Determine the unit-step response of the discrete-time LTI systems

In summary, the unit-step response of the discrete-time LTI systems described by the given impulse response can be determined by convolving u[n] with h[n], resulting in a summation from k = -∞ to ∞ of (0.9)^k*e^(j(pi/2)k)*u[n-k]. This can be simplified using the property of summation and defining a as (0.9)^k, resulting in a final expression of (1-(0.9)^(n+1))/(1-0.9).
  • #1
Mr.Tibbs
24
0
Determine the unit-step response of the discrete-time LTI
systems described by the following impulse responses:

h[n]=(0.9)[itex]^{n}[/itex]e[itex]^{j(\pi/2)n}[/itex]u[n]So I am completely confused. . . I don't even know how to start. . . I want to say that I need to do a summation but the more examples and text I look up the more I'm in the dark. . . any help is appreciated.

The only thing I can think to start is you assume

x[n] = u[n]
 
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  • #2
How about convolving u[n] with your h[n]?
 
  • #3
My apologies for taking so long to reply. I was able to talk to my professor and this is what I have now.

[itex]\sum[/itex](0.9)[itex]^{k}[/itex]e[itex]^{j(\pi/2)k}[/itex]u[n-k] from k = -∞ to ∞.

this turns into

[itex]\sum[/itex](0.9)[itex]^{k}[/itex]e[itex]^{j(\pi/2)k}[/itex] from k = 0 to n.

using the property of summation :

[itex]\sum[/itex] a[itex]^{k}[/itex] = [itex]\frac{1-a^{n+1}}{1-a}[/itex]

my new snag is what do I define as a?
 

1. What is a discrete-time LTI system?

A discrete-time LTI (Linear Time-Invariant) system is a mathematical model used to describe the behavior of a system that operates on discrete time signals. It is characterized by its linearity, which means that the output is directly proportional to the input, and its time-invariance, which means that the system's response does not change over time.

2. What is the unit-step response of a discrete-time LTI system?

The unit-step response of a discrete-time LTI system is the output of the system when the input is a unit step function, also known as the unit step signal. This signal has a value of 1 for all time values greater than or equal to 0, and 0 for all time values less than 0. The unit-step response is used to analyze the behavior of a system in response to a sudden change or step in the input signal.

3. How is the unit-step response of a discrete-time LTI system determined?

The unit-step response of a discrete-time LTI system can be determined using the system's impulse response function. This function describes the output of the system when the input is a unit impulse, which is a signal that has a value of 1 at time 0 and 0 at all other times. By convolving the unit step function with the impulse response function, the unit-step response can be calculated.

4. What does the shape of the unit-step response tell us about the system?

The shape of the unit-step response provides information about the system's stability, time delay, and frequency response. A stable system will have a unit-step response that approaches a steady-state value over time, while an unstable system will have a unit-step response that grows without bound. The time delay of the unit-step response indicates the time it takes for the system to respond to changes in the input signal. The frequency response can be determined by analyzing the amplitude and phase of the unit-step response at different frequencies.

5. Can the unit-step response of a discrete-time LTI system be used to predict the system's behavior for other input signals?

Yes, the unit-step response can be used to predict the system's behavior for other input signals by applying the principle of superposition. This principle states that the response of a system to a sum of input signals is equal to the sum of the individual responses to each input signal. Therefore, by decomposing a complex input signal into a series of unit steps, the overall response can be determined by adding the individual unit-step responses together.

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