How to decide system response using step response?

Hi all,

Suppose the forced input to the system is f(t), the step response of
the system is a(t) and the output is y(t).

Now we want to find y(t),

I am confused:

Which of the following is the correct output y(t)?

(1)       y(t)=convolution(differentiate(a(t)), f(t))

and

(2)       y(t)=differentiate(convolution(a(t), f(t))

???

All "differentiate" and "convolve" operations are w.r.t. "t"...

Using infinite summation of piecewise response to the input f(t) and
then take limit as n->infinity,

I can obtain

y(t)=differentiate(convolution(a(t), f(t));


But when think about the diff(step-response of the system)=impulse
response of the system,

I obtained

y(t)=convolution(differentiate(a(t)), f(t));

Please zoom in the following picture to see the detailed derivations...

http://www.yourupload.com//uploads/losemind/dc46a-Capture9.JPG

--------------------------

Note there the convolution is defined as integration(a(t-u)*f(u), u
from 0 to t), instead of the integration(a(t-u)*f(u), u from -infinity
to +infinity)...

I think this makes a big difference... for the above two equations (1)
and (2)...

If the convolution is defined as

integration(a(t-u)*f(u), u from -infinity to +infinity)

the above two equations should be the same...

Am I right?

lucy wrote:
> Hi all,
> 
> Suppose the forced input to the system is f(t), the step response of
> the system is a(t) and the output is y(t).
> 
> Now we want to find y(t),
> 
> I am confused:

   ...

I won't answer your question because I think you can answer it yourself. 
One way to discern the correct method is applying both to a simple case 
with known result and observing which method gives that result. When you 
know which is correct, I think you will easily see why it must be so.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

I still don't understand it...

Please tell me if you know which derivation is correct...

Thanks a lot!

lucy wrote:
> I still don't understand it...
> 
> Please tell me if you know which derivation is correct...
> 
> Thanks a lot!

Your functions are all in time. Since y(t_step) --the step response-- is

              t
           integral[y(t_impulse)]dt
              0

(where t_impulse is the impulse response and the impulse itself occurs 
at t=0), Obtain the step response by direct integration and then decide 
whether convolution or multiplication makes sense.

Did I meet you half way?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

lucy wrote:
> I still don't understand it...
>
> Please tell me if you know which derivation is correct...

Although trying a known example will not tell you if a
derivation is correct, it might tell you if a derivation
is incorrect.  Do you know of any example step responses?
Have you tried them using the methods of each derivation?


-- 
rhn

The problem is that I could not find a good example to test them...

My feeling tells that for convolution defined from -inf to +inf,

Cov(diff(f(t)), a(t))
=Cov(f(t), diff(a(t)))
=diff(Cov(f(t), a(t)))

am I right?

For convolution defined from 0 to t,
the above do not hold,
we have to take initial conditions into consideration,
diff(Cov(f(t), a(t)))
=f(0)*a(t)+Cov(diff(f(t)), a(t))
=f(t)*a(0)+Cov(f(t), diff(a(t)))

am I right?

I think I am near that point which clarifies everything...

lucy wrote:
> The problem is that I could not find a good example to test them...

A unit impulse is the simplest test probe for impulse response. By 
definition.

> My feeling tells that for convolution defined from -inf to +inf,
> 
> Cov(diff(f(t)), a(t))
> =Cov(f(t), diff(a(t)))
> =diff(Cov(f(t), a(t)))
> 
> am I right?
> 
> For convolution defined from 0 to t,
> the above do not hold,
> we have to take initial conditions into consideration,
> diff(Cov(f(t), a(t)))
> =f(0)*a(t)+Cov(diff(f(t)), a(t))
> =f(t)*a(0)+Cov(f(t), diff(a(t)))
> 
> am I right?

You can think of the initial conditions at t = 0 being the result of 
convolution from -infinity to zero, if you like.

> I think I am near that point which clarifies everything...

I'm really pleased. I hope you are too.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������