Fourier Series and Transform (I)

Fourier SeriesBefore we begin the study of Fourier, we will go thru one of the most famous series: Taylor Series, which approximate function by polynomial.Taylor Formula

f(x) is Cn+1 over x[a,b]; let x, x0 [a,b] then

f(x) = Pn(x) + Rn+1(x) where

(nth Taylor polynomial)

Note: is called the nth variation ofat

for some[x,x0] (nth remainder)

Taylor Series: (expanding at point x0=a)

Ex.: (expanding at x0=0)

(expanding at x0=0)

DEMO w/ taylortool in MATLABA function is periodic if f(x+T)=f(x) for all x, and T is called the period of f(x).If T is the period of f(x), then f(x+nT)=f(x), i.e., 2T, 3T, 4T, are also period of f(x). (The smallest positive period of a periodic function is its fundamental period.)Moreover, if f(x) and g(x) have period of T, then h(x)=af(x)+bg(x) (a, b const) has period T.

For example, cos(x), sin(x), cos(2x), sin(2x), , cos(nx), sin(nx) all have period 2 (figure)

And the series: with an, bn coefficients also has period 2.

For periodic function f(x), piecewise continuous, with period 2 which can be represented by a trigonometric series:

(Fourier Series of f(x))then

Pf.:

and

similar

Ex.: and (figure)

(S1 to S3)

(S1 to S7 and S21)

From above representation:

(Leibniz, 1673 from geometrical analyses)

Fourier Series: If a periodic function f(x) with period 2 is piecewise continuous and has a left- and right-side derivates at each point (not necessary equal) then its Fourier series exist. The function value of the Fourier series equals to the function value of f(x) except at a point, x0, at which f(x) discontinuous and the value of the series is the average of the left- and right-side limits of f(x) at x0, i.e., .(Note: In the figure above, even in the S21 (the right bottom one), there is a jump at the discontinuous points, -pi and pi. This jump will not disappear, flatten out, even as the terms increased to infinite. Instead, the peak maintain roughly the same height, but move closer to the vertical axis as term increased. This is known as the Gibbs Phenomenon.)For function f(t) w/ a period T, not necessary 2, then we introduce a new variable x such that:

has a period of 2.

where

( Euler Formulas:

(Fourier Series of function w/ period T)

Ex.: T=4

bn=0

Ex.: A sinusoidal voltage is passed through a half-way rectifier which clip the negative portion off. (Figure)

More often we would write, now the Fourier series of f on [-T/2, T/2] is:

where

and sometimes we would write the Fourier series as:

(superposition of cosine wave)

where and

This is called the harmonic form or phase angle form in which the term is called the nth harmonic of f. The cn is the nth harmonic amplitude, andis the nth phase angle of f.Complex Number:The Fourier series could also be expressed as complex number form. Before we introduce this form, let's briefly go thru some basic concept in complex number.

Let , then the conjugate of a complex number is. Viewing in the complex (number) plane is represented by the point (a,b) while is represented as (a,-b). The magnitude , or modulus, of is . It is useful to observed that:

Moreover, by introduce the polar coordinate (figure) the complex number can be written as (polar form):

where and (the Euler's formula) are the argument of z. It is noted that this Euler's formula is a definition which can simplify the representation and computation of complex number or trigonometric functions and match the operations of exponential such as: ;

Again this is just a definition because using complex number as exponent has not definitive meanings.

Ex.: (figure)

Moreover,

Let f(x) be a real-valued periodic function w/ fundamental period p. Then complex Fourier series of f(x) is: where and

Pf.: The Fourier series is:

Ex.: Full-wave rectification of

The amplitude spectrum plot (figure)

Note: The coefficient could be viewed as the amplitude associated with frequency of . (figure)

For even function g(-x)=g(x) the Fourier series contains only cosines terms while for odd function h(-x)=-h(x), the Fourier series contains only sines terms. (Figure)

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