SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Fourier integral, transform. Magnitude and phase spectrum. Theorems. Impulse, rectangle, triangle, Heaviside unit step, sign functions. Convolution. Correlation, autocorrelation.

The Fourier integral. Let *f *(t) satisfy the Dirichlet conditions

(a) it is continuous except for a finite number of discontinuities

(b) it has only a finite number of maxima and minima.

on any finite interval

and assume that the integral

exists. Then Fourier’s integral theorem states that

(This is the complex, exponential form of the Fourier integral.)

Fourier transform. If we define

equation 1) becomes

The functions *f *(t) and *F*(ω) are called a Fourier transform pair. *F*(ω) is the Fourier
transform of *f* (t) and *f* (t) is the inverse Fourier transform of *F*(ω). A sufficient condition for
the existence of the Fourier transform *F*(ω) is that

i.e. that the integral exists. This condition is not a necessary condition, however, as functions exist which don’t meet the condition but do have Fourier transforms.

Theorem 1. If *f(t)* is real, then Fourier’s Integral Theorem states

Magnitude spectrum and phase spectrum of a function *f(t*). Let F(ω) be
the Fourier transform of a function *f *(t). The function F(ω) is, in general, complex. Let us write it
as

5) F(ω) = R(ω) + i X(ω) = |F(ω)| e ^{i φ(ω)}

where |F(ω)| is called the magnitude spectrum of *f *(t) and φ(ω) is called the phase spectrum
of *f *(t). Since for any complex number x + iy, x + iy = re^{iθ}, the magnitude spectrum is given by

and the phase spectrum is given by

General Fourier transform theorems

Theorem 2. If *f *(t) is real, then the real and imaginary parts of the Fourier transform F(ω) of
*f*(t) are

and

10) R(ω) = R(-ω)

11) X(ω) = -X(-ω)

12) F(-ω) = F*(ω)

where F*(ω) denotes the complex conjugate of F(ω).

From 10) and 11), we see that R(ω) and X(ω) are even and odd functions of ω, respectively.

From Theorem 2, we conclude:

Theorem 3. If *f *(t) is real, the Fourier transform F(ω) of *f*(t) is

where the real portion of F(ω) is an even function and the imaginary portion is an odd function.

Theorem 4. If F(ω) is the Fourier transform of *f *(t),

F(-ω) = F*(ω)

is a necessary and sufficient condition for *f *(t) to be real.

Theorem 5. Let F(ω) be the Fourier transform of *f *(t). If *f *(t) is real, its magnitude spectrum
|F(ω)| is an even function of ω and its phase spectrum φ(ω) is an odd function of ω.

Theorem 6. If the Fourier transform of a real function *f *(t) is real, then *f *(t) is an even
function of t, and if the Fourier transform of a real function *f *(t) is pure imaginary, then *f *(t) is
an odd function of t.

Fourier sine and cosine transforms

Theorem 6. If *f *(t) is given only for 0 < t < ∞, *f *(t) can be represented by

where F_{C}(ω) is the Fourier cosine transform of *f *(t)

Procedure. If *f *(t) is given only for 0 < t < ∞, we can define *f *(t) for negative t by the equation *f *(-t) = *f *(t), so the resulting function is even. We may then utilize the Fourier cosine transform and
its inverse to represent the function. In interpreting results we must, of course, remember that *f *(t)
is defined only for t greater than zero.

Theorem 7. If *f *(t) is given only for 0 < t < ∞, *f *(t) can be represented by

where F_{S}(ω) is the Fourier sine transform of *f *(t)

Procedure. If *f *(t) is given only for 0 < t < ∞, we can define *f *(t) for negative t by the equation *f *(-t) = *-f *(t), so the resulting function is odd. We may then utilize the Fourier sine transform and its
inverse to represent the function. In interpreting results we must, of course, remember that *f *(t) is
defined only for t greater than zero.

Another form of the Fourier integral and Fourier transform

Let us make the substitution

ω = 2πs

into equation 1) above where ω represents angular velocity and s represents frequency. With this substitution, the Fourier integral theorem becomes

Let us now define the Fourier transform as

and 18) becomes

where *F(s) *and *f(t) * constitute a Fourier transform pair.

Fourier transform theorems

Following is a concise summary of important Fourier transform theorems.

Fourier transform. The Fourier transform of the function *f(x) *is *F(s). We can write F[f(x)]
= F(s) and F ^{-1}[F(s)]= f(x).*

and

Theorem *f *(x) *F(s)*

_____________________________________________________________________________

Linearity a *g(x) *+ b *h(x) * a *G(x) *+ b *H(x)*

(a and b constants)

Similarity *f *(ax)

Shift *f(x-a)* *F(s)e ^{-i2πas}*

Modulation f(x)cos ωx *½ F(s-[ω/2π]) +½ F(s+[ω/2π])*

Convolution *f(x)*g(x)* *F(s)G(s)*

Autocorrelation *f(x)*[f*(-x)]* *|F(s)| ^{2}*

Derivative *f’(x)* *i2πsF(s)*

(providing *f(x)* →0 as x → + ∞)

_____________________________________________________________________________

Derivative of convolution

Rayleigh

Power

(*f *and *g* real)

_____________________________________________________________________________

Special functions. There are a number of special functions that are widely used in Fourier analysis. The most important ones with definitions and graphs are shown in Tables 1 and 2 below.

Def. Rectangle function. A function [see Fig. 1 (a)] of unit height and unit base length defined by

This function is used to select segments of some particular function, masking out the rest. For example, the function f(x) = II(x) cos x selects and displays the portion of the cos x function in the interval [-½, ½] and masks out the rest as shown in Fig.1 (b), providing a compact notation for

It should be noted that hII[(x - c)/b] is a displaced rectangle function of height h and base b, centered at x = c. See Fig 1 (c). Using a suitably displaced rectangle function one can select any segment of a given function, masking out the rest (i.e. reducing it to zero).

Def. Triangle function. A function defined by

See Fig. 2. Note that h∧(x/½b) is a triangle function of height h, base b, and area ½hb. The main importance of the triangle function derives from it being the self-convolution of II(x).

Def. Heaviside’s unit step function. A function defined by

See Fig. 3 (a). As with the rectangle function, the Heaviside unit step function is used to select segments of some function, masking out the rest. The rectangle function II(x) can be expressed in terms of step functions as

25) II(x) = H(x + ½) - H(x - ½)

See Fig. 3(b).

Def. Sign function. The function sgn x (pronounced signum x) is equal to +1 or -1 , according to the sign of x.

Def. Impulse
function (also
called Delta
function). The
impulse function
*δ(x) *(which is not actually a function in the usual sense of the word) has the following two
properties:

It can be defined as:

The τ^{-1}II(x/τ) product represents a rectangle function of height 1/τ and base τ and with unit area.
As τ approaches zero a sequence of unit-area pulses of ever-increasing height are generated. The
limit of the integral is, of course, equal to 1.

Fourier transform of functions built up from rectangle functions

If

it can be shown that its transform is given by

29) F(s) = ace^{-2πbs }sinc cs

If now we have a function built up from a sequence of rectangle functions

the transform is given by

Such functions occur frequently in engineering and can simulate, as closely as desired, any kind of variation.

Convolution of two functions. The convolution of two functions *f* (x) and *g*(x) is
defined as

and denoted by *f
***g*.

Syn. bilateral convolution

The convolution
of two functions
occurs widely in
mathematics,
science and
engineering in
many contexts
and under
various names
(including the
German term
*Faltung)*. One
place where it
occurs is in
connection with
data smoothing.
An example
involving data
smoothing is
shown in Fig. 4
where *f(x), g(x),*
and *h(x)* are
shown. The
functions *g(x-u)
*and* f(u)g(x-u) *are
shown for one particular value of x, x= x_{i}. h(x_{i}) represents the area under *f(u)g(*x_{i}*-u). *

Fig. 5 shows the reason for the German term Faltung (which means “folded back”) for the
convolution operation. Here *g(u) *is shown folded back on itself about the line u = ½x to give *g(x-u)*. For each value of x= x_{i} there is a function *g(*x_{i}*-u) *that can be obtained by a fold at u = ½x —
along with the function *f(u)g(*x_{i}*-u). * h(x_{i}) represents the area under *f(u)g(*x_{i}*-u). *

The Convolution theorem. The Fourier transform of the convolution of *f* and *g* is the
product of the Fourier transforms of *f* and *g* i.e.

32) *F
*(*f* **g*)
=*F* (*f* )
*F*(*g*)

Theorem 8.
Convolution obeys
the
*commutative law*.

33) *f(t)*g(t) = g(t)*f(t)*

Theorem 9. Convolution obeys the *associative law*.

34) [*f _{1}(t)**

Theorem 10. The convolution of a function *f(t) *with a unit impulse function *δ(t) *yields the
function *f(t) *itself.

35) *f(t)*δ(t) = f(t)*

Theorem 11. The following relationships hold:

36) *f(t)*δ(t - T) = f(t - T)*

37) *f(t - t _{1})*δ(t - t_{2}) = f(t - t_{1} - t_{2})*

Correlation functions. In signal processing, the functions

and

(where *f _{1}(t)* and

is called the *autocorrelation* of *f _{1}(t)*. In Fig. 6. the shaded area under the curve f(u)f(u - x)
represents the autocorrelation

for a particular value of x,
x = x_{i}.

Note. The cross-correlation integral is
similar to the convolution
integral but has different
properties. For example, in the case of convolution *f _{1}* f_{2} = f_{2}*f_{1}*, whereas with cross-correlation
this is not the case. When the functions are complex it is customary to define the complex cross-correlation function by

where the * denotes the conjugate.

By making a change of variable from t to t + τ, it can be readily seen that

Theorem 12. The following relationships hold:

44) R_{12}(τ) = R_{21}(-τ)

45) R_{11}(τ) = R_{11}(-τ)

Equation 45) states that the autocorrelation function is an even function of τ.

Theorem 13. If *F*[*f _{1}(t)*]

46) *F*[R_{12}(τ)] = *F*_{1}[(s)] *F*_{2}[(-s)]

47) *F*[R_{21}(τ)] = *F*_{1}[(-s)] *F*_{2}[(s)]

48) *F*[R_{11}(τ)] = *F*_{1}[(s)] *F*_{1}[(-s)]

49) *F*[R_{11}(τ)] = *|F*_{1}[(s)]|^{2} if *f _{1}(t)* is real

From 49) we see that the Fourier transform of the autocorrelation function R_{11}(τ) yields the
energy spectrum *|F*_{1}[(s)]|^{2} . Thus the autocorrelation function R_{11}(τ) and the energy spectral
density *|F*_{1}[(s)]|^{2} constitute a Fourier transform pair.

If *f(t) *is complex it is customary to define the complex autocorrelation of *f(t) *as

where the * denotes the conjugate.

References

Hsu. Fourier Analysis

Bracewell. The Fourier Transform and Its Applications

James, James. Mathematics Dictionary

Spiegal. Laplace Transforms. (Schaum) Chap. 6

Wikipedia

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]