Peak voltage of a modulating signal is 2 V. The carrier wave is represented by C(t) = 4sin(8πt)V. The modulation index of the modulated signal is:
0.5
Understanding Modulation Index in Communication Systems
Modulation is a process where a characteristic (like amplitude, frequency, or phase) of a high-frequency carrier wave is varied in accordance with the instantaneous amplitude of a low-frequency modulating signal. This process is essential for transmitting information over long distances efficiently.
For amplitude modulation (AM), the amplitude of the carrier wave is changed based on the strength of the modulating signal. The modulation index is a key parameter in AM that tells us how much the carrier amplitude is varied by the modulating signal.
Calculating the Modulation Index
The modulation index ($\mu$) is defined as the ratio of the amplitude of the modulating signal ($A_m$) to the amplitude of the carrier wave ($A_c$). Mathematically, it is expressed as:
\begin{equation}\mu = \frac{A_m}{A_c}\end{equation}
Let's identify the given values from the problem:
- Peak voltage of the modulating signal, $A_m = 2$ V. This represents the maximum amplitude of the signal we want to transmit.
- The carrier wave is represented by the equation $C(t) = 4\sin(8\pi t)$ V.
The standard form of a carrier wave for amplitude modulation is $C(t) = A_c \sin(\omega_c t)$, where $A_c$ is the amplitude of the carrier wave and $\omega_c$ is its angular frequency.
Comparing the given carrier wave equation $C(t) = 4\sin(8\pi t)$ with the standard form $C(t) = A_c \sin(\omega_c t)$, we can identify the amplitude of the carrier wave:
$A_c = 4$ V
Now we have both the amplitude of the modulating signal ($A_m$) and the amplitude of the carrier wave ($A_c$). We can substitute these values into the formula for the modulation index:
\begin{equation}\mu = \frac{A_m}{A_c} = \frac{2 \text{ V}}{4 \text{ V}}\end{equation}
\begin{equation}\mu = 0.5\end{equation}
So, the modulation index of the modulated signal is 0.5.
Understanding the Result
A modulation index of 0.5 means that the amplitude of the carrier wave varies by 50% above and below its original amplitude when modulated by the given signal. A modulation index between 0 and 1 (or 0% and 100%) is typical for standard AM broadcasting. A value of 0 means no modulation, and a value of 1 (or 100%) means the carrier amplitude varies from zero to twice its original amplitude.
Based on our calculation, the modulation index is 0.5.
| Parameter | Value | Source |
|---|---|---|
| Modulating Signal Peak Voltage ($A_m$) | 2 V | Given |
| Carrier Wave Equation | $C(t) = 4\sin(8\pi t)$ V | Given |
| Carrier Wave Amplitude ($A_c$) | 4 V | From Carrier Equation |
| Modulation Index ($\mu$) | 0.5 | Calculated ($\mu = A_m/A_c$) |
Revision Table: Key Concepts for Modulation Index
| Term | Definition/Formula | Significance |
|---|---|---|
| Modulation Index ($\mu$) | Ratio of Modulating Signal Amplitude ($A_m$) to Carrier Amplitude ($A_c$). $\mu = A_m/A_c$. | Indicates the degree of amplitude variation in AM. |
| Modulating Signal | The information signal (e.g., audio) that modifies the carrier. | Carries the actual message. |
| Carrier Wave | A high-frequency wave (e.g., radio wave) that is modulated. | Used to transport the modulating signal over distance. |
| Amplitude Modulation (AM) | A modulation technique where the carrier wave's amplitude is varied. | A common method for broadcasting. |
Additional Information on Modulation Index
The value of the modulation index is crucial in amplitude modulation systems:
- $\mu < 1$ (Under-modulation): The carrier amplitude never goes to zero. This is the desired condition for AM broadcasting as it avoids distortion. Our calculated value of 0.5 falls into this category.
- $\mu = 1$ (Critical modulation or 100% modulation): The carrier amplitude varies exactly between zero and twice the original carrier amplitude. This provides the maximum signal power without distortion.
- $\mu > 1$ (Over-modulation): The carrier amplitude goes to zero or becomes negative during parts of the modulation cycle. This leads to severe distortion of the modulated signal, causing the demodulated signal to be unclear. Over-modulation also creates additional frequencies (harmonics) that can interfere with other channels.
Therefore, maintaining the modulation index within the range $0 \le \mu \le 1$ is important for proper operation and quality in AM communication.
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