A) Show that the Fourier Transform is lineal...

a) Show that the Fourier Transform is lineal meaning: F(af(x) + bg(x)) = a F(f(x)) + b F(g(x)) b) Computer the Fourier transform of f(x - xo) as a function of the Fourier transform of f(x) c) Compute the Fourier Transform of the following functions: f(t) = {1, if - 1 < t < 1; 0, otherwise g(t) = a cos(wot) h(t) = ? sin(wot) + ? cos (4wot) p(t) = exp(iwot) (sin(t/2) / (? t)) d) Proof the convolution theorem for the Fourier transform e) Apply the Fourier transform to the following convolutions: z(t) = h(t) * p(t)

Transcript

00:01 Hello everyone, let's see the questions.

00:02 According to the question in part a, f of a of fx plus bg of x is equals to integration minus infinite to infinite a fx plus bgx e minus i omega t dt is equals to a minus infinite to infinite f of x e minus iota omega t dt plus b minus infinite to infinite g x e minus iota omega t dt is equals to a f of fx plus b f of g x.

00:57 Now moving to part b of this question, then according to the part b, f is equals to integration minus infinite to infinite f of x minus x naught e minus iota omega x dx.

01:17 Now it is let x minus x naught is equals to t, then dx is equals to dt and then we get x is equals to t plus x naught.

01:31 So by this we get f of fx minus x naught is equals to integration minus infinite to infinite f of t e minus iota omega t plus x naught dt.

01:49 On solving this, we get it equals to e minus iota omega x naught minus infinite to infinite f x e minus iota omega x dx which is equals to e minus iota omega x naught f of f x.

02:10 Now as according to this moving to part c of this question, then in the first part of this question is f t is equals to 0 to 1 for minus 1 less than t less than 1 and this is for otherwise.

02:31 Now f of f t is equals to integration minus infinite to f of t e minus iota omega t dt.

02:43 On solving this, we get it equals to e minus iota omega divided minus iota omega plus e iota omega divided iota omega.

02:55 So it is equals to the 2 divided omega sin omega.

03:02 Now moving to part second of this question, then according to this we are having g of t is equals to a cos omega naught t.

03:13 By this we get f of g t is equals to the 2 pi delta omega minus omega naught which is equals to a pi delta omega minus omega naught plus delta omega plus omega naught...

Question

Please give Ace some feedback