Bandwidth Decision Method

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Figure 4.5: The same bandwidth distributions shown in Figure 4.1 after power normal-ization.

4.2.2 Bandwidth Decision Method

Because the number of all possible bandwidth distributions we have to detect is limited (as Figure 4.1 shows), we can make use of the concept of multiple hypotheses testing to detect them. First, we use (4.7) to calculate the mean of each point of all possible band-width distributions. In order to apply to the fading condition, we normalize the power to make the mean power of the 2048 points of the bandwidth distributions equal to one.

After we have received 2048 points and calculated the final normalized power spectrum, compute the distance between the received power spectrum and that of all possible band-width distributions. Here, the distance between A and B means the sum of the squared differences between each point of A and B. Then, bandwidth distribution that has the smallest distance is our determination. Figure 4.1 shows the mean of each point of some bandwidth distributions, and Figure 4.5 shows that after power normalization. Due to the fact that some points of the bandwidth distributions are almost the same, we present two ways to calculate the distance. The main different of the two way is the number of points to calculate distance. Here we call the number of points to calculate distance as the number of dimensions of distance. So one of the two way is just calculates the distance

in 2048 dimensions (2048D), and the other reduces the dimension to 23 (23D). The 23D structure is as shown in Figure 4.6, where we combine the points that are adjacent and have the same value no matter which distribution it belongs to, resulting in 23 segments.

In each segment, we calculate the mean of all spectrum points in it to transfer the received 2048 points and the 2048 points of each type of bandwidth distribution to 23D. And then, like 2048D, we calculate the distance with new 23D, and choose the lowest distance as the determination. We analyze the performance of these two methods in the following.

Detection Error Probability Analysis

Consider two arbitrary bandwidth distributions A and B, and the known means of power of each points of them are |A|² and |B|², where |A|² and |B|² can be calculate by (4.7).

Assume that the bandwidth distribution of the signal that we received is A. If we detect A, then that means the distance between the power of received signal and |A|² is lower than the distance between the power of received signal and |B|². Conversely, if we detect B, then that means some added noise has rendered the power of received signal closer to

|B|²than |A|². Therefore, the distance between |A|²and |B|², and the variance of the dis-tance between received signal and |A|²are the key to analyze the detect error rate. Since

|A|² and |B|² are the given value, we can calculate dist(|A|², |B|²)² easily. Following, we compute the variance of distance between any received signal and its bandwidth dis-tribution type. For convenience, we calculate the squared distance instead of the distance.

For 2048D, let R_A(k), k = 0, ..., 2047, denote the 2048 samples of the signal which original bandwidth distribution is A. The variance of the squared distance between R_A and |A|² are (since previous assuming R(k) normal distributed,

|R_A(k)|² exponential distributed) (4.9)

Figure 4.6: Each segment in the 23D structure calculates the mean of all spectrum points Similar to 2048D, the variance of the squared distance between R⁰_Aand |A|²

V ar[dist(R⁰_A, |A|²)²] =

Theoretical Performance Analysis of 23D and 2048D

As preceding analyzed, the performance of bandwidth detection depends on the mean and variance of distance between the power of received signal and different bandwidth distribution.

We already known the squared power of bandwidth distributions is exponential dis-tribution, but we cannot find the probability distribution of the squared distance of the squared power of bandwidth distributions. So we try using the Q function at this stage to calculate the probability that original A but detect B is

P_e(B|A) = Q( dist(|A|², |B|²)² q

V ar[dist(|R_A|², |A|²)²]

), (4.13)

where Q(x) = _2π¹ R_∞

x e^−t22 dt. Then, the totally error probability of original A_kis P_e= 1 −

Y49 i=1,i6=k

{1 − P_e(A_i|A_k)}, (4.14)

where A_i, i = 1–49 and i 6= k are the 48 types of bandwidth distribution other than A_k. Consider the AWGN channel case. We just add the variance of noise to V ar[<{R}]

and V ar[={R}], and use the new |R|² to compute as above. So we can calculate the theoretical performance of 2048D and 23D in AWGN channel as Figure 4.7 shows.

However, the above analysis is assumes usage of all subchannels and no pilot sub-carrier. If we want to apply this method to a true 802.16e OFDMA system, we need to modify the means of all possible bandwidth distributions and the resulting formula is still similarly to (4.7). We can even collect several 2048 points and average them in one de-tection. If we collect n times 2048 points and average, the variance in squared distance will be divided by n and the detection error rate would decrease.

In this section, we have considered the bandwidth structure in 20 MHz. There may be systems that straddle two 20 MHz bandwidths, which follow the rules for the center frequency of a system of a certain bandwidth specified by the system profile in IEEE 802.16e [2]. If we want to consider this condition, we have to calculate the mean power spectrum of the straddling bandwidth distribution and add the straddle condition to all possible bandwidth distributions. This would result in higher error rate and

computa-−2 0 2 4 6 8 10

The Theoretical Performance of a, b, and c System in AWGN channel

2048D a

Figure 4.7: The theoretical performance of the detection of type a, b, and c in AWGN channel.

tional complexity, because the number of distances needed to calculate and compare is increased.