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

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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^-1II(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^-1II(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

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Note that the quantity δ(t-t₀) represents a unit impulse at t = t₀.

Theorem 5

Derivation. Let τ = t - t₀. Then t = τ + t₀ and dt = dτ.

Then by Theorem 1

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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.

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Theorem 7. Let g(t) be continuous at t = t₀ and a < b. Then

Theorem 8. Let a < b. Then

Theorem 9. Let f(t) be continuous at t = t₀. 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 ω₀ 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 ω₀t 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 ω₀t and sin ω₀t

Theorem 14. The Fourier transforms of cos ω₀t and sin ω₀t expressed in terms of the impulse function are

15) F[cos ω₀t] = π δ(ω - ω₀) + π δ(ω + ω₀)

16) F[sin ω₀t] = -iπ δ(ω - ω₀) + iπ δ(ω + ω₀)

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

Fig. 1 (a) shows the function f(t) = cos ω₀t 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

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

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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.

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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 f₀(t) be that function obtained by translating the basic cycle of f(t) so as to center it over the origin and retaining only the basic cycle as shown in Fig. 4 (b) i.e.

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 F₀(ω) of the function f₀(t) at ω = nω₀ = n2π/T i.e.

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

where

Now

Substituting nω₀ for ω in 28) gives

Substituting 29) into 27) gives

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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 f₀(t) be that function obtained by translating the basic cycle of f(t) so as to center it over the origin and retaining only the basic cycle as shown in Fig. 4 (b) i.e.

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

By Theorem 17 the Fourier transform of the function f₀(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 F₀(ω) of the function f₀(t) at ω = nω₀ = n2π/T i.e.

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, ±ω₀, ±2ω₀, ±3ω₀, ..... , etc. The strength of the impulse located at ω = nω₀ 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