Question 2

A                                _t

t(1)…t(N)

Figure 3

There is a sin- wave F(t) = Asin(ωt+ θ) as shown in Figure 3. The frequency is =2000Hz, ω= 2πf, and θ is the initial phase angle. You are asked to find the amplitude A and phase θ based on the data (with noise) recorded at N time instants t(1),…t(N) (N=10, see table 1).

Table 1

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Time instants (sec)       Recorded data at each time instant

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5.010000e-08                         -0.706660742882478709603333300037775188684463500976562500000

5.020000e-08                         -0.706659646227875248847283273789798840880393981933593750000

5.030000e-08                         -0.706658953831649672139292306383140385150909423828125000000

5.040000e-08                         -0.706658046948851814583747454889817163348197937011718750000

5.050000e-08                         -0.706657500337526500722162836609641090035438537597656250000

5.060000e-08                         -0.706656779024575243397521262522786855697631835937500000000

5.070000e-08                         -0.706655608523821032740386272053001448512077331542968750000

5.080000e-08                         -0.706655021153845264159087946609361097216606140136718750000

5.090000e-08                         -0.706653508699094934364381970226531848311424255371093750000

5.100000e-08                         -0.706652799008767318333923412865260615944862365722656250000

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• For equation fit, we suppose the desired criterion is that the fitted data exhibit a minimum of the sum of the absolute deviations between the recorded data and its prediction.

Please develop a Linear Programming (LP) model to minimise the sum of the absolute deviations, and then solve the formed LP problem using the MATLAB function-linprog(). To describe the constructed LP model, you need to provide solutions details and write down the tabular form.

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(25 marks)

• Suppose the desired criteria for equation fit is that the fitted data exhibit a minimum sum of the squared deviations between the parts price and its prediction. You are then asked to solve the formed least square (LS) problem.

Set up the linear system equation (Ax=B) of the LS problem, and solve it using the Normal equations approach.

(20 marks)

Hint:

To solve both questions, you need to develop a linear model.

Ref:

This math question comes from the following paper: ref.pdf.

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