Visible Light Communications (VLC) rely on LED light sources to transmit information wirelessly, offering an alternative air interface to contemporary mobile networks using Radio Frequency bands for wireless connectivity. Most commercial LEDs found in the market nowadays have been designed for illumination, showing a notable non-linear response when performing the Electrical-to-Optical (E/O) conversion, particularly when using deep Intensity Modulation (IM) indexes for increased useful radiated power. Several different OFDM-based waveforms, adapted to the unipolarity requirement that IM demands, have been proposed in the literature. However, the performance of most of these Optical OFDM waveforms is seriously affected by the non-linear magnitude distortion of the low-cost white LEDs, particularly when the Peak-to-Average Power Ratio (PAPR) is high. To tackle this issue, this paper analyzes the performance of Constant-Envelope (CE) OFDM, a waveform that modulates the phase of an Intermediate Frequency (IF) carrier with a real-valued OFDM signal before its E/O conversion. The Bit Error Rate of CE-OFDM is compared to the one of DC-biased Optical OFDM and passband OFDM using a software-defined VLC demonstrator. CE-OFDM allows the VLC system to work in energy-efficient regimes, thanks to its robustness to the nonlinear magnitude distortion introduced by the LED.