Expectation of terms in double summation

In summary, we discussed finding the time-average of an equation involving double summation, specifically in the context of deriving PAPR for QAM OFDM signals. The equation is simplified if the f's and θ's are time-independent, and the cosine terms have a time-average of 0 except when j=k. However, since j is never equal to k in this summation, the overall average is 0.
  • #1
singhofmpl
15
0
Does anybody help me how to find the average (expectation) of terms involving double summation? Here is the equation which I'm trying solve.

[\tex]E\Big[2\sum_{k=0}^{N-2}\sum_{j=k+1}^{N-1}f(k,j)\cos[2\pi(j-k)t-\theta_{k,j}]\Big][\tex]
where f(k,j) and [\tex]\theta_{k,j}[\tex] are some function of k and j.
 
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  • #2
singhofmpl said:
does anybody help me how to find the average (expectation) of terms involving double summation? Here is the equation which I'm trying solve.

[itex]e\big[2\sum_{k=0}^{n-2}\sum_{j=k+1}^{n-1}f(k,j)\cos[2\pi(j-k)t-\theta_{k,j}]\big][/itex]
where f(k,j) and [itex]\theta_{k,j}[/itex] are some function of k and j.

[ latex ] expresion [ /latex ]
 
  • #3
Are you wanting to take the time-average? Have you been given the equation for doing that -- it involves doing an integral with respect to time...
 
  • #4
Redbelly98 said:
Are you wanting to take the time-average? Have you been given the equation for doing that -- it involves doing an integral with respect to time...

Yes I want to take the time average of the said expression. Its not part of any assignment. I came across this equation while deriving PAPR for the QAM OFDM signal. It does not involve any integral.
 
  • #5
The cosine terms all have a time-average of 0, except when j=k. As long as the f's and θ's are time-independent, this simplifies things greatly.

When j=k, the time-average of cos[2π(j-k)t - θk,j]=cos(θk,j). However, it appears that j is never equal to k in this summation, so we are left with 0 for the average.
 
Last edited:
  • #6
Redbelly98 said:
The cosine terms all have a time-average of 0, except when j=k. As long as the f's and θ's are time-independent, this simplifies things greatly.

When j=k, the time-average of cos[2π(j-k)t - θk,j]=cos(θk,j). However, it appears that j is never equal to k in this summation, so we are left with 0 for the average.

Dear Sir
Thanks a lot for your prompt response. After your comment now I've got the confirmation of my answer. Thanks a lot once again.
 

1. What is the purpose of using double summation in expectation of terms?

The purpose of using double summation in expectation of terms is to calculate the expected value of a function that has two variables. This is often used in statistics or probability to determine the average value of a function over a range of values for two different variables.

2. How is the double summation notation used in expectation of terms?

The double summation notation is written as E[X,Y] or E[XY] and represents the expected value of a function with two variables, X and Y. It is calculated by summing the product of the function and the probability of each possible value of X and Y.

3. What is the difference between single summation and double summation in expectation of terms?

The main difference between single summation and double summation in expectation of terms is the number of variables involved. Single summation involves only one variable, while double summation involves two variables. Single summation is used to calculate the expected value of a function with one variable, while double summation is used for functions with two variables.

4. What is the significance of expectation of terms in double summation in data analysis?

Expectation of terms in double summation is significant in data analysis as it allows for the calculation of the average value of a function with two variables. This can be useful in determining patterns and relationships between two variables in a dataset, and can also be used to make predictions based on the expected value.

5. Can double summation be extended to more than two variables in expectation of terms?

Yes, double summation can be extended to more than two variables in expectation of terms. This is known as multiple summation and involves summing the product of the function with the probabilities of each possible value for each variable. However, as the number of variables increases, the calculations become more complex and may require advanced mathematical techniques.

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