Let {x(t), t greaterthanorequalto 0} be a signal that grows with time. We sample

ID: 3229830 • Letter: L

Question

Let {x(t), t greaterthanorequalto 0} be a signal that grows with time. We sample the signal at times t = 0, 1, 2, 3, 4 and observe the values x(0) = 1, x(1) = 5, x(2) = 10, x(3) = 18, x(4) = 30. We want to determine if the signal growth is closer to linear or quadratic in time. (a) Using the observed data, find the best linearly growing approximation to the signal: that is, find the constants a and b in the model x_L(t) = at + b to fit the data with the least squared error. (b) Now find the best quadratically growing approximation; that is, find the constants alpha and beta in the model x_Q(1) = alpha t^2 + beta to fit the data with the least squared error. (c) Which model - linear or quadratic - is better?

Explanation / Answer

Part (a)

Given the linear model. xL(t) = at + b, the least square estimate of a = acap = Sxt/Stt = 71/10 = 7.1

And the least square estimate of b = bcap = x(t)bar – acap x tbar = 12.8 – (7.1 x 2) = - 1.4

[x(t)bar and tbar are respective means, Sxt = [1,5]{(xi – xbar)(ti - tbar)} and Stt = [1,5]{ti - tbar)2}]

So, the linear estimate of signal w.r.t time is xL(t) = 7.1t – 1.4 ANSWER

Part (b)

Given the quadratic model. xQ(t) = t2 + , let for convenience, t2 be represented by T.

Then, least square estimate of = cap = SxT/STT = 303/174 = 1.741

And the least square estimate of = cap = x(t)bar – cap x Tbar = 12.8 – (1.741 x 6) = 2.352

[x(t)bar and Tbar are respective means, SxT = [1,5]{(xi – xbar)(Ti - Tbar)} and STT = [1,5]{Ti - Tbar)2}]

So, the quadratic estimate of signal w.r.t time is xL(t) = 1.741t2 + 2.352 ANSWER

Part (c)

One of the best measure of fit is the sum of squared deviations of the estimate from the actual.

The table below presents this measure for the two estimates.

x(t)

xL(t)

xQ(t)

d1 = x(t) - xL(t)

d2 = x(t) - xQ(t)

- 1.4

2.352

2.4

- 1.352

5.7

4.093

- 0.7

0.907

12.8

9.316

- 2.8

0.684

19.9

18.021

- 1.9

- 0.021

30.208

- 0.208

Sum d12 = 26.7 and Sum d22 = 3.162. So, quadratic fit is better. ANSWER