SolitaryRoad.com

Website owner: James Miller

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

Theorems involving Impulse function. Complex Fourier series representation and Fourier transform of a train of rectangular pulses of width d with period T.

Impulse function (also called Delta function). The impulse function *δ(x) *(which is not
really 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.

Theorems involving the impulse function

Theorem 1. Given any function *f(x), *the following holds:

which is deemed to mean

where the τ^{-1}II(x/τ) product represents a rectangle, centered at the origin, of height 1/τ, base τ, and
with unit area. As τ →0, f(x) →* f(0)* and the limit is equal to

Theorem 2. The Fourier transform of *δ(x) *is given by (using the definition of the Fourier
transform and Theorem 1):

Thus the Fourier transform of the unit impulse function is unity.

Theorem 3

where ω = 2πs.

Derivation. Applying the inverse Fourier transform to 4) gives

Theorem 4

Derivation.

Because *δ(x) *is an even function, the sin integral is equal to zero and

Thus we have

___________________________________

Note that the quantity* δ(t-t _{0}) *represents a unit impulse at t = t

Theorem 5

Derivation. Let τ = t - t_{0}. Then t = τ + t_{0} and dt = dτ.

Then by Theorem 1

___________________________________

Theorem 6. Given function *f(t)*

Derivation. Let τ = at. Then t = τ/a, dt = (1/a) dτ.

Case 1. a > 0.

Case 2. a < 0.

___________________________________

Theorem 7. Let *g(t) *be continuous at t = t_{0} and a < b. Then

Theorem 8. Let a < b. Then

Theorem 9. Let *f(t) *be continuous at t = t_{0}. Then

1) *f(t) δ(t) = f(0) δ(t)*

2) *t δ(t) *= 0

3) *δ(at) *= [1/|a|] *δ(t)*

4) *δ(-t) = δ(t)*

Theorem 10. The Fourier transform of a constant function

*f(t) *= a

expressed as a function of the impulse function is

Theorem 11. Let ω_{0} be a real constant and let *F(ω) *be the Fourier transform of *f(t)* i.e. *F(ω) *=
*F*[f(t)]. Then

Derivation.

Theorem 12. Let *F(ω) *be the Fourier transform of *f(t).* Then the Fourier transform of *f(t) *cos
ω_{0}t is given by

Derivation. Using the identity

and Theorem 11 we get

Theorem 13. The Fourier transform of

expressed in terms of the impulse function is

Derivation. From Theorem 10 we have

F[*f(t) *=1] = 2π δ(ω)

and from Theorem 11 we have

Thus

Fourier transforms of cos ω_{0}t
and sin ω_{0}t

Theorem 14. The Fourier transforms
of cos ω_{0}t and sin ω_{0}t expressed in terms
of the impulse function are

15) F[cos ω_{0}t] = π δ(ω - ω_{0}) + π
δ(ω + ω_{0})

16) F[sin ω_{0}t] = -iπ δ(ω - ω_{0}) + iπ δ(ω + ω_{0})

Derivation. The formulas follow from the use of Theorem 10 and Theorem 12.

Fig. 1 (a) shows the function *f(t) *= cos ω_{0}t and
Fig. 1 (b) shows its Fourier transform.

Fourier transform of a unit step function. The Fourier transform of a unit step function u(t) defined by

is

If the Fourier transform is written as

F(ω) = R(ω) + i X(ω)

then

R(ω) = π δ(ω)

X(ω) = -1/ω

See Fig. 2. It can be seen that it contains an impulse at ω = 0.

The Fourier transform of a
periodic function. A periodic function
*f(t*) with a period T can be expressed as

Theorem 15. The Fourier transform of the periodic function *f(t*) with a period T

is given by

This means that the Fourier transform of a periodic function consists of the sum of a sequence of equidistant impulses located at the harmonic frequencies of the function.

Derivation. Taking the Fourier Transform of both sides of 19) gives

From Theorem 13

so the Fourier transform of f(t) is

___________________________________

Fourier transform of a unit impulse train. The unit impulse train function is defined by

δ_{T}(t) = ..... + δ(t + 3T) + δ(t + 2T) + δ(t + T) + δ(t) + δ(t - T) + δ(t - 2T) + δ(t - 3T) + .....

and consists of the sum of an infinite sequence of equally spaced unit impulses.

Theorem 16. The Fourier transform of the unit impulse train function is

Thus the Fourier transform of a unit impulse train is a similar impulse train. See Fig. 3. Consequently, we can say that the impulse train function is its own transform.

Fourier transform of a
rectangular pulse. Let *P _{d}(t) *denote
the function

consisting of a single pulse of unit height and width d, centered at the origin, as shown in Fig. 4(a).

Theorem 17. The Fourier transform of the function* P _{d}(t) *is given by

where snc (x_{n}) = (sin x_{n})/x_{n} is one variation of the sinc function and

x_{n} = ωd/2

See Fig. 4(b).

Derivation.

___________________________________

Complex coefficients c_{n} of the
Fourier series expansion of a
periodic function. Let* f(t) *be an
arbitrary periodic function with period T such
as the one shown in Fig 5 (a). Let *f _{0}(t) *be that
function obtained by translating the basic cycle
of

Theorem 18. The complex coefficients c_{n} of the Fourier series expansion of the periodic
function *f(t) *are equal to the values of the Fourier transform *F _{0}(ω) *of the function

Proof. The periodic function* f(t) *with period T can be written as

where

Now

Substituting nω_{0} for ω in 28) gives

Substituting 29) into 27) gives

___________________________________

Complex Fourier series
representation of a train of
rectangular pulses of width d with period T. Let *f(t) *be a train of rectangular
pulses of width d with period T as shown in Fig. 6 (a). Let *f _{0}(t) *be that function obtained by
translating the basic cycle of

The complex Fourier series representation of *f(t) *is given by

By Theorem 17 the Fourier transform of the function* f _{0}(t) *is given by

and by Theorem 18 the complex coefficients c_{n} of *f(t) *are equal to the values of the Fourier
transform *F _{0}(ω) *of the function

Thus the complex coefficients c_{n} of *f(t) *are given by

Fourier transform of a train of rectangular pulses of width d with
period T. Let *f(t) *be a train of rectangular pulses of width d with period T as shown in Fig. 6
(a). The complex Fourier series representation of *f(t) *is given by

where the complex coefficients c_{n} of *f(t) *are given by

where snc x = (sin x)/x. It is a variation of the sinc function. It now follows from Theorem 15 that the Fourier transform of f(t) is given by

Equation 30) reveals that the Fourier transform of a function consisting of a train of
rectangular pulses such as that shown in Fig. 6(a) consists of impulses located at ω = 0, ±ω_{0},
±2ω_{0}, ±3ω_{0}, ..... , etc. The strength of the impulse located at ω = nω_{0} is given by (2πd/T) snc
(nπd/T). A spectrum is shown in Fig. 7 for a d/T = 1/5 case.

References

Hsu. Fourier Analysis

Bracewell. The Fourier Transform and Its Applications

James, James. Mathematics Dictionary

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 ]