Digital Communication Systems - Autocorrelation

In summary, the problem involves a white noise process with a unity power spectral density being input into a linear system. The output of the linear system is X(t), which is equal to W(t) - W(t-1). The task is to determine the autocorrelation of X(t) and sketch it. The solution involves using the definition of autocorrelation and setting the other three terms to different autocorrelation functions. It then simplifies the expressions to Dirac delta functions. The solution is unclear on how it goes from the second line to the third line, as well as from the third line to the fourth line. Further clarification is needed.
  • #1
GreenPrint
1,196
0

Homework Statement



A white noise process W(t) with unity (N_0/2 = 1) power spectral density is input to a linear system. The output of the linear system is X(t), where

X(t) = W(t) - W(t - 1)

Determine the autocorrelation of X(t) and sketch it.

Homework Equations



Capture.png

Let τ denote a time shift; that is, t = t_2 and τ = t_1 - t_2
Capture.png


The Attempt at a Solution



Capture.png

I understand that the first term on the last line is indeed equal to
Capture.png
. I'm however unsure what to do with the other three terms. The solution sets the other three terms to different autocorrelation functions and I'm not sure how these other three terms are autocorrelation functions as well based off of the definition.

Here's what the solution is. I don't understand how it went from the second line to the third line.
Capture.png

Any help would be greatly appreciated. I also don't understand how how the solution goes from the third line to the fourth line. It seems to just simply replacing the autocorrelation functions with dirac delta functions. I'm not sure how these are equal in any way.
 
Physics news on Phys.org
  • #2
sorry I didn't mean to post
 

1. What is autocorrelation in digital communication systems?

Autocorrelation is a mathematical measure of how similar a signal is to itself over time. In digital communication systems, it is used to determine the level of self-similarity in a signal and is an important tool for analyzing the performance of a system.

2. How is autocorrelation used in digital communication systems?

Autocorrelation is used to measure the level of noise and distortion in a signal, which can affect the quality of communication. It is also used for channel estimation, equalization, and synchronization in the receiver.

3. What is the autocorrelation function?

The autocorrelation function is a mathematical representation of the correlation between a signal and a delayed version of itself. It is used to calculate the autocorrelation coefficient, which indicates the level of similarity between the signal and its delayed version.

4. How does autocorrelation affect communication system performance?

Autocorrelation can affect the performance of a communication system in various ways. A high level of autocorrelation can lead to interference and distortion in the received signal, resulting in a high bit error rate. On the other hand, a low level of autocorrelation can improve the reliability and accuracy of the received signal.

5. What are the advantages of using autocorrelation in digital communication systems?

Using autocorrelation in digital communication systems can help in detecting and correcting errors in the received signal. It can also improve the performance of the system by reducing the effects of noise and distortion. Additionally, autocorrelation can provide valuable information about the channel characteristics, which can be used for channel estimation and equalization.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
8
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Differential Equations
Replies
5
Views
591
  • Classical Physics
Replies
1
Views
674
Back
Top