Determining the Highest Carrier Frequency for a New AM Radio Station Without Interference

This problem involves calculating the maximum permissible carrier frequency for a new AM radio station based on existing broadcasting regulations and the properties of the currently licensed station. Key factors include the modulation frequency, carrier frequency, and the bandwidth required to prevent adjacent channel interference.

Understanding AM Radio Bandwidth and Interference

In Amplitude Modulation (AM), a signal's bandwidth is primarily determined by the range of frequencies it modulates onto the carrier wave. Specifically, the bandwidth ($BW$) is twice the highest modulation frequency ($f_m$).

• Bandwidth Formula: $BW = 2 \times f_m$

Broadcasting regulations ensure that different stations do not interfere with each other. This means the frequency spectrum occupied by one station (its bandwidth) must not overlap with the spectrum occupied by another. For adjacent channels, this typically means the upper edge of one station's bandwidth must be below or exactly at the lower edge of the next station's bandwidth.

Calculating the Existing Station's Carrier Frequency

We are given that the existing AM station transmits with a modulation frequency ($f_m$) of $250 \text{ kHz}$. This frequency represents $10\%$ of its carrier wave frequency ($f_{c,existing}$). We can set up an equation to find the existing carrier frequency:

$f_m = 0.10 \times f_{c,existing}$

Substituting the given modulation frequency:

$250 \text{ kHz} = 0.10 \times f_{c,existing}$

To find $f_{c,existing}$, we rearrange the equation:

$f_{c,existing} = \frac{250 \text{ kHz}}{0.10}$

$f_{c,existing} = 2500 \text{ kHz}$

So, the carrier frequency of the existing AM station is $2500 \text{ kHz}$.

Calculating the Bandwidth of the Existing Station

Using the bandwidth formula ($BW = 2 \times f_m$) and the given modulation frequency:

$BW_{existing} = 2 \times 250 \text{ kHz}$

$BW_{existing} = 500 \text{ kHz}$

This bandwidth extends $f_m$ below and $f_m$ above the carrier frequency. Therefore, the frequency range occupied by the existing station is:

• Lower frequency limit: $f_{c,existing} - f_m = 2500 \text{ kHz} - 250 \text{ kHz} = 2250 \text{ kHz}$
• Upper frequency limit: $f_{c,existing} + f_m = 2500 \text{ kHz} + 250 \text{ kHz} = 2750 \text{ kHz}$

The existing station occupies the spectrum from $2250 \text{ kHz}$ to $2750 \text{ kHz}$.

Applying Non-Interference Regulations for the New Station

The regulations state that the new station's bandwidth must not overlap with the existing station's bandwidth. We are also told that the new station requires a *lower* carrier frequency ($f_{c,new} < f_{c,existing}$).

To find the *highest possible* carrier frequency ($f_{c,new}$) for the new station without causing interference, the upper limit of the new station's frequency range must be less than or equal to the lower limit of the existing station's frequency range.

Upper limit of new station: $f_{c,new} + f_{m,new}$

Lower limit of existing station: $2250 \text{ kHz}$

The condition for non-interference is:

$f_{c,new} + f_{m,new} \le 2250 \text{ kHz}$

The problem doesn't explicitly state the modulation frequency for the new station. However, the context implies similar bandwidth requirements are expected, or a standard channel width is used. Assuming the new station also requires a bandwidth related to a $250 \text{ kHz}$ modulation frequency (making its total required bandwidth $500 \text{ kHz}$, meaning $f_{m,new} = 250 \text{ kHz}$), we can proceed:

$f_{c,new} + 250 \text{ kHz} \le 2250 \text{ kHz}$

Calculating the New Station's Highest Carrier Frequency

To find the highest possible $f_{c,new}$, we solve the inequality:

$f_{c,new} \le 2250 \text{ kHz} - 250 \text{ kHz}$

$f_{c,new} \le 2000 \text{ kHz}$

Therefore, the highest possible carrier frequency that can be allotted to the new station without causing interference is $2000 \text{ kHz}$.

Summary of Calculation Steps

1. Calculate the existing station's carrier frequency using the given modulation frequency and its percentage of the carrier: $f_{c,existing} = 250 \text{ kHz} / 0.10 = 2500 \text{ kHz}$.
2. Determine the bandwidth of the existing station: $BW_{existing} = 2 \times 250 \text{ kHz} = 500 \text{ kHz}$.
3. Identify the frequency range of the existing station: $2500 \text{ kHz} \pm 250 \text{ kHz}$, which is $2250 \text{ kHz}$ to $2750 \text{ kHz}$.
4. Apply the non-interference rule: The new station's highest frequency ($f_{c,new} + f_{m,new}$) must not exceed the existing station's lowest frequency ($2250 \text{ kHz}$). Assume $f_{m,new} = 250 \text{ kHz}$.
5. Calculate the maximum new carrier frequency: $f_{c,new} = 2250 \text{ kHz} - 250 \text{ kHz} = 2000 \text{ kHz}$.

This result aligns with the requirement that the new station's carrier frequency must be lower than the existing one ($2000 \text{ kHz} < 2500 \text{ kHz}$) and ensures no overlap in bandwidth.