Question 10.8: Again consider the situation in which an ideal input signal ......
Again consider the situation in which an ideal input signal\ x_1(t) = \sin t + \sin 2t is contaminated by uniformly distributed white noise over the time horizon [0, 10]. We wish to study the effects of the white noise variance\ σ^2_w and the two major objective functions outlined above, assuming the critical damping hypothesis.
Listing 10.5 performs a uniform search by graphing the objective function given by Equation (10.17) with W = 1. This minimizes the deviation of the signal value against the ideal signal,\ x_1(t) . By varying β from 0 to 3.5 in steps of 0.01 and computing α within the loop, it is possible to construct a graph of the objective function f(α, β) as a function of β. The results of this procedure are shown in Figure 10.14 for white noise variances of\ σ^2_w = 0.4, 0.6, 0.8, and 1.0 . It is clear from the graph that optimal values exist, and vary as a function of β. Assuming convexity, which seems reasonable from the graph, the intervals of uncertainty show an underlying convexity, even though the function is obviously somewhat noisy. With a proper filter, this can be eliminated and a sequential search such as the golden ratio algorithm implemented.
\ f(\alpha,\beta)=\int_{t_{\min}}^{t_{max}}{\{ W[x_1(t)-u(t)]^2+(1-W)[\dot{x}_1(t)-v(t)]^2 \} dt}. (10.17)
If an objective function incorporating both the signal value and the signal derivative\ (W= \frac{1}{2}) is used, the graph changes. Figure 10.15 shows a graph with a similar shape but some significant differences too. In the signal-value-only objective function, the acceptable white noise variance is higher. Also, the optimal β is generally larger for the signal-value-only objective than it is for the signal value and derivative combined objective. In any case, there is an apparent (and for some variances) obvious optimum β-value for the α/β tracker.
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-given formulas and data \ x_1(t) = sin (t) + sin (2t) -optimization procedure \ x_2(t)=\sqrt{3}(2*RND-1) \ \alpha = 2\sqrt{\beta}-\beta t = δk next k next β LISTING 10.5 Program segment to train the α/β tracker according to the specifications of Example 10.8. |
