An AM signal is written as xc(t) = Ac(1+µx(t))cos(2pifct) where fc is the carrie

Question

An AM signal is written as xc(t) = Ac(1+µx(t))cos(2pifct) where fc is the carrier frequency, Ac is the carrier amplitude, µ is the
modulation index, and x(t) is the baseband message signal. We assume
that x(t) has absolute bandwidth W << fc, and that its amplitude has
been normalized so that |x(t)| <= 1.
If x(t) is a cosine of amplitude 1 and frequency fm << fc:

• Obtain an expression for the amplitude spectrum X c(f) of the
AM signal xc(t).

• Determine the power in the carrier and in the sidebands. Express
the powers in units of dBm into a 50 ohm load. (Remember that
the spectrum analyzer input impedance is 50 ohm) • Determine the ratio of the power in the sidebands to the power in the carrier
An AM signal is written as xc(t) = Ac(1+µx(t))cos(2pifct) where fc is the carrier frequency, Ac is the carrier amplitude, µ is the
modulation index, and x(t) is the baseband message signal. We assume
that x(t) has absolute bandwidth W << fc, and that its amplitude has
been normalized so that |x(t)| <= 1.
If x(t) is a cosine of amplitude 1 and frequency fm << fc:

• Obtain an expression for the amplitude spectrum X c(f) of the
AM signal xc(t).

• Determine the power in the carrier and in the sidebands. Express
the powers in units of dBm into a 50 ohm load. (Remember that
the spectrum analyzer input impedance is 50 ohm) • Determine the ratio of the power in the sidebands to the power in the carrier

Explanation / Answer

XAM(t) = Ac [ 1 + µ x(t) ] cos (?ct) Given The modulating signal x(t) = cos(?mt) XAM(t) = Ac [ 1 + µcos(?mt)] cos (?ct) XAM(t) = Ac cos (?ct) + µAc cos(?mt)cos (?ct) XAM(t) = Ac cos (?ct) + µAc cos(?mt)cos (?ct) XAM(t) = Ac cos (?ct) + [(µAc /2) [cos(?m +?c)t ] + [(µAc /2)[cos(?m -?c)t ] XAM(t) = Ac cos (2pfct) + [(µAc /2) [cos2p(fc +fm)t ] + [(µAc /2)[cos(2pfc - 2pfm)t ] XAM(t) = Ac cos (2pfct) + [(µAc /2) [cos2p(fc +fm)t ] + [(µAc /2)[cos(2pfc - 2pfm)t ] The above equation shows that the modulated signal has three frequenc componenets 1.    Ac cos (2pfct) 2.   [(µAc /2)[cos(2pfc - 2pfm)t ] 3.   [(µAc /2) [cos2p(fc +fm)t ] Thus the spectrum consists of center frequenc fc, upper side band frequency (fc+fm) and lower side band frequency (fc - fm) Power P = V2 / R V is the rms value Form the XAM(t) equation PT = [ (Ac/v2)2 + (µAc/2v2)2 + (µAc/2v2)2 ] / R PT = Pc + PUSB + PLSB PT = PC + PSB PT / PSB = (Ac/v2)2 / (µAc/v2)2 PT / PSB = (Ac/v2)2 / (µAc/v2)2

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