2.4.1 Correlation Based Technique for Differential TOF Estimation

The correlation based TOF estimation involves the following steps:

1. A pulse train is transmitted from transducer 1 in Figure 2.
2. At a certain pre-determined time, the ADC is started to capture the data at the receive transducer. Let the received data be indicated as Equation 5.
Equation 5. ${\overline{\mathrm{r}}}_2=\left\{{\mathrm{r}}_2^1,{\mathrm{r}}_2^2,{\mathrm{r}}_2^3,...,{\mathrm{r}}_2^{\mathrm{N}}\right\}$
where

• 
• f_s = sampling rate of the ADC
• i = index of the sample
3. Similarly, transmit the pulse train from transducer 2 and receive the signal at the transducer. Let the sampled signal at transducer 1 be given by Equation 6.
Equation 6. ${\overline{\mathrm{r}}}_1=\left\{{\mathrm{r}}_1^1,{\mathrm{r}}_1^2,{\mathrm{r}}_1^3,...,{\mathrm{r}}_1^{\mathrm{N}}\right\}$
4. Based on r₁ and r₂, a correlation value is calculated by Equation 7.
Equation 7. $\mathrm{corr}\left(\mathrm{k}\right)=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{r}}_1^{\mathrm{i}=\mathrm{k}}{\mathrm{r}}_2^{\mathrm{k}}$
where

• k = {–m, –(m–1), ..., (m–1), m}
• for i < 1 and i > N
5. The maximum of the correlation is calculated by Equation 8.
Equation 8. $\hat{\mathrm{k}}=\underset{\mathrm{k}}{\max }\left(\mathrm{corr}\left(\mathrm{k}\right)\right)$
6. Let Z_–1 = corr(k̂–1), Z₀ = corr(k̂), and Z₁ = corr(k̂+1) be the values of correlation at and around the maximum.
7. The real maximum of the correlation is now given by the interpolation in Equation 9.
Equation 9.
8. The net different time of flight is now given by Equation 10.
Equation 10. ${\mathrm{T}}_{12}^{\mathrm{corr}}=\left(\hat{\mathrm{k}}–\mathrm{m}+\mathrm{\delta }\right)$
9. In USS Software Library implementations, m is typically chosen to be +1, implying that only 3 correlations must be computed most of the time.

This correlation-based TOF calculation has been reported in the literature previously as given in ^{reference [12]} and ^{reference [13]}. The details of the equations for δ in Equation 10 can be found in ^{reference [14]} and provided in Section 2.4.2.1 for completeness. Figure 3 shows a block diagram of the receiver for the correlation.

Efficient interpolation techniques have been given in ^{reference [4]}. As previously mentioned, for efficiency of implementation, the correlation is computed over few points leading to a low-power implementation.

To ensure that Z₀ is the maximum point, the correlation peak runs through a search algorithm sequentially computing the three adjacent correlation terms (Z_–1, Z₀, and Z₊₁) until it finds the maximum Z₀ that is larger than Z_–1 and Z₊₁ and the earlier correlation terms. Typically, only an additional 1 or 2 adjacent correlation points are computed in addition to the 3 correlation points.