It is observed from (4) that the mean of the denominator of the timing metric is 2N times the sum of variance of signal and noise. So, the mean of the denominator value is high at correct time. This results in the low value of timing metric at correct time. Thus the probability of miss detection (PM) is more. Further, low timing metric value at correct time leads to lower value in the difference between timing metric value at correct time and side lobe value. This results in the increase of false detection (PF). Due to several false detections, and miss detections, the probability of detection failure (PD) increases. In order to decrease PD, it is required to reduce probability of miss detection PM. This can be achieved by increasing the value of timing metric at correct time. In order to increase timing metric value at correct time, we have proposed a new timing metric with modified denominator in (5).

The timing metric at αth instant is given by:

$${M}_{Proposed}\left(\alpha \right)=\frac{{\left|{P}_{Proposed}\left(\alpha \right)\right|}^{2}}{{{R}_{Proposed}}^{2}\left(\alpha \right)}$$

5

where, \({ P}_{Proposed}\left(\alpha \right)={\sum }_{k=0}^{\frac{N}{2}-1}r\left(\alpha -k\right).r\left(\alpha +k\right)\) (6)

$${R}_{Proposed}\left({\alpha }\right)={\sum }_{k=0}^{\frac{N}{2}-1}{\left(\left|r\left(\alpha -k\right)\right|-\left|r\left(\alpha +k\right)\right|\right)}^{2}$$

7

It is observed that the denominator of the proposed timing metric is the square of difference of absolute value over N/2 samples. In order to determine the performance of the proposed timing metric, we have determined the mean of denominator of timing metric.

Expectation or Mean of the denominator R Proposed can be written as,

E\(\left[{\text{R}}_{\text{P}\text{r}\text{o}\text{p}\text{o}\text{s}\text{e}\text{d}}\left({\alpha }\right)\right]\) =2NσN2 (8)

It is observed from the above equation that the mean of denominator of the proposed scheme depends as the variance of noise only. Since, variance of noise is low; the value of denominator is low. This low value of denominator leads to higher value of timing metric at correct time compared to Zhang’s scheme. The difference between high value of timing metric at correct time and side lobe value increases. This results in the lower value of PM and PF compared to Zhang’s scheme and subsequently leads to lower probability of detection failure PD.

The mean of denominator of the timing metric for the Zhang’s scheme and the proposed timing metric is represented in Fig. 1. It represents mean of the denominator v/s SNR for the timing metric due to Zhang and the proposed timing metric. It is observed that the mean of both the schemes decreases with SNR. However, the mean of the denominator for the Zhang’s timing metric is higher than the proposed timing metric by 57.6.

The timing metric of Park, Zhang and the proposed method is shown in the Fig. 2 for AWGN channel distortion. Here Total Subcarrier (N) is 1024 samples, a cyclic prefix (Ncp) is 128 samples and the AWGN transmission Channel is used. The correct Timing index is 1152.

Since at correct timing new timing metric yields a sharp peak which is greater than 1 and negligible side lobe as compared to peak value, it has low miss detection probability. Unlike Park and Zhang scheme where a side lobe appeared at a 640-time index causing false alarm detection, our proposed method has an insignificant side lobe at 640-time index. So it's better than Park and Zhang scheme.