Peak to Average Power Ratio (PAPR)

Before the signal transmitted into the physical channel, it needs to pass through a power amplifier. Because the orthogonal frequency division modulation (OFDM) is one

kind of multitone modulation system, the power of each subcarrier changes at different time. The phenomenon results in the power amplifier operated in large linear region. In order to operate the power amplifier in high efficient region, the transmitted signals are clipped to make errors.

Therefore, In practice, the transmitted system is peak power restrict. We expect the system operated in the perfect linear region that often means the average power level below the maximum power available. If the gap between the peak and average power values of transmitted signals is large, the power amplifier works on the low efficiency zone.

In order to operate the power amplifier with higher efficiency, we expect to decrease the gap between the peak and average power values of the transmitted signals. The PAPR of the discrete-time signal is defined as,

P AP R = maxn| x(n) |²

E[| x(n) |²] . (3.1)

The x is obtained by unblocking the IDFT output vector and adding a cyclic prefix.

Therefore, the transmitter output has the same peak power and average power as the IDFT outputs xn and we can consider the ratio

maxn | x(n) |²

E[| x(n) |²] (3.2)

Instead of the PAPR of x. The output vector of the IDFT matrix xn = W + s, and its autocorrelation matrix Rx is given by

Rx= E[xx^†] = W^†RsW = εsIM (3.3)

Because the input vector s is uncorrelated, xn is also uncorrelated and their variances are the same, equal to εs. That can be express as

E[max

The maximum value can be attained. For example, when all input symbols have the same value, we can obtain the maximum value √

M maxk | s^k |. And the value of inputs are set as | sⁿ |= max^k | s^k |. Then turning to compute the PAPR value by the expression above,

P AP R = Mmaxk| s(k) |²

E[| s(k) |²] . (3.6)

The PAPR of OFDM system is M times the PAPR of the input modulation symbol.

For some applications of OFDM system, the IDFT matrix size M is large, for instance, M can be as larger as 2048 for fixed broadband wireless access systems. Because M is usually large for the OFDM systems, the PAPR value is large. That is why the PAPR reduction is the popular issue for the OFDM systems. There are many methods proposed to reduce PAPR, e.g. partial transmitted sequence (PTS) method, selective mapping (SLM), Clipping, and so on.

In fact, when we discuss the PAPR problem, we use the more precise PAPR mea-surement in practice. That is done with continuous-time transmitted signal, that is

P AP Rt= maxt| x^a(t) |²

E[| x^a(t) |²] . (3.7)

Where the subscript t denotes the PAPR value is computed from continuous-time transmitted signals.

But we can not get the through the continuous-time transmitted signals directly. We can only predict the P AP Rt by P AP RL. The P AP RL value is come from the discrete-time signal x. The subscript L means the oversampling rate. When L is 2, the sampling period is one half of the original sampling period. Then the sampling rate is twice. By the way, oversampling factor L is 1 means there is no oversampling, and the sampling rate is the same as the Niquist rate. Define P AP RL the as

P AP RL= maxn| x^L(n) |²

E[| x^L(n) |²] . (3.8)

xL is produced by sampling the continuous-time xa(t) transmitted signal .The sam-pling period is T. The expression of xL is given as

xL(n) = xa(nT

The process from discrete-time signal to is shown in Figure 3.1.

)

Figure 3.1: The Process of producing xL(n)

The data stream x(n) is the form ±1±j. Therefore, from (3.8), we can view as linear combinations of sampled points of transmitting pulse p1(t) with unit gain. Then the im-pulse response of transmitting im-pulse affects the PAPR performance. If the combination

of the sampled points of transmitting pulse has large peak to average power ratio, the PAPR performance is worse.

Pay attention that the part of producing xL(n) by oversampling the continuous-time signal xa(t) is only for computing the PAPR value. The oversampling component is not really a part of our system model.

As increasing the oversampling factor, the P AP RL value is more and more close to the P AP RL value. Then we use this method the get the approximate P AP RL value.

The trade-off is the larger oversampling factor waste computation. We expect the use the least oversampling factor to use the lower computation to get the desired information.

By definition, xa(t)is produced from the D/C converter reconstruction filter p1(t), so the PAPR value depends on the reconstruction filter p1(t) of the D/C converter at the transmitter, too. Therefore, choosing different reconstruction filter p1(t), result in some difference of the PAPR values.

Generally speaking, the pulses at the transmitter and the receiver sides would be taken ideal lowpass filters for analyzing OFDM systems. Therefore, the affect coming from transmitting and receiving pulses is ignored as computing the PAPR value. The oversampling factor L is taken 4 for most PAPR analysis papers. But, from the definition of the , we know the choice of the D/C reconstruction filter affects the value. But general PAPR analysis paper set the reconstruction filter p1(t) as ideal low-pass, and ignoring the effect comes from the transmitting pulse.

Besides, we usually compute the PAPR value based on the analog OFDM system which introduced in chapter 2. But the practical implemented OFDM system is digital ODFM system. These two kinds of OFDM systems result in different continuous-time signal xa(t) that is mentioned in chapter 2. Owing to the distinct continuous-time signal xa(t), the P AP RL value is also different. There must be some error when we using the analog OFDM system to compute the P AP RL value. In order to get the more precise P AP RLvalue in the practical environment, we will use the digital OFDM system and

some cases of realizable analog filter to simulate the transmitting and receiving pulses in the thesis.

The complementary cumulative distribution function (CCDF) is the most used method for measuring the performance of PAPR. The CCDF of the PAPR denotes that the prob-ability that the PAPR of a data stream exceeds a given threshold (P AP R₀). The CCDF is defined as,

CCDF (P AP R0) = prob(P AP R > P AP R0). (3.10) Under the same threshold P AP R0, the smaller value of CCDF means the PAPR performance is better.