Once the frame header of the burst signal is detected, the following work will be the synchronization parameter (frequency offset, phase offset, and timing error) estimation.
2.2.1. Frequency Offset Estimation
We start our derivation of the estimation of frequency offset in Equation (
4). Specifically, the term:
In order to facilitate the derivation, we temporarily ignore the influence of the noise term
and define
. Equation (
7) can be written as follows:
As we mentioned earlier, the value of
indicates the judgment time, which only represents the starting position of the burst signal and does not affect the parameter estimation. Thus, we can set take the value of
to zero. Then, incorporating Equation (
8) into Equation (
4):
In order to obtain
in Equation (
6), we first simplify
:
where:
Equation (
12) implies that the value of
A is a real number, and accordingly, it will not affect the result of the complex angle of Equation (
11). Obviously, the following results can be obtained:
The result of Equation (
13) is in an ideal case, that is no noise is considered. Unfortunately, noise has a non-negligible effect on the frequency offset estimation. In order to improve the performance of the estimation, we accumulate the value of
to smooth the effect of noise. We should also note that in order to facilitate the derivation, we defined
above. Consequently, the final frequency offset can be calculated as:
Equation (
14) indicates that the frequency offset estimation range is inversely proportional to
L, and its estimation accuracy is directly proportional to it. It is particularly important to note that
in Equation (
14) equals Equation (
6) after removing the modulus. Therefore, the burst signal detection and parameter estimation algorithm are connected here. In this case, we can directly use the results of burst signal detection to estimate the frequency offset, which is why the complexity of the algorithm in this article is greatly reduced.
Table 2 is a comparison of the complexity of the frequency offset estimation algorithms.
in L&R and Fitz is usually
.
2.2.3. Timing Error Estimation
After the burst signal detection is completed, in addition to estimating the above carrier synchronization parameters, it is also necessary to estimate the timing error. The signal model with timing error is:
where
represents the independent and identically distributed information sequence and g(t) and
denote the baseband shaping function and timing error, respectively. When the received signal is sampled at
, the sampled data
can be obtained. The ML-based data-aided timing error estimation formula given in [
20] is as follows:
If removing the square, Equation (
18) is accurate for the burst signal detection based on cross-correlation. The process of capturing the frame header is the process of searching for correlation peaks. When Equation (
18) achieves the maximum value, the timing error can be calculated as:
As mentioned above, the cross-correlation algorithm has the fatal disadvantage of no anti-frequency offset. Fortunately, we found that the segment correlation used in this article is also the process of capturing the frame header, and its maximum value position is the same as Equation (
18), that is when the local
is completely aligned with the
in the received signal. Therefore, the cross-correlation peak in Equation (
19) can be replaced by the segment correlation peak in Equation (
6), and the timing error estimation algorithm at this time obtains the ability of anti-frequency offset. Thus, burst signal detection and timing error estimation are linked via:
The interpolation method is usually adopted to calculate the maximum value of Equation (
20); here, we chose a simple and highly accurate trigonometric interpolation method [
21]. It should be noted that the number of points used must be an even number. In this article, there were four points employed for interpolation, and each symbol took two points; thus, a total of two symbols were required. The four-point interpolation formula given in [
21] is as follows:
To find the timing error, we only need to find out what the value of
is to maximize
:
Therefore, finding the timing error is transformed into the problem of finding the maximum value of
. We define:
For the convenience of calculation, we define
, where
,
. According to the Euler formula,
,
can be obtained; thus, Equation (
24) can be rewritten as:
The derivative of the above equation:
Take zero for the above equation, then solve it; we can obtain:
According to Equation (
27), the corresponding timing error can be calculated as:
where
P is the number of sampling points used for each symbol during interpolation, and
in this article. The value of
cannot be calculated by Equation (
28) because the equation is already in its simplest form; thus, we have to do approximate processing. The four points adopted in the interpolation are approximately symmetrical, in which the values of
and
in the middle are larger and the values of
and
on both sides are smaller. Therefore, we can get a larger
A and a smaller
, and further,
can be ignored:
Although we obtain Equation (
29) by ignoring
, simulation indicates that it still reaches high accuracy.