Effective Modulation Index For Multi-Tone Modulation MCQ Quiz - Objective Question with Answer for Effective Modulation Index For Multi-Tone Modulation - Download Free PDF

Explanation:

In AM modulation Index is defined as:

$\mu =\frac{A_m}{A_c}=\frac{A_{max}-A_{min}}{A_{max}+A_{min}}$

Where Am = amplitude of modulating wave

Ac = amplitude of carrier wave

Clearly, it is independent of fm.

• The value of modulation varies from 0 to 1.   0 ≤ μ ≤ 1
• For μ < 1, the AM wave is said to be under modulated.
• For μ > 1, the AM wave is said to be over-modulated.
• For μ = 1, the AM wave is said to be critically modulated.
• 100% modulation is where the modulating signal drives the carrier to zero and is theoretically the maximum modulation that can be successfully demodulator by a regular AM envelope detector.

A general Amplitude modulation process is as shown:

Concept:

When a carrier is modulated by different waves having different modulation indexes, the effective (total) modulation index is given by:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2+...}$

Calculation:

With μ₁ = 0.25, μ2 = 0.50, μ3 = 0.75, the effective modulation index will be:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{(0.25{)}^2+(0.50{)}^2+(0.75{)}^2}$

μ_eff = 0.935

Important Note:

The total power of the modulated signal for the given effective modulation index is given by:

$P_t=P_c\left(1+\frac{{\mu }_{eff}^2}{2}\right)$

Pc = Carrier Power

μeff = Effective modulation Index

When a carrier is modulated by different waves having different modulation indexes,

The effective modulation index is given by:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mu }_1^2+{\mu }_2^2}$

μ₁ = 0.6

μ₂ = 0.8

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{0.36+0.64}=1$

Concept: Modulation index is also called Depth of modulation.

Application: The general AM modulated signal expression is $\mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\left(1+\mu \cos {\omega }_{\mathrm{m}}\mathrm{t}\right)\cos {\omega }_{\mathrm{c}}\mathrm{t}$

$\begin{array}{l}\Rightarrow \mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\left[2\cos {\omega }_{\mathrm{m}}\mathrm{t}\cos {\omega }_{\mathrm{c}}\mathrm{t}\right]\\ ={\mathrm{A}}_{\mathrm{c}}\cos {\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\left[\cos \left({\omega }_{\mathrm{n}}+{\omega }_{\mathrm{c}}\right)\mathrm{t}\cos \left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)\mathrm{t}\right]\\ \Rightarrow \mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\cos \left({\omega }_{\mathrm{m}}+{\omega }_{\mathrm{c}}\right)\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\cos \left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)\mathrm{t}\end{array}$

Thus, we see that two terms have same amplitudes. Comparing with

$\mathrm{s}\left(\mathrm{t}\right)=5\cos 1600\pi \mathrm{t}+20\cos 1800\pi \mathrm{t}+5\cos 2000\pi \mathrm{t}$

we have $A_c=20$

Explanation:

In AM modulation Index is defined as:

$\mu =\frac{A_m}{A_c}=\frac{A_{max}-A_{min}}{A_{max}+A_{min}}$

Where Am = amplitude of modulating wave

Ac = amplitude of carrier wave

Clearly, it is independent of fm.

• The value of modulation varies from 0 to 1.   0 ≤ μ ≤ 1
• For μ < 1, the AM wave is said to be under modulated.
• For μ > 1, the AM wave is said to be over-modulated.
• For μ = 1, the AM wave is said to be critically modulated.
• 100% modulation is where the modulating signal drives the carrier to zero and is theoretically the maximum modulation that can be successfully demodulator by a regular AM envelope detector.

A general Amplitude modulation process is as shown:

When a carrier is modulated by different waves having different modulation indexes,

The effective modulation index is given by:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mu }_1^2+{\mu }_2^2}$

μ₁ = 0.6

μ₂ = 0.8

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{0.36+0.64}=1$

Concept:

When a carrier is modulated by different waves having different modulation indexes, the effective (total) modulation index is given by:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2+...}$

Calculation:

With μ₁ = 0.25, μ2 = 0.50, μ3 = 0.75, the effective modulation index will be:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{(0.25{)}^2+(0.50{)}^2+(0.75{)}^2}$

μ_eff = 0.935

Important Note:

The total power of the modulated signal for the given effective modulation index is given by:

$P_t=P_c\left(1+\frac{{\mu }_{eff}^2}{2}\right)$

Pc = Carrier Power

μeff = Effective modulation Index

Concept: Modulation index is also called Depth of modulation.

Application: The general AM modulated signal expression is $\mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\left(1+\mu \cos {\omega }_{\mathrm{m}}\mathrm{t}\right)\cos {\omega }_{\mathrm{c}}\mathrm{t}$

$\begin{array}{l}\Rightarrow \mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\left[2\cos {\omega }_{\mathrm{m}}\mathrm{t}\cos {\omega }_{\mathrm{c}}\mathrm{t}\right]\\ ={\mathrm{A}}_{\mathrm{c}}\cos {\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\left[\cos \left({\omega }_{\mathrm{n}}+{\omega }_{\mathrm{c}}\right)\mathrm{t}\cos \left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)\mathrm{t}\right]\\ \Rightarrow \mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\cos \left({\omega }_{\mathrm{m}}+{\omega }_{\mathrm{c}}\right)\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\cos \left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)\mathrm{t}\end{array}$

Thus, we see that two terms have same amplitudes. Comparing with

$\mathrm{s}\left(\mathrm{t}\right)=5\cos 1600\pi \mathrm{t}+20\cos 1800\pi \mathrm{t}+5\cos 2000\pi \mathrm{t}$

we have $A_c=20$

Explanation:

In AM modulation Index is defined as:

$\mu =\frac{A_m}{A_c}=\frac{A_{max}-A_{min}}{A_{max}+A_{min}}$

Where Am = amplitude of modulating wave

Ac = amplitude of carrier wave

Clearly, it is independent of fm.

• The value of modulation varies from 0 to 1.   0 ≤ μ ≤ 1
• For μ < 1, the AM wave is said to be under modulated.
• For μ > 1, the AM wave is said to be over-modulated.
• For μ = 1, the AM wave is said to be critically modulated.
• 100% modulation is where the modulating signal drives the carrier to zero and is theoretically the maximum modulation that can be successfully demodulator by a regular AM envelope detector.

A general Amplitude modulation process is as shown:

When a carrier is modulated by different waves having different modulation indexes,

The effective modulation index is given by:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mu }_1^2+{\mu }_2^2}$

μ₁ = 0.6

μ₂ = 0.8

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{0.36+0.64}=1$

Concept:

When a carrier is modulated by different waves having different modulation indexes, the effective (total) modulation index is given by:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mu }_1^2+{\mu }_2^2+{\mu }_3^2+...}$

Calculation:

With μ₁ = 0.25, μ2 = 0.50, μ3 = 0.75, the effective modulation index will be:

${\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{(0.25{)}^2+(0.50{)}^2+(0.75{)}^2}$

μ_eff = 0.935

Important Note:

The total power of the modulated signal for the given effective modulation index is given by:

$P_t=P_c\left(1+\frac{{\mu }_{eff}^2}{2}\right)$

Pc = Carrier Power

μeff = Effective modulation Index

Concept: Modulation index is also called Depth of modulation.

Application: The general AM modulated signal expression is $\mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\left(1+\mu \cos {\omega }_{\mathrm{m}}\mathrm{t}\right)\cos {\omega }_{\mathrm{c}}\mathrm{t}$

$\begin{array}{l}\Rightarrow \mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\left[2\cos {\omega }_{\mathrm{m}}\mathrm{t}\cos {\omega }_{\mathrm{c}}\mathrm{t}\right]\\ ={\mathrm{A}}_{\mathrm{c}}\cos {\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\left[\cos \left({\omega }_{\mathrm{n}}+{\omega }_{\mathrm{c}}\right)\mathrm{t}\cos \left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)\mathrm{t}\right]\\ \Rightarrow \mathrm{s}\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{c}}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{\mathrm{c}}\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\cos \left({\omega }_{\mathrm{m}}+{\omega }_{\mathrm{c}}\right)\mathrm{t}+\frac{{\mathrm{A}}_{\mathrm{c}}\mu }{2}\cos \left({\omega }_{\mathrm{c}}-{\omega }_{\mathrm{m}}\right)\mathrm{t}\end{array}$

Thus, we see that two terms have same amplitudes. Comparing with

$\mathrm{s}\left(\mathrm{t}\right)=5\cos 1600\pi \mathrm{t}+20\cos 1800\pi \mathrm{t}+5\cos 2000\pi \mathrm{t}$

we have $A_c=20$

Transmitter current is $I_t=I_c\sqrt{1+\frac{{\mu }^2}{2}}$               ; where $I_c$is carrier current

$\begin{array}{rl}& 5=I_c\sqrt{1+\frac{{0.2}^2}{2}}-------(1)\\ & 5.5=I_c\sqrt{1+\frac{{{\mu }_t}^2}{2}}-------(2)\end{array}$

By dividing (2) by (1) -

$\begin{array}{rl}& \frac{5.5}{5}=\sqrt{\left(1+\frac{{{\mu }_t}^2}{2}\right)/\left(1+0.02\right)}\\ & \Rightarrow 1+\frac{{{\mu }_t}^2}{2}=1.02\times {\left(\frac{5.5}{5}\right)}^2\\ & \Rightarrow 1+\frac{{{\mu }_t}^2}{2}=1.02\times 1.21=1.2342\\ & \Rightarrow {{\mu }_t}^2=2\times \left(1.2342-1\right)=0.468\\ & \text{And}{{\mu }_t}^2={{\mu }_1}^2+{{\mu }_2}^2\\ & \Rightarrow {{\mu }_2}^2={{\mu }_t}^2-{{\mu }_1}^2=0.468-0.04=0.4284\\ & \Rightarrow {\mu }_2=0.6545=65.45\mathrm{\%}\end{array}$

 Now for the function g(t) can be represented as

             g(t)= $\{\begin{array}{c}tfor0\le t\le 1\\ 2-tfor1\le 2\end{array}$

          Modulating index is 0.5

           Modulating signal is t