00:01
Hello students, in this question we are asked to find the fourier transform of phi function.
00:05
So the first function is f of t is equal to 5 cos phi t plus 3 sin phi t.
00:13
So what we can do is that we can now write down the fourier transform of cos a t.
00:24
The fourier transform of cos a t is equal to pi times delta of w minus a plus delta of w minus w plus a.
00:40
And sin a t fourier transform is equal to minus j pi times delta of w minus a minus delta of w plus a.
00:58
So this is how you can take the, now you can use this result to simplify this.
01:03
So the fourier transform of this function f of omega will be equal to phi times cos will be equal to pi times delta of w minus 5 plus delta of w plus 5 plus 3 times minus j pi times delta of w minus 5 minus delta of w plus 5.
01:37
That is the function.
01:39
So we can simplify this as f of omega will be equal to, so this will be simplified as this 5 pi times delta w minus 5 plus delta w plus 5 plus or you can simply say it's minus 3 times pi j times 8 w minus 5 minus delta w plus 5.
02:14
Okay.
02:15
This will be the simplification.
02:17
Now the second one will be is given as x of t is equal to cos square 6 t.
02:26
Right? so fourier transform, we can write down cos square 6 t as half into 1 plus cos 2 t.
02:35
So that is cos 12 t.
02:41
So now we can substitute the transform x of w is equal to half x of omega or x of w, whatever.
02:48
So x of omega will be equal to 1 plus cos of 12 t can be transformed as pi times delta of w minus 12 w minus 12 plus delta of w plus 12 omega plus 12 or w plus 12.
03:16
So this is, this is it 1 plus pi times that.
03:23
So that is the solution for that.
03:26
Now c part is that fourier transform of y of t is equal to e raised to minus t u t times cos 2 t.
03:39
So the e raised to the fourier transform of e raised to minus t u t is equal to 1 by i omega plus 1.
03:55
Okay.
03:55
1 by i omega plus 1...