Question

The system under consideration is an RC low-pass filter with R = 1 kΩ and C = 1 µF. Let H (f) denotes the frequency response of the RC, low-pass filter. Let f1 be the highest frequency, such that 0≤|f|≤f1,  \(\frac{|H(f1)|}{H(0)}\)≥0.95 Then f1 is ___________

(a) 327.8

(b) 163.9

(c) 52.2

(d) 104.4

I got this question in examination.

My question is taken from Discrete Fourier Transform in section Fourier Transform of Signals and Systems

Answer 1

The correct choice is (c) 52.2

The explanation is: H (ω) = \(\frac{\frac{1}{jωC}}{R+(\frac{1}{jωC})} = \frac{1}{1+jωRC}\)

H (f) = \(\frac{1}{1+j2πfRC}\)

|H (f)| = \(\frac{1}{\sqrt{1+4π^2 f_1^2 R^2 C^2}}\)

H (0) = 1

Given that \(\frac{|H(f1)|}{H(0)}\)≥0.95

Or, 1 + 4π^2 f1^2 R^2 C^2 ≤ 1.108

Simplifying, f1  ≤  \(\frac{0.329}{2πRC}\)

∴f1 ≤ 52.2 Hz.