Cross-Correlation Periodograms with Decaying Noise Floor for Power Spectral Density Estimation

Abstract

We present a statistical analysis of a variant of the periodogram method that forms power spectral density estimates by cross-correlating the discrete Fourier transforms of adjacent time windows. The proposed estimator is closely related to cross-power spectral methods and to a technique introduced by Nelson, which has been observed empirically to improve detection of sinusoidal components in noise. We show that, under a white Gaussian noise model, the expected contribution of noise to the proposed estimator is zero and that the estimator is unbiased under certain window alignment conditions. This contrasts with classical estimators where averaging reduces variance but not expected noise. Moreover, we derive closed-form expressions for the variance and prove an upper bound on the expected magnitude of the estimator that decreases as the number of windows increases. This establishes that the proposed method achieves a noise floor that decays with averaging, unlike standard nonparametric spectral estimators. We further analyze the effect of taking the absolute value to enforce nonnegativity, providing bounds on the resulting bias, and show that this bias also decreases with the number of windows. Theoretical results are validated through numerical simulations. We demonstrate the potential sensitivity to phase misalignment and methods of realignment. We also provide empirical evidence that the estimator is robust to other types of noise.

Figures (5)

• Figure 1: PSD estimation using the proposed CCP method, Bartlett's method, and Welch's method (Hann window, no overlap) on a 200 Hz signal with additive Gaussian white noise ($\sigma^2 = 1)$. Top subplot uses 10 windows and bottom uses 100 windows. Going from 10 to 100 windows does not decrease the 0 dB noise floor in Bartlett's method or the -4 dB noise floor in Welch's method. In contrast, the CCP method achieves a noise floor decreasing from -8 dB (10 windows) to -13 dB (100 windows). Notably, both are lower than the noise floor from Bartlett's and Welch's method.
• Figure 2: Table of empirical expected values compared with the predicted bound from Theorem 1. 100 samples are taken per interval from $N(0,1)$. Empirical expected value is the mean of the 10th frequency component over 10,000 trials.
• Figure 3: 121 Hz sinusoidal with additive white Gaussian noise divided into 10 windows of length 1 s. Gaps of 20, 40, 60, and 80 samples are introduced between windows. Because of the gaps, a slight phase mismatch reduces the 121 Hz signal slightly but enough of the signal is preserved to detect the frequency in all four cases.
• Figure 4: 121 Hz sinusoidal with additive white Gaussian noise divided into 10 windows of length 1.25 s. Because of the mismatched window length and the target frequency the cross-correlation periodogram (CCP) method completely eliminates the 121 Hz signal in its PSD estimation while Bartlett's method does not encounter this issue. Despite this, multiplying cross-correlations by $i$ before computing the real part restores the signal.
• Figure 5: Table of empirical expected noise contribution when the cross-correlation periodogram is used on non-Gaussian noise. All three exhibit the same noise contribution decay proportional to the square root of the number of windows.

Theorems & Definitions (15)

• Theorem 1
• Corollary 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of \ref{['main thm']}.
• Theorem 2