Note 31. Welch’s Periodogram
The Welch periodogram can be viewed as a generalization of the Bartlett periodogram. As with the Bartlett, the Welch periodogram is based on the notion of creating a pseudoensemble of sample sequences by dividing a long sequence of samples into a set of shorter segments. However, these shorter segments can be allowed to overlap with their neighbors for some portion of their length. Furthermore, a data window, w[n], is applied to each segment before the segment’s sample spectrum is computed. The specific steps for computing a Welch periodogram are listed in Recipe 31.1.
Example 31.1. Side Lobe Reduction in the Welch Periodogram
The MATLAB code in Computer Listing 31.1 generates a signal consisting of two cosine waves, x1 and x2, in AWGN, and then computes the Welch periodogram for this signal. The amplitude of x1 is four times larger than the amplitude of x2. This code was run for the combinations of P, overlap, and window type listed in Table 31.1, with the results plotted in the figures indicated by the table entries.
Table 31.1. Cases for Example 31.1
Case 1, with a rectangular window and P = 1, is equivalent to an unmodified periodogram. Case 2, with P = 128, is equivalent to a Bartlett periodogram. The low side lobe levels of the Hann window used in Case 3 and of the Hamming window used in Cases 4 and 5 make it possible to distinguish the spectral peak corresponding to the weaker signal, x2.
Recipe 31.1. Computing the Welch Periodogram
- Divide the available sample sequence into P (possibly) overlapping segments of D samples each, with a shift of S < D samples between consecutive segments. If the original sequence is x[k], the pth segment can be expressed as
xp[n] = x[pS + n]
- Apply a data window, w[n], to each segment:
yp[n] = w[n]xp[n] P = 0, 1, ..., P – 1
- Compute the discrete-frequency sample spectrum for each of the P windowed segments:
where
- Compute the arithmetic average of the P different sample spectra at each frequency:
The result, SW[m], is the Welch periodogram.
Computer Listing 31.1. Welch Periodogram for Two Unequal Sinusoids
The following segment of MATLAB code computes the Welch periodogram for a signal consisting of two unequal-amplitude cosine waves.
Figure 31.1. Periodogram for Example 31.1, Case 1
Figure 31.2. Periodogram for Example 31.1, Case 2
Figure 31.3. Periodogram for Example 31.1, Case 3
Figure 31.4. Periodogram for Example 31.1, Case 4
Example 31.2. Spectral Nulls in the Welch Periodogram
A PracSim simulation [1] was used to generate a 2048-sample segment of binary CPFSK signal with Tsymb = 8, Tsamp = 1, and fd = 0.04375. The segment of MATLAB code in Computer Listing 31.2 computes the Welch periodogram of this signal. The code was run for the different combinations of P and the window types listed in Table 31.2, with the results plotted in the figures indicated by the table entries. In each case, there is no overlap between consecutive applications of the window. Each plot of results also includes the theoretical PSD for CPFSK that was discussed in Note 28.
Table 31.2. Cases for Example 31.2
Figure 31.5. Periodogram for Example 31.1, Case 5
Figure 31.6. Periodogram for Example 31.2, Case 1
Figure 31.7. Periodogram for Example 31.2, Case 2
Figure 31.8. Periodogram for Example 31.2, Case 3
Figure 31.9. Periodogram for Example 31.2, Case 4
References
1. C. B. Rorabaugh, Simulationg Wireless Communication Systems: Practical Models in C++, Prentice Hall, 2004.