This dissertation introduces a new class of Doppler estimation capable radar and communications waveforms called Logarithmic Frequency Waveforms (LFWs). These waveforms are created by mapping the magnitudes, phases, and spacing of good digital radar codes onto a set of carriers defined over a logarithmically warped frequency axis. This mapping preserves the digital code's autocorrelation when matched filtering is done over the warped frequency variable. The logarithm converts the multiplicative Doppler shift into a linear translation of the code's autocorrelation and thus is also used to estimate the waveform's Doppler shift.

There are two main classes of Logarithmic Frequency Waveforms. The first class comprises the Logarithmic Frequency Codes (LFCs) and the second class encompasses the Logarithmic Frequency Rulers (LFRs). LFCs are created by mapping phased radar codes, such as Barker codes, Frank Codes, and Px codes, onto the magnitudes and phases of a set of carriers that are equally spaced after logarithmically warping the frequency axis. The main subtype of LFCs is made up of the Multicarrier Logarithmic Frequency Codes (MC-LFCs) which are waveforms created by transmitting all the carriers simultaneously. LFRs are created by spacing the carriers' logarithmic frequencies according to a Golomb Ruler. The first subtype constitutes the Multicarrier Logarithmic Frequency Rulers (MC-LFRs), created by transmitting all the carriers simultaneously. The next subtype is formed from Hopping Logarithmic Frequency Rulers (H-LFRs), which are transmitted by hopping the carriers sequentially in time. Extended Logarithmic Frequency Rules (E-LFRs) are created by using an Extended Golomb Ruler in the logarithmic frequency domain as the carrier frequency spacing mechanism. Finally, we introduce for the first time a hybrid type called Phased Logarithmic Frequency Rulers (P-LFRs), which are created by mapping phased radar codes onto the carriers of an LFR.

Chapter 1 introduces the general model for an LFW, along with LFCs, LFRs, and their various subtypes. Examples of how these waveforms perform Doppler shift estimation are given along with plots of the waveforms, their frequency spectra, and their time and logarithmic frequency autocorrelations. Chapter 2 derives the logarithmic frequency domain Neyman-Pearson detectors for LFCs in Additive White Gaussian Noise (AWGN) and illustrates their performance numerically. The probability of a Doppler error is also derived and illustrated numerically. Chapter 3 follows the same blueprint for LFRs. Chapter 4 numerically illustrates the ambiguity functions for the various LFW subtypes along with their Doppler estimation performance in AWGN. The Doppler estimation performance is then compared with that of the Extended Matched Filter (EMF). Chapter 5 concludes the dissertation and discusses future work.