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For an LTI system with impulse response h(t),determine system responses to the f...

Mar 16, 2024

For an LTI system with impulse response h(t),determine system responses to the following inputs (1)f(t)=e^{s_{0}t} (2)f(t)= \sum \limits _{k=- \infty }^{ \infty }a_{k}e^{s_{k}t}where so, Sk, and ak (for all k) are known constants.

To determine the system response of an LTI (Linear Time-Invariant) system with impulse response

h(t)

to the given inputs, we can use the convolution theorem. The output

y(t)

of an LTI system to an input

f(t)

is given by the convolution of

f(t)

with the system's impulse response

h(t)

, which is mathematically represented as

y(t) = f(t) * h(t)

For the first input

f(t) = e^{s_0t}

, the system response

y(t)

can be found by convolving

f(t)

with

h(t)

, which gives us

y(t) = h(t) * e^{s_0t}

. In the Laplace domain, this convolution turns into multiplication, so if

H(s)

is the Laplace transform of

h(t)

, the Laplace transform of the output

Y(s)

Y(s) = H(s) \cdot E(s)

where

E(s)

is the Laplace transform of

e^{s_0t}

, which is

\frac{1}{s - s_0}

. Therefore,

Y(s) = H(s) \cdot \frac{1}{s - s_0}

For the second input

f(t) = \sum \limits _{k=- \infty }^{ \infty }a_{k}e^{s_{k}t}

, the system response

y(t)

is the sum of the individual responses to each term in the input sum. Since the system is linear, we can find the response to each term

a_{k}e^{s_{k}t}

separately and then sum them up. The response to each term is

a_{k}h(t) * e^{s_{k}t}

, and in the Laplace domain, this is

a_{k}H(s) \cdot \frac{1}{s - s_k}

. The total response is the sum of these individual responses, so

Y(s) = \sum \limits _{k=- \infty }^{ \infty }a_{k}H(s) \cdot \frac{1}{s - s_k}

In summary, the system responses are found by convolving the input functions with the impulse response of the system, and the convolution operation simplifies to multiplication in the Laplace domain. For the first input, the response is

Y(s) = H(s) \cdot \frac{1}{s - s_0}

, and for the second input, the response is

Y(s) = \sum \limits _{k=- \infty }^{ \infty }a_{k}H(s) \cdot \frac{1}{s - s_k}