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This paper proves a novel sampling theorem with constant amplitude and variable width pulses. The theorem states that any bandlimited baseband signal within ±0.637 can be represented by a pulsewidth modulation (PWM) waveform with unit amplitude. The number of pulses in the waveform is equal to the number of Nyquist samples and the peak constraint is independent of whether the waveform is two-level or three-level. The proof of the sampling theorem uses a simple iterative technique that is guaranteed to converge to the exact PWM representation whenever it exists. The paper goes on to develop a practical matrix based iterative technique to generate the PWM waveform that is guaranteed to converge exponentially. The peak constraint in the theorem is only a sufficient condition. In fact, many signals with higher peaks, e.g., lower than Nyquist frequency sinusoids, can be accurately represented by a PWM waveform.