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1

saying that one has recorded the convolution ;1 r( )s(t )d of the earth response

;

with the source signal, but that one is only interested in the earth response r(t). One

would like to \undo" this convolution, this process is called deconvolution. Carrying out

the deconvolution seems trivial in the frequency domain. According to (11.51) one only

needs to divide the data in the frequency domain by the source spectrum S(!) to obtain

R(!). The problem is that in practice one often does not know the source spectrum S(!).

This makes seismic deconvolution a di cult process, see the collection of articles compiled

by Webster 65]. It has been strongly argued by Ziolkowski 69] that the seismic industry

should make a larger e ort to record the source signal accurately.

The convolution of two signal was obtained in this section by taking the product

F(!)H(!) and carrying out a Fourier transform back to the time domain. The same steps

can be taken by multiplying F(!) with the complex conjugate H (!) and by applying a

Fourier transform to go the time domain.

Problem g: Take the similar steps as in the derivation of the convolution to show that

Z1 Z

;i!t d! = 1 1 f(t + )h ( )d :

F(!)H (!)e (11.52)

;1 2 ;1

The right hand side of this expression is called the correlation of the functions f(t) and

h (t). Note that this expression is very similar to the convolution theorem (11.50). This

result implies that the Fourier transform of the product of a function and the complex

conjugate in the frequency domain corresponds with the correlation in the time domain.

Note again the constant 1=2 in the right hand side. This constant again depends on the

scale factors used in the Fourier transform.

Problem h: Set t = 0 in expression (11.52) and let the function h(t) be equal to f(t).

Show that this gives:

Z1 Z

2 d! = 1 1 (t)j2 dt :

(!)j (11.53)

;1 2 ;1

jF jf

R 1

This equality is known as Parseval's theorem. To see its signi cance, note that ;1 (t)j2 dt =

jf

(f f), with the inner product of equation (11.20) p t as integration variable and with

with

R 1 (!)j2 d! Since (foff) is normnorm of f measuredthe

the integration extending from to the in

;1 1.

of F measured in

the time domain, and since ;1 is square the

jF

frequency domain, Parseval's theorem states that with this de nition of the norm, the

norm of a function is equal in the time domain and in the frequency domain (up to the

scale factor 1=2 ).

11.7 Linear lters and the convolution theorem

Let us consider a linear system that has an output signal o(t) when it is given an input

signal i(t), see gure (11.2). There are numerous examples of this kind of systems. As an

example, consider a damped harmonic oscillator that is driven by a force, this system is

_2

described by the di erential equation x+2 x+!0 x = F=m, where the dot denotes a time

derivative. The force F(t) can be seen as the input signal, and the response x(t) of the

11.7. LINEAR FILTERS AND THE CONVOLUTION THEOREM 141

oscillator can be seen as the output signal. The relation between the input signal and the

output signal is governed by the characteristics of the system under consideration, in this

example it is the physics of the damped harmonic oscillator that determines the relation

between the input signal F(t) and the output signal x(t).

Note that we have not de ned yet what a linear lter is. A lter is linear when an input

c1 i1 (t)+c2i2 (t) leads to an output c1 o1(t)+c2 o2 (t), when o1(t) is the output corresponding

to the input i1 (t) and o2 (t) is the output the input i2 (t).

input output

Filter

Figure 11.2: Schematic representation of a linear lter.

Problem a: Can you think of another example of a linear lter?

Problem b: Can you think of a system that has one input signal and one output signal,

where these signals are related through a nonlinear relation? This would be an

example of a nonlinear lter, the theory of this section would not apply to such a

lter.

It is possible to determine the output o(t) for any input i(t) if the output to a delta

function input is known. Consider the special input signal (t ) that consists of a delta

;

function centered at t = . Since a delta function has \zero-width" (if it has a width at

all) such an input function is very impulsive. Let the output for this particular input be

denoted by g(t ). Since this function is the response at time t to an impulsive input at

time this function is called the impulse response:

The impulse response function g(t ) is the output of the system at time t due

to an impulsive input at time .

How can the impulse response be used to nd the response to an arbitrary input function?

Any input function can be written as:

Z1

i(t) = )i( )d :

(t (11.54)

;1

;

This identity follows from the de nition of the delta function. However, we can also look

at this expression from a di erent point of view. The integral in the right hand side of

(11.54) can be seen as a superposition of in nitely many delta functions (t ). Each;

delta function when considered as a function of t is centered at time . Since we integrate

over these di erent delta functions are superposed to construct the input signal i(t).

Each of the delta functions in the integral (11.54) is multiplied with i( ). This term plays

the role of a coe cients that gives a weight to the delta function (t ). ;

CHAPTER 11. FOURIER ANALYSIS

142

At this point it is crucial to use that the lter is linear. Since the response to the

input (t ) is the impulse response g(t ), and since the input can be written as the

;

superposition (11.54) of delta function input signals (t ), the output can be written

;

as the same superposition of impulse response signals g(t ):

Z1

o(t) = g(t )i( )d : (11.55)

;1

Problem c: Carefully compare the expressions (11.54) and (11.55). Note the similarity

and make sure you understand the reasoning that has led to the previous expression.

You may nd this \derivation" of (11.55) rather vague. The notion of the impulse response

will be treated in much greater detail in chapter (14) because it plays a crucial role in

mathematical physics.

At this point we will make another assumption about the system. Apart from the

linearity we will also assume it is invariant for translations in time. This is a complex

way of saying that we assume that the properties of the lter do not change with time.

This is the case for the damped harmonic oscillator used in the beginning of this section.

However, this oscillator would not be invariant for translations in time if the damping

parameter would be a function of time as well: = (t). In that case, the system would

give a di erent response when the same input is used at di erent times.

When the properties of lter do not depend on time, the impulse response g(t )

depends only on the di erence t; . To see this, consider the damped harmonic oscillator

again. The response at a certain time depends only the time that has lapsed between the

excitation at time and the time of observation t. Therefore, for a time-invariant lter:

g(t ) = g(t ): (11.56)

;

Inserting this in (11.55) shows that for a linear time-invariant lter the output is given by

the convolution of the input with the impulse response:

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