Signals & Systems (EC-402) - Important Questions

Explain and distinguish the classification of signals: continuous-time vs discrete-time, unit step vs unit ramp, periodic vs aperiodic, deterministic vs random, energy vs power, and even vs odd, with examples.

Determine/check whether given systems or impulse responses are linear, time-invariant, causal, static/memoryless, dynamic, and stable.

Define a system and define/describe system properties: linearity (additivity and homogeneity), time-invariance, causality and realizability, stability, static vs dynamic, and even signal.

Define a signal and classify signals as continuous-time vs discrete-time with examples.

Explain the basic operations performed on Continuous-Time and Discrete-Time Signals.

Determine whether the given signals are energy, power, or neither signals, for a single-tone periodic signal $x(t)=\cos t$ or $e^{j(2t+\pi/4)}$ and a decaying exponential $x(n)=(1/3)^n u(n)$.

Determine whether the given sinusoidal continuous-time (and discrete-time) signal(s) are periodic and find the fundamental period.

Define signal and classify the different types of signals

Determine whether the signals are energy or power signals, for $x(t)=t u(t)$ and $x(t)=e^{-at}u(t)$.

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Compare linear vs non-linear systems with examples.

Check linearity and causality for given continuous-time and discrete-time systems.

Explain Direct Form-I, Direct Form-II, Cascade Form and Parallel Form realization structures with block diagrams.

Find the impulse response of a continuous-time causal LTI system described by a linear constant-coefficient differential equation with zero initial conditions.

Find the continuous-time convolution / LTI output $y(t) = x(t) * h(t)$ for given $x(t)$ and $h(t)$ using the convolution integral.

Explain the impulse response of a discrete-time LTI system, its properties, and derive the convolution sum.

Explain the properties of the impulse response representation for LTI systems.

Block diagram representation of linear time-invariant discrete-time systems and their properties.

Define an LTI system, explain how discrete-time LTI systems differ from continuous-time LTI systems, and explain the importance of the impulse response to LTI systems.

Determine the impulse response of a causal LTI system described by a linear constant-coefficient difference equation.

What is convolution? Find the convolution integral of given exponential signals $x(t)$ and $h(t)$.

Discuss the impulse response representation of an LTI system and describe LTI systems by difference equations.

Determine causality and stability from a given impulse response $h(t) = (2 + e^{-3t}) u(t)$ and find the step response of the system.

Show the Direct Form-I realization of the transfer function $H(s) = \frac{s+5}{s^{2}+2s+4}$.

State and prove/explain the properties of Z-transform including time shifting, convolution, initial value theorem, differentiation and integration

Explain the properties of the Z-Transform, including unilateral Z-Transform, and describe methods for obtaining the Inverse Z-Transform.

Define the Z-Transform and explain the Region of Convergence (ROC) for finite-duration and infinite-duration sequences, including properties of the ROC.

Define the Region of Convergence (ROC) of the z-transform and explain its properties.

Explain the relationship between the Discrete-Time Fourier Transform (DTFT) and the Z-Transform.

Explain the Unilateral Z-Transform. State its advantages in the analysis of causal systems.

Using Z-transform properties, find the Z-transform and ROC of a given signal

Derive the Z-transform of the accumulation/summation signal $y(n)=\sum_{k=-\infty}^{n}x(k)$ in terms of $X(z)$

Find the inverse Z-transform for an interior ROC $|z| < r$ (anti-causal / left-sided sequence).

Prove that $x(n) = a^n u(n)$ and $x(n) = -a^n u(-n-1)$ have the same $X(z)$ but different ROCs, and plot their ROCs.

Define ROC and its properties and explain the inverse z-transform.

Find the inverse Z-transform for an exterior ROC $|z| > r$ (causal sequence), e.g., via partial fraction expansion.

Explain the properties and applications of the Discrete-Time Fourier Series (DTFS) / Fourier series.

What is the (continuous-time) Fourier transform? Explain its properties.

Explain the Fourier Transform of Discrete-Time Aperiodic Signals. State its convergence conditions.

Explain the Fourier Transform of Periodic Discrete-Time Signals. State its practical applications.

Discuss the DTFT and its properties.

Find the DTFT of $x(n)=\alpha^n u(n),\, |\alpha|<1$ (and $\delta(n)$) and draw its spectra.

Find the Fourier transform of $e^{-3t}\sin w_0 t\, u(t)$.

Find the Fourier series coefficients of the given discrete-time signal $x(n) = 1 + \sin\left(\frac{2\pi}{5}n\right) + 3\cos\left(\frac{2\pi}{5}n\right) + \cos\left(\frac{4\pi}{5}n + \frac{\pi}{2}\right)$.

Discuss convergence of discrete time Fourier transform and write applications of DTFT.

Obtain the DTFT of $x[n] = a^{n} \sin(w_{o}n) u(n)$ and $x[n] = n3^{-n} u(-n)$.

If $X(e^{j\omega})$ is the DTFT of $x[n]$, find the DTFT of $(n-1)^2 x[n]$ in terms of $X(e^{j\omega})$.

Define the state transition matrix and explain its role / importance in dynamic system analysis.

State and prove the sampling theorem for band-limited signals, including effect of under-sampling

Explain state-space representation of Multi-Input Multi-Output (MIMO) systems, e.g. 3-input 2-output system

Discuss sampling theorem and its implications along with reconstruction of a signal from its samples

Explain state-space analysis and state-variable / matrix representation of dynamic systems

Numerical to find minimum sampling rate and maximum sampling interval for a given bandpass spectral range

Write properties and role of the state transition matrix and describe methods to determine it.

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State and explain the sampling theorem with equations, illustration and its implications

Explain various methods of evaluation / determination of the state transition matrix with suitable example.

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Find the exponential Fourier series representation of the signal $x(t) = \cos^2 t$

Compute the Fourier transform of the signal x(t) applying differentiation-in-time property of the Fourier transform.

Find the exponential Fourier series and plot the magnitude and phase spectra of the following triangular wave form.

Use the Fourier series analysis to calculate the coefficients $a_{k}$ for the continuous time periodic signal. $$x(t)=\begin{cases} 1.5 & 0 \le t < 1 \\ -1.5 & 1 \le t < 2 \end{cases}$$

Find the Fourier Transform of Rectangular pulse. Sketch the signal and Fourier transform.

Compute the Fourier transform of the following signals.

Demonstrate time shifting property and time-scaling property of Fourier transform.

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State and prove any three properties of Fourier Transform

Using the properties of Fourier Transform find the $X(j\omega)$ and $G(j\omega)$