Given a composite signal with a bandwidth of 200 kHz, what are the extreme frequencies if the middle frequency is 140 kHz?
Understand the Problem
The question is asking us to find the extreme frequencies of a composite signal given its bandwidth and middle frequency. To do this, we will use the formula for calculating the lower and upper frequencies using the middle frequency and half of the bandwidth.
Answer
Lower Frequency: $f_{low} = f_m - \frac{BW}{2}$; Upper Frequency: $f_{high} = f_m + \frac{BW}{2}$.
Answer for screen readers
The extreme frequencies are:
- Lower Frequency: $f_{low} = f_m - \frac{BW}{2}$
- Upper Frequency: $f_{high} = f_m + \frac{BW}{2}$
Steps to Solve
- Identify given values
From the problem, identify and define the middle frequency ($f_m$) and the bandwidth ($BW$). Let's denote:
- $f_m$ = middle frequency
- $BW$ = bandwidth
- Calculate half of the bandwidth
We need to compute half of the bandwidth using the formula: $$ \text{Half Bandwidth} = \frac{BW}{2} $$
- Calculate the lower frequency
The lower frequency ($f_{low}$) can be calculated using this formula: $$ f_{low} = f_m - \frac{BW}{2} $$
- Calculate the upper frequency
The upper frequency ($f_{high}$) is determined as follows: $$ f_{high} = f_m + \frac{BW}{2} $$
- Summarize results
Now, we summarize the calculated lower and upper frequencies.
The extreme frequencies are:
- Lower Frequency: $f_{low} = f_m - \frac{BW}{2}$
- Upper Frequency: $f_{high} = f_m + \frac{BW}{2}$
More Information
The extreme frequencies demarcate the range of the frequencies present in a composite signal. The midpoint or center frequency indicates where the signal's energy is concentrated, while the bandwidth represents the total spread of frequencies.
Tips
- Miscalculating half of the bandwidth. It's important to divide the bandwidth by 2 before using it in the calculations for lower and upper frequencies.
- Confusing the signs in the formulas. Remember that the lower frequency is calculated by subtracting half of the bandwidth, while the upper frequency adds it.
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