AM modulation concept

Take the baseband spectrum of the pulse and tack it on to each side of the carrier with the zero frequency end of the base band spectrum lined up with the carrier. For instance, the DC component of the base band spectrum will be at the carrier frequency and a base band component at 1 MHz will be at carrier + 1 MHz and carrier - 1 MHz.

What is modulation in time ?
It is the multiplication of the modulating signal by the modulator signal in the time domain.

What is its equivalent in frequency domain?

We know that multiplication in time correspond to convolution in frequency.

What is the fourier transform of an periodic pulse ?

It is a periodic sinc function.

What happen when we convolve a spectrum with an impulse?

The spectrum is displaced by the impulse position .

What is the fourier transform of an sinusoidal funtion?

It is two impulse placed in + and - the sinusoidal frequency.

What happen if we convolve the periodic sinc signal with the two impulse quoted above ?

The periodic sinc signal we be displaced + and - by the sinusoidal frequency. Conversely the sincs displaced by the sinusoidal frequency will keep their original frequency.

What does it mean in time domain ?

It means that we are sampling the carrier ( sinusoidal signal) whith the periodic pulse .

So it is the principle of the sampling theory ?

Exactly!

sorry, but it is confused... could show an numerical example of how to calculate the spectrum of the carrier modulated by a periodic pulse?

tks

for example, if I have a pulse train, with ζ pulse width, and T = 5*ζ
in Fourier series:

f(t) = Co + ∑ [Cn * cos (nωot + Φn)]

and

Cn = (ζ/T)*A*Sa(n*pi*ζ/T)

where:
Co: DC component
n: number of the harmonic
Φ: phase
ζ: pulse width
T: period of the pulse trains or 1/frequence
Sa: Sample function (sen x)/x

The Amplitud modulation is:

[Eo + K*a(t)]*cos ωct

where:
Eo: Carrier amplitud
K: modulator constant
ωc: carrier frequency
a(t): information signal

Do I have to substitute a(t) by f(t) and make the calculus? How many harmonics do I have to consider for represent the right frequency spectrum and recover the same pulse trains at the demodulator?