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

  1. 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
  1. Calculate half of the bandwidth

We need to compute half of the bandwidth using the formula: $$ \text{Half Bandwidth} = \frac{BW}{2} $$

  1. Calculate the lower frequency

The lower frequency ($f_{low}$) can be calculated using this formula: $$ f_{low} = f_m - \frac{BW}{2} $$

  1. Calculate the upper frequency

The upper frequency ($f_{high}$) is determined as follows: $$ f_{high} = f_m + \frac{BW}{2} $$

  1. 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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