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!










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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
















0












$begingroup$


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!










share|cite|improve this question









$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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!










share|cite|improve this question









$endgroup$




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






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










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


















  • $begingroup$
    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










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