by searching through the samples sequentially. A match is found when a first block of NCP = Δ/T samples (over the cyclic prefix) correlates with a second block that is NFFT samples later (over the symbol end). The peak location of the complex correlation points to the start of an OFDM symbol, used as an estimate of the integer STO, while the phase of the complex correlation provides a coarse estimate of the fractional CFO because the phase is only measured within ±π. The fractional CFO thus estimated is then removed from the samples by phase rotation (multiplying the samples by a complex exponential of the CFO estimate).
Since the coarse estimate of the symbol start via cyclic prefix matching may be off by ±50 samples, to ensure that the FFT window starts within a safe zone of cyclic prefix, the FFT window is purposely adjusted ahead of the peak location by a certain number of samples. This adjustment introduces an extra phase due to the circular shift property of cyclic prefix, which is readily absorbed into the channel model together with a fractional STO. They are ultimately removed in channel equalization, and thus data demodulation is not affected. Note that the start sample of each sliding FFT window, plus the number of advanced samples, and the fractional STO together constitute the TOA estimate (the start of an OFDM symbol) in the receiver’s local time. However, the TOA estimates are coarse and are typically not tracked over time in a wireless communications receiver. It is one of the reasons a refined TOA estimation and tracking process is needed for PNT.
FFT is applied to the samples within the sliding window, yielding a frequency‐domain representation of an OFDM symbol. After the fractional CFO correction, the spectrum (the received frequency‐domain signal) may still be subject to a shift by a number of frequency bins (subcarriers) equal to an integer CFO if present. Since the continual pilots are shifted by the same amount, the integer CFO can be estimated by determining where the continual pilots reside in the spectrum. Correlation between the continual pilot subcarriers of two consecutive symbols reaches the maximum value when the nominal continual pilot indexes are adjusted up or down by an amount equal to the integer CFO. Furthermore, the phase of the correlation peak between the continual pilots of two consecutive symbols can be used to estimate the residual fractional CFO and the SCO. The integer CFO is then easily corrected by spectrum shift (re‐indexing).
Figure 40.10 Architecture of a DVB‐T OFDM signal processor with TOA tracking.
There are four insertion patterns of scattered pilots in successive symbols, and each pattern therefore repeats once every four symbols. The particular insertion pattern of a received symbol can be determined by correlating its scattered pilot subcarriers with those at the indexes of four possible patterns. Four correlations are therefore calculated, one for each possible insertion pattern, and the one that produces the maximum correlation value is the pattern present in the current symbol.
Once the pattern of scattered pilots is found for the current symbol, the scattered pilots together with continual pilots extracted from the current symbol (as received) are scaled by those from a local replica (as transmitted) to afford an estimate of the channel frequency response (transfer function) at the pilot subcarrier frequencies. As shown in Figure 40.9, the spacing is 12 subcarriers between scattered pilots. As a result, a linear frequency interpolation can be applied to extend the estimated frequency response from the pilot subcarriers to the full used OFDM subcarriers.
At this point, a communications receiver goes on with channel equalization, which scales the received symbol frequency response with the inverse of the estimated channel frequency response at all data OFDM subcarriers to obtain the equalized symbol subcarriers, from which information data bits are demodulated after de‐mapping, de‐interleaving, and decoding, among other necessary steps. On the other hand, a PNT receiver can apply a suitable method to obtain TOA measurements for ranging and positioning as described below.
An open‐loop TOA estimation scheme consists of applying the inverse FFT (IFFT) to the estimated frequency response at all data OFDM subcarriers, zero‐padded to cover the guard bands, to produce the channel impulse response (CIR), which describes the multipath signals in terms of their strength and delay relative to the start of the sliding window. The peak location of either the earliest arrival (above a detection threshold) or the strongest arrival can be taken for coarse TOA estimation through interpolation via a quadratic or sinc‐function curve fitting to within a fractional of a sample (about 0.11 μs for the 8K mode or 30 m) [49].
A more elaborate method to estimate the multipath signal parameters is to apply the matching pursuit (MP) algorithm [50] to the CIR [9, 10, 51, 52] in the time domain or the order‐recursive least‐square matching pursuit algorithm [53] to the channel transfer function [54] in the frequency domain. The estimated multipath signal parameters are then used to initialize a number of DLLs to track the delay of dominant signals for multipath resolution and refined TOA estimation [9, 10, 51, 52, 54]. Similar methods are used for 4G LTE signals [55, 56] and GNSS signals [57–59].
Although individual OFDM symbols are generated and processed almost independently, the signal stream is continuous in a framed OFDM system. As a result, the TOA of OFDM symbols can be tracked from symbol to symbol over time, which can be implemented either based on the pilot component or on the full symbol. In pilot‐carriers‐based delay tracking for refined TOA estimation (the lower‐middle part of Figure 40.10), the received pilot carriers are correlated with early, prompt, and late versions of the locally generated pilots. The normalized early minus late (EML) correlation power serves as the delay error discriminator, which drives a low‐pass loop filter. The filtered delay error is then used to correct the received pilot components so as to align up with the locally generated ones, thus closing the tracking loop [51, 52, 54].
In decision‐directed delay tracking for refined TOA estimation (the lower‐left part of Figure 40.10), the correlation is made between the received full OFDM symbol and the one reconstructed from the demodulated data (the signal path to @ in the right part of Figure 40.10) [60, 61]. However, the latency in decoding and de‐interleaving may degrade the tracking performance. A simpler method is to use hard data decisions on equalized symbols (the signal path in the dashed line to @ in Figure 40.10). Decision‐directed delay tracking offers two advantages. First, the use of full OFDM symbols in correlation involves more subcarriers, particularly high‐frequency components, which tend to sharpen the correlation peak while lowering the side lobes. Second, it allows for a time‐domain implementation of tracking in addition to the frequency‐domain implementation (the dashed line from the frequency‐domain OFDM symbols) similar to the above‐described pilot‐carriers‐based delay tracking. In the time domain, a joint time and frequency tracking loop can be implemented, operating on the time samples independent of the communications receiver except for data bits.
Sample results of processing in‐the‐air DVB‐T signals are presented below. The correlation functions of various components of an OFDM symbol are shown in Figure 40.11. In Figure 40.11(a), the correlation function for the scattered pilots of an OFDM symbol is shown. Since the scattered pilots are inserted once every 12 carriers, their time‐domain waveform has a periodicity of NFFT/12, and so does their correlation function. The details of the correlation peak are shown in Figure 40.11(b). As long as the initial STO is less than 1/12th
