Assume a surveillance system calculates the time differences (\tau_i for i=1,2,...,m-1) of wavefronts touching each receiver. The TDOA equation for receivers i and 0 is (where the wave propagation speed is c and the true vehicle-receiver ranges are R_0 and R_i) {{NumBlk|:| \begin{align} c \, \tau_i &= c \, T_i - c \, T_0, \\ c \, \tau_i &= R_i - R_0. \end{align} The quantity c \, T_i is often termed a pseudo-range. It differs from the true range between the vehicle and station i by an offset, or bias, which is the same for every signal. Differencing two pseudo-ranges yields the difference of the same two true-ranges. Figure 4a (first two plots) show a simulation of a pulse waveform recorded by receivers P_0 and P_1. The spacing between E, P_1 and P_0 is such that the pulse takes 5 time units longer to reach P_1 than P_0. The units of time in Figure 4 are arbitrary. The following table gives approximate time scale units for recording different types of waves: The red curve in Figure 4a (third plot) is the
cross-correlation function (P_1 \star P_0). The cross-correlation function slides one curve in time across the other and returns a peak value when the curve shapes match. The peak at time = 5 is a measure of the time shift between the recorded waveforms, which is also the \tau value needed for equation . Figure 4b shows the same type of simulation for a wide-band waveform from the emitter. The time shift is 5 time units because the geometry and wave speed is the same as the Figure 4a example. Again, the peak in the cross-correlation occurs at \tau_1 = 5. Figure 4c is an example of a continuous, narrow-band waveform from the emitter. The cross-correlation function shows an important factor when choosing the receiver geometry. There is a peak at time = 5 plus every increment of the waveform period. To get one solution for the measured time difference, the largest space between any two receivers must be closer than one wavelength of the emitter signal. Some systems, such as the
LORAN C and
Decca mentioned at earlier (recall the same math works for moving receiver and multiple known transmitters), use spacing larger than 1 wavelength and include equipment, such as a
phase detector, to count the number of cycles that pass by as the emitter moves. This only works for continuous, narrow-band waveforms because of the relation between phase \theta, frequency f and time T: : \theta = 2 \pi f \cdot T. The phase detector will see variations in frequency as measured
phase noise, which will be an
uncertainty that propagates into the calculated location. If the phase noise is large enough, the phase detector can become unstable. Navigation systems employ similar, but slightly more complex, methods than surveillance systems to obtain delay differences. The major change is DTOA navigation systems cross-correlate each received signal with a stored replica of the transmitted signal (rather than another received signal). The result yields the received signal time delay plus the user clock's bias (pseudo-range scaled by 1/c). Differencing the results of two such calculations yields the delay difference sought (\tau_i in equation ). TOT navigation systems perform similar calculations as TDOA navigation systems. However, the final step, subtracting the results of one cross-correlation from another, is not performed. Thus, the result is m received signal time delays plus the user clock's bias (T_i in equation ). ==See also==