The classical method for sampling a smooth non-bandlimited signal requires a lowpass anti-aliasing filter. In applications like distributed sampling where sampling and quantization operations precede filtering, aliasing-error is inevitable. Motivated by such applications, the sampling of smooth and bounded non-bandlimited signals whose spectra have a finite absolute first moment, without the use of an analog anti-alias lowpass filter, is studied in a centralized setup. Upper bounds for the distortion-rate function are derived by first upper-bounding the distortion with a linear combination of errors due to aliasing and quantization and then balancing their contributions by selecting an appropriate reconstruction bandwidth. For a class of dithered sampling methods, it is shown that a lower quantizer-precision can be traded for a higher sampling-rate without affecting the realizable high-rate asymptotic distortion-rate characteristics. These results are applied to signals with exponentially and polynomially decaying spectral characteristics and truncated bandlimited signals to uncover the realizable distortion-rate characteristics for these signal classes.