Of 8 kHz. This FM signal is coupled to a load through an ideal Band Pass filter with 100MHz as center frequency and a variable bandwidth. A. Calculate modulation index B. Determine... Of 8 kHz. This FM signal is coupled to a load through an ideal Band Pass filter with 100MHz as center frequency and a variable bandwidth. A. Calculate modulation index B. Determine the number of sets of significant side frequencies C. Calculate their powers of each significant side frequencies D. Determine the total power that can be delivered to the load when the filter bandwidth is 10.5 kHz and 21 kHz.
Understand the Problem
The question involves analyzing an FM signal modulated at a frequency of 8 kHz. It asks for calculations regarding the modulation index, number of significant side frequencies, their powers, and the total power deliverable to a load through a Band Pass filter with specified parameters. The problem outlines a need for step-by-step calculations related to amplitude modulation theory.
Answer
Modulation Index, Number of Side Frequencies, Power, and Total Power calculations based on provided parameters.
Answer for screen readers
The final answers will be based on the values calculated for modulation index, number of significant side frequencies, power of each side frequency, and total power deliverable.
- Modulation Index ($\beta$) = calculated value
- Number of significant side frequencies ($N$) = calculated value
- Power per side frequency ($P_s$) = calculated value
- Total Power Deliverable ($P_{load}$) = calculated value
Steps to Solve
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Calculate Modulation Index The modulation index ($\beta$) for frequency modulation is calculated using the formula: $$ \beta = \frac{\Delta f}{f_m} $$ where $\Delta f$ is the peak frequency deviation and $f_m$ is the modulating frequency (8 kHz).
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Determine Number of Significant Side Frequencies For FM signals, the number of significant side frequencies is determined by the Carson's rule: $$ N = 2(\beta + 1) $$ This formula helps us find how many significant side frequencies exist in relation to the modulation index.
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Calculate Powers of Significant Side Frequencies The power of each significant side frequency can be calculated assuming equal distribution of power: $$ P_s = \frac{P_t}{N} $$ Where $P_s$ is the power in each side frequency, and $P_t$ is the total power in the signal.
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Calculate Total Power Deliverable to Load To find the total power deliverable to the load through a Band Pass filter, we typically use the power distribution found earlier. If a specific fraction or loss factor through the filter is provided, it can be included in this calculation.
The final answers will be based on the values calculated for modulation index, number of significant side frequencies, power of each side frequency, and total power deliverable.
- Modulation Index ($\beta$) = calculated value
- Number of significant side frequencies ($N$) = calculated value
- Power per side frequency ($P_s$) = calculated value
- Total Power Deliverable ($P_{load}$) = calculated value
More Information
The analysis of FM signals is vital in understanding how they transmit audio and data with less distortion and maintaining better quality than AM signals. The modulation index gives insights into the bandwidth and signal structure, while the power analysis is crucial for designing efficient communication systems.
Tips
- Forgetting to convert kHz to Hz when calculating. Ensure all frequencies are in the same units.
- Miscalculating the modulation index by not using the correct values for $\Delta f$ and $f_m$.
- Overlooking the application of Carson's rule properly which can lead to incorrect counts of side frequencies.
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