Fourier Transform Pairs The Fourier transform transforms a function PDF

Table of Fourier Transform Pairs

Fourier Transform F(w) 2. 1. )( Definition of Fourier Transform ... Inversion formula. (2). ?f(?t). 2?f(?). Duality property. (3) e.

Fourier Transform Pairs The Fourier transform transforms a function

Fourier Transform of Gaussian. Let f(t) be a Gaussian: f(t) = e. ?? t. 2 . By the definition of Fourier transform we see.

The Fourier transform

= ??. A exp(?[k/. ?. 2A]2/2) = ??. A exp(?k2/(4A)). Page 8. Examples. Description. Function Transform. Delta function in x.

Appendix A: Fourier Transform

functions of one and two variables. A.I. One-Dimensional Fourier Transform. The harmonic function F exp(j2rvt) plays an important role in science and

Lecture 11 The Fourier transform

f(t)e. ?j?t dt very similar definitions with two differences: • Laplace transform integral is over 0 ? t < ?; Fourier transform integral.

Chapter 1 The Fourier Transform

1 mar. 2010 F(x) exp(itx)dx. From the above we deduce a uniqueness result: Theorem 2 Let f

TRANSFORMÉE DE FOURIER DISCRÈTE

2. Transformée de Fourier Rapide TFR Fast Fourier transform FFT . x(nTe)e. ? j2? f nTe. Ce qui donne pour les valeurs de fréquences f k = k fe/N:.

1 Fourier Transform

17 août 2020 ?f(x)e?ikx dx. Remark 2. Technically the Fourier inversion theorem holds for almost everywhere if f is discontinuous. In fact one can show ...

Integral Transformation Methods 1. Fourier transforms. 1.1

Example 2. (§12.18 1 a)) The most important Fourier transform formula in practice is. F( e? ax2 )( ?) = 12 a. ? e? ? 2. 4 a . (3). In particular.

4 Fourier transform

(ii) Let c be a positive real number. Compute the Fourier transform of f(x) = e?cx2 sin(bx). Solution: Using the fact that sin(bx) = ?i1. 2.

Fourier Transform Pairs

The Fourier transform transforms a function of

time,f(t), into a function of frequency,F(s):

F{f(t)}(s) =F(s) =Z

-¥f(t)e-j2pstdt. tion of frequency,F(s), into a function of time, f(t): F -1{F(s)}(t) =f(t) =Z -¥F(s)ej2pstds. form is the identity transform: f(t) =Z Z¥ -¥f(t)e-j2pstdt? e j2pstds.

Fourier Transform Pairs (contd).

Because the Fourier transform and the inverse

Fourier transform differ only in the sign of the

exponential's argument, the following recipro- cal relation holds betweenf(t)andF(s): f(t)F-→F(s) is equivalent to

F(t)F-→f(-s).

This relationship is often written more econom-

ically as follows: f(t)F←→F(s) wheref(t)andF(s)are said to be aFourier transform pair.

Fourier Transform of Gaussian

Letf(t)be a Gaussian:

f(t) =e-pt2.

By the definition of Fourier transform we see

that:

F(s) =Z

-¥e-pt2e-j2pstdt Z -¥e-p(t2+j2st)dt.

Now we can multiply the right hand side by

e -ps2eps2=1:

F(s) =e-ps2Z¥

-¥e-p(t2+j2st)+ps2dt =e-ps2Z¥ -¥e-p(t2+j2st-s2)dt =e-ps2Z¥ -¥e-p(t+js)(t+js)dt =e-ps2Z¥ -¥e-p(t+js)2dt

Fourier Transform of Gaussian (contd.)

F(s) =e-ps2Z¥

-¥e-p(t+js)2dt

After substitutingufort+jsanddufordtwe

see that:

F(s) =e-ps2Z¥

-¥e-pu2du 1.

It follows that the Gaussian is its own Fourier

transform: e -pt2F←→e-ps2.

Fourier Transform of Dirac Delta Function

To compute the Fourier transform of an impulse

we apply the definition of Fourier transform:

F{d(t-t0)}(s) =F(s) =Z

-¥d(t-t0)e-j2pstdt which, by the sifting property of the impulse, is just: e -j2pst0.

It follows that:

d(t-t0)F-→e-j2pst0.

Fourier Transform of Harmonic Signal

What is the inverse Fourier transform of an im-

pulse located ats0? Applying the definition of inverse Fourier transform yields: F -1{d(s-s0)}(t)=f(t)=Z -¥d(s-s0)ej2pstds which, by the sifting property of the impulse, is just: e j2ps0t.

It follows that:

e j2ps0tF-→d(s-s0).

Fourier Transform of Sine and Cosine

We can compute the Fourier transforms of the

sine and cosine by exploiting the sifting prop- erty of the impulse:Z¥ -¥f(x)d(x-x0)dx=f(x0). •QuestionWhat is the inverse Fourier trans- form of a pair of impulses spaced symmetri- cally about the origin? F -1{d(s+s0)+d(s-s0)} form: f(t) =Z -¥[d(s+s0)+d(s-s0)]ej2pstds.

Fourier Transform of Sine and Cosine (contd.)

Expanding the above yields the following ex-

pression forf(t):Z¥ -¥d(s+s0)ej2pstds+Z -¥d(s-s0)ej2pstds

Which by the sifting property is just:

f(t) =ej2ps0t+e-j2ps0t =2cos(2ps0t).

Fourier Transform of Sine and Cosine (contd.)

It follows that

cos(2ps0t)F←→1

2[d(s+s0)+d(s-s0)].

A similar argument can be used to show:

sin(2ps0t)F←→j

2[d(s+s0)-d(s-s0)].

Fourier Transform of the Pulse

To compute the Fourier transform of a pulse we

apply the definition of Fourier transform:

F(s) =Z

-¥P(t)e-j2pstdt Z 1 2 1

2e-j2pstdt

1 -j2pse-j2pst????1 2 1 2 =1 -j2ps?e-jps-ejps? 1 ps? ejps-e-jps?2j

Using the fact that sin(x) =(ejx-e-jx)

2jwe see

that:

F(s) =sin(ps)

ps.

Fourier Transform of the Shah Function

Recall the Fourier series for the Shah function:

2pIII?t2p?

=12p¥å w=-¥ejwt.

By the sifting property,

III?t 2p? w=-¥Z -¥d(s-w)ejstds.

Changing the order of the summation and the

integral yields III?t 2p? =Z w=-¥d(s-w)ejstds.

Factoring outejstfrom the summation

III?t 2p? =Z -¥ejst¥å w=-¥d(s-w)ds Z -¥ejstIII(s)ds.

Fourier Transform of the Shah Function

Substituting 2ptfortyields

III(t) =Z

-¥III(s)ej2pstds =F-1{III(s)}(t).

Consequently we see that

F{III}=III.

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[PDF] Fourier Transform Pairs The Fourier transform transforms a function

Fourier Transform Pairs

The Fourier transform transforms a function of

F{f(t)}(s) =F(s) =Z

Fourier Transform Pairs (contd).

Because the Fourier transform and the inverse

Fourier transform differ only in the sign of the

F(t)F-→f(-s).

This relationship is often written more econom-

Fourier Transform of Gaussian

Letf(t)be a Gaussian:

By the definition of Fourier transform we see

F(s) =Z

Now we can multiply the right hand side by

F(s) =e-ps2Z¥

Fourier Transform of Gaussian (contd.)

F(s) =e-ps2Z¥

After substitutingufort+jsanddufordtwe

F(s) =e-ps2Z¥

It follows that the Gaussian is its own Fourier

Fourier Transform of Dirac Delta Function

To compute the Fourier transform of an impulse

F{d(t-t0)}(s) =F(s) =Z

It follows that:

Fourier Transform of Harmonic Signal

What is the inverse Fourier transform of an im-

It follows that:

Fourier Transform of Sine and Cosine

We can compute the Fourier transforms of the

Fourier Transform of Sine and Cosine (contd.)

Expanding the above yields the following ex-

Which by the sifting property is just:

Fourier Transform of Sine and Cosine (contd.)

It follows that

2[d(s+s0)+d(s-s0)].

A similar argument can be used to show:

2[d(s+s0)-d(s-s0)].

Fourier Transform of the Pulse

To compute the Fourier transform of a pulse we

F(s) =Z

2e-j2pstdt

Using the fact that sin(x) =(ejx-e-jx)

2jwe see

F(s) =sin(ps)

Fourier Transform of the Shah Function

Recall the Fourier series for the Shah function:

2pIII?t2p?

By the sifting property,

Changing the order of the summation and the

Factoring outejstfrom the summation

Fourier Transform of the Shah Function

Substituting 2ptfortyields

III(t) =Z

Consequently we see that

F{III}=III.