Proposed Stop Criterion

A stop criterion is proposal in this subsection to further reduce the computational complexity by dropping unnecessary calculations. Before introducing this criterion, two points have to be highlighted.

First, recall the definition of the PAPR of a signal

P AP R =

0≤n<N L−1max |x_n|²

E [|xn|²] , (3.15)

where x_n|n = 0, 1, · · · , NL − 1, are time-domain samples and L is the factor of over-sampling. The set of allowed phase factors is written as P = {ej2πl/W |l=0,1,··· ,W −1 from which elements of B^(u) can choose, where W is the number of allowed phase factors (angles). It can be noticed that all allowed phase factors locate on the unit circle, i.e.

their magnitudes are all 1. We can show that for all U candidate sequences x^(u) which are the IDFT of X^(u) derived via element-wise multiplication (3.7) or (3.13), their ex-pected power Eh

|x^(u)n |²i

is a constant if elements of B^(u) are drawn from the polyphase set P . Therefore, we can focus only on the peak value of each candidate sequence (the numerator of (3.15)) instead of calculating (3.15) wholely when choosing the one with the lowest PAPR in our proposed algorithms.

Second, as discussed in the previous subsections, the basic structure in our proposed algorithms is the radix-r DIT IFFT algorithm which owns a particular character that can be utilized to reduce the amount of calculations. Besides, from Fig. 3.3, it can be noted that the original NL-point IDFT is decomposed into r × (NL/r)-point IDFTs with linear combinations at the first stage and the output sequence of each (NL/r)-point

IDFT is a fraction of the length-NL oversampled time-domain samples. Therefore, the fact that in our proposed algorithms, these r × (NL/r)-point IDFT can be computed sequentially enables us to early terminate the procedure for determining a candidate’s PAPR immediately if its intermediate peak value already exceeds the minimum one calculated from the previous candidate sequences. In this way, needless calculations are avoided and thus the computational complexity decreased. In the following discussion, the detail of the proposed stop criterion is introduced.

Figure 3.3: Flow graph of the radix-r DIT decomposition of an NL-point IDFT com-putation into r × (NL/r)-point IDFT comcom-putations.

Our proposed stop criterion is executed by comparing and terminating. To launch this process, it is necessary to have a reference value for comparison. We take the corresponding SLM scheme with radix-4 at the first stage as an example and give some figures to illustrate the proposed stop criterion. Hence, in the beginning, the peak value of the candidate sequence without phase rotation X⁽¹⁾= X, i.e. the original sequence, is taken as the reference value P_ref to which P_i,j, denotes the peak value of the jth

(NL/4)-point IDFT output of the ith candidate sequence for i = 2, 3, · · · , U and j = 1, 2, · · · , 4, is compared to.

We start from the second candidate sequence (i = 2), NL/4 time-domain samples are obtained after calculating the first (NL/4)-point IDFT. The peak value among these samples are found out and called P_2,1 as shown in the frame of Fig. 3.4. Then,

Figure 3.4: Illustration of the proposed stop criterion (I).

P_2,1 is compared to the reference peak value P_ref. If P_2,1 ≥ P_ref, it can be figured out from (3.15) that the PAPR value of the second candidate sequence, generated by the second phase sequence, must be larger than that of the time-domain samples of the original input data block. It is obvious that the second candidate sequence cannot achieve better performance of PAPR reduction, so the process to generate this candidate

sequence is terminated and the rest operations are dropped. Otherwise, the PAPR of this candidate sequence may be lower than that of the original input data, so the peak value of the second (NL/4)-point IDFT output samples P_2,2 is necessary to be computed and compared with P_ref as depicted in Fig. 3.5.

Figure 3.5: Illustration of the proposed stop criterion (II).

As the same account mentioned above, comparing P_2,2 with P_ref is required to decide whether this candidate sequence can achieve better PAPR reduction performance or not.

If P_2,2 ≥ P_ref, we terminate the current process immediately, drop the rest processes to generate this candidate sequence, and move on to the computation of the next candidate sequence. Otherwise, it is requested to calculate the peak value of the third (NL/4)-point IDFT output P2,3 and so on. Finally, if P2,4 is still smaller than Pref, it can be

Table 3.1: The Proposed Algorithm

X⁽¹⁾ = [X₀, X₁, · · · , X_{N −1}] x = IDF T {X}

Pref = max(|x|²) For u = 2 to U

compute P_u,1; let s = 2;

while((P_u,j < P_ref, j < s)&&(s ≤ r)) do compute Pu,s;

s++;

end

if((s − 1 = r)&&(Pu,r < Pref))

update P_ref = max(P_u,j|j = 1, . . . , r);

end end

concluded that the PAPR of this candidate sequence is lower than that of the original input data block and P_ref is updated by the largest among P_2,j, j = 1, 2, 3, 4, and PAPR of the second candidate sequence is recorded.

In general, based on the proposed stop criterion, r (NL/r)-point IDFTs are executed sequentially so that we can derive the peak value a part of the time-domain samples at a time. Therefore, the process of generating candidate sequence can be terminated once the peak value of a part of this candidate sequence already exceeds the reference value P_ref. The complete procedure is summarized in Table (3.1). As we can see that unnecessary computations are neglected to lower the amount of computations.

Accordingly, the computational complexity is reduced by employing the proposed stop criterion.