Meteorological Doppler radars have typically utilized constant pulse-repetition intervals (PRI) to facilitate clutter filtering and estimation of weather echo spectral moments via pulse-pair or periodogram-based algorithms. Utilization of variable PRIs to support resolution of velocity ambiguities has been discussed, for example by Banjanin and Zrnic, but not implemented owing to difficulties associated with clutter filter design. Recent work by Chornoboy presents design algorithms for time-varying finite impulse response (FIR) filters that achieve Chebyshev or mean-squared error (MSE) optimality when processing multi-PRI waveforms. This paper is a follow-on to that work, treating techniques for post-clutter filter processing (e.g. periodogram estimation) that are appropriate for such waveforms. Our approach involves a least-squares fitting of the signal - sampled at a nonuniform rate - to a weighted sum of uniformly spaces sinusoids. The sinusoids or "basis functions" are chosen to span a Nyquist interval consistent with the longest PRI in the transmitted waveform, and need not be centered at zero Doppler. Determination of the sinusoid weightings - effectively a discrete Fourier transformation (DFT) - and the associated residual between the harmonic fit and the data area accomplished via multiplications of the signal vector with precomputed matrices. The resulting spectrum estimate can be used directly for weather echo moment calculations, or can be inverse-Fourier transformed using conventional techniques to generate a time-domain signal representation. This work has been motivated by a specific application - estimation of weather spectrum moments for a Wind Shear Processor (WSP) modification to the Federal Aviation Administration's Airport Surveillance Radar (ASR-9). Our approach supports candidate low-altitude radial wind estimation algorithms that operate on frequency-domain signal representations and require that the radar's block-stagger PRI and the possibility of velocity ambiguities be accounted for in generating the spectrum estimates. In principle, however, these processing techniques are also applicable to weather radar systems such as WSR-88D and Terminal Doppler Weather Radar (TDWR) where range and Doppler ambiguities are an operational concern.