Response of LTI discrete time causal system

In summary, this conversation discusses a discrete time causal LTI system with an impulse response of h(n) = (-1/2)^n for n≥0. The transfer function for this system is given by H(z) = z/(z+1/2) and its frequency characteristics can be found using z=exp(jΩ) in H(z). The response for a given input of u(n) = μ(n)-μ(n-4) can be found using convolution sum, resulting in a response of y(n) = -5(-1/2)^n.
  • #1
crom1
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Homework Statement


Discrete time causal LTI system has impulse response h(n) = (-1/2)^n, n≥0.
a)Find transfer function of given system.
b)Find frequency characteristics of system.
c) Find response for u(n) = μ(n)-μ(n-4)

The Attempt at a Solution


a) Either from definition, or from table transformation, $$\frac{z}{z+\frac{1}{2}} $$
$$H(z) = \sum_{n=0}^{\infty} \left( -\frac{1}{2} \right)^n z^{-n} = \sum_{n=0}^{\infty} \left( -\frac{1}{2z} \right) = \frac{ 1}{1 + \frac{1}{2z}} = \frac{z}{z+\frac{1}{2}} $$

b) We get those for z=exp(jΩ) in H(z) . $$H(\exp(jΩ)) = \frac{\exp(j\Omega)}{\exp(j\Omega)+1/2} \Rightarrow |H(\exp(j\Omega))| = \frac{|\exp(j\Omega)|}{|\exp(j\Omega)+1/2|} = \frac{1}{\sqrt{\frac{5}{4} + \cos \Omega}} =\frac{2}{\sqrt{5 + 4 \cos \Omega}} , \angle H(\exp(j\Omega)) = \Omega - \arctan\left(\frac{\sin \Omega}{cos\Omega+\frac{1}{2}} \right)$$
c) The signal u(t) is finite, and equal to 1 for n=0,1,2,3. Response can be found with convolution sum

$$y(n) = \sum_{m=-\infty}^{\infty} u(m)h(n-m) = \sum_{m=0}^{3} h(n-m) = \sum_{m=0}^{3} \left(-\frac{1}{2} \right)^{n-m} = \left(-\frac{1}{2} \right)^{n} \sum_{m=0}^{3} \left(-\frac{1}{2} \right)^{-m} = -5 \left(-\frac{1}{2} \right)^{n} $$

I need someone to check if these are correct. Thanks!
 
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  • #2
This is really good work.
THe only thing I could fault was under a) where you missed a power of n which I'm sure was a typo since your math looks fine after that.

Congrats for finding the closed-form of Z{h(n)}! At first I thought you got it wrong but it's correct!

I didn't check part c) since I got discouraged for finding nothing wrong up to that point so I'm sure it's OK also. :smile:
Nicely done!
 

Related to Response of LTI discrete time causal system

What is a LTI discrete time causal system?

A LTI (Linear Time-Invariant) discrete time causal system is a system in which the output at any given time depends only on the current and past inputs, and the system's behavior remains the same over time. This means that the system is both linear and time-invariant, and the output does not depend on future inputs.

What is the difference between a continuous time system and a discrete time system?

A continuous time system is a system in which both the input and output signals are continuous functions of time, while a discrete time system is a system in which both the input and output signals are only defined at discrete time intervals. In other words, a continuous time system operates on signals that are continuous in time, while a discrete time system operates on signals that are represented by a sequence of discrete values.

How do you know if a system is causal?

A system is considered causal if the output at any given time depends only on the current and past inputs, and not on any future inputs. This means that the system does not predict future behavior and is not affected by any future inputs.

What are some examples of LTI discrete time causal systems?

Some examples of LTI discrete time causal systems include digital filters, signal processing systems, and control systems. These systems are commonly used in various applications such as audio and video processing, communication systems, and robotics.

How do you analyze the response of a LTI discrete time causal system?

The response of a LTI discrete time causal system can be analyzed using various techniques such as convolution, difference equations, and z-transforms. These techniques allow us to determine the output of the system for a given input, and to understand the behavior and characteristics of the system.

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