Find the periodicity with the help of Laplace transform
I have a function
$$x(t) = picos(21omega_0t)+0.1cos(39omega_0t)$$
that I want to solve T from the periodicity identity, $x(t)=x(t+T)$.
What I have tried now is basically just solving $x(t)=x(t+T)$, but this gets really complex. So I tried to do the laplace transform of the function in order to maybe find out a way of solving it more simple, the laplace transform is,
$$X(s) = pi frac{s}{s^2+(21omega_0)^2}+0.1frac{s}{s^2+(39omega_0)^2}$$
What struck me here is that I do not really know how to find the periodicity when I am in the frequency plane. I get a feeling it should be a better way of doing it than the latter, but how?
laplace-transform periodic-functions
asked Jan 4 at 7:10
SalviatiSalviati
224111
add a comment |
I have a function
$$x(t) = picos(21omega_0t)+0.1cos(39omega_0t)$$
that I want to solve T from the periodicity identity, $x(t)=x(t+T)$.
What I have tried now is basically just solving $x(t)=x(t+T)$, but this gets really complex. So I tried to do the laplace transform of the function in order to maybe find out a way of solving it more simple, the laplace transform is,
$$X(s) = pi frac{s}{s^2+(21omega_0)^2}+0.1frac{s}{s^2+(39omega_0)^2}$$
What struck me here is that I do not really know how to find the periodicity when I am in the frequency plane. I get a feeling it should be a better way of doing it than the latter, but how?
laplace-transform periodic-functions
asked Jan 4 at 7:10
SalviatiSalviati
224111
1
They are just sums of cosines with periods $T_1 = frac{2pi}{21omega_0}$ and $T_2 = frac{2pi}{39omega_0}$ so it is definitely periodic of period at most $T$ such that $T = nT_1 = mT_2$ for some integers $n$ and $m$. Simplifying a bit, we have $39n = 21m implies 3 cdot 13n = 3 cdot 7m implies 13n = 7m$ so the optimal integers are $n = 7, m = 13$. So $T = frac{2pi}{3omega_0}$.
– Pratyush Sarkar
Jan 4 at 8:52
Ohh, talking about making it more simple. Thank you!
– Salviati
Jan 4 at 8:55
add a comment |
I have a function
$$x(t) = picos(21omega_0t)+0.1cos(39omega_0t)$$
that I want to solve T from the periodicity identity, $x(t)=x(t+T)$.
What I have tried now is basically just solving $x(t)=x(t+T)$, but this gets really complex. So I tried to do the laplace transform of the function in order to maybe find out a way of solving it more simple, the laplace transform is,
$$X(s) = pi frac{s}{s^2+(21omega_0)^2}+0.1frac{s}{s^2+(39omega_0)^2}$$
What struck me here is that I do not really know how to find the periodicity when I am in the frequency plane. I get a feeling it should be a better way of doing it than the latter, but how?
laplace-transform periodic-functions
asked Jan 4 at 7:10
SalviatiSalviati
224111
I have a function
$$x(t) = picos(21omega_0t)+0.1cos(39omega_0t)$$
that I want to solve T from the periodicity identity, $x(t)=x(t+T)$.
What I have tried now is basically just solving $x(t)=x(t+T)$, but this gets really complex. So I tried to do the laplace transform of the function in order to maybe find out a way of solving it more simple, the laplace transform is,
$$X(s) = pi frac{s}{s^2+(21omega_0)^2}+0.1frac{s}{s^2+(39omega_0)^2}$$
What struck me here is that I do not really know how to find the periodicity when I am in the frequency plane. I get a feeling it should be a better way of doing it than the latter, but how?
laplace-transform periodic-functions
laplace-transform periodic-functions
asked Jan 4 at 7:10
SalviatiSalviati
224111
asked Jan 4 at 7:10
SalviatiSalviati
224111
asked Jan 4 at 7:10
SalviatiSalviati
224111
asked Jan 4 at 7:10
SalviatiSalviati
224111
asked Jan 4 at 7:10
SalviatiSalviati
224111
224111
1
They are just sums of cosines with periods $T_1 = frac{2pi}{21omega_0}$ and $T_2 = frac{2pi}{39omega_0}$ so it is definitely periodic of period at most $T$ such that $T = nT_1 = mT_2$ for some integers $n$ and $m$. Simplifying a bit, we have $39n = 21m implies 3 cdot 13n = 3 cdot 7m implies 13n = 7m$ so the optimal integers are $n = 7, m = 13$. So $T = frac{2pi}{3omega_0}$.
– Pratyush Sarkar
Jan 4 at 8:52
Ohh, talking about making it more simple. Thank you!
– Salviati
Jan 4 at 8:55
add a comment |
1
They are just sums of cosines with periods $T_1 = frac{2pi}{21omega_0}$ and $T_2 = frac{2pi}{39omega_0}$ so it is definitely periodic of period at most $T$ such that $T = nT_1 = mT_2$ for some integers $n$ and $m$. Simplifying a bit, we have $39n = 21m implies 3 cdot 13n = 3 cdot 7m implies 13n = 7m$ so the optimal integers are $n = 7, m = 13$. So $T = frac{2pi}{3omega_0}$.
– Pratyush Sarkar
Jan 4 at 8:52
Ohh, talking about making it more simple. Thank you!
– Salviati
Jan 4 at 8:55
1
1
They are just sums of cosines with periods $T_1 = frac{2pi}{21omega_0}$ and $T_2 = frac{2pi}{39omega_0}$ so it is definitely periodic of period at most $T$ such that $T = nT_1 = mT_2$ for some integers $n$ and $m$. Simplifying a bit, we have $39n = 21m implies 3 cdot 13n = 3 cdot 7m implies 13n = 7m$ so the optimal integers are $n = 7, m = 13$. So $T = frac{2pi}{3omega_0}$.
– Pratyush Sarkar
Jan 4 at 8:52
They are just sums of cosines with periods $T_1 = frac{2pi}{21omega_0}$ and $T_2 = frac{2pi}{39omega_0}$ so it is definitely periodic of period at most $T$ such that $T = nT_1 = mT_2$ for some integers $n$ and $m$. Simplifying a bit, we have $39n = 21m implies 3 cdot 13n = 3 cdot 7m implies 13n = 7m$ so the optimal integers are $n = 7, m = 13$. So $T = frac{2pi}{3omega_0}$.
– Pratyush Sarkar
Jan 4 at 8:52
Ohh, talking about making it more simple. Thank you!
– Salviati
Jan 4 at 8:55
Ohh, talking about making it more simple. Thank you!
– Salviati
Jan 4 at 8:55
add a comment |
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They are just sums of cosines with periods $T_1 = frac{2pi}{21omega_0}$ and $T_2 = frac{2pi}{39omega_0}$ so it is definitely periodic of period at most $T$ such that $T = nT_1 = mT_2$ for some integers $n$ and $m$. Simplifying a bit, we have $39n = 21m implies 3 cdot 13n = 3 cdot 7m implies 13n = 7m$ so the optimal integers are $n = 7, m = 13$. So $T = frac{2pi}{3omega_0}$.
– Pratyush Sarkar
Jan 4 at 8:52
Ohh, talking about making it more simple. Thank you!
– Salviati
Jan 4 at 8:55