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Determining if a function is periodic

Yup, this is homework. So, here are my answers and thoughts. Am I correct, incorrect or in the wrong ballpark ;)

Determine if the following function is periodic and if so, find the fundamental frequency. (The use of "j" is for imaginary numbers)

1) x(t) = 3cos(4t + pi/3)

  This function is periodic (the amplitude doesn't matter..?..).
  The fundamental frequency is T = pi/2

2) x(t) = exp( j (pi * t - 1))

  From euler form (  exp[x] = cos x - jsin x )

  x(t) = cos(pi t - 1) + j sin(pi t - 1)

  I do not think this is periodic due to the complex portion.

3) x(t) = [ cos(2t - pi/3) ] ^ 2

  Using a Half-Angle identity:

  x(t) = .5 [ 1 + cos( 2 ( 2t - pi/3 ) ) ]
        = .5 [ 1 + cos(4t - 2pi/3) ]

  Period with fundamental period T = pi/2

4) x(t) = Even { cos(4pt*t)) } u(t)

   This is not periodic due to the unit step function (as the amplitude changes at t=0)
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bingie
Asked:
bingie
3 Solutions
 
ozoCommented:
1) correct
2) with the complex portion, is x(t) = x(t+T) for some T!=0?
3) correct
4) I'm not sure how you define Even{} or u(t), but cos(4pt*t) is
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bingieAuthor Commented:
2) Not sure....

I made a table and substituted values for t (starting at 0 and increasing by 90) and the real portion increased by 281 each time and the angle remained constant at 57 degrees.

Does this indicate a periodic function with fundamental frequency of 2?

4) Even {} is defined where x(t) = x(-t) and then x(t) = Even{x(t)} + Odd{x(t)} where Odd{x(t)} -> x(t) = -x(-t)
    u(t) is a unit step function ( where x = 0 for t <0 and x = 1 for t>=0)
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Harisha M GCommented:
Hi bingie,

1) You are right. You know that all trigonometric funcitons are periodic with the period 2*PI (And TAN is periodic with PI)

You have cos(4t + pi/3) and pi/3 is a constant phase. So, cos(4t). Now, cos(4t+2pi) = cos(4t) = cos(4t'), (say)

So, 4t' = 4t+2pi => t' = t + pi/2

So, fundamental frequency is pi/2

2) x(t) = exp(j(pi*t-1)) = exp(j*pi*t-j) = exp(j*pi*t) * exp(-j) = ( cos (pi*t) - j sin(pi*t) ) * exp(-j)

which is clearly periodic, with period = 2

3) You are right, very similar to 1)

4) You are right.

But check whether it is given as Even { cos(4pt*t)) } u(t) or Even { cos(4pt*t)) * u(t) }

In the latter case, it will be periodic


ozo,
Even{x(t)} = ( x(t) + x(-t) ) / 2
Odd{x(t)} = ( x(t) - x(-t) ) / 2
u(t) = 1 for t >= 0, 0 otherwise



---
Harish
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aburrCommented:
But more fundamentally, what is your definition of periodic?
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bingieAuthor Commented:
harish,

>>Even { cos(4pt*t)) } u(t) or Even { cos(4pt*t)) * u(t) }

What is the difference in these two functions?
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Harisha M GCommented:
Only thing is u(t) is outside the even() function
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bingieAuthor Commented:
I noticed that, but how does that affect the function?
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NovaDenizenCommented:
For 1, the _period_ is pi/2, so the _frequency_ is 2/pi.

2 is periodic.  The complexity doesn't enter into it.  After every period the function generates the same value.  x(t) = x(t+delta) for all t and some particular deltas.

3 looks right.

4.  Consider the definition of periodicity, where x(t) = x(t+delta) for all t and some delta.  What is the minimum positive delta for which this definition works?  The fact that there is a unit step function doesn't rule out periodicity.
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