Equal Gain Combining (EGC)
6.3.4 Discrete Time Estimation by Energy Maximiza- Maximiza-tion
Single-frame estimation
In this section, we present and analyze a discrete time energy detection based estimator of the time delay. The energy detector scheme is shown in fig. (6.3).
The received signalr(t) during an observation period [0, Tf] is first filtered by an ideal band pass filter and then squared and integrated over a time
interval Td to produce a measure of the received energy. The output of the integrator is then sampled at rate δ. The produced samples will act as the test statistics of the time delay estimator.
According to the sampling theorem, the filtered signal can be expressed as
b
The output of the integrator can be then approximated as [118, 119]
g(t) = 1
Where ⌊x⌋ denotes integer part ofx.
The produced integrator samples at rate ∆ can be written as
Gk = 1
The time delay estimate is then defined as
bθ0=k0∆ Wherek0=k∈{0,...,N−M−1}arg max L(k) (6.60)
Using the bound in theorem 6.3.1, the MSE of this estimator is upper-bounded as
Figure 6.3: Discrete Time Energy Detector
6.3 Performance of Sub-optimal estimators 137
As Pd is independent from the arrival delay θ0, we get
Pd(j∆) = 1
In Fig. 6.5 we plot upper bounds on the RMSEs achieved by the mis-matched ML estimator, the EGC, and the energy maximization estimator, and we compare them to the IZZLB. The signal is of duration Td = 20ns, Wb = 1Ghz, Tf = 100ns, Nf = 1, PDF α = 2, and discretization step for the upper bound and the energy detector is ∆ = 0.5ns. The mis-matched ML estimator builds its information on Ky by observing K = 10 realizations of the signal.
We observe then that the Mis-matched ML perform nearly as the perfect ML, taking into account that the gap in the performances of the two schemes is due also to the fact that we are considering an upper bound for the mis-matched ML. The EGC estimator achieves the worst performance while the energy detector begin to really approaches the lower bound for SNR greater than 20dB. In Fig .6.5 we plot the same performance, except the perfect ML for which we plot its upper bound, forNf = 10. We observe then clearly that the mis-matched ML approaches the ML, and that the energy detector scheme exploits better the signal repetition than the EGC estimator. However, even with 10 frames repetition, the gain in the performance is not too significant.
To investigate on the impact of the integration window length of the energy detection scheme, we plot in Fig. 6.6 the achieved RMSE vs SNR for various
−40 −30 −20 −10 0 10 20 30 40 10−11
10−10 10−9 10−8 10−7
W=1Ghz,Tf=100ns,Td=20ns,Tf=1ns
SNR in dB
RMSE
IZZLB EGC
Noisy statistics, K=10 Energy maximization
Figure 6.4: IZZLB and upper bounds on RMSE of single frame estimators Vs. average SNR
−40 −30 −20 −10 0 10 20 30 40
10−10 10−9 10−8 10−7
W=1Ghz,Tf=100ns,Td=20ns, Nf=10
SNR in dB
Estimation Error STDV
Perfect 2nd ord. stat., UB EGC, UB
Noisy 2nd ord. stat., K=10, UB Energy Detector, UB
Figure 6.5: Upper bounds on RMSE of muliframe estimators Vs. average SNR
6.3 Performance of Sub-optimal estimators 139
Figure 6.6: Upper bound on RMSE of the energy maximization scheme Vs.
SNR for various integration window lengths and PDF α= 2
value of the integration window W. The discretization step for the upper bound is again ∆ = 0.5ns. In this case, the estimator function is given as
L(k) = 1
We observe then that estimation error decreases significantly for decreasing window length W. We can see also that for W = 4 and W = 8, the error achieved in single frame (one shot) estimation is similar to that achieved by the multiframe one. This means that the energy detector scheme can provide precise estimation of the signal delay without a reliable estimation procedure.
This is a desirable result for ranging or localization applications that may be offered by UWB networks as extra features.
Given the last result, we look now on the impact of the power decay factor on the choice of the integration window length. Fig. 6.7 illustrates the RMSE of energy maximization estimator vs. integration window lengthW, for various discretization step ∆, various power decay factor α, Tf=100ns, Td=20ns, and SNR=-10dB. We then observe a tradeoff behavior where the optimal integration depends on the value of the decay factor. Indeed, for W greater than the optimal window length, loss is due to the fact that the integrator
collect no more significant signal components. While in the contrary case, the integrator misses significant signal components.
0 5 10 15 20
10−10 10−9 10−8 10−7
W=1Ghz,Tf=100ns,Td=20ns, SNR=−10dB
Integration window length in ns
RMSE ∆=1ns, Nf=1
W=1Ghz,Tf=100ns,Td=20ns, SNR=−10dB
Integration window length in ns
RMSE ∆=1ns, Nf=1
Integration window length in ns
RMSE ∆=1ns, Nf=1
Integration window length in ns
RMSE ∆=1ns, Nf=1
Figure 6.7: RMSE of energy maximization estimator Vs. integration win-dow length, for various integrating step ∆, various power decay factor α, Tf=100ns, Td=20ns, and SNR=-10dB
6.4 Conclusion
In this chapter, we give fundamental limitations on the performance of some coherent and non-coherent time-delay estimation schemes of UWB signals.
For this purpose, we use he IZZLB to express the lower bound and we derive a new upper bouund suited for sub-optimal estimators. The upper bound is simple to derive, and was shown to provide good indications of estimation
6.4 Conclusion 141 performances.
The IZZLB was then used to set a lower bound of a ML estimator based on the 2nd. order statistics information, and performance of sub-optimal schemes was given by the upper bounds.
The obtained estimation results of the energy detection scheme are moti-vating to consider the same procedure for reception purpose. Nevertheless, the obtained performance account only for single user transmission, so the multi-user case is still to be investigated.