  This work establishes the design, analysis, and fine-tuning of a
Peak-to-Average-Power-Ratio (PAPR) reducing system, based on compressed sensing
at the receiver of a peak-reducing sparse clipper applied to an OFDM signal at
the transmitter. By exploiting the sparsity of the OFDM signal in the time
domain relative to a pre-defined clipping threshold, the method depends on
partially observing the frequency content of extremely simple sparse clippers
to recover the locations, magnitudes, and phases of the clipped coefficients of
the peak-reduced signal. We claim that in the absence of optimization
algorithms at the transmitter that confine the frequency support of clippers to
a predefined set of reserved-tones, no other tone-reservation method can
reliably recover the original OFDM signal with such low complexity.
  Afterwards we focus on designing different clipping signals that can embed a
priori information regarding the support and phase of the peak-reducing signal
to the receiver, followed by modified compressive sensing techniques for
enhanced recovery. This includes data-based weighted {\ell} 1 minimization for
enhanced support recovery and phase-augmention for homogeneous clippers
followed by Bayesian techniques.
  We show that using such techniques for a typical OFDM signal of 256
subcarriers and 20% reserved tones, the PAPR can be reduced by approximately
4.5 dB with a significant increase in capacity compared to a system which uses
all its tones for data transmission and clips to such levels. The design is
hence appealing from both capacity and PAPR reduction aspects.
