The correlation based TOF estimation involves the following steps:

1. A pulse is transmitted from transducer 1 in Figure 1-2.
2. Immediately a clock counter is started with respect to the reference clock. The reference clock can be either 8 or 16 MHz.
3. 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 7.
   Equation 7. ${\overline{)r}}_2=\left(r_2^1,r_2^2,r_2^3,...,r_2^N\right)$

   where

   • 
   • f_s = sampling rate of the ADC
   • i = index of the sample
4. Similarly, transmit the pulse from transducer 2 and receive the signal at the transducer. Let the sampled signal at transducer 1 be given by Equation 8.
   Equation 8. ${\overline{)r}}_1=\left(r_1^1,r_1^2,r_1^3,...,r_1^N\right)$
5. Based on r₁ and r₂, a correlation value is calculated by Equation 9.
   Equation 9. $\mathrm{corr}\left(k\right)=\sum _{i=1}^N{r}_1^{i+k}{r}_2^k$

   where

   • k = {–m, –(m–1), ..., (m–1), m}
   • for i < 1 and i > N
6. The maximum of the correlation is calculated by Equation 10.
   Equation 10. $\hat{k}=\underset{k}{\max }\left(\mathrm{corr}\left(k\right)\right)$
7. Let Z_–1 = corr(k̂–1), Z₀ = corr(k̂), and Z₁ = corr(k̂+1) be the values of correlation at and around the maximum.
8. The real maximum of the correlation is now given by the interpolation in Equation 11.
   Equation 11. $\delta =inter{p}_{max}[Z_{–1},Z_0,Z_1]$
9. The net different time of flight is now given by Equation 12.
   Equation 12. $T_{12}^{\mathrm{corr}}=\left(\hat{k}–m+\delta \right)$
10. 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 2 and 3. The details of the equations for δ in Equation 12 can be found in 4 and provided in Section 2.2.1 for completeness. Figure 2-2 shows a block diagram of the receiver for the correlation. Figure 2-3 shows a block diagram of the signal processing algorithms.

Figure 2-2 Block Diagram for Correlation-Based Differential TOF Estimator

Figure 2-3 Block Diagram of Signal Processing for Differential TOF Measurement Using the Correlation Method

Figure 2-4 Interpolation Step in the ADC-Based Correlation Technique for Differential TOF Estimation

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. The search window for finding the correlation peak can be set using USS_ALG_NUM_CYCLES_SEARCH_CORR in USS_userConfig.h or ussAlgConfig.numCycleSearchCorrPeak. Typically, only an additional 1 or 2 adjacent correlation points are computed in addition to the 3 correlation points.