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A nonsinusoidal waveform called speedy delivery (SD) that does not undergo shape distortion during propagation on lossy transmission lines having frequency dependent parameters (LFTL) is the subject of this paper. An outline of the proof that the SD waveform propagates on LFTL without shape distortion is included. This waveform also has other unusual propagation properties on LFTL, namely controllable propagation velocity and controllable attenuation. Experimental measurements illustrating some of these special propagation properties of the nonsinusoidal SD signal on cable are discussed in a companion paper in this workshop.