Question

Consider two periodic signals x(t) and δ_(T_0 ) (t) as shown in Fig. 1, where δ_(T_0 ) (t)=∑_(k=-[infinity])^[infinity]▒〖δ(t-kT_0)〗.
(a) Determine the complex exponential Fourier series of both x(t) and δ_(T_0 ) (t).
(b) Determine the trigonometric Fourier series of both x(t) and δ_(T_0 ) (t).
COmplete question in file

Answer 1

Answer:

Caveat: I'm using the normalization f^(ω)=∫∞−∞f(t)e−itωdt.

A cute way to to derive the Fourier transform of f(t)=e−t2 is the following trick: Since

f′(t)=−2te−t2=−2tf(t),

taking the Fourier transfom of both sides will give us

iωf^(ω)=−2if^′(ω).

Solving this differential equation for f^ yields

f^(ω)=Ce−ω2/4

and plugging in ω=0 finally gives

C=f^(0)=∫∞−∞e−t2dt=π−−