LTI Systems, Convolution and Correlation - Signals and Systems - Biomedical Engineering Previous Year Questions

1

2020 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2020

\(m_1\) and \(m_2\) are the roots of the characteristic equation of a linear second order physical system. Match the nature of the roots with the natural response of the system.

A

P-L, Q-M, R-K

B

P-M, Q-L, R-K

C

P-L, Q-K, R-M

D

P-M, Q-K, R-L

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2

2020 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2020

A second order low pass filter is being constructed by cascading two first order low pass filters with the following transfer functions \(H_1(j\omega) = \frac{1}{1 + j\omega/\omega_1}\), \(H_2(j\omega) = \frac{1}{1 + j\omega/\omega_2}\), where \(\omega_1\) and \(\omega_2\) are the respective 3dB cut off frequencies. The undamped natural frequency \(\omega_c\) of the resulting second order low pass filter is

A

\(\omega_c = \sqrt{\omega_1\omega_2}\)

B

\(\omega_c = \omega_1 + \omega_2\)

C

\(\omega_c = \frac{\omega_1\omega_2}{\omega_1 + \omega_2}\)

D

\(\omega_c = \sqrt{\omega_1^2 + \omega_2^2}\)

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3

2020 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2020

A cell is injected with a current i(t) = u (t) to produce a change in the intracellular membrane voltage v (t). The cell-membrane is modeled as a linear system with impulse response h (t) = A e^−t/τ u(t). The cell membrane voltage output at 5 ms is __________ mV.
Use A = −34 V/s; τ = 3 ms.

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4

2021 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2021

A unit step input is applied to a system with impulse response \(H(s) = \frac{1-s/\omega_z}{1+s/\omega_p}\) at \(t = 0\). The output of the system \(y(t)\) at \(t = 0^+\) is ______.

A

1

B

\(-\omega_z/\omega_p\)

C

\(-\omega_p/\omega_z\)

D

0

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5

2021 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2021

\(x[n]\) is convolved with \(h[n]\) to give \(y[n]\). If \(y[2] = 1\) and \(y[3] = 0\), \(h[0]\) = ______. (Graphs are not uniformly scaled)

A

1.85

B

−2.50

C

−1.90

D

2.38

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6

2021 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2021

The closed-loop characteristic equation of a system is given by
\[ s^4 + 2s^3 + 8s^2 + 8s + 16 = 0 \]
The frequency of oscillations of this closed-loop system at steady state is ______ rad/s.

A

1

B

2

C

4

D

8

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7

2022 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2022

An input \(x(t)\) is applied to a system with a frequency transfer function given by \(H(j\omega)\) as shown below. The magnitude and phase response of the transfer function are shown below. If \(y(t_d) = 0\) for \(x(t) = u(t)\), the time \(t_d (> 0)\) is ______ \(\mu s\).

A

100 \(\ln(2)\)

B

10 \(\ln(2)\)

C

1000 \(\ln(2)\)

D

\(\ln(2)\)

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8

2022 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2022

The block diagram of a two-tap high-pass FIR filter is shown below. The filter transfer function is given by \(H(z) = Y(z)/X(z)\).
If ratio of the maximum to minimum value of \(H(z)\) is 2 and \(|H(z)|_{max} = 1\), the coefficients \(\beta_0\) and \(\beta_1\) are ______ and ______, respectively.

A

0.75, –0.25

B

0.67, 0.33

C

0.60, –0.40

D

–0.64, 0.36

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9

2022 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2022

The block diagrams of an ideal system and a real system with their impulse responses are shown below. An auxiliary path is added to the delayed impulse response in the real system.
For a unit impulse input (\(x(t) = \delta(t)\)) to both systems, gain \(\beta\) is chosen such that \(y(4T)\) is same for both systems. The value of \(\beta\) is ______.

A

\(e^{-3aT} (1 - e^{-2aT})\)

B

\(-e^{-aT} (1 - e^{-3aT})\)

C

\(-e^{-3aT} (1 - e^{-aT})\)

D

\(e^{-2aT} (1 - e^{-2aT})\)

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10

2023 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2023

A causal, discrete time system is described by the difference equation
\[ y[n] = 0.5 y[n-1] + x[n], \text{ for all } n, \]
where \( y[n] \) denotes the output sequence and \( x[n] \) denotes the input sequence. Which of the following statements is/are TRUE?

A

The system has an impulse response described by \(0.5^n u[-n] \) where \(u[n] \) is the unit step sequence.

B

The system is stable in the bounded input, bounded output sense.

C

The system has an infinite number of non-zero samples in its impulse response.

D

The system has a finite number of non-zero samples in its impulse response.

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11

2023 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2023

Two sequences \(x_1[n]\) and \(x_2[n]\) are described as follows:
\(x_1[0] = x_2[0] = 1\)
\(x_1[1] = x_2[2] = 2\)
\(x_1[2] = x_2[1] = 1\)
\(x_1[n] = x_2[n] = 0\) for all \(n < 0\) and \(n > 2\)
If \(x[n]\) is obtained by convolving \(x_1[n]\) with \(x_2[n]\), which of the following equations is/are TRUE?

A

\(x[2] = x[3]\)

B

\(x[1] = 2\)

C

\(x[4] = 3\)

D

\(x[2] = 5\)

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12

2023 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2023

The continuous time signal \(x(t)\) is described by
\(x(t) = \begin{cases} 1, & 0 \le t \le 1 \\ 0, & \text{elsewhere} \end{cases}\)
If \(y(t)\) represents \(x(t)\) convolved with itself, which of the following statements is/are TRUE?

A

\(y(t) = 0\) for all \(t < 0\)

B

\(y(t) = 0\) for all \(t > 1\)

C

\(y(t) = 0\) for all \(t > 3\)

D

\(\int_{0.1}^{0.75} \frac{dy(t)}{dt} dt \neq 0\)

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13

2023 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2023

A system is described by the following differential equation
\( 0.01 \frac{d^2y(t)}{dt^2} + 0.2 \frac{dy(t)}{dt} + y(t) = 6x(t) \),
where time (t) is in seconds. If \( x(t) \) is the unit step input applied at \( t = 0 \) s to this system, the magnitude of the output at \( t = 1 \) s is ______. (Round off the answer to two decimal places.)

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14

2024 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2024

The input \(x(t)\) and the output \(y(t)\) of a linear time invariant system are related as follows: \[y(t) + \frac{dy(t)}{dt} + 0.5\frac{d^2y(t)}{dt^2} = x(t) + 0.1\frac{dx(t)}{dt}\] What is the Laplace transform of the impulse response of the system?

A

\(\frac{0.5s^2 + s + 1}{0.1s + 1}\)

B

\(\frac{0.1s + 1}{0.5s^2 + s + 1}\)

C

\(\frac{0.1s + s^2}{s^2 + s + 0.5}\)

D

\(\frac{s^2 + s + 0.5}{0.1s^2 + s}\)

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15

2024 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2024

If \(x[n] = u[n] - u[n - 5]\), and \(h[n] = \delta[n] - \delta[n - 1]\) and \(y[n] = x[n] * h[n]\), then the value of \(\sum_{n=-\infty}^{\infty} y[n]\) is ______. Give your answer rounded off to the nearest integer.

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16

2025 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2025

Two systems \( U \) and \( V \) are defined as shown below. Which of the following statement(s) is/are CORRECT?

A

System \(U \) is nonlinear; System \(V \) is linear

B

System \(U \) is causal; System \(V \) is noncausal

C

System \(U \) is noncausal; System \(V \) is causal

D

System \(U \) is nonlinear; System \(V \) is nonlinear

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17

2025 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2025

A discrete linear time invariant system has an impulse response function given by \( h[n] = \delta[n] + \frac{1}{2} \delta[n-1] + \frac{1}{3} \delta[n-2] \). For input signal \( x[n] = \delta[n] + \delta[n-1] \), which of the following option(s) is/are CORRECT for the output signal \( y[n] \)?

A

\(y[2] = 11/6 \)

B

\(y[2] = 5/6 \)

C

\(y[k] = 0 \) for all \(k \geq 3 \)

D

\(y[k] = 0 \) for all \(k \geq 4 \)

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18

2026 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2026

The output \(y[n]\) of a digital moving average filter for an input \(x[n]\) is given by

\(y[n] = \frac{1}{4}(x[n] + 2x[n-1] + x[n-2])\)

Which one of the following options is a correct statement about the transfer function of this filter?

A

It has two poles and two zeros

B

It has two poles and no zeros

C

It has no poles and two zeros

D

It has one pole and one zero

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19

2026 · Biomedical Engineering · Signals and Systems · LTI Systems, Convolution and Correlation

Biomedical Engineering (BM) 2026

The figure below shows the impulse response of a linear time-invariant (LTI) system, \(h[n] = [0, 0.6, 0.8, 0.5, 0.3, 0]\). For an input \(x[n] = [1, 0.6, 0, 0, 0, 0]\), the maximum value of its output \(y[n]\) is __________
(Round off to one decimal place)

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