Lab6 Fourier Series Approximation Using Ordinary Least Squares

# Lab6 Fourier Series Approximation Using Ordinary Least Squares

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6 F OURIER S ERIES A PPROXIMATION U SING O RDINARY L EAST S QUARES 6.1 O BJECTIVES The objective of this lab is to familiarize you with the basics of the Fourier series analysis using artificial signals. 6.2 I NTRODUCTION The Fourier series analysis is one of the foundations of signal processing. The assumption here is that the signal has been present from –inf to + inf. The basic principle is that assume that any signal can be written as a sum of sine and cosine waves with coefficients. In formula form: () 0 11 cos sin nn f ta a n t b n ∞∞ == =+ + ∑∑ t (1.) The term is simply the mean value of the signal (type help mean). Also, the cos and sin terms can be estimated separately using the OLS method. From the previous lab, we know that the parameter vector can be estimated based on the regressor matrix A and the data vector 0 a y as follows: ( ) 1 TT AA Ay θ = (2) If we substitute the following model 1 cos n n f = = n t (3.) into the OLS estimator equation we get a coefficient (parameter) vector and a regressor matrix as follows: 1 2 3 4 5 a a a a a ⎛⎞ ⎜⎟ ⎝⎠

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(4.) 1111 cos cos2 cos3 cos4 cos5 cos mmmm tttt A ⎛⎞ ⎜⎟ = ⎝⎠ 1 m t t The combination becomes a unity matrix, and therefore the OLS estimator can be written as follows. () 1 T AA 2 T Ay N θ = (5.) Where N is the number of data points. Now the coefficient estimation boils down to taking the
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Lab6 Fourier Series Approximation Using Ordinary Least Squares

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