This course will introduce you to SQL Server Core 2016, as well as teach you about SSMS, data tools, installation, server configuration, using Management Studio, and writing and executing queries.

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)

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)

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)

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

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

What is the difference in these two functions?

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.

Question has a verified solution.

Are you are experiencing a similar issue? Get a personalized answer when you ask a related question.

Have a better answer? Share it in a comment.

All Courses

From novice to tech pro — start learning today.

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