Signals and Systems: Response of LTI systems to Complex Exponentials

In summary, to evaluate the integral for calculation of Amplitude Factor for an LTI system given the input and output are related by a time shift of 3, we use the impulse response of the LTI system, represented by the function H(s) = e^(-3s), which is equal to the input transformed by a factor of e^(-3s). This allows us to understand the Mathematics behind the evaluation of the integral.
  • #1
mayan
2
0
How will we evaluate the integral for calculation of Amplitude Factor for an LTI system for which the input and output are related by a time shift of 3, i.e.,

y(t) = x(t - 3)


The answer is: H(s) = e^(-3s)

I want to understand the Mathematics behind the evaluation of the integral.

Thanks.
 
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  • #2
i'm not sure how the title of your post is related to the content.

by definition, on the left side you have,

[tex] Y(s) = \mathcal{L} \left\{y(t)\right\} = \int_{-\infty}^{+\infty} e^{-st} y(t) \,dt [/tex]

and, also by definition,

[tex] X(s) = \mathcal{L} \left\{x(t)\right\} = \int_{-\infty}^{+\infty} e^{-st} x(t) \,dt [/tex]

and, it turns out that for LTI systems, after you show that the convolution operator and some impulse response is what relates the input x(t) to the output y(t), then

[tex] Y(s) = H(s) X(s) [/tex]

where

[tex] H(s) = \mathcal{L} \left\{h(t)\right\} = \int_{-\infty}^{+\infty} e^{-st} h(t) \,dt [/tex]

and h(t) is the impulse response of the LTI system and completely characterizes the input/output relationship of the LTI system.

now, in your specific case,

[tex] Y(s) = \mathcal{L} \left\{y(t)\right\} = \mathcal{L} \left\{x(t-3)\right\} = \int_{-\infty}^{+\infty} e^{-st} x(t-3) \,dt [/tex]

which, after you do a trivial substitution of variable of integration is

[tex] Y(s) = \int_{-\infty}^{+\infty} e^{-s(t+3)} x(t) \,dt = e^{-s \cdot 3} \int_{-\infty}^{+\infty} e^{-st} x(t) \,dt = e^{-s \cdot 3} \cdot X(s) [/tex]

which means that

[tex] H(s) = e^{-s \cdot 3} [/tex].
 
Last edited:
  • #3


To understand the mathematics behind the evaluation of the integral for the amplitude factor of an LTI system with a time shift of 3, we need to first understand the concept of complex exponentials in the context of signals and systems.

Complex exponentials are functions of the form e^(st), where s is a complex number and t is the independent variable. These functions are important in the field of signals and systems because they can model a wide range of physical systems, including electrical circuits, mechanical systems, and even biological systems.

In the context of an LTI (linear time-invariant) system, the input-output relationship can be described by a transfer function H(s), which is a function of the complex variable s. This transfer function describes the system's response to a complex exponential input, and it is typically represented in terms of its magnitude and phase response.

Now, let's consider the specific case of a time shift of 3 in the input signal. This means that the input signal x(t) is delayed by 3 units of time, resulting in the output signal y(t) = x(t-3). In the frequency domain, this time shift corresponds to a phase shift of -3s, which can be represented as e^(-3s).

To evaluate the integral for the amplitude factor, we use the property of linearity of the integral. This means that we can split the integral into two parts: one for the input signal and one for the output signal. Since the input signal is a complex exponential, the integral can be easily evaluated using the definition of the complex exponential function.

The integral for the input signal can be written as:

A = ∫ e^(st) dt = (1/s) * e^(st)

Similarly, the integral for the output signal can be written as:

B = ∫ e^(-3s) * e^(st) dt = (1/s) * e^(st-3s)

Using the property of linearity, we can combine these two integrals to get the overall amplitude factor:

A/B = (1/s) * e^(3s)

Substituting the transfer function H(s) = e^(-3s), we get the final expression for the amplitude factor as:

A/B = H(s)/s

This is the mathematical explanation behind the evaluation of the integral for the amplitude factor of an LTI system with a time shift of 3. It is important to note that this is just
 

1. What is the definition of a complex exponential?

A complex exponential is a function of the form e^(αt), where e is the base of the natural logarithm, α is a complex number, and t is the independent variable.

2. How are complex exponentials used in signals and systems?

Complex exponentials are used to represent signals in the time or frequency domain. In systems analysis, they are used to model the input and output responses of linear time-invariant (LTI) systems.

3. What is the response of an LTI system to a complex exponential input?

The response of an LTI system to a complex exponential input is another complex exponential with the same frequency, but with an amplitude and phase shift determined by the system's transfer function.

4. Can the response of an LTI system to a complex exponential be written in a simpler form?

Yes, the response can be written as a product of the input complex exponential and the system's frequency response function, which is a complex-valued function of frequency.

5. How does the response of an LTI system to a complex exponential differ from its response to a real exponential?

The response to a complex exponential includes both an amplitude and phase shift, while the response to a real exponential only includes an amplitude change. Additionally, the complex exponential response may exhibit oscillatory behavior, while the real exponential response is always monotonic.

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