<< Chapter < Page Chapter >> Page >
(Blank Abstract)

Before diving into a more complex statistical analysis of random signals and processes , let us quickly review the idea of correlation . Recall that the correlation of two signals or variables is the expectedvalue of the product of those two variables. Since our main focus is to discover more about random processes, a collectionof random signals, we will deal with two random processes in this discussion, where in this case we will deal with samplesfrom two different random processes. We will analyze the expected value of the product of these two variables and how they correlate to one another, where theargument to this correlation function will be the time difference. For the correlation of signals from the same randomprocess, look at the autocorrelation function .

Crosscorrelation function

When dealing with multiple random processes, it is also important to be able to describe the relationship, if any,between the processes. For example, this may occur if more than one random signal is applied to a system. In order to dothis, we use the crosscorrelation function , where the variables are instances from two different wide sensestationary random processes.

if two processes are wide sense stationary, the expected value of the product of a random variable from one randomprocess with a time-shifted, random variable from a different random process
Looking at the generalized formula for the crosscorrelation, we will represent our two random processes by allowing U U t and V V t . We will define the crosscorrelation function as
R u v t t U V u v u v f u v
Just as the case with the autocorrelation function, if ourinput and output, denoted as U t and V t , are at least jointly wide sense stationary, then the crosscorrelation does not depend on absolute time; it is justa function of the time difference. This means we can simplify our writing of the above function as
R u v U V
or if we deal with two real signal sequences, x n and y n , then we arrive at a more commonly seen formula for the discrete crosscorrelation function. See the formula belowand notice the similarities between it and the convolution of two signals:
R x y n n m R x y m n x n y n m

Properties of crosscorrelation

Below we will look at several properties of the crosscorrelation function that hold for two wide sense stationary (WSS) random processes.

  • Crosscorrelation is not an even function; however, it does have a unique symmetryproperty:
    R x y R y x
  • The maximum value of the crosscorrelation is not always when the shift equals zero; however, we can prove thefollowing property revealing to us what value the maximum cannot exceed.
    R x y R x x 0 R y y 0
  • When two random processes are statistically independent then we have
    R x y R y x


Let us begin by looking at a simple example showing the relationship between two sequences. Using , find the crosscorrelation of the sequences x n 0 0 2 -3 6 1 3 0 0 y n 0 0 1 -2 4 1 -3 0 0 for each of the following possible time shifts: m 0 3 -1 .

  • For m 0 , we should begin by finding the product sequence s n x n y n . Doing this we get the following sequence: s n 0 0 2 6 24 1 -9 0 0 and so from the sum in our crosscorrelation function we arrive at the answer of R x y 0 22
  • For m 3 , we will approach it the same was we did above; however, we will now shift y n to the right. Then we can find the product sequence s n x n y n 3 , which yields s n 0 0 0 0 0 1 -6 0 0 and from the crosscorrelation function we arrive at the answer of R x y 3 -6
  • For m -1 , we will again take the same approach; however, we will now shift y n to the left. Then we can find the product sequence s n x n y n 1 , which yields s n 0 0 -4 -12 6 -3 0 0 0 and from the crosscorrelation function we arrive at the answer of R x y -1 -13

Got questions? Get instant answers now!

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of signal processing' conversation and receive update notifications?