System Description
The system the authors studied /4-DQPSK modulated with differential detection and a bit rate of 36 Kbps. Transmission and reception filters are both square-root raised cosine filters with a roll-off factor equal to 0.35. The burst has two information blocks (216 bits each) and a sequence training block (22 bits) between them. This structure belongs to the TETRA system. We obtain channel impulse-response estimation with the LSSE (Least Sum Squared Errors) algorithm. We consider four samples for every symbol, using TETRA propagation models TU50 and HT200 for this study. We also assume that frame synchronisation was done previously, so that we expect only small changes in the optimum sampling instant around the initial value.

This article considers two receiver structures (Figure 1):

1. Timing recovery and decimation before differential demodulation
2. Timing recovery and decimation after differential demodulation.

The second possibility has more computational cost since differential demodulation is done with four samples per symbol. We chose differential demodulation because it is easier to implement and offers better performance for high Doppler shifts.

Figure 1:  (a) Differential demodulation after timing recovery and decimating. (b) Differential demodulation before timing recovery and decimating.

Timing Recovery
We propose six schemes for timing recovery in every slot. These schemes are divided into two groups depending on the receiver structure.

1. With timing recovery and decimating before differential demodulation.
   
   
   • Case 1—From the central training sequence, channel impulse response is estimated after the square-root raised cosine filter and before demodulating, getting
     
     
     
     and timing (t₀) is recovered using
     
     
     
     
   • Case 2—In this case, the absolute value of the four samples per symbol before demodulating are averaged over the full slot. The timing is obtained by means of
     
     
     
     
   • Case 3—This scheme is similar to Case 2, but the time slot is divided into several windows of equal length with averaging done over each window. This method will obtain as many t₀ values as the number of windows created. These t₀ values can be different, making decimation process-independent for each window.
     
     
   • Case 4—A t₀ for every received symbol is obtained, choosing the sample with higher absolute value from the four possibilities.
   
   
   
2. With timing recovery and decimating after differential demodulation.
   
   
   • Case 5—In this case, from the four samples per symbol after differential demodulation, we can obtain four values averaging the absolute value of the signal samples over the full slot. The timing is obtained by
     
     
     
     
   • Case 6—From the four samples per symbol obtained after the training sequence differential demodulation, we compute the distance from samples to the transmitted phases (± /4, ±3 /4). The best timing is the one providing minimum distance. This scheme can result in worse performance for high Doppler-shift values due to channel variations propagating along the slot, since this method is only based on the training sequence, which is in the middle of the slot. Applicable expressions are:
     
     

Results
Figures 2 and 3 show the performance of the techniques described in this article for typical propagation models such as TU50 (mobile speed of 50 km/h) and HT200 (mobile speed of 200 Km/h). TU50 and HT200 have Doppler shifts of 25 and 100 Hz, respectively. We present results in terms of BER (Bit Error Rate) vs. SNR (Signal-to-Noise Ratio). The TETRA standard defines some quality requirements in terms of BER vs. SNR for propagation models. These requirements are: for a SNR=40 dB, BER < 3*10^-2 for HT200 and BER 4*10^-3 for TU50, which we will check. You obtain the results for Case 3 with the slot divided in six equal length windows.

You can see from Figure 2 that for TU50, Case 4 is clearly the worst, around 6 dB for BER=10^-2 and it does not achieve the TETRA quality specification of BER less than 4*10^-3 for SNR=40 dB). The remaining cases achieve the TETRA quality specification. Cases 1 and 6 exhibit the same behavior for SNR < 29 dB, improving for Case 1 around 2-3 dB for SNR > 29 dB. Cases 2, 3, and 5 show similar performances for over the entire SNR range, improving 1-2 dB for Case 1.

For HT200, Case 6 is the worst due to high Doppler shift, which leads to significant variations along the slot and does not meet TETRA specifications. Cases 1 and 4 exhibit similar performance and have a BER near the limit of 3*10^-2 at 40 dB. Cases 2, 3, and 5 again show the best results with Case 3 slightly better than Cases 2 and 5.

From these results we can conclude that from the proposed schemes for timing recovery, Case 3 is the best. However, the results of Cases 3 and 2 are very similar. This means that we would select Case 2 due to its simplicity—there are no windows and timing recovery is performed before differential demodulation.

Figure 2:  BER vs. SNR for the TU50 propagation model

Figure 3:  BER vs. SNR for the HT200 propagation model