Difficult discrete-time periodicity problem

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In summary, The task is to determine if x[n] is periodic and to find the fundamental period if it is. The given equation is x[n]=cos(π/6 * n^2) and the task is to check if x(n) = x(n +N) holds true for all n and find the lowest possible N, the fundamental period.
  • #1
jti5017
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Homework Statement



Determine whether x[n] is periodic, if so find the fundamental period.

ps forgive my notation, I'm new to physics forums and haven't had a chance to figure out the exact notation syntax yet.x[n]=cos(π/6 * n^2)

The Attempt at a Solution



x(n) is periodic when x(n) = x(n +N)x(n+N) = cos(π/6 * (n+N)^2)

π/6(2*n*N + N^2) = k*2*π

this is kinda where i got stuck

(2*n*N + N^2) = 12 * k
 
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  • #2
welcome to pf!

hi jti5017! welcome to pf! :smile:

try using the X2 button just above the Reply box :wink:

is it true for the same N, for all n ? :wink:
 
  • #3
thanks for the reply!

it has to be true for all n to be periodic by definition i believe.
i mean, I've personally never heard of a sinusoidal period being a function of time (in this case discrete time)

And there are an infinite number of 'N's (provided it is periodic) but I am specifically looking for the lowest N, the fundamental period
 

1. What is the "Difficult discrete-time periodicity problem"?

The difficult discrete-time periodicity problem is a mathematical problem that involves determining whether a given sequence of numbers is periodic or not. This problem is considered difficult because it is known to be NP-complete, meaning that it cannot be solved efficiently by computers.

2. How is this problem related to computer science?

The difficult discrete-time periodicity problem is related to computer science because it is a central problem in the field of computational complexity theory. It is used to classify problems according to their difficulty and to study the limits of what computers can efficiently solve.

3. What are the applications of the "Difficult discrete-time periodicity problem"?

The difficult discrete-time periodicity problem has applications in various fields, including cryptography, signal processing, and data compression. It is also used in the design and analysis of algorithms for solving other difficult problems.

4. How is this problem different from the "Easy discrete-time periodicity problem"?

The easy discrete-time periodicity problem differs from the difficult one in that it only asks whether a given sequence is periodic with a known period, while the difficult problem aims to determine the period (if it exists) of an unknown periodic sequence.

5. Are there any known solutions to the "Difficult discrete-time periodicity problem"?

Currently, there is no known general solution to the difficult discrete-time periodicity problem. However, there are some algorithms that can efficiently solve special cases of the problem, such as when the sequence is restricted to a certain range or when the period is known to be relatively small.

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