GATE E&C Engineering Networks, Signals and Systems Syllabus

Write the time constant and natural response form of a source-free RL circuit.

Time constant $\tau = \dfrac{L}{R}$. The current decays as $i(t) = i(0)\,e^{-t/\tau} = i(0)\,e^{-Rt/L}$.

Write the time constant and natural response form of a source-free RC circuit.

Time constant $\tau = RC$. The capacitor voltage decays as $v(t) = v(0)\,e^{-t/\tau} = v(0)\,e^{-t/RC}$.

For a series RLC circuit, define the resonant frequency $\omega_{0}$ and the quality factor $Q$.

$\omega_{0} = \dfrac{1}{\sqrt{LC}}$ and $Q = \dfrac{\omega_{0}L}{R} = \dfrac{1}{R}\sqrt{\dfrac{L}{C}}$.

What is the damping condition for an RLC circuit in terms of damping ratio $\zeta$ (or comparison of damping coefficient $\alpha$ and $\omega_{0}$)?

Overdamped: $\zeta>1$ ($\alpha>\omega_{0}$); critically damped: $\zeta=1$ ($\alpha=\omega_{0}$); underdamped: $\zeta<1$ ($\alpha<\omega_{0}$). For series RLC, $\alpha = \dfrac{R}{2L}$.

At resonance in a series RLC circuit, what is the impedance and the phase between source voltage and current?

Impedance is purely resistive and minimum, $Z = R$, since $X_{L}=X_{C}$. The current is maximum and in phase with the source voltage (power factor unity).

State the continuity conditions for inductor current and capacitor voltage in a circuit.

Inductor current cannot change instantaneously: $i_{L}(0^{+}) = i_{L}(0^{-})$. Capacitor voltage cannot change instantaneously: $v_{C}(0^{+}) = v_{C}(0^{-})$ (assuming finite voltages/currents).

Give the s-domain (Laplace) impedances of R, L, and C including initial conditions ignored.

$Z_{R}(s) = R$, $\quad Z_{L}(s) = sL$, $\quad Z_{C}(s) = \dfrac{1}{sC}$. Initial conditions add a series voltage source $Li_{L}(0^{-})$ for L and a series source $\dfrac{v_{C}(0^{-})}{s}$ for C.

Why is the Laplace transform useful for solving network differential equations?

It converts linear constant-coefficient differential equations into algebraic equations in $s$, automatically incorporates initial conditions, and lets the solution be obtained by algebra plus inverse transform (partial fractions).

Give the Laplace transform of the unit step $u(t)$, unit impulse $\delta(t)$, and $e^{-at}u(t)$.

$\mathcal{L}\{u(t)\} = \dfrac{1}{s}$, $\quad \mathcal{L}\{\delta(t)\} = 1$, $\quad \mathcal{L}\{e^{-at}u(t)\} = \dfrac{1}{s+a}$.

Write the wye-to-delta ($Y\to\Delta$) transformation formulas for resistances $R_{1},R_{2},R_{3}$ (star) to $R_{a},R_{b},R_{c}$ (delta).

Each delta resistance equals the sum of pairwise products of the star resistances divided by the opposite star resistance, e.g. $R_{a} = \dfrac{R_{1}R_{2}+R_{2}R_{3}+R_{3}R_{1}}{R_{1}}$ (and similarly using $R_{2}$, $R_{3}$ in the denominators).

Write the delta-to-wye ($\Delta\to Y$) transformation for delta resistances $R_{a},R_{b},R_{c}$ to star resistances.

Each star resistance equals the product of the two adjacent delta resistances divided by the sum of all three: $R_{1} = \dfrac{R_{b}R_{c}}{R_{a}+R_{b}+R_{c}}$, and similarly for $R_{2}$ and $R_{3}$.

For a balanced network with all delta resistances equal to $R_{\Delta}$, what is the equivalent wye resistance $R_{Y}$?

$R_{Y} = \dfrac{R_{\Delta}}{3}$, or equivalently $R_{\Delta} = 3R_{Y}$.

Write the trigonometric (real) Fourier series of a periodic signal $x(t)$ with period $T_{0}$ and fundamental frequency $\omega_{0} = 2\pi/T_{0}$.

$x(t) = a_{0} + \sum_{n=1}^{\infty}\left[a_{n}\cos(n\omega_{0}t) + b_{n}\sin(n\omega_{0}t)\right]$, with $a_{n} = \dfrac{2}{T_{0}}\int_{T_{0}} x(t)\cos(n\omega_{0}t)\,dt$ and similarly for $b_{n}$ with $\sin$.

Write the exponential (complex) Fourier series and its coefficient formula.

$x(t) = \sum_{n=-\infty}^{\infty} c_{n}e^{jn\omega_{0}t}$, where $c_{n} = \dfrac{1}{T_{0}}\int_{T_{0}} x(t)e^{-jn\omega_{0}t}\,dt$.

What symmetry property does the Fourier series have for an even function and for an odd function?

An even function has only cosine terms (all $b_{n}=0$); an odd function has only sine terms (all $a_{n}=0$, and $a_{0}=0$). Half-wave symmetry eliminates all even harmonics.

Write the Fourier transform and inverse Fourier transform pair.

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

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

$\int_{-\infty}^{\infty}|x(t)|^{2}\,dt = \dfrac{1}{2\pi}\int_{-\infty}^{\infty}|X(j\omega)|^{2}\,d\omega$. Total signal energy is preserved between time and frequency domains.

What is the Fourier transform property for time shifting, $x(t-t_{0})$?

$x(t-t_{0}) \longleftrightarrow X(j\omega)e^{-j\omega t_{0}}$: a time shift introduces a linear phase shift but does not change the magnitude spectrum.

Define the Discrete-Time Fourier Transform (DTFT) of a sequence $x[n]$.

$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n]e^{-j\omega n}$. The DTFT is continuous and periodic in $\omega$ with period $2\pi$.

Write the inverse DTFT synthesis equation.

$x[n] = \dfrac{1}{2\pi}\int_{-\pi}^{\pi} X(e^{j\omega})e^{j\omega n}\,d\omega$, integrating over any $2\pi$ interval.

Define the $N$-point Discrete Fourier Transform (DFT) of $x[n]$.

$X[k] = \sum_{n=0}^{N-1} x[n]e^{-j2\pi kn/N}$, for $k = 0,1,\dots,N-1$. It samples one period of the DTFT at $N$ equally spaced frequencies.

Write the inverse DFT (IDFT).

$x[n] = \dfrac{1}{N}\sum_{k=0}^{N-1} X[k]e^{j2\pi kn/N}$, for $n = 0,1,\dots,N-1$.

How does the DFT relate to the DTFT of a finite-length sequence?

The $N$-point DFT equals samples of the DTFT taken at frequencies $\omega_{k} = \dfrac{2\pi k}{N}$: $X[k] = X(e^{j\omega})\big|_{\omega = 2\pi k/N}$.

What kind of convolution does multiplication of two DFTs correspond to in the time domain?

Multiplication of DFTs corresponds to circular (periodic) convolution of the length-$N$ sequences, not linear convolution. Linear convolution requires zero-padding to length $\geq N_{1}+N_{2}-1$.

Define the bilateral Z-transform of $x[n]$ and its relationship to the DTFT.

$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$. The DTFT is the Z-transform evaluated on the unit circle, $z = e^{j\omega}$, provided the unit circle lies in the region of convergence (ROC).

Give the Z-transform and ROC of $x[n] = a^{n}u[n]$ (right-sided).

$X(z) = \dfrac{1}{1 - az^{-1}} = \dfrac{z}{z-a}$, with ROC $|z| > |a|$.

State the Z-transform condition for a causal LTI system to be stable in terms of poles and the ROC.

For a causal system the ROC is the exterior of the outermost pole. The system is stable if and only if the ROC includes the unit circle, i.e. all poles lie strictly inside the unit circle ($|z|<1$).

In discrete-time processing of continuous-time signals, state the Nyquist sampling theorem.

A band-limited signal with maximum frequency $f_{m}$ can be perfectly reconstructed from its samples if the sampling rate satisfies $f_{s} > 2f_{m}$ (Nyquist rate $= 2f_{m}$). Otherwise aliasing occurs.

What is aliasing, and at what frequency does a sampled component at $f_0$ (with $f_0>f_s/2$) appear?

Aliasing is the overlap of spectral replicas when $f_{s}<2f_{m}$, causing distinct frequencies to become indistinguishable. A component folds to an alias frequency $f_{alias} = |f_{0} - k f_{s}|$ that falls in $[0, f_{s}/2]$ for the appropriate integer $k$.

Define a causal system, and state the time-domain condition on the impulse response $h(t)$ (or $h[n]$).

A causal system's output depends only on present and past inputs, not future inputs. The condition is $h(t) = 0$ for $t < 0$ (continuous-time) or $h[n] = 0$ for $n < 0$ (discrete-time).

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 LTI system is stable iff $\sum_{n=-\infty}^{\infty}|h[n]| < \infty$ (absolutely integrable/summable impulse response).