The Fourier technique is used for the spectral analysis or frequency domain analysis of a signal.

1] Time shifting property : The time shifting property states that of

X(t) and X(f) form a Fourier transform pair then,

Here the signal x(t-td) is a time shifted signal, it is the same signal x(t) only shifted in time.

Proof :

$f[x(t-td)] = \int_\infty^\infty x(t-td) e^-\zeta2\pi ft_{dt}$

Let t - td = z

dt = dz

$\therefore$ $f [ x(t - td)] = \int_\infty^\infty x(z)_e^{-j 2} \ \pi F(z+td)_{dz}$

$x_1 (+) * x_2 (+) = \int_\infty^\infty \ x_1 \ ( \lambda ) \ x_2 \ (t-\lambda) \ d \ \lambda$

$F[x_1 (+) * x_2 (+) ] = \int_\infty^\infty [ \int_\infty^\infty x_1 (\lambda) x_2 (t - \lambda) d \lambda]$

Multiply and divide by $e^{-j\pi} \ f \ \lambda$

$f [ x_1 (+) * x_2 (+) ] = \int_\infty^\infty x_1 \ (\lambda) e^{-j} \ 2n \ f \lambda d \lambda - \int_\infty^\infty x_2 (t - \lambda)$

$= \int_\infty^\infty x_1 (\lambda) e^{- j2} \pi f \lambda_d\lambda \int_\infty^\infty x_2 (t - \lambda) e^{-j2nf} (t - \lambda) dt$

Let $t - \lambda = m$

$eFm = s(+) = Ec \ sin \ wct + \frac{mf}{2} Ec \ sin \ (wc + wm) t$

$\frac{mf \ Ec}{2} \ sin \ (wc - wm) t$

Practically NBFM have mf less than 1, the system is used in FM mobile communication. For larger values of modulation index mf, the FM wave ideally contains the carrier and infinite number of side bands around the carrier, such a FM wave has infinite BW and hence called as wide band FM, the modulation index is higher than 1.