The method of least absolute deviations has been successfully used in construction of robust tools for spectral analysis under the conditions of outliers and heavy-tailed statistical distributions. This article compares two such spectral analyzers, called the Laplace periodograms of the first and second kind, which can be viewed as robust alternatives to the ordinary periodogram. The article proves that the Laplace periodogram of the second kind has a similar asymptotic distribution to the Laplace periodogram of the first kind which established its association with the zero-crossing spectrum in a recent publication. The article also demonstrates that the Laplace periodogram of the second kind has a smoother sample path which is more suitable for visualization, identification, and estimation of arbitrarily located narrowband components. Extensions of the results to complex time series are discussed. © 2011 Blackwell Publishing Ltd.