Intrigued about this question on convolution

[curious_mind]

This is an elementary question on signal processing with analogy. To increase the quantity of anything, we may use either add or mul operator. Similarly, to reduce a quantity of something we can use either subtraction or division. Now what is the corresponding function or attribute we are looking when using convolution as an operator?

Here is the table which we need to fill
-------------------------------------------
Function/attribute | operator
-------------------------------------------
Increase quantity | add or mul
Decrease quantity | sub or div
? | Convolution
---------------------------------------------

In other words what are we trying to achieve with convolution. OR putting in another way is, when to decide that convolution is needed to solve a particular problem.

 

[KlausST]

Hi,

To increase the quantity of anything, we may use either add or mul operator.

not generally true:
* If we add "a negative number" the result becomes smaller
* if we multiply a positive with a "negative number" the result becomes smaller
* if we multiply a negative number with a "positive number >1" the result becomes smaller

.. same is with your other statement.

Klaus

 

[curious_mind]

As you know that analogies or examples are incomplete as they do have corner cases. But please explain the role of convolution. When to decide that we need convolution to solve a problem.

 

[kaz1]

convolution is sum of products involving a sequence of samples all products added up to produce one output. In effect it is adding scaled contribution of a number of input past samples towards each output sample

 

[D.A.(Tony)Stewart]

Convolution - Wikipedia

en.m.wikipedia.org

Convolution is used when you want to correlate inputs, either digital or analogue to a particular operating optimum point by integrating the error. This could apply to digital sync words where the bit patterns correlate to increasing values towards sync or optimizing the bandwidth to the signal in a spectral density analyzer. (FFT)

There are different shaping filters for auto correlation that helped suppress intermodulation side bands

 

[oscillationeer]

As you know that analogies or examples are incomplete as they do have corner cases. But please explain the role of convolution. When to decide that we need convolution to solve a problem.

Much of real information comes in scattered bits and pieces; a record of birth dates, for instance,
might be summarized by a bar chart with one bar for each month. This kind of data, a 'distribution',
has features (like peaks, valleys, slopes) which allow u s to consider the distribution to be a kind
of function.

Convolution applies a functional distribution to an existing distribution.
Thus, a fuzzy picture of a furry animal may convolve the resolution of an imperfect camera
with the blur of the moving critter to make an effect on the image.

Or, a measured data set may have an instrument-effect that makes its response to a transient blip
into a wobbly oscillation; it's common to check an oscilloscope response by viewing the edges and tables
of a square wave, and adjust the probe to compensate.

It is also possible to treat time delay by multiple echoes as a distribution, and deconvolve (invert the convolution)
it in order to make a poor transmitted signal into a cleaner copy of the original (and
DSL modems are doing this somewhere near you).

 

[curious_mind]

Convolution applies a functional distribution to an existing distribution.

Ok, it is partially clear. I wanted to understand the reason for summation in the equation of convolution. Summation alters the entire distribution. OR can we take it as the same consequence of multiplying two polynomials, which results in multiplication and addition

 

[D.A.(Tony)Stewart]

It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted.

Wikiwand - Convolution

In mathematics, convolution is a mathematical operation on two functions that produces a third function. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the...

www.wikiwand.com