Fourier transform of cos(at) u(t)
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Ashamed to say I've been trying for hours.
Specifically have to use the integration of f(t)e^(-jwt).
I'm pretty comfortable with identities and integration. Just can't figure out for the life of me what to do with that unit step. Can't simplify down the cos(at) to a form that works with the limits of +inf and 0.
Any ideas?
Revising for a signal analysis exam tomorrow, so any kind of response would be appreciated!
fourier-transform
asked Jan 7 at 14:02
RichardRichard
1
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add a comment |
$begingroup$
Ashamed to say I've been trying for hours.
Specifically have to use the integration of f(t)e^(-jwt).
I'm pretty comfortable with identities and integration. Just can't figure out for the life of me what to do with that unit step. Can't simplify down the cos(at) to a form that works with the limits of +inf and 0.
Any ideas?
Revising for a signal analysis exam tomorrow, so any kind of response would be appreciated!
fourier-transform
asked Jan 7 at 14:02
RichardRichard
1
$endgroup$
$begingroup$
Remember $cos(at)=frac12[exp(iat)+exp(-iat)]$ so $mathcal{F}[cos(at) u(t)](omega)=dfrac{hat{u}(omega-a)+hat{u}(omega+a)}{2}$ and you should know $hat{u}(omega)$ is a sum of (multiples of) $1/omega$ and $delta$.
$endgroup$
– user10354138
Jan 7 at 14:23
add a comment |
$begingroup$
Ashamed to say I've been trying for hours.
Specifically have to use the integration of f(t)e^(-jwt).
I'm pretty comfortable with identities and integration. Just can't figure out for the life of me what to do with that unit step. Can't simplify down the cos(at) to a form that works with the limits of +inf and 0.
Any ideas?
Revising for a signal analysis exam tomorrow, so any kind of response would be appreciated!
fourier-transform
asked Jan 7 at 14:02
RichardRichard
1
$endgroup$
Ashamed to say I've been trying for hours.
Specifically have to use the integration of f(t)e^(-jwt).
I'm pretty comfortable with identities and integration. Just can't figure out for the life of me what to do with that unit step. Can't simplify down the cos(at) to a form that works with the limits of +inf and 0.
Any ideas?
Revising for a signal analysis exam tomorrow, so any kind of response would be appreciated!
fourier-transform
fourier-transform
asked Jan 7 at 14:02
RichardRichard
1
asked Jan 7 at 14:02
RichardRichard
1
asked Jan 7 at 14:02
RichardRichard
1
asked Jan 7 at 14:02
RichardRichard
1
asked Jan 7 at 14:02
RichardRichard
1
1
$begingroup$
Remember $cos(at)=frac12[exp(iat)+exp(-iat)]$ so $mathcal{F}[cos(at) u(t)](omega)=dfrac{hat{u}(omega-a)+hat{u}(omega+a)}{2}$ and you should know $hat{u}(omega)$ is a sum of (multiples of) $1/omega$ and $delta$.
$endgroup$
– user10354138
Jan 7 at 14:23
add a comment |
$begingroup$
Remember $cos(at)=frac12[exp(iat)+exp(-iat)]$ so $mathcal{F}[cos(at) u(t)](omega)=dfrac{hat{u}(omega-a)+hat{u}(omega+a)}{2}$ and you should know $hat{u}(omega)$ is a sum of (multiples of) $1/omega$ and $delta$.
$endgroup$
– user10354138
Jan 7 at 14:23
$begingroup$
Remember $cos(at)=frac12[exp(iat)+exp(-iat)]$ so $mathcal{F}[cos(at) u(t)](omega)=dfrac{hat{u}(omega-a)+hat{u}(omega+a)}{2}$ and you should know $hat{u}(omega)$ is a sum of (multiples of) $1/omega$ and $delta$.
$endgroup$
– user10354138
Jan 7 at 14:23
$begingroup$
Remember $cos(at)=frac12[exp(iat)+exp(-iat)]$ so $mathcal{F}[cos(at) u(t)](omega)=dfrac{hat{u}(omega-a)+hat{u}(omega+a)}{2}$ and you should know $hat{u}(omega)$ is a sum of (multiples of) $1/omega$ and $delta$.
$endgroup$
– user10354138
Jan 7 at 14:23
add a comment |
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$begingroup$
Remember $cos(at)=frac12[exp(iat)+exp(-iat)]$ so $mathcal{F}[cos(at) u(t)](omega)=dfrac{hat{u}(omega-a)+hat{u}(omega+a)}{2}$ and you should know $hat{u}(omega)$ is a sum of (multiples of) $1/omega$ and $delta$.
$endgroup$
– user10354138
Jan 7 at 14:23