1.4 Simulation Results
1.4.1 Effect of IF Estimation Error on the Performance of the ASTFT-tf 19
The standard deviation in the ASTFT-tf is dependent on the chirp rate of the signal. There-fore, accuracy of IF estimation would influence the performance. In this chapter, a low-complexity CM5-based ASTFT is adopted for IF estimation in the ASTFT-tf. To analyze the effect of the IF estimation error on the energy concentration, the ASTFT-tf with perfect IF estimation is compared with the original ASTFT-tf (using the CM5) and the ASTFT-tf substituting the CM5 for the CM3. Consider a synthetic signal given by
x(t) = cos(200πt− 20πt2)+ cos (4π sin(5πt) + 80πt) ,
with ∆t= 1/256 and ∆f = 1. The ridges shown in Fig. 1.6(a) are the exact IFs, and the corresponding ASTFT-tf is depicted in Fig. 1.6(d). The ridges shown in Figs. 1.6(b) and
0 0.5 1
Figure 1.6: Effect of IF estimation error on the performance of the ASTFT-tf: (a) exact ridges; (b) detected ridges from the CM3-based ASTFT (c) detected ridges from the CM5-based ASTFT; (d) ASTFT-tf using the ridges in (a); (e) ASTFT-tf using the ridges in (b); and (f) ASTFT-tf using the ridges in (c). Subfigure (e) shows that partial serious IF estimation error would only induce partial ASTFT performance loss. Subfigure (f) shows that small IF estimation error is tolerable.
1.6(c) are respectively obtained from the IF estimation methods based on the CM3 and the CM5. The ASTFT-tf corresponding to the CM3 and the ASTFT-tf corresponding to the CM5 are depicted in Figs. 1.6(e) and 1.6(f), respectively. It is shown that higher IF estimation error would lead to lower energy concentration. By comparing Figs. 1.6(d) and 1.6(f), the performance loss induced by the CM5-based IF estimation is tolerable. This explains why the low-complexity CM5-based ASTFT rather than other more involved methods is adopted for IF estimation.
1.4.2 Energy Concentration Analysis of the ASTFT-f, the ASTFT-t and the ASTFT-tf
Energy concentration of the ASTFT-f, the ASTFT-t and the ASTFT-tf is examined by using a multicomponent signal consisting of two linear FM components,
x(t) = exp
where f1 = 0.05, f2 = 0.5, f3 = 0.15, f4 = 2. Fig. 1.7 shows these three TFRs of the sig-nal with ∆t= 1/2 and ∆f = 1/256. The energy concentration of the ASTFT-tf is higher than that of the ASTFT-f. This is because the standard deviation of the Gaussian kernel in the ASTFT-f is time-independent. Therefore, observing the ASTFT-f from f = 0.15 to f = 0.5 in Fig. 1.7(a), the obtained standard deviation cannot be simultaneously the op-timal for both the components. Similarly, the ASTFT-tf has higher energy concentration than the ASTFT-t since the standard deviation in the ASTFT-t is frequency-independent.
Thus, observing the ASTFT-t from t = 0 to t = 120 in Fig. 1.7(b), the obtained standard deviation can not be simultaneously the optimal for both the components. Since these two components have different chirp rates, it is better to use time-frequency-varying standard deviation adapted to each component. This example verifies that the ASTFT-tf is supe-rior to the ASTFT-f and the ASTFT-t for signals having multiple chirp rates at some time instant or frequency.
Consider another signal which comprises one nonlinear FM component,
x(t) = exp(j2π(100t5− 25t4− 85t3+ 8t2− 62t)).
The ASTFT-f, the ASTFT-t and the ASTFT-tf of the signal with ∆t= 1/256 and ∆f = 1 are depicted in Fig. 1.8. At any frequency between −102Hz and −62Hz, there are two different chirp rates along the time axis. Therefore, the ASTFT-f is no doubt inferior to the ASTFT-tf in this frequency band. It has been illustrated in Fig. 1.5 and Section 1.3.4 that the time-frequency-varying standard deviation is still a better choice even though the signal has single chirp rate at any time instant or frequency. Therefore, at any fre-quency larger than−62Hz in Figs. 1.8(a) and 1.8(c), it can be found that the ASTFT-tf somewhat outperforms the ASTFT-f. From Fig. 1.8(b), it is shown that the ASTFT-t suf-fers from poor energy concentration for two main reasons: first, the standard deviation is time-varying but not frequency-varying; second, the relationship between the standard de-viation and the chirp rate in (1.8) is not adequate. Besides, as mentioned in Section 1.2.2, there’s no criterion for determining the threshold ξ used in this relationship. ξ = 0.07 is used for the signal in Fig. 1.7, while ξ = 25 is applied to the signal in Fig. 1.8. These
Time (sec)
Figure 1.7: The TFRs of a multicomponent signal: (a) ASTFT-f; (b) ASTFT-t; and (c) ASTFT-tf. In this case, the ASTFT-tf has the highest energy concentration.
Figure 1.8: The TFRs of a nonlinear FM signal: (a) ASTFT-f; (b) ASTFT-t; and (c) ASTFT-tf. In this case, the ASTFT-tf has the highest energy concentration.
0 5 10
(a) Envelopes of x(t) and the two components
Envelope
Figure 1.9: TFRs of a more general multicomponent signal: (a) envelopes of the signal x(t) and its two components (A1(t) is the sinusoidal envelope of the linear FM component, while A2(t) is the positive random envelope of the nonlinear component); (b) ASTFT-f;
(c) ASTFT-t; and (d) ASTFT-tf. The ASTFT-tf is somewhat better than the ASTFT-f and the ASTFT-t, especially for the TF regions within the dashed rectangles.
values of ξ are obtained by means of try and error such that most part of the ASTFT-t has high energy concentration. In contrast, the energy distribution in Fig. 1.8(c) shows that the nonparametric relationship used in the ASTFT-tf is capable of achieving much higher energy concentration, even though the ASTFT-t and ASTFT-tf use the same estimated chirp rate.
Considering the more general signal model x(t) =∑
k
Ak(t) exp (jφk(t)) where Ak(t)
≥ 0, another simulation result is given in Fig. 1.9. In this simulation, the signal under analysis consists of a linear FM component with sinusoidal envelope and a nonlinear FM component with positive random envelope,
x(t) = A1(t) exp(j2π(−0.3125t2+ 2t))+A2(t) exp (j2π (13 cos 0.1πt + 5 cos 0.2πt)) ,
where A1(t) =− cos 0.2πt+3 and A2(t) is the absolute value of a Gaussian random signal with unit variance. The envelope of x(t), A1(t) and A2(t) are shown in Fig. 1.9(a). The ASTFT-f, the ASTFT-t and the ASTFT-tf of the signal with ∆t = 1/16 and ∆f = 1/16 are depicted in Figs. 1.9(b), 1.9(c) and 1.9(d), respectively. The ASTFT-tf is somewhat better than the ASTFT-f and the ASTFT-t, especially for the TF regions within the dashed rectangles shown in Fig. 1.9.