Lower peak to average power ratio by phase shifting A particular, frequency-domain sequence of +1 and -1  as follows:

$X=[-1, -1, +1, +1, +1, +1, +1, +1, +1, -1, +1, -1, -1, +1, +1, -1, -1, -1, +1, +1, -1, -1, -1, -1, +1, -1, +1, -1, +1, -1, -1, +1]$

is being transformed to the time-domain and transmitted over a wireless channel. It was stated that if $X$ is phase shifted i.e. multiplied by the following sequence

$w= [1, j,-1,-j,1, j,-1,-j,1, j,-1,-j,1, j,-1,-j,1, j,-1,-j,1, j,-1,-j..]$

The resulting time-domain sequence would have a lower peak to average power ratio.

Can anyone tell me why?
Each symbol (in the narrow sense - bit) in your frame modulates a sine wave of frequency depends on symbol's number and phase depends of symbol's value. It can be watched from the Inverse Fourier transform. Then these sin waves are added to each other to create time domain signal. If you change phase relations in your data frame, you change phase relations in sine waves in the sum. By this multiplication you actually do this. So obviously it alter PAPR. OFDM precoding technique (e.g. based on Hadamard orthogonal transform) is an example of using this Fourier property.