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We experimentally demonstrate, for the first time, a method for arbitrary waveform generation (AWG) with time-bandwidth product (TBP) expansion beyond the limitations set by a spectral encoder. We show that, by using an anamorphic stretch transform with a specific group delay (GD) profile, one can boost the TBP of the generated electrical waveform in time-stretch AWG. In particular, we show that the TBP expansion can be achieved, depending on the signal sparsity and proper choice of the warped GD dispersion profile. The new waveform generation system is implemented here using a nonlinearly chirped fiber Bragg grating with a warped GD profile. Experimental demonstrations in this paper show a TBP expansion of more than three times above the maximum achievable using conventional methods. Details of the signal dependence and spectral redundancy/sparsity requirement for time-bandwidth compression and expansion are also studied. [ABSTRACT FROM AUTHOR]