Thread Subject: conv doesn't agree with analytical solution!

Hi all!
I've encountered somewhat of a problem in Matlab... that is when I try to convolve two functions! I've tried to convolve two exponentials, a convolution I can solve analytically. The problem is that when I plot the analytical solution to the solution from matlabs "conv" function, they don't agree! I also tried to make the convolution from the definition:
a(t) x b(t) = int(a(tau)*b(t-tau)dtau)
using the trapz function. This result agrees with the analytical result, but conv doesn't! Here's a bit of the code I used for testing conv:
%---------------------------------------
t = 0:20; %times
a = exp(-t); %first function
b = exp(-t); %second function
c = t.*exp(-t); %Analytical solution to (a x b)

%%%%%%%% Matlabs conv-function %%%%%%%%%%%%%%
test1 = conv(a,b);
test1 = test1(1:numel(t)); %should equal c...
%%%%%%%% Matlabs trapz-function %%%%%%%%%%%%%%
test2 = zeros(1,numel(t));
for i = 2:numel(t)
    test2(i) = trapz(t(1:i), a(1:i).*exp(-(t(i)-t(1:i))));
end

plot(t,c,'-o',t,test1,'r',t,test2,'g');
legend('Analytical','conv','trapz');
%--------------------------------------
What is actually happening??? Can I trust conv? Thanks!
/Ida

Conv returns a discrete sum and not integral. If you intend to use it to compute integral convolution (it is *not* its purpose), you won't get anything accurate with a time step as coarse as 1: your function a and b decrease by 2.71828 from one point to another.

dt=0.01;
t = 0:dt:20; %times
a = exp(-t); %first function
b = exp(-t); %second function
c = t.*exp(-t); %Analytical solution to (a x b)

%%%%%%%% Matlabs conv-function %%%%%%%%%%%%%%
test1 = dt*conv(a,b);
test1 = test1(1:numel(t)); %should equal c...
%%%%%%%% Matlabs trapz-function %%%%%%%%%%%%%%
test2 = zeros(1,numel(t));
for i = 2:numel(t)
    test2(i) = trapz(t(1:i), a(1:i).*exp(-(t(i)-t(1:i))));
end

plot(t,c,'-o',t,test1,'r',t,test2,'g');
legend('Analytical','conv','trapz');

Bruno

Hi Bruno, and thanks for your time!
I see what you mean, and the results are good when I try your function. However, the reason I'm trying this (conv and trapz) is because I have physical data, with a nonlinear time vector like:
t = [0 5 10 15 ... 60 70 ... 150 180 ...240 300 ... 480 600 ...1620];
When I convolve a and b (as below) using conv, the results are not the same as when I use the trapz approach below. The trapz approach agrees with the analytical solution (using the same t vector of course) whereas the conv solution doesn't. Can I use conv in any way for my problem? Cheers,
/Ida


"Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <ge22bt$een$1@fred.mathworks.com>...
> Conv returns a discrete sum and not integral. If you intend to use it to compute integral convolution (it is *not* its purpose), you won't get anything accurate with a time step as coarse as 1: your function a and b decrease by 2.71828 from one point to another.
>
> dt=0.01;
> t = 0:dt:20; %times
> a = exp(-t); %first function
> b = exp(-t); %second function
> c = t.*exp(-t); %Analytical solution to (a x b)
>
> %%%%%%%% Matlabs conv-function %%%%%%%%%%%%%%
> test1 = dt*conv(a,b);
> test1 = test1(1:numel(t)); %should equal c...
> %%%%%%%% Matlabs trapz-function %%%%%%%%%%%%%%
> test2 = zeros(1,numel(t));
> for i = 2:numel(t)
> test2(i) = trapz(t(1:i), a(1:i).*exp(-(t(i)-t(1:i))));
> end
>
> plot(t,c,'-o',t,test1,'r',t,test2,'g');
> legend('Analytical','conv','trapz');
>
> Bruno

"Ida Haggstrom" <ida_haggstrom@yahoo.se> wrote in message <ge43s5$pcj$1@fred.mathworks.com>...
> Hi Bruno, and thanks for your time!
> I see what you mean, and the results are good when I try your function. However, the reason I'm trying this (conv and trapz) is because I have physical data, with a nonlinear time vector like:
> t = [0 5 10 15 ... 60 70 ... 150 180 ...240 300 ... 480 600 ...1620];
> When I convolve a and b (as below) using conv, the results are not the same as when I use the trapz approach below. The trapz approach agrees with the analytical solution (using the same t vector of course) whereas the conv solution doesn't. Can I use conv in any way for my problem? Cheers,

I don't think so, conv function only approximates the mathematical integral convolution when the time-step is constant (and fine enough). In your case the first criteria is not fulfilled.

However you could preprocess the data by interpolating them on the regular grid.

Bruno

On 27 Okt, 11:07, "Ida Haggstrom" <ida_haggst...@yahoo.se> wrote:
> Hi Bruno, and thanks for your time!
> I see what you mean, and the results are good when I try your function. However, the reason I'm trying this (conv and trapz) is because I have physical data, with a nonlinear time vector like:
> t = [0 5 10 15 ... 60 70 ... 150 180 ...240 300 ... 480 600 ...1620];
> When I convolve a and b (as below) using conv, the results are not the same as when I use the trapz approach below. The trapz approach agrees with the analytical solution (using the same t vector of course) whereas the conv solution doesn't. Can I use conv in any way for my problem?

No. CONV requires that the data are regularly sampled.
It is not obvious how to process irregularly sampled
data. One place where I have seen this discussed is

Press & al: "Numerical recipies in C" (1992)

The book contains very good overview articles and many
good references. Well worth a try.

Rune

Hi Rune!
Hm, I guess conv is not such a good idea for my problem... I will try another approach to solving it instead! Thanks all for your valuable tips! Cheers,
/Ida

Rune Allnor <allnor@tele.ntnu.no> wrote in message <92eaeb03-3727-40dd-9b61-d7ab96174fd8@u28g2000hsc.googlegroups.com>...
> On 27 Okt, 11:07, "Ida Haggstrom" <ida_haggst...@yahoo.se> wrote:
> > Hi Bruno, and thanks for your time!
> > I see what you mean, and the results are good when I try your function. However, the reason I'm trying this (conv and trapz) is because I have physical data, with a nonlinear time vector like:
> > t = [0 5 10 15 ... 60 70 ... 150 180 ...240 300 ... 480 600 ...1620];
> > When I convolve a and b (as below) using conv, the results are not the same as when I use the trapz approach below. The trapz approach agrees with the analytical solution (using the same t vector of course) whereas the conv solution doesn't. Can I use conv in any way for my problem?
>
> No. CONV requires that the data are regularly sampled.
> It is not obvious how to process irregularly sampled
> data. One place where I have seen this discussed is
>
> Press & al: "Numerical recipies in C" (1992)
>
> The book contains very good overview articles and many
> good references. Well worth a try.
>
> Rune