GATE Biomedical Engineering Signals and Systems Syllabus

Signals and Systems syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Signals and Systems in GATE Biomedical Engineering, not a summary of it.

1. 1

   Continuous and Discrete Signal and Systems

   8 topics

   • Periodic, aperiodic and impulse signals
   • Sampling theorem
   • Laplace and Fourier transforms
   • Impulse response of systems
   • Transfer function
   • Frequency response of first and second order linear time invariant systems
   • Convolution
   • Correlation
2. 2

   Discrete time systems

   5 topics

   • Impulse response
   • Frequency response
   • DFT
   • Z-transform
   • Basics of IIR and FIR filter

Signals and Systems flashcards for GATE Biomedical Engineering

23 of 51 cards from the Signals and Systems deck — real questions with worked answers.

1. What defines a periodic signal in continuous time, and what is its fundamental period?

   A signal $x(t)$ is periodic if $x(t) = x(t+T)$ for all $t$. The fundamental period $T_0$ is the smallest positive value of $T$ for which this holds; the fundamental frequency is $f_0 = \frac{1}{T_0}$.

2. What is the condition for a discrete-time signal $x[n]$ to be periodic?

   $x[n] = x[n+N]$ for some positive integer $N$. A sinusoid $x[n]=\cos(\Omega_0 n)$ is periodic only if $\frac{\Omega_0}{2\pi}$ is a rational number.

3. Define the continuous-time unit impulse (Dirac delta) function by its two key properties.

   The impulse $\delta(t)$ satisfies $\delta(t)=0$ for $t\neq 0$ and $\int_{-\infty}^{\infty}\delta(t)\,dt = 1$. Its sifting property is $\int_{-\infty}^{\infty} x(t)\delta(t-t_0)\,dt = x(t_0)$.

4. What is the discrete-time unit impulse $\delta[n]$?

   $\delta[n] = 1$ for $n=0$ and $\delta[n]=0$ otherwise. Sifting property: $\sum_{k=-\infty}^{\infty} x[k]\delta[n-k] = x[n]$.

5. What is the relationship between the unit step $u(t)$ and the unit impulse $\delta(t)$?

   The impulse is the derivative of the step and the step is the integral of the impulse: $\delta(t) = \frac{d\,u(t)}{dt}$ and $u(t) = \int_{-\infty}^{t}\delta(\tau)\,d\tau$.

6. Distinguish an energy signal from a power signal.

   An energy signal has finite, nonzero total energy $E=\int_{-\infty}^{\infty}|x(t)|^{2}\,dt$ and zero average power. A power signal has finite, nonzero average power $P=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^{2}\,dt$ and infinite energy. Periodic signals are power signals.

7. State the Nyquist–Shannon sampling theorem.

   A band-limited signal with maximum frequency $f_m$ can be perfectly reconstructed from its samples if the sampling frequency satisfies $f_s > 2 f_m$. The minimum rate $2 f_m$ is the Nyquist rate and $\frac{f_s}{2}$ is the Nyquist frequency.

8. What is aliasing and when does it occur?

   Aliasing is the overlapping of spectral replicas that causes high-frequency components to be misrepresented as lower frequencies. It occurs when $f_s < 2 f_m$ (undersampling). It is prevented by an anti-aliasing low-pass filter before sampling.

9. Define the bilateral (two-sided) Laplace transform.

   $X(s) = \int_{-\infty}^{\infty} x(t) e^{-st}\,dt$, where $s = \sigma + j\omega$ is the complex frequency. The unilateral version integrates from $0^{-}$ to $\infty$.

10. What is the Region of Convergence (ROC) of a Laplace transform and why does it matter?

    The ROC is the set of values of $s$ for which $X(s)$ converges. It is needed to uniquely determine $x(t)$ from $X(s)$ and to assess stability: an LTI system is stable iff its ROC includes the $j\omega$-axis.

11. Give the Laplace transforms of $\delta(t)$, $u(t)$, and $e^{-at}u(t)$.

    $\mathcal{L}\{\delta(t)\}=1$ (all $s$); $\mathcal{L}\{u(t)\}=\frac{1}{s}$, $\mathrm{Re}(s)>0$; $\mathcal{L}\{e^{-at}u(t)\}=\frac{1}{s+a}$, $\mathrm{Re}(s)>-a$.

12. State the time-differentiation and time-integration properties of the Laplace transform.

    $\mathcal{L}\{\frac{dx}{dt}\} = sX(s) - x(0^{-})$ and $\mathcal{L}\{\int_{0}^{t} x(\tau)d\tau\} = \frac{X(s)}{s}$.

13. Define the continuous-time Fourier transform and its inverse.

    $X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t}\,dt$ and $x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(\omega) e^{j\omega t}\,d\omega$.

14. State the Dirichlet conditions for the existence of the Fourier transform.

    The signal must be absolutely integrable ($\int_{-\infty}^{\infty}|x(t)|\,dt < \infty$), have a finite number of maxima/minima in any finite interval, and a finite number of finite discontinuities in any finite interval.

15. How is the Fourier transform obtained from the Laplace transform?

    By evaluating the Laplace transform on the imaginary axis, $X(\omega) = X(s)\big|_{s=j\omega}$, provided the ROC of $X(s)$ includes the $j\omega$-axis.

16. State Parseval's theorem for the continuous-time Fourier transform.

    $\int_{-\infty}^{\infty}|x(t)|^{2}\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}|X(\omega)|^{2}\,d\omega$. Energy is conserved between time and frequency domains.

17. What is the duality property of the Fourier transform?

    If $x(t) \leftrightarrow X(\omega)$, then $X(t) \leftrightarrow 2\pi\, x(-\omega)$. The roles of time and frequency can be interchanged with a scaling and reflection.

18. State the time-shift and frequency-shift (modulation) properties of the Fourier transform.

    Time shift: $x(t-t_0) \leftrightarrow X(\omega) e^{-j\omega t_0}$. Frequency shift: $x(t) e^{j\omega_0 t} \leftrightarrow X(\omega-\omega_0)$.

19. Define the impulse response of an LTI system.

    The impulse response $h(t)$ (or $h[n]$) is the output of the system when the input is a unit impulse $\delta(t)$ (or $\delta[n]$) with zero initial conditions. It completely characterizes an LTI system.

20. How is the output of an LTI system computed from its impulse response?

    By convolution of the input with the impulse response: $y(t) = x(t) * h(t)$ in continuous time, or $y[n] = x[n] * h[n] = \sum_{k} x[k] h[n-k]$ in discrete time.

21. State the BIBO stability condition for an LTI system in terms of its impulse response.

    A continuous-time LTI system is BIBO stable iff $\int_{-\infty}^{\infty}|h(t)|\,dt < \infty$. A discrete-time system is stable iff $\sum_{n=-\infty}^{\infty}|h[n]| < \infty$ (absolutely summable).

22. What condition on the impulse response makes an LTI system causal?

    The system is causal iff $h(t)=0$ for $t<0$ (continuous time) or $h[n]=0$ for $n<0$ (discrete time); the output depends only on present and past inputs.

23. Define the transfer function of an LTI system.

    The transfer function $H(s) = \frac{Y(s)}{X(s)}$ is the Laplace transform of the impulse response, equal to the ratio of output to input transforms with zero initial conditions. For discrete systems, $H(z)=\frac{Y(z)}{X(z)}$.

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