Note 30. Bartlett’s Periodogram
The Bartlett periodogram is based on the notion of creating a pseudo-ensemble of sample sequences by dividing a long sequence of samples into P non-overlapping segments of D samples each. The individual sample spectra for these segments are then averaged to form the periodogram. There is an implicit assumption that the random process under consideration is ergodic and that therefore this pseudo ensemble is an adequate substitute for the unobservable “true” ensemble. The steps for computing a Bartlett periodogram are listed in Recipe 30.1.
Example 30.1. Noise Reduction in the Bartlett Periodogram
The MATLAB code in Computer Listing 30.1 generates a pair of cosine waves in AWGN and then computes the Bartlett periodogram for different combinations of SNR and P, as listed in Table 30.1.
Table 30.1. Cases for Example 30.1
Case 1, with P = 1, is equivalent to the unmodified periodogram, and the signal is noise-free. Two distinct peaks corresponding to the two sinusoids are visible in the plotted results, shown in Figure 30.1. In Case 2, the SNR is reduced to -10 dB, and the presence of the sinusoidal signal components cannot be discerned in the periodogram results (Figure 30.2). Case 3, with P = 8, demonstrates the advantage provided by the Bartlett periodogram. A peak is clearly visible in the plotted result (Figure 30.3). However, it is not possible to resolve two distinct peaks corresponding to the two distinct sinusoidal frequencies.
Figure 30.1. Periodogram result for Example 30.1, Case 1
Figure 30.2. Periodogram result for Example 30.1, Case 2
Figure 30.3. Periodogram result for Example 30.1, Case 3
Recipe 30.1. Computing the Bartlett Periodogram
- Divide the available sample sequence into P non-overlapping segments of D samples each. If the original sequence is x[k], the pth segment can be expressed as
xp[n] = x[pD + n]
- Compute the discrete-frequency sample spectrum for each of the P segments
where Xp[m] is the DFT of the segment xp[n].
- Compute the arithmetic average of the P different sample spectra at each frequency:
The result, SB[m], is the Bartlett periodogram.
Computer Listing 30.1. Bartlett Periodogram
The following segment of MATLAB code computes the Bartlett periodogram for a signal consisting of two equal-amplitude cosine waves in AWGN.
Example 30.2. Variance Reduction in the Bartlett Periodogram
A PracSim simulation[3] was used to generate a 2048-sample segment of binary CPFSK signal with Tsymb = 8, Tsamp = 1, and fd = 0.04375. The properties of CPFSK signals are summarized in Math Box 28.1 of Note 28. The Bartlett periodogram was computed for the cases of P = 1 (equivalent to unmodified periodogram) and P = 8. The results are shown in Figures 30.4 and 30.5 respectively. The averaging provided by the Bartlett technique reduces the noise-like variations in the periodogram at low frequencies, but the periodogram’s bias error at higher frequencies is relatively unaffected by the averaging.
Figure 30.4. Periodogram result for Example 30.2, with P = 1
Figure 30.5. Periodogram result for Example 30.2, with P = 8
References
1. M. S. Bartlett, “Smoothing Periodograms from Time Series with Continuous Spectra,” Nature, vol. 161, pp. 686–687.
2. M. S. Bartlett, “Periodogram Analysis and Continuous Spectra,” Biometrika 37, 1950, pp. 1–16.
3. C. B. Rorabaugh, Simulating Wireless Communication Systems: Practical Models in C++, Prentice Hall, 2004.