How does DFT-s-OFDM precoding reduces PAPR?

Question

I would like to understand how does DFT-s-OFDM reduces PAPR compared to CP-OFDM. I am not able to locate any mathematical explanation for this. If anybody can provide an explanation it will be great.

Thanks.

Answer 1

Since there are no answers yet, I'll provide what I believe is being done, and others or the OP can confirm if this makes sense or not. I have worked with OFDM but have not had the opportunity to work on DFT-Spread-OFDM (DFT-s-OFDM) specifically, so my answer is not from experience.

Bottom Line:

I believe DFT-s-OFDM essentially sends single carrier modulations using the OFDM structure. DFT-s-OFDM maps a single carrier by taking an M-point DFT of the time domain sequence of modulated data, and using those frequency bins as the DFT input to a standard N-bin OFDM transmitter. The OFDM transmitter then takes the N-point IFFT to create the time domain sequence that is sent. This utilizes features of OFDM including efficient frequency domain equalization, but is really just sending a single carrier. A single carrier, in comparison to standard OFDM has a lower peak to average ratio (PAPR).

Further Related Details:

The standard OFDM transmit process is to use each data symbol as an FFT bin (and as such each bin represents a separate and independent single-carrier modulation, typically QAM as I demonstrate here). Once a complete FFT frame is filled, we use the inverse FFT to create the time domain waveform to transmit (representing identically what we would transmit if we did actually implement a multi-carrier QAM system with the carrier spacing given by the FFT bin spacing). There are a lot more details and considerations such as adding the cyclic prefix, but that is not necessary for understanding the peak to average affects.

Because of this, OFDM on its own has a very high PAPR: the distribution of the amplitude in time is well approximated as a Gaussian distribution, consistent with the central limit theorem, which has a high PAPR compared to single carrier modulations: random data is assigned to each FFT bin, which is then rotated (with no statistical consequence, it is still random after rotation) and summed (this has statistical consequence- the sum of samples from independent random processes will converge to a Gaussian). I have also confirmed this experimentally as demonstrated at this post. For comparison to that post which shows the Gaussian distribution as well as OFDM even for a small number of bins with a PAPR close to 10.5 dB at 1E-5 probability. Below is a simulation I have done for 16QAM with a relatively aggressive root-raise-cosine alpha (which increases PAPR) of 0.1, where we see the comparative PAPR is only 8 dB at the same 1E-5 probability. The 2.5 dB power savings is of very high interest when the Power Amplifier in your transmitter is dominating your DC power budget, or limiting your transmit range!

Reducing PAPR is referred to as "crest factor reduction". It may be helpful in terms of understanding crest factor reduction strategies to envision a worst case PAPR condition which occurs if we were to assign the same data symbol to every FFT bin: this would be a constant in the frequency domain which represents an "impulse" in the time domain-- meaning one dominant bins will all the rest at 0. This is worst case peak to average, and well beyond the typical Gaussian distribution in magnitude we would expect with the random data condition. (This also illuminates other peak cancellation strategies for crest factor reduction where we can do the exact opposite! That isn't what is happening however with DFT-s-OFDM).

With DFT-s-OFDM we can choose a time domain modulation with even lower PAPR since the transmitter is just recreating the same time domain waveform at the output in contrast to random data at each FFT bin (associated with subsequent data samples in time in standard OFDM). The bin mapping can also be modified through interleaving and grouping of bins to better approximate a more distributed time domain waveform (lower PAPR). Ultimately we have a lot more control of the time domain characteristics of the output of the transmitter when we provide the time domain characteristics at the input, for the case when we want to sent a single carrier only (or smaller subset of carriers).