Physics AMPLITUDE AND FREQUENCY MODULATION

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AMPLITUDE AND FREQUENCY MODULATION

Amplitude Modulation

In amplitude modulation the amplitude of the carrier is varied in accordance with the information signal. Here we explain amplitude modulation process using a sinusoidal signal as the modulating signal.

Let carrier wave `=>` `c(t) = A_c sinomega_ct`
modulating signal `=>` `m(t) = A_m sinomega_mt`

where angular frequency `omega_m = 2pif_m`

The modulated signal `c_m (t)` can be written as

`c_m (t) = (A_c + A_m sinomega_mt) sinomega_ct`
`A_c(1+(A_m)/(A_c)sinomega_mt)sinomega_ct--(1)`

Note that the modulated signal now contains the message signal. This can also be seen from Fig. (c). From Eq. (1), we can write,

`c_m( t) =A_c sinomega_ct +muA_csinomega_mtsinomega_ct-..(2)`
Here modulation index(`mu`) = `(A_m)/(A_c)`

`mu` is kept `le1` to avoid distortion.

we can write `c_m (t)` of Eq. (2) as

`c_m(t)=A_csinomega_ct+(muA_c)/2 cos(omega_c-omega_m)t-(muA_c)/2 cos(omega_c+omega_m)t-..(3)`

`omega_c-omega_m=>` Lower side frequency
`omega_c+omega_m=>` Upper side frequency

The modulated signal now consists of the carrier wave of frequency `omega_c` plus two sinusoidal waves each with a frequency slightly different from, known as side bands.

As long as the broadcast frequencies (carrier waves) are sufficiently spaced out so that sidebands do not overlap, different stations can operate without interfering with each other.

In amplitude modulation the amplitude of the carrier is varied in accordance with the information signal. Here we explain amplitude modulation process using a sinusoidal signal as the modulating signal.

Let carrier wave `=>` `c(t) = A_c sinomega_ct`
modulating signal `=>` `m(t) = A_m sinomega_mt`

where angular frequency `omega_m = 2pif_m`

The modulated signal `c_m (t)` can be written as

`c_m (t) = (A_c + A_m sinomega_mt) sinomega_ct`
`A_c(1+(A_m)/(A_c)sinomega_mt)sinomega_ct--(1)`

Note that the modulated signal now contains the message signal. This can also be seen from Fig. (c). From Eq. (1), we can write,

`c_m( t) =A_c sinomega_ct +muA_csinomega_mtsinomega_ct-..(2)`
Here modulation index(`mu`) = `(A_m)/(A_c)`

`mu` is kept `le1` to avoid distortion.

we can write `c_m (t)` of Eq. (2) as

`c_m(t)=A_csinomega_ct+(muA_c)/2 cos(omega_c-omega_m)t-(muA_c)/2 cos(omega_c+omega_m)t-..(3)`

`omega_c-omega_m=>` Lower side frequency
`omega_c+omega_m=>` Upper side frequency

The modulated signal now consists of the carrier wave of frequency `omega_c` plus two sinusoidal waves each with a frequency slightly different from, known as side bands.

As long as the broadcast frequencies (carrier waves) are sufficiently spaced out so that sidebands do not overlap, different stations can operate without interfering with each other.

Production of Amplitude Modulated Wave

Amplitude modulation can be produced by a variety of methods. A conceptually simple method is shown in the block diagram of Fig.

`x(t)=m(t)+c(t)`

`x(t)=A_m sinω_mt + A_c sinω_ct`

`y (t) = Bx(t) + Cx^2(t)..........(4)`

where B and C are constants. Thus,

`y (t)= BA_m sin ω_mt + BA_c sin ω_ct + C[A_m^2sin^2ω_mt + A_c^2sin^2 ω_ct + 2A_mA_csinω_mtsinomega_ct].....(5)`

`y(t)=BA_m sin ω_mt + BA_c sin ω_ct + (CA_m^2)/2 + A_c^2 - (CA_m^2)/2 cos(2omega_mt) - (CA_c^2)/2 cos(2omega_ct) + CA_mA_c cos (ω_c - ω_m) t - CA_mA_c cos (ω_c+ ω_m) t....(6)`

In Eq. (6), there is a dc term `C//2 (A_m^2 + A_c^2)` and sinusoids of frequencies `ω_m, 2ω_m, ω_c, 2ω_c, ω_c - ω_m` and `ω_c + ω_m`. As shown in Fig.(a).

This signal is passed through a band pass filter which rejects dc and the sinusoids of frequencies `ω_m , 2ω_m` and `2 ω_c` and retains the frequencies `ω_c, ω_c - ω_m` and `ω_c + ω_m`.

The output of the band pass filter therefore is of the same form as Eq. (3) and is therefore an AM wave.

It is to be mentioned that the modulated signal cannot be transmitted as such. The modulator is to be followed by a power amplifier which provides the necessary power and then the modulated signal is fed to an antenna of appropriate size for radiation as shown in Fig.(b).

Amplitude modulation can be produced by a variety of methods. A conceptually simple method is shown in the block diagram of Fig.

`x(t)=m(t)+c(t)`

`x(t)=A_m sinω_mt + A_c sinω_ct`

`y (t) = Bx(t) + Cx^2(t)..........(4)`

where B and C are constants. Thus,

`y (t)= BA_m sin ω_mt + BA_c sin ω_ct + C[A_m^2sin^2ω_mt + A_c^2sin^2 ω_ct + 2A_mA_csinω_mtsinomega_ct].....(5)`

`y(t)=BA_m sin ω_mt + BA_c sin ω_ct + (CA_m^2)/2 + A_c^2 - (CA_m^2)/2 cos(2omega_mt) - (CA_c^2)/2 cos(2omega_ct) + CA_mA_c cos (ω_c - ω_m) t - CA_mA_c cos (ω_c+ ω_m) t....(6)`

In Eq. (6), there is a dc term `C//2 (A_m^2 + A_c^2)` and sinusoids of frequencies `ω_m, 2ω_m, ω_c, 2ω_c, ω_c - ω_m` and `ω_c + ω_m`. As shown in Fig.(a).

This signal is passed through a band pass filter which rejects dc and the sinusoids of frequencies `ω_m , 2ω_m` and `2 ω_c` and retains the frequencies `ω_c, ω_c - ω_m` and `ω_c + ω_m`.

The output of the band pass filter therefore is of the same form as Eq. (3) and is therefore an AM wave.

It is to be mentioned that the modulated signal cannot be transmitted as such. The modulator is to be followed by a power amplifier which provides the necessary power and then the modulated signal is fed to an antenna of appropriate size for radiation as shown in Fig.(b).

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