The time at which this leading edge leaves the transmitter antenna is the TOT. The successive times of transmit are related by
where n = 0, 1, 2, … is the number of fields, and Tfield is the nominal period of a field, which is about 24.2 ms (at a field rate of 41.32 Hz) for ATSC‐8VSB signals.
Assume that the receiver’s time ticks at a sampling rate of, say, 10 MHz. The TOA of the leading edge of the field sync segments is estimated by determining the location of the correlation peaks as detailed in Section 40.2.1. Referring to the RX time, we estimate the TOA by counting the samples between successive correlation peaks (denoted by Pn) and the first peak relative to the first sample (denoted by P0).
The first sample is set to be zero for the receiver clock, which differs from the transmitter time by an offset, denoted by t0. As a result, the TOA can be expressed in terms of the correlation peak locations as
(40.2)
where t0 is different for each transmitter using an independent clock.
If we calculate the pseudorange for and at each time of arrival TOAn, the measurements will not be on a uniform scale due to the random nature of the TOA caused by relative movement and noise. Hence, they are called aperiodic pseudoranges, denoted by APRn, and given by
The time of measurement for the aperiodic pseudoranges is the same as the TOA. But aperiodic pseudoranges are not available regularly on a uniform time scale. In order to integrate these pseudoranges with other sensor measurements, interpolation may be required. Alternatively, we can form periodic pseudoranges [23].
In addition to the initial clock offset t0, the clocks may drift in frequency, leading to Tfield and Ns (the number of samples per field) off their nominal values. For the stationary transmitter and receiver, there is no Doppler frequency shift. The changes in symbol rate and sampling rate are due to the clock frequency instability, and the combined effect is observed at the receiver.
Figure 40.16 Relationship of timelines at transmitter and receiver and aperiodic pseudoranges.
For asynchronous transmitters, each pseudorange equation contains at least an unknown of its own related to the transmitter (i.e. the initial clock offset t0). No instantaneous position fixing is possible with such pseudorange measurements for a stand‐alone solution unless additional information such as TOT and LOT is encoded on broadcasting signals (add‐on services). Nevertheless, there are different positioning mechanisms that can be employed to deal with the unknowns in pseudoranges, including differential ranging, relative ranging, and self‐calibration, among others.
Differential ranging involves a reference receiver at a known location that provides an estimate of the TOT or TOA of the same event via a data link to a user in order to cancel out the common TOT at the user receiver, leading to spatial difference of pseudoranges [7, 19, 20]. Relative ranging accumulates changes in range to a transmitter from a starting location [25]. As long as the signal tracking is maintained, the displacement from the starting point can be estimated from the temporal differences of pseudoranges to several transmitters in a process known as radio dead reckoning [23, 24, 82]. If the transmitter locations are known and the receiver starts from a known initial location, the method of self‐calibration can be used to estimate the unknown TOT [17].
As an example, consider the case of self‐calibration with an aperiodic pseudorange (Eq. 40.3). We first form the range between a known transmitter and our receiver at the initial known location as APRn, and count the samples between successive correlation peaks Pn. Since we do not know t0 and Tfield (except its nominal value), we can reformulate Eq. 40.3 as
(40.4)
Assume that the receiver is stationary (or its location known if it is moving). We collect N+1 measurements of APRn = APR and Pn and obtain the following matrix equation:
The least squares solution applied to Eq. 40.5 gives
(40.6a)
(40.6b)
Note that due to the transmitter clock frequency instability, the actual field period may differ from the nominal one, which is thus estimated as part of the calibration process. In Eqs. 40.5 and 40.6c, the scaling by the number of measurements N is to ensure numerical stability of the solution when N becomes very large. Similar equations can be formulated for periodic pseudoranges [23].
Two field test examples with ATSC‐8VSB [23, 29] and one example with DVB‐T [9] are presented next. The test environment with ATSC‐8VSB signals is in the San Francisco Bay area shown on Google Earth in Figure 40.17. The test site is in Foster City; DTV transmitters are located around the Bay at Sutro Tower, Mount San Bruno, Monument Peak, and Mount Allison, respectively; and one CDMA cell tower is along SR92 near the San Mateo Bridge across the Bay. The first test example with ATSC‐8VSB shows the effect of fast fading on mobile ranging, and the second test shows the effect of clock errors on the range bias and their possible calibration.
Mobile Test 1: Slow and Fast Fading. Severe Rayleigh fading occurs for mobile users in urban environments [42, 83], creating “holes” in data streams, which cannot be easily corrected by conventional coding schemes. Only 1 out of 313 segments per data field (about 24 ms) contains pseudorandom (PN) codes that can be used for timing and ranging. Such a low‐duty cycle (0.3%) requires specially designed correlators and code tracking loops for mobile users, particularly when low‐quality clocks are used in both transmitters and receivers. Although subject to Rayleigh fading, tracking of the PN codes is less devastating for DTV‐based ranging than for DTV viewing. In the latter case, interruption prevents continuous reception of ATSC‐8VSB