Question

3. Compute the Fourier transform of the following functions and sketch them (a) f(x)-cos2 (2πχ) (b) f(x) = ramp(x) rect(x/2) Comment on the similarity with Fourier series coefficients in 1(a) and 1(b)

Answer 1

"Fourier transform f(x) = cos^2 2 pi x. the Fourier transform of f(x) is f(w) = integral_-infinity^infinity f(t) e^-j wt dt f(w) = integral_-infinity^infinity cos_2^2 pi t e^-jwt dt 1/2 integral_-infinity^infinity(1 + cos 4 pi t) e^-jwt dt = 1/2 integral_-infinity^infinity e^-jwt + e^-jwt cos 4 pi t)dt since cos^2 2 pi x is even function f(w) = 2/2 integral_0^infinity [e^-jwt + e^-jwt cos 4 pi t) dt [e^jwt/- jw]_0^infinity + integral_0^infinity e^-jwt cos 4 pi t dt f(w) [1/jw++ jw/(jW)^2 + (4 pi)^2] since integral_0^0 e^-ax cos bx = a/a^2 + b^2 = [1/jw + jw /16 pi^2 + (jw)^2 f(w) = 16 pi^2 + 2 = (pi w)^2/(jw) [16 pi^2 + (jw)^2] (b) the unit ramp function, ramp (x) = {x x > 0 0x lessthanorequalto0}. Rect (x/2) = {1