How to find transfer function for this :

[anjana.rafta]

so here is an LTi system, i/p is given by:

(1/k1) exp(-t/K1)

ouput being:

(1/K2)exp(-t/K2)

i could take Fourier transform and then divide the two and find out H(w), then try to take inverse transform of it, to find h(t). but that somehow looks little complicated to me. Is there any other way around?

thanks.

 

[0xDEADBEEF]

This looks like homework.

 

[anjana.rafta]

not exactly...
i encountered this problem in one qs paper, tried solving..but couldn't, so need help...

 

[uart]

not exactly...
i encountered this problem in one qs paper, tried solving..but couldn't, so need help...

Ok well here's some hints. This question is about generalized complex frequency response.

The system has an input at one "frequency" (1/k1) and has an output at a different generalized frequency. We know that in general this doesn't happen in a linear system, so what's going on?

Well for one thing it tells us that the "exp(-t/k2)" output component can ONLY be due to the systems natural response.

Secondly the fact that the input term "exp(-t/k1)" produces no output component at all tells us that there must be a zero in the transfer function at this generalized frequency.

Together these two facts are enough to find a plausible transfer function for the system.

 

[anjana.rafta]

okay... one very basic doubt:
exp(j*t/k1) is an eigenfunction vector to LTI system i understand, but exp(t/k1) ?
is that a eigenfunction to LTI system?
i my understanding is that its not,...!

 

[uart]

okay... one very basic doubt:
exp(j*t/k1) is an eigenfunction vector to LTI system i understand, but exp(t/k1) ?
is that a eigenfunction to LTI system?
i my understanding is that its not,...!

Sure it is. If you feed $e^{\lambda t}$ into a linear DE then every term will be a multiple of $e^{\lambda t}$. It's true for $\lambda$ real or complex.

 

[anjana.rafta]

OK.. so if i could understand your statemenet correctly then,

response = x(t)*h(t) + exp(t/k2) , where h(t) have zero at 1/k1, x(t) = exp(t/K1)

now i don;t undestand how to write h(t) in time domain??
h(s) = (s-1/K1) => h(t) = d/dt - 1/K1*delta(t) ?
also how to represent natural response part?

 

[uart]

OK.. so if i could understand your statemenet correctly then,

response = x(t)*h(t) + exp(t/k2) , where h(t) have zero at 1/k1, x(t) = exp(t/K1)

now i don;t undestand how to write h(t) in time domain??
h(s) = (s-1/K1) => h(t) = d/dt - 1/K1*delta(t) ?
also how to represent natural response part?

Don't work it in the time domain (where you need convolution), work in the complex frequency domain where it's just simple multiplication.

We figured out there is a zero at 1/k1 and a pole at 1/k2 so try,

$$H(s) = \frac{1 + k_1 s}{1 + k_2 s}$$

If the input is,

$$x(t) = \frac{1}{k_1} e^{-t/k_1} \, u(t)$$

(Where u(t) is the unit step function.) Then in the frequency domain we have,

$$X(s) = \frac{1}{1 + k_1 s}$$

The output is therefore,

$$Y(s) = H(s) X(s) = \frac{1}{1 + k_2 s}$$

And in the time domain,

$$y(t) = \frac{1}{k_2} e^{-t/k_2} \, u(t).$$