the vicinity of the spectral nulls, and it also appears to reduce the periodogram bias error at higher frequencies. However, the noise-like variations in the periodogram are relatively unaffected by windowing. The properties of CPFSK signals are summarized in Math Box 28.1 of Note 28.

Figure 29.4. Periodogram results for Example 29.2. The dotted trace is the theoretical PSD, and the solid trace is the corresponding unmodified periodogram.

Figure 29.5. Periodogram results for Example 29.2. The dotted trace is the theoretical PSD, and the solid trace is the corresponding modified periodogram using a Hann window.

Figure 29.6. Periodogram results for Example 29.2. The dotted trace is the theoretical PSD, and the solid trace is the corresponding modified periodogram using a Hamming window.

Computer Listing 29.1. Modified Periodogram for Sinusoids of Disparate Powers

The following segment of MATLAB code computes the modified periodogram for a 256-sample segment of two unequal-amplitude cosine waves.