To express the magnitude response in terms of frequency in Hertz (\( f \)), we need to convert the angular frequency (\( \omega \)) to frequency (\( f \)).

The relationship between angular frequency and frequency is given by:

\[ \omega = 2\pi f \]

Given the magnitude response in terms of angular frequency:

\[ |H(j\omega)| = \frac{\omega}{\sqrt{\omega^2 + \left(\frac{1}{RC}\right)^2}} \]

Substitute \( \omega = 2\pi f \):

\[ |H(j2\pi f)| = \frac{2\pi f}{\sqrt{(2\pi f)^2 + \left(\frac{1}{RC}\right)^2}} \]

Simplify the expression:

\[ |H(j2\pi f)| = \frac{2\pi f}{\sqrt{4\pi^2 f^2 + \left(\frac{1}{RC}\right)^2}} \]

To make the equation clearer, we can factor out \( 2\pi \) inside the square root in the denominator:

\[ |H(j2\pi f)| = \frac{2\pi f}{\sqrt{4\pi^2 \left( f^2 + \left(\frac{1}{4\pi^2 R^2 C^2}\right) \right)}} \]

\[ |H(j2\pi f)| = \frac{2\pi f}{2\pi \sqrt{ f^2 + \left(\frac{1}{4\pi^2 R^2 C^2}\right)}} \]

The \( 2\pi \) terms cancel out:

\[ |H(j2\pi f)| = \frac{f}{\sqrt{ f^2 + \left(\frac{1}{4\pi^2 R^2 C^2}\right)}} \]

This is the magnitude response of the system in terms of frequency \( f \) in Hertz (Hz).

Would you like more details or have any questions about this process? Here are 5 related questions you might consider:

1. How do you derive the phase response of the given transfer function in terms of frequency \( f \) in Hertz?
2. What is the significance of the cut-off frequency in the magnitude response in terms of \( f \)?
3. How do you determine the bandwidth of the system from the magnitude response in Hertz?
4. Can you explain the behavior of the magnitude response at low and high frequencies in terms of \( f \)?
5. How would the magnitude response change if the resistance \( R \) or capacitance \( C \) is altered, specifically in terms of frequency \( f \)?

**Tip:** Converting angular frequency to frequency in Hertz is crucial when dealing with practical applications since most measurements and specifications in signal processing and communication systems are given in Hertz.

Understand how to convert angular frequency to frequency in Hertz for solving magnitude response in signal processing and communication systems. Learn the key formulas and relationships essential for this conversion. Suitable for advanced undergraduate level and beyond.

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