Single-tone AM

single tone AM:-

The expression for conventional AM is S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t

now if the message signal is a single-tone    i.e, m(t) = A_{m}cos 2\pi f_{m}t

S_{Single-tone AM}(t)=A_{c}(1+k_{a}A_{m}cos2\pi f_{m}t)cos 2\pi f_{c}t

where \mu =k_{a}A_{m} is called as modulation index 

S_{Single-tone AM}(t)=A_{c}cos 2\pi f_{c}t+\mu A_{c}cos2\pi f_{m}tcos 2\pi f_{c}t

this equation can be further simplified as follows     S_{Single-tone AM}(t)=A_{c}cos 2\pi f_{c}t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}+f_{m})t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}-f_{m})t

that is by taking the fourier transform 

\dpi{150} S_{Single-tone AM}(f)=\frac{A_{c}}{2}\left \{ \delta (f-f_{c})+\delta (f+f_{c}) \right \}+\frac{\mu A_{c}}{4}\left \{ \delta (f-(f_{c}+f_{m}))+\delta (f+(f_{c}+f_{m})) \right \}+\frac{\mu A_{c}}{4}\left \{ \delta (f-(f_{c}-f_{m}))+\delta (f+(f_{c}-f_{m})) \right \}

from the above expression the amplitude spectrum can be drawn as follows

from the spectrum single tone AM consists of 6 impulse functions located at frequencies \pm f_{c} , \pm (f_{c} + f_{m}) and \pm (f_{c} - f_{m}) respectively.

Power content in AM/ Conventional AM:-

S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t represents the AM signal , here m(t) is  some arbitrary signal , then the power of this signal can be calculated from its Mean Square value \overline{}{m^{2}(t)}

 i.e, message signal power = \overline{}{m^{2}(t)} Watts.

Carrier signal is  C(t)=A_{c}cos 2\pi f_{c}t and it’s power is \frac{A_{c}^{2}}{2} Watts.

Now the total power available in the signal S_{AM}(t)=A_{c}(1+k_{a}m(t))cos 2\pi f_{c}t   will be  P_{TOTAL} .

S_{AM}(t)=A_{c}cos 2\pi f_{c}t +A_{c}k_{a}cos 2\pi f_{c}t . m(t)

P_{TOTAL} =\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}k_{a}^{2}}{2} X message signal power

P_{TOTAL} =\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}k_{a}^{2}}{2} X\overline{m(t)^{2}} Watts.

Total Side Band power can be calculated from the term   A_{c}k_{a}cos 2\pi f_{c}t . m(t) can be denoted as P_{SB} that would be \frac{A_{c}^{2}k_{a}^{2}}{2} X\overline{m(t)^{2}} Watts.

from these power calculations transmission efficiency of AM can be obtained as \eta = \frac{P_{SB}}{P_{Total}} X100 %

\eta = \frac{\frac{A_{c}^{2}k_{a}^{2}}{2} .\overline{m(t)^{2}}}{\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}k_{a}^{2}}{2} .\overline{m(t)^{2}}} X 100%.

\eta = \frac{k_{a}^{2}.\overline{m(t)^{2}}}{1+k_{a}^{2}.\overline{m(t)^{2}}} X 100%.

Power content in Single-tone AM:-

In single tone AM message signal is i.e, m(t) = A_{m}cos 2\pi f_{m}t, then power of the message signal is \frac{A_{m}^{2}}{2} watts

carrier signal is C(t) = A_{c}cos 2\pi f_{c}t implies the carrier power is \frac{A_{c}^{2}}{2} watts.

S_{Single-tone AM}(t)=A_{c}cos 2\pi f_{c}t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}+f_{m})t+\frac{\mu A_{c}}{2}cos2\pi (f_{c}-f_{m})t

then the  total power of the  single-tone AM signal is from the  above equation given as

 P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu ^{2}}{8}+\frac{A_{c}^{2}\mu ^{2}}{8}

PTotal = Pc +PUSB+PLSB

P_{Total}=\frac{A_{c}^{2}}{2}+\frac{A_{c}^{2}\mu ^{2}}{4}

P_{Total}=\frac{A_{c}^{2}}{2}(1+\frac{\mu ^{2}}{2}) Watts.

USB and LSB has same power P_{USB}=P_{LSB}=\frac{A_{c}^{2}\mu ^{2}}{8} watts.

Now total side band power is P_{SB}=P_{USB}+P_{LSB}=\frac{A_{c}^{2}\mu ^{2}}{4}

from these power calculations transmission efficiency of AM can be obtained as \eta = \frac{P_{SB}}{P_{Total}} X100 %

\eta = \frac{\frac{A_{c}^{2}\mu ^{2}}{4}}{\frac{A_{c}^{2}}{2}(1+\frac{\mu ^{2}}{2})} X 100%

\eta = \frac{\mu ^{2}}{(\mu ^{2}+2)} X100%.

Note:- Effeciency (or) Transmission efficiency of AM is only 33.3% only i.e, \eta value  calculated when \mu =1.

 

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Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.