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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?

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    $\begingroup$ The sequence already has the smallest possible peak to average power ratio. Maybe you should clarify in which domains these sequences are in and in which domain the multiplication is to be done and in which domain the peak to average power ratio is to be calculated. $\endgroup$ – Olli Niemitalo Jun 16 '15 at 15:04
  • $\begingroup$ @OlliNiemitalo you are right, I shouldnt have missed those details. The sequences defined here are in the frequency domain and they are multiplied in the frequency domain. The peak to average ratio is to be calculated for their product in the time domain $\endgroup$ – Henry Jun 16 '15 at 15:26
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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.

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Not true. If you calculate the inverse FFT for each frequency sequence and the resulting peak to average power ratio in the time domain, you will find that they are the same, $\sqrt 2$ in either case.

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The frequency-domain sequence w corresponds to a circular rotation of the time-domain sequence by 8 samples (1/4 of the frame size). Therefore I think the peak-to-average ratio is the same.

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