# use of convolution?

#### gani

Joined May 15, 2005
46
hello

Why shud one use convolution ?Inorder to get the state transition matrix
from
x'=Ax+Bu

x(t)=exp(at)x(0) + integral[exp(a(t-tau)) bu(tau) dtau

where tau is the variable of convolution.
i have read that convolution is used when integration is reqd,for instance,in capacitors and inductors which dont respond to an input immediately.....but can someone PLEASEEEEEEEEEE help me understand how convolution really works?

Thanks for any help always!

#### gani

Joined May 15, 2005
46
hi 9M
thanx for ur help.this is good too

www.swarthmore.edu/NatSci/ echeeve1/Ref/Convolution/SysConvolve.html

but it still doesn answer why its used in obtaining the state transition matrix.i was actually reading control systems.but i have dsp is pretty rotten.dono yet.hahaheehe

#### bennitech

Joined Jul 26, 2005
3
You need to understand convolution from the most basic perspective - it's no good using a tool if you have not been taught what it actually does.
Go back to poor old analogue - ie. continuous time and amplitude and phase domain of a signal.
Imagine two functions in time, like say, a speech signal (relatively random over a limited range but that is not relevant) and a single square pulse (let's call this one the 'window' function) . It doesn't really matter which signal is used as the "windowing" signal. If you swap them the result is the same graph flipped horizontally (in time in this case).
A convolution in time of the two functions, taken across the whole time domain (-ve infinity to +ve infinity) is like sliding one function over the other and at every point, you integrate the total area under the product (this equates to energy) of the two functions and plot this as a point on a third graph, the horizontal position being the time difference between the two functions (ie. the horizontal axis for the convolution signal is the offset between the two signals). It is easier to imagine it in small increments. Place one function where it normally is in time (referenced at zero, say, or wherever you have defined it to be), and place the reference of the other one at -ve infinity (or in practical terms, at some point where the first has a significant amplitude). Take the product of the two functions, integrate under the product curve from -ve inf. to +ve inf., plot the result on the convolution graph at the offset in time between the two functions. Now slide the second (windowing) function along by a small increment (+dt). Do the whole thing again over and over until the windowing function reaches +ve inf. (or in practical terms where the first function's amplitude is sufficiently diminished and will not increase any more). Voila, you have a convolution.
Now, if you think carefully about this, it is like the effect of a filter on a signal over -ve to +ve infinity in time (or frequency if that is where you are doing the convolution). Product of two signals in the frequency domain (one representing the filter function, the other the signal being filtered) can be calculated in time as the convolution of the signal in time with the filter's impulse response (ie. natural response). In other words, it is a TRANSFORM. That is the key word - it transforms a signal via the filter it represents - ALL circuits that alter a signal are fundamentally just some kind of filter (irrespective of the nature of that transform - could be non-linear, could be time-varying adaptive, could be a transmission line, could be a neural network, etc.)
If I have made any errors here, it's probably because it has been about twenty years since I dealt with this stuff so I apologise.
I hope I have not confused you any further. Just give it time - these things have a way of sinking in if you don't let it worry you - just absorb what you can and move on - I found that it works well because neural nets (brains) are very good at behind-the-scenes processing. Then one day you get the 'Aha!' response as the understanding bursts forth apparently all on its own!!! It works!
Cheers.