The transmit signals in an orthogonal frequency-division multiplexing (OFDM) system can have high peak values in the time domain since many subcarrier components are added via an inverse fast Fourier transformation (IFFT) operation. As a result, OFDM systems are known to have a high peak-to-average power ratio (PAPR) when compared to single-carrier systems. In fact, the high PAPR is one of the most detrimental aspects in an OFDM system as it decreases the signal-to-quantization noise ratio (SQNR) of the analog-digital convertor (ADC) and digital-analog convertor (DAC) while degrading the efficiency of the power amplifier in the transmitter. As a side note, the PAPR problem is more of a concern in the uplink since the efficiency of the power amplifier is critical due to the limited battery power in a mobile terminal.

Let’s start by showing why PAPR problems are an important problem to take care of in an OFDM system. The PAPR of a signal is expressed by the following formula:

Where ()* corresponds to the conjugate operator. Since an OFDM symbol can be express as a sum of complex tones equally spaced in frequency, let’s start by calculating the PAPR of a single complex tone. Consider a complex tone signal:

with a period T. The peak value of the signal is:

The mean square value of the signal is:

This gives us a PAPR of 0 dB. Consider that an OFDM time signal is made of *K* complex tones (usually called subcarriers). Our signal can be represented by the following formula:

For simplicity, let’s assume *a _{k}*=1 for any

*k*. In this scenario, the peak value of the signal is:

The mean square value of the signal is:

Given this, the PAPR of an OFDM symbol with *K* subcarriers, with each subcarrier having the same modulation, is simply *K*.

Still, if we take a look at the 802.11a numbers, an OFDM symbol would have a PAPR around 17 dB (since *K*=52), which is fairly high. However, this number is the worst-case scenario and the probability of getting this high number is very unlikely as the modulated data on each subcarrier is theoretically random and uncorrelated. For this reason, the PAPR of an OFDM system is usually interpreted as a random variable with a distinct probability density function (PDF). So, when we are talking about PAPR reduction, we are usually talking about increasing the probability of getting low PAPR values overall.

Here is an example of the PDF for an OFDM signal with 8 subcarriers, using a QPSK modulation:

From the above figure, we can conclude that the PDF of the real and imaginary parts of an OFDM signal follows a Gaussian distribution (also called normal distribution). They also show that the power of an OFDM signal is Rayleigh-distributed.

Knowing this, our main goal is to reduce the actual variance of that Rayleigh distribution. There are many ways to achieve this. For some examples, see the following blog series:

https://nutaq.com/blog/papr-reduction-techniques-mimo-ofdm-systems-part-1-impact-papr-performance-mimo-ofdm-systems

You can also check out the following blog post written by Dr. Younes Jabrane, from the École Nationale des Sciences Appliquées (ENSA) of Marrakech:

https://nutaq.com/blog/reduction-power-envelope-fluctuations-ofdm-signals

It shows how to use Nutaq’s platforms to implement an OFDM PAPR reduction technique.