Weighted SS(E) = w 2 1( y 1 y1) y n. w n. Wy W y 2 = Wy WAx 2. WAx = Wy. (WA) T WAx = (WA) T b

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1 6.8 - Applications of Inner Product Spaces Weighted Least-Squares Sometimes you want to get a least-squares solution to a problem where some of the data points are less reliable than others. In this case, you want to weight some points more heavily than others. Then the weighted sum of the squares for error is A weighted least-squares problem can be transformed into an equivalent ordinary leastsquares problem by multiply the Weighted SS(E) = w ( y y) + + w n( y n yn) y vector by the following positive diagonal matrix: w y Wy = w y = w n y n w y w y w n y n SS(E) This weighted is the square of or So we seek the solution to the problem Wy Wy Wy W y = Wy WAx WAx = Wy The normal equation for the least-squares solution is (WA) T WAx = (WA) T b Find the least-squares line that best fits the data (, 3), (, 5), (, 5), (, 4) and (, 3). Suppose the errors in measuring the y-values of the last two data points are greater than for the other points. Weight these data half as much as the rest of the data. y = β + β x 3 β 5 A =, β =, y = [ β ] 5 4 3

2 W,,, W A y For a weighting matrix, choose with diagonal entries and. Left multiplication by scales the rows of and. 4 WA =, Wy = (WA) T 4 9 (WA) = and (WA) T 59 Wy = [ 9 5 ] [ 34 ] So, 4 9 β [ 9 5 ] [ ] = 59 β [ 34 ] β [ ] = 4.3 β [. ] Fourier Series Sine and cosine functions enable modern scientists and engineers to analyze the frequency response of structures like bridges, dams and skyscrapers to predict how they will respond to earthquakes, to design computers which can talk to you, read text to you and carry out tasks in response to your voice commands, to remove noise from old audio recordings or to synthesize the sound of any particular instrument or even to generate sounds which no physical instrument can make, to remove scratches and improve contrast from old or faded photographs or to generate special effects for movies to make the impossible seem believable. Many of these later applications are based on the work of Jean Baptiste Joseph Fourier, who worked closely with Napoleon and several famous mathematicians around 8. See the following link for some historical information about Fourier: Background You are probably familiar with the idea that sine and cosine functions are phase-shifted versions of each other. This is expressed mathematically as sin ( x + ) = cos(x) and cos ( x ) = sin(x)

sin(x) Graphically this corresponds to the observation that the function, when shifted to the left by radians (or 9 ) turns into cos(x), and the function cos(x), when shifted to the right by radians

sin(x) Another property of sine and cosine functions you may have learned about in physics or differential equations or a circuits class, is that a linear combination of sine and cosine functions of

3 sin(x) Graphically this corresponds to the observation that the function, when shifted to the left by radians (or 9 ) turns into cos(x), and the function cos(x), when shifted to the right by radians turns into. sin(x) Another property of sine and cosine functions you may have learned about in physics or differential equations or a circuits class, is that a linear combination of sine and cosine functions of the same frequency can be written as a single sine or cosine function of the same frequency but with a different amplitude and an appropriate phase shift. For example, 3 sin(x) + 4 cos(x) = 5 sin ( x + ta n 4 3 ) In general, and A sin(x) + B cos(x) = A + B sin ( x + ta n B ( A )) A sin(x) + B cos(x) = A + B cos ( x tan A ( B )) Furthermore, a linear combination of sine and cosine functions (with frequencies which

4 have a ratio of integers to each other) always produces a periodic function. For example, the graphs of two functions whose frequencies have a rational ratio such as are shown below 3 sin(x) and 4 cos(3x), and the graph of their sum is shown below to be periodic. 3 sin(x) + 4 cos(3x)

Fourier Series What Fourier discovered is the converse of this last observation, that any periodic function can be written as a linear combination of sines and cosines with frequencies which are

5 Fourier Series What Fourier discovered is the converse of this last observation, that any periodic function can be written as a linear combination of sines and cosines with frequencies which are integer multiples of a fundamental frequency. Such a linear combination is called a Fourier series. Fourier series take advantage of the fact that the inner product defined as < f, g >= f (x) g(x) dx makes basis vectors orthogonal for integer and. In fact, This means that a periodic function, such as the square wave commonly used in circuit applications, can be constructed by adding together sines and cosines of appropriate frequencies and amplitudes. Suppose we wanted to approximate the square-wave function SQ(x) shown below {sin mx, cos nx} m n if m n < sin(nx), sin(mx) >= sin(nx) sin(mx) dx = { if m = n < sin(nx), cos(mx) >= sin(nx) cos(mx) dx = if m n < cos(nx), cos(mx) >= cos(nx) cos(mx) dx = { if m = n with a linear combination of sines and cosines from the following set {, cos(x), cos(x), cos(3x),, cos(nx),, sin(x), sin(x), sin(3x),, sin(nx), } In other words, we want to find f (x) such that f (x) = a + a cos(x) + a cos(x) + + b sin(x) + b sin(x) +

6 In effect, we just need to find values for the coefficients a k and b k. In this particular case, since the square-wave function SQ(x) we re trying to approximate is an odd function, f ( x) = f (x), we ll find that all the a k s are zero. To find the b k s, we take the integral in(kx) k =,, 3,, n from to of the product of SQ(x) with the functions s, where. b k SQ(x), sin(kx) = = sin(kx), sin(kx) SQ(x) sin(kx) dx sin(kx) sin(kx) dx SQ(x) = { if x < if < x < b k = ( sin(kx) dx sin(kx) dx ) b k = [ cos(kx) ] = k [ cos(kx) k ] [ cos(k)] k Hence the first four non-zero terms of the linear combination of cosine and sine functions which approximate SQ(x) are given by f (x) = sin(x) + sin(3x) + sin(5x) + sin(7x) and the full Fourier series is given by the infinite series f (x) = [ ( ) k ] k k= sin(kx) The graphs below show the square-wave function SQ(x) overlaid with the first term, the first two nonzero terms, the first three nonzero terms, the first nonzero four terms, and the first ten nonzero terms of the Fourier series. f undamental (k = ) f undamental and f irst overtone (k =, 3)

.9) You can see from the sequence of graphs above that as higher frequency terms are

7 f undamental and f irst two overtones (k =, 3, 5) f undamental and f irst three overtones (k =, 3, 5, 7) f undamental and f irst nine overtones(k =..9) You can see from the sequence of graphs above that as higher frequency terms are included, the sides get more vertical and the tops and bottoms of the square wave approximation get flatter.

8 Practice:. Write the following sum of sine and cosine functions as a single sine function with a suitable phase shift:.. Write the sum of sine functions with frequencies which have integer ratios to each other:. (HINT:.) 3. Which of the series below are Fourier series? a) b) 5 sin(x) + cos(x) sin(3x) + sin(5x) sin(3x) = sin(4x x) and sin(5x) = sin(4x + x) + cosx + cos x + cos 3 x + cos 4 x + sin(x) + sin(x + ) + sin(x + ) + c) d) cos(x) + sin(x) cos(x) sin(x) + cos(3x) + sin(3x) sin(x) + sin(x) sin(3x) Find the first four non-zero terms of the Fourier Series to approximate the function and show graphs of the function below overlaid with the first term, the first two nonzero terms, the first three nonzero terms, the first four nonzero terms, and the first ten nonzero terms. f (x) = { x if x < x if x <

Fourier Series and the Discrete Fourier Expansion

2 2.5.5 Fourier Series and the Discrete Fourier Expansion Matthew Lincoln Adrienne Carter sillyajc@yahoo.com December 5, 2 Abstract This article is intended to introduce the Fourier series and the Discrete