@Avinash: In mathematical usage function and transformation are synonymous, and convolution is a name for multiplication (and certain transformations constructed from it) in a group algebra. So convolution has a much narrower meanig than transformation. This is probably not what you read in the books dealing with the problem field you are working in. Gyorgy and Adwait explained the usage in their fields and I could add such explanations for just a few other fields. I think the best you can do is to figure out the textbooks in your field that don't take the wide-spread approch to ignore all professional mathematics and to cook their own soup. Engineering sciences have very good text books, which in their contexts, by properly introducing concepts, would have prevented your questions from coming up.

First of all I declare that I am not a specialist in this field, but I met the problem in several applied aspects: e.g. resolving overlapping spectrum lines, step response - dynamic response in mechanical and dielectric relaxation spectroscopy or delayed response of visco-elastic bodies.

In spectrum resolution the physical meaning of the deconvolution approact is the insturment distortion (e.g. finite slig width in the monochormator). If you know the distortion function, you can use it backwards to get a shape closer to the original, which might "de-smear" the overlapping lines. If you use this in consecutice steps, you soon get "ghost peaks" which are not real.

In dielectric spectroscopy the physical meaning of the Fourier transformation is that the step response and the dynamic spectrum are Fourier and inverse Fourier transforms of each other.

In viscoleastic response the deformation can be obtained for any loading function of you know teh step-response by convolution (using a kern function).

The problem is that the transformation or convolution approaches are used severaltimes only for thechnicla purposes, where the phyical meaning may be missing - that makes the interpretation more complicated. Spectral rsolution e.g. can be solved by convolution-deconvolution and also by transform approaches. (Convolution becomes multiplication in the Fourier space). Another problem is that in most cases the transform and/or the convolution/deconvolution process should be done numerically and there are distortions coming from the finite data set.

Maybe all I aid is well known to you and your question is not answered - in this case you can neglect my answer.

The Fourier Transform is by far the most important used in seismology. Fourier's theory states that a given signal can be synthesised as a summation of sinusoidal waves of various amplitudes, frequencies and phases. Using the Fourier Transform a time domain signal is transformed to the frequency domain where it is equivalent to an Amplitude Spectrum and a Phase Spectrum. The adjacent figure shows a simple time domain wavelet transformed into it's frequency domain components. The F-X or w -x domain is also shown. Basically this is a one-dimensional Fourier Transform over time of usually of a gather or group of traces (hence the x-dimension). A process operating in this domain will mix or alter the amplitude spectra of the group of traces before the inverse transform is applied. Since there is only a single trace in the figure the F-X domain here just represents a different view of the amplitude spectrum. The figure was created with PROMAX routine Interactive Spectral Analysis.

Many operations e.g. bandpass frequency filtering are easier to understand in the frequency domain. The Discrete Fourier Transform (DFT) is applied to a digitised time series, and the Fast Fourier Transform (FFT) is a computer algorithm for rapid DFT computations. The latter imposes the restriction that the time series must be a power of two samples long

CONVOLUTION: Is a mathematical way of combining two signals to achieve a third, modified signal. The signal we record seems to respond well to being treated as a series of signals superimposed upon each other that is seismic signals seem to respond convolutionally. The process of DECONVOLUTION is the reversal of the convolution process. Convolution in the time domain is represented in the frequency domain by a multiplying the amplitude spectra and adding the phase spectra. The adjacent figure shows a spike series representing an acoustic impedance response from the earth which is convolved (*) with a source wavelet to produce a resulting seismic signal which is measured. In principal by deconvolving the source wavelet we could obtain the earth's reflectivity. However, noise (unwanted signal) and other features are also present in the recorded trace and the source wavelet is rarely known with any accuracy. In the figure (a) the spikes are sufficiently separated that the convolution just results in a duplication of the input wavelet at the spike times and with the spike amplitudes. The convolution process just involves multiplying every sample of the spike series by the input wavelet and adding all the results. In (b) the spikes are closer together and interference occurs in the resulting trace. If the wavelet were known the input spike series could be discovered by the deconvolution process. The convolutional model of the seismic trace states that the trace we record is the result of the earth's reflectivity (what we want) convolved with the source wavelet (and it's ghosts), multiples, the recording system and some noise.

@Avinash: In mathematical usage function and transformation are synonymous, and convolution is a name for multiplication (and certain transformations constructed from it) in a group algebra. So convolution has a much narrower meanig than transformation. This is probably not what you read in the books dealing with the problem field you are working in. Gyorgy and Adwait explained the usage in their fields and I could add such explanations for just a few other fields. I think the best you can do is to figure out the textbooks in your field that don't take the wide-spread approch to ignore all professional mathematics and to cook their own soup. Engineering sciences have very good text books, which in their contexts, by properly introducing concepts, would have prevented your questions from coming up.

Physical significance of a convolution is such that it is present in numerous measurements. Say, you are observing the readings of a voltmeter connected somewhere. Do you think what you see is the current voltage? Wrong, you see a peculiar combination of the current and past voltages, a convolution mathematically speaking. This is because every voltmeter is characterized by its own "time constant". By the way: deconvolution is not well defined mathematical operation, except for special cases.

Various transforms (not only Fourier or Laplace) - that's another and rather rich story. They were invented to make the description of many phenomena simpler. Some of them make possible "measurements" of physical quantities not directly accessible otherwise. A Kramers-Kronig transform, for example, is of this kind. It is commonly applied for finding the absorption spectrum having the optical reflectance data.

Answers above, and with due respect because all people sound as knowledgable in the subject, are not clear enough to my admitted narrow understanding. They tend to introduce more confusion in a subject that from its very set up is straight:

When a given signal gets into a linear system, the system "operates" on it using its own describing funtion (its characteristics) to produce an output.

Conversely: Knowing the input and the output offer enough information to obtain the describing system's function by the inverse convolution (deconvolution), which is an operation showing a certain degree of uncertainty, as all inverse operations do.

Why then all that rather obscure verbosity?

Kindly,

Max Valentinuzzi

Institute of Biomedical Engineering

University of Buenos Aires, Argentina

Oh Max, be happy with what you appropriately characterize as 'narrow understanding'.

Hello,

to tell you simply -

# Trasformation is basically changing domains. if you draw the graphs, then it most generally means that you are going to change your x-axis. The classic example can be given of changing from the time domain to the frequency domain. that is performing the Fourier Transformation.

# Convolution is in basic multiplication and to describe it simply, if you do a convolution in the time domain then it is equivalent to the multiplication in the frequency domain, and vice-versa.

Hope this helps in simple words, for details are are always many books and literature.

@Max: I think that the requester may already have some idea, maybe a bit hazy, what is convolution and what is transform. But he also asks: "explain the physical significance of convolution". Theory is one thing and the practical use of purely mathematical concept is another one. This is how I understand Avinash's question.

I hope we all have contributed to satisfy his greedy willingness for better understanding of the commonly used phrases and why we need them at all.

Besides, I'm also trying follow the KISS principle (Keep It Simple, Stupid) whenever possible. But I strongly object to limit the answer to a single, particular case, good for signal processing only (but even there there also is a Walsh Transform, perfectly invertible, just like FT).

are you solving a a homework? Think here is for solving troubles...

I dont know which came earlier egg or hen but

Convolution is the time domain processing for defining the filtering operation on any input signal to transform it to some altered output signal and depicts a system.

whereas to relax on complexity of convolution operation in time domain or to understand convolution better or with ease transforms may be used as in time domain convolution is replaced with multiplication in Fourier or Laplace or Z transforms.

@Imran Shafi,

To my view egg came earlier thann hen. The logic is simple, egg has only one cell and hen have many cells. 2 cannot come before 1 because to make 2 we need two 1's.

Regarding convolution and transformation, I would rather say that transformation is any operation or a set of operations you on something to change the original form to someother form. The operation may also be a convolution operation or some simple addition or subtraction operation. For example, you have an image matrix (say gray image) and you subtract every pixel of the image from 255. Then you will get another image, called negative image which is a simple example of transfromation. Transformation may also be done using convolution operation, for example, you have an image matrix, and you perform convolution operation of it with a Fourier kernel of appropriate size, then you will get Fourier transform. The comvolution operation may also be combined with some other operations in a some other kind of transforms, for example, wavelet transform in which you perform convolution of the signal with wavelet transform kernel (popularly called wavelet filter coefficients) then you perform downsampling operations.

About convolution, it is nothing but a polynomial multiplication. Why we see convolution operation as simple multiplication in Z tranforms is becuase in Z tranforms, the signals are in polynomial form in Z. In Fourier domain, the signals are represneted in the form of polynomial in exponential. So, in the Fourier domain also, we convolution means multiplication.

More about convolution and multiplication can be obtained from the following link

www.oocities.org/kiranisingh/multi.html .

Hope it may help you.

Y. K. Singh