# Fourier transforms of cos and sin

I have the function of time $f(t)=\cos(t10\pi) + \sin(t10\pi)$ and i wish to transform it. By using the tables, i have $\pi [\delta(w-10\pi) + \delta(w+10\pi)] + (\frac{\pi}{j})[ \delta (w-10\pi) = \delta(w+10\pi)]$

I just want to know if its correct. Also i see another answer and the solution goes by this

He first factors out the $2\pi$ (Why? For what reason?)

and it became

$$F\{\cos[2\pi(5t)] + \sin[2\pi(5t)]\}$$

$$(j[\delta(w-5) + \delta(w+5)] + \delta(w-5) - \frac{\delta(w+5)}{2j}$$

Which is correct? and i didn't understand why he factored the $2\pi$. Thanks

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You really should use TeX – Andrea Mori Feb 20 '12 at 10:07

Both are "correct", provided that in your first paragraph one of the = signs is actually a + sign.

The basic problem is: there isn't just one single definition of the Fourier transform! The Fourier transform of a function $f(t)$ generally looks like this

$$\mathcal{F}f(\omega) = A \int_{-\infty}^{\infty} f(t) \exp ( - C i \omega t) \mathrm{d}t$$

where $C$ and $A$ are fixed non-zero constants. Common choices of $C$ includes $1$ or $2\pi$. While common choices of $A$ include $1$, $1/2\pi$, or $1/\sqrt{2\pi}$. Wikipedia has some more discussion about this. The different choices of $C$ and $A$ have different desireable properties. If $C$ is set to be $1$, then under the Fourier transform the derivative $\frac{d}{dt}$ becomes $-i$. However, the frequency $\omega$ should be then interpreted as an "angular frequency" instead of a frequency in terms of signal processing. Conversely, setting $C$ to $2\pi$ you get that $\omega$ is an actual frequency, but the relation with the derivatives pick up extra factors of $2\pi$.

Similarly, setting $A$ to different values makes the Fourier inversion formula look different. Some of which are intuitively more natural when you consider the Fourier transform as breaking down a signal into sines and cosines, and some of which are more "aesthetically pleasing" (the choice of $A = 1/\sqrt{2\pi}$ makes the Fourier and Fourier-inverse transforms look more similar to each other).

In short, whenever you consult a reference, you should first double check the convention the author uses for the Fourier transform. Between different conventions there will be different factors of $2\pi$ floating around. And as long as you keep track of them and make sure you only write using one fixed convention, you shouldn't run into problems.

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