Question 10.7: Suppose an ideal signal x1 (t) = sin t + sin 2t is combined ......
Suppose an ideal signal\ x_1 (t) = \sin t + \sin 2t is combined with uniformly distributed white noise with unit variance to form an observed signal. This observed signal is sampled at a sampling frequency f = l/δ = 10 samples per unit time. Compute and compare the estimates u(k) and v(k) against the ideal functional value and the derivative of the signal.
From the specifications, the noise is produced by\ x_2(t) = \sqrt{3}(2 RND – 1) . This, combined with the ideal signal\ x_1(t) , forms the observed signal
\ x(t) = \sin t + \sin 2t + \sqrt{3}(2 RND – 1).
This is incorporated in Listing 10.4, which is a program segment implementing the α/β tracker using α = 0.5 and β = 0.2 for 10 time units. Since no initial conditions were specified, u(0) and v(0) are set to zero, since this will affect only the transient phase anyway.
The results comparing the estimated signal u(k) with the ideal\ x_1(t) are given in Figure 10.12. It will be noted that u has a smoothing effect on the noise, and yet serves as an estimate of\ x_1 . This is in contrast to the derivative estimator v(k) which is shown along with\ \dot{x}_1(t) = \cos t + 2 \cos 2t in Figure 10.13. Clearly v(k) follows\ \dot{x}_1(t) , but still retains high-frequency components, and there appears to be a small phase shift to the right. Since the α/β tracker acts as a filter, it should be possible to reduce these components by a more judicious choice of the parameters α and β.
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– given formulas and data \ x(t) = sin (t) + sin (2t) + \sqrt{3}(2 * RND – 1) -optimization procedure n = 10 / δ t = δk next k LISTING 10.4 Implementation of the α/β tracker for α = 0.5 and β = 0.2, Example 10,7. |
