# Signals and Systems Questions and Answers – Ideal LPF, HPF, BPF and BSF Characteristics

This set of Tricky Signals & Systems Questions and Answers focuses on “Ideal LPF, HPF, BPF and BSF Characteristics”.

1. In the circuit given below, C = C1 = C2. The gain of the multiple-feedback band-pass filter is ___________

a) $$A_0 = \frac{R_2}{R_1}$$
b) $$A_0 = \frac{R_1}{R_2}$$
c) $$A_0 = \frac{R_2}{2R_1}$$
d) $$A_0 = \frac{R_1}{2R_2}$$

Explanation: The total output C = CINPUT + COUTPUT that is the gain capacitor.
∴ The total Resistance is equal to the Resistance input and Resistance output.
Again, the total resistance gain = $$\frac{R_1 R_2}{R_1+ R_2}$$
Hence, the gain = $$A_0 = \frac{R_2}{2R_1}$$.

2. Two network functions are given below.
H = $$\frac{1}{s^2 + \sqrt{2} s + 1}$$, H2 = $$\frac{1}{s^3+2s^2+2s+1}$$
The frequency response indicates that the filter is ___________
a) Low-pass
b) Band-pass
c) High-pass
d) Band reject

Explanation: | H1 |2 = $$\frac{1}{[(jω)^2 + \sqrt{2} jω + 1] [(jω)^2 + \sqrt{2} jω + 1]}$$
Similarly, | H2 |2 = $$\frac{1}{1+ ω^6}$$
Therefore at ω = 0, 1 and∞, we have | H |2 = 1, $$\frac{1}{2}$$ and 0 respectively.
Hence, the filter is a Low-pass filter.

3. A particular band-pass function has a network function as H(s) = $$\frac{3s}{s^2+4s+3}$$ then, its quality factor Q is ___________
a) $$\frac{3}{4}$$
b) $$\frac{2}{\sqrt{3}}$$
c) $$\frac{\sqrt{3}}{2}$$
d) $$\frac{\sqrt{3}}{4}$$

Explanation: H(s) = $$\frac{Ks}{s^2+as+b}$$
Then, quality factor is given as $$\frac{\sqrt{b}}{a}$$
Here, b = 3, a = 4
∴ Q = $$\frac{\sqrt{3}}{4}$$.

4. The filter which passes all frequencies above fc by attenuating significantly, all frequencies below fc is _______________
a) Low-pass
b) High-pass
c) Band-pass
d) Band-stop

Explanation: A high-pass filter is one which passes all frequencies above fc by attenuating significantly, all frequencies below fc.

5. For the circuit given below, the cut-off frequency of the filter is ________________

a) 5283 kHz
b) 5283 Hz
c) 2653.1 kHz
d) 2653.1 Hz

Explanation: We know that, F = $$\frac{1}{2π×R×C}$$
Where, R = 1200 Ω, C1 = 0.05 × 10-6 F
∴ F = $$\frac{1}{2π×1200×C}$$
= $$\frac{1}{2π×60×10^{-6}}$$
= $$\frac{10^6}{2π×60}$$ = 2653.1 Hz.
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6. For a Band Pass Filter, the slope of the filter is given as 40dB/decade. The order of the Band Pass Filter is __________
a) 2
b) 3
c) 4
d) 6

Explanation: The Bode plot is a logarithmic plot which helps in fitting a large scale of values into a small scale by the application of logarithm. Plotting the slope 40dB/decade on the Bode plot, we get n = 4. Hence the order of the Band Pass Filter is 4.

7. The circuit given below represents which type of filter circuit?

a) Low-pass Filter
b) High-pass Filter
c) Band-pass Filter
d) Band-stop Filter

Explanation: We know that, the position of Resistance (R) and Capacitance (C) determines whether it is Low-pass Filter or High-Pass Filter. If R is connected directly to source and the capacitor connected in parallel to it, then it is a Low-pass Filter and if the position of R and C are inter change then high pass filter is formed.
Here since R and C are in series and also R is not connected directly to the power source, hence the filter is a High-pass Filter.

8. The circuit given below represents which type of filter circuit?

a) Low-pass Filter
b) High-pass Filter
c) Band-pass Filter
d) Band-stop Filter

Explanation: The given circuit is a first order Band Pass Filter. Also, the Roll-off of the filter depends upon the order of the filter. For a first order it is 20dB/decade, for second order it is 40dB/decade, and so on.

9. For the circuit given below, the Roll-off value of the filter is _____________

Explanation: The given filter is a first order Band Pass Filter. Also, the Roll-off of the filter is depends upon the order of the filter. For a first order it is 20dB/decade, for second order it is 40dB/decade, and so on. Therefore, the Roll-off of the filter = 20 dB/decade. Roll of first order low pass Butterworth filter is 20dB/decade. Now here two stages of second order Low-pass Butterworth filter are cascaded.
∴ Roll-off = 20*4 = 80 dB/decade.

10. The circuit given below, represents which filter?

a) Low-pass
b) High-pass
c) Band-pass
d) Band-stop

Explanation: From the given circuit, we can infer that Roll off of the Filter circuit is 80dB/decade. This Roll-off value is obtained as second order high pass filter followed by another 2nd order HPF results in an HPF.
Therefore the circuit represents a High-pass Filter.

11. In the circuit given below, the Roll-off of the filter is _______________

Explanation: The given filter is a first order Band Pass Filter. Also, the Roll-off of the filter depends upon the order of the filter. For a first order it is 20dB/decade, for second order it is 40dB/decade, and so on.
Therefore, the Roll-off of the filter = 20 dB/decade.

12. In which of the filter circuits given below, will the bandwidth be equal to the critical frequency?
a) Low-pass
b) High-pass
c) Band-pass
d) Band-stop

Explanation: Bandwidth can be calculated by considering,
Largest positive value – Smallest Positive Value
Here, in case of the Low-pass filter only, the largest positive value will of course be the critical frequency, beyond which frequencies have to be blocked. Hence, the bandwidth in a Low-pass filter equals the critical frequency.

13. For the circuit given below, the cut-off frequency of the filter is ________________

a) 3225.8 Hz
b) 7226 Hz
c) 3225.8 kHz
d) 7226 kHz

Explanation: We know that, F = $$\frac{1}{2π×(R_1×R_2)×(C_1×C_2)}$$
Where, R1 = 4700 Ω, R2 = 4700 Ω, C1 = 0.047×10-6 F, C2 = 0.047 × 10-6 F
∴ F = $$\frac{1}{2π×(4700×4700)×(C_1×C_2)}$$
= $$\frac{1}{2π×48796.81×10^{-12}}$$
= $$\frac{10^6}{2π×0.04879681}$$
= $$\frac{10^6}{0.30654156042}$$ = 3225.8 kHz.

14. For providing a Roll-off greater than 20dB/decade/pole, filters with which characteristics are useful?
a) Butterworth
b) Chebyshev
c) Bessel
d) Butterworth & Bessel

Explanation: Roll off is a term commonly refers to the steepness of the transmission function wrt to the frequency.
For a Chebyshev filter, the Roll-off value greater than 20. This characteristic feature is useful when a rapid roll-off is required because it provides a Roll-off rate is more than 20.
On the other hand, both Butterworth and Bessel have the Roll-off rate less than or equal to 20 dB/decade/pole.

15. A Low-pass filter circuit has a cut-off frequency of 1.23 kHz. The bandwidth of the filter is ______________
a) 2.46 kHz
b) 1.23 kHz
c) 0.615 kHz
d) 3.69 kHz

Explanation: The bandwidth is defined as the highest cut-off frequency to the lowest cut-off frequency. Here the lowest cut-off frequency is Zero.
For a Low-pass filter, Cut-off Frequency = Bandwidth of the filter
∴ Bandwidth = 1.23 kHz.

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