Fourier Analysis, definition of convolution

in summary, the convolution of two functions is a mathematical operation that produces a new function that is the product of the two original functions.
  • #1
Narcol2000
25
0
I having a hard time understanding an aspect of the definition of the convolution of two functions. Here is the lead up to its definition...

It is apparent that any attempt to measure the value of a physical quantity is
limited, to some extent, by the finite resolution of the measuring apparatus used.
On the one hand, the physical quantity we wish to measure will be in general a
function of an independent variable, x say, i.e. the true function to be measured
takes the form f(x). On the other hand, the apparatus we are using does not give
the true output value of the function; a resolution function g(y) is involved. By
this we mean that the probability that an output value y = 0 will be recorded
instead as being between y and y +dy is given by g(y) dy.

It goes on to discuss what the observed distribution h(z) will be if we try to measure f(x) with an apparatus with resolution function g(y). And tries to justify why h(z) is defined at the convolution of the functions f and g, however i have a problem with one of its statements. The book says:

The probability that a true reading lying between x and x + dx, and so having
probability f(x) dx of being selected by the experiment, will be moved by the
instrumental resolution by an amount z − x into a small interval of width dz is
g(z − x) dz. Hence the combined probability that the interval dx will give rise to
an observation appearing in the interval dz is f(x) dx g(z − x) dz. Adding together
the contributions from all values of x that can lead to an observation in the range
z to z + dz, we find that the observed distribution is given by

[tex]
h(z) = \int^{\infty}_{-\infty} f(x)g(z-x)dx
[/tex]

here is the part I have an issue with.

The probability that a true reading lying between x and x + dx, and so having probability f(x) dx of being selected by the experiment,

Why is the probability of a reading being between x and x+dx, equal to f(x) dx? I thought f(x) was an arbitrary function representing a relation between an independant variable x and the observable f(x), why is it being treated as a probability density function :confused:

If i just accept f(x) dx as being the probability of a true reading lying between x and x + dx. Then I can understand how the definition of h(z) as the convolution follows from this, but i just don't understand how an arbitrary function can be treated as a probabilty density. If f(x) = x (or any function that doesn't have a finite integral between - and + infinity, surely this argument doesn't hold, meaning that f(x) can't be arbitrary, yet if f(x) is an observable quantity surely it must be possible for it to have an arbitrary dependancy on x.
 
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  • #2
Yeah, that stuff about probability seems kind of hand-wavey and unnecessary. Generally speaking, convolution is defined over a much larger class of functions than pdf's. In the case that both f(x) and g(x) are positive and integrable, though, this derivation is fine, and is an interesting interpretation of convolution.

The derivations of the convolution integral that I'm more familiar with start out with linear, time-invariant systems, then work out impulse response, and then put the whole thing together. This is also kinda hand-wavey, since it typically uses Dirac deltas, but is still more general and explicit than the derivation in the OP (which doesn't even state the linearity assumptions used in the measurement model).
 
  • #3
I find the wording in your text quite confusing, but the concept is sound and worth understanding. It might be easier to think of a concrete example--capturing ("measuring") an image with a lens. f is intensity of the scene as a function of position x (actually we need x and y). The lens forms an image with a resolution given by its "point spread function" g(x,y), which is an Airy disk--the little round diffraction pattern that arises from the circular aperture.

If we turn our lens to the skies, then a delta-function or point excitation (starlight) produces exactly g. But an extended object or scene f(x,y) produces an image f*g, where * is a convolution. This is the sense in which instruments introduce a response function.
 
  • #4
Thanks for clearing that up, saying that f(x) is an experimental observable yet having a definition that restricts f(x) to being a pdf is a recipe for confusion!

I sought out a derivation http://cnx.org/content/m10085/latest/ along the lines of what quadraphonics suggested and i must say i prefer that more general derivation. Would have been far better for the book to lead out with the more general definition then provide the example as marcusl mentioned.

thx again.
 
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  • #5
One comment on the [tex] f(x) \, dx [/tex] line - in introductory mathematical statistics, we introduce the idea that if [tex] f(x) [/tex] is a density for a continuous random variable, the quantity [tex] f(x) \, dx [/tex] can be considered to be the probability of finding an observation between [tex] x [/tex] and [tex] dx [/tex]. It then leads to a heuristic explanation of why [tex] \Pr(a \le X \le B) [/tex] is given by

[tex]
\int_a^b f(x) \, dx
[/tex]

If the snippet provided by the OP relates to some probability-related application, that could be the tie-in. I must say, however, that I've never seen it used as a lead-in to convolution; I agree that without greater context it is odd.
 
  • #6
Yeah, what book was the material in the OP quoted from? It seems pretty ham-fisted compared to what I'm used to (which are books on systems engineering), but it may not be that bad if this is a side-blurb in a book on applied physics or something...
 
  • #7
The book is Mathematical Methods for Physics and Engineering by Riley.
Pretty decent book overall but this particular section on convolution is a bit naff to be honest, it makes more sense after having understood the more general definition in the link i posted above. As a stand alone intro to convolution it definitely comes up a bit short... :/
 
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1. What is Fourier analysis?

Fourier analysis is a mathematical method used to break down a function into its individual frequency components. It is based on the idea that any periodic function can be represented as a sum of simple sine and cosine functions with different frequencies.

2. What is a Fourier transform?

A Fourier transform is a mathematical operation that converts a function from its original domain (usually time or space) to a representation in the frequency domain. It allows us to analyze the frequency content of a signal and is an important tool in Fourier analysis.

3. What is the definition of convolution?

Convolution is a mathematical operation that combines two functions to produce a third function. It is commonly used in signal processing to model the effects of a system on a signal. In Fourier analysis, convolution is used to calculate the frequency response of a system.

4. How is convolution related to Fourier analysis?

Convolution is an important tool in Fourier analysis as it allows us to combine two functions in the time domain and analyze the resulting function in the frequency domain. This is useful in understanding the effects of a system on a signal and in solving differential equations in the frequency domain.

5. What is the significance of convolution in real-world applications?

Convolution has many practical applications in fields such as signal processing, image processing, and engineering. It is used in tasks such as filtering, noise reduction, and pattern recognition. In addition, convolution is a fundamental concept in the development of many mathematical models and equations used in various scientific and engineering disciplines.

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