Presentation on theme: "中華大學 資訊工程系 Fall 2002 Chap 5 Fourier Series. Page 2 Fourier Analysis Fourier Series Fourier Series Fourier Integral Fourier Integral Discrete Fourier Transform."— Presentation transcript:
Page 5 Fourier, Joseph In 1807, Fourier submitted a paper to the Academy of Sciences of Paris. In it he derived the heat equation and proposed his separation of variables method of solution. The paper, evaluated by Laplace, Lagrange, and Lagendre, was rejected for lack rigor. However, the results were promising enough for the academy to include the problem of describing heat conduction in a prize competition in 1812. Fourier’s 1811 revision of his earlier paper won the prize, but suffered the same criticism as before. In 1822, Fourier finally published his classic Theorie analytique de la chaleur, laying the fundations not only for the separation of variables method and Fourier series, but for the Fourier integral and transform as well.
Page 6 Periodic Function Definition: Periodic Function A function f(x) is said to be periodic with period T if for all x T f(x) x
Page 7 Periodic Function f(x+p)=f(x), f(x+np)=f(x) If f(x) and g(x) have period p, the the function H(x)=af(x)+bg(x), also has the period p If a period function of f(x) has a smallest period p (p >0), this is often called the fundamental period of f(x)
Page 8 Periodic Function Example Cosine Functions: cosx, cos2x, cos3x, … Sine Functions: sinx, sin2x, sin3x, … e ix, e i2x, e i3x, … e -ix, e -i2x, e -i3x, …
Page 9 Fourier Cosine and Sine Series Lemma: Trigonometric System is Orthogonal
Page 10 Fourier Cosine and Sine Series A function f(x) is periodic with period 2 and
Page 11 Fourier Cosine and Sine Series (Euler formulas) Then
Page 19 Odd and Even Functions A function f(x) is said to be even if A function f(x) is said to be odd if
Page 20 Odd and Even Functions f(x) x x Even Function Odd Function
Page 21 Odd and Even Functions Property The product of an even and an odd function is odd.
Page 22 Odd and Even Functions Fourier Cosine Series Fourier Sine Series
Page 23 Sun of Functions The Fourier coefficients of a sum f 1 +f 2 are the sum of the corresponding Fourier coefficients of f 1 and f 2. The Fourier coefficients of cf are c times the corresponding Fourier coefficients of f.
Page 24 Examples Rectangular Pulse The function f*(x) is the sum in Example 1 of Sec.10.2 and the constant k. Sawtooth wave Find the Fourier series of the function
Page 25 Half-Range Expansions A function f is given only on half the range, half the interval of periodicity of length 2L. even periodic extention f 1 of f odd period extention f 2 of f