GATE Electronics and Communication Engineering (EC) Syllabus

State the BIBO stability and causality conditions for an LTI system in terms of its impulse response $h(t)$.

BIBO stable iff the impulse response is absolutely integrable: $\int_{-\infty}^{\infty}|h(t)|\,dt < \infty$ (discrete: $\sum|h[n]| < \infty$). Causal iff $h(t) = 0$ for $t < 0$.

List three key properties of convolution.

Commutative: $x*h = h*x$. Associative: $(x*h_1)*h_2 = x*(h_1*h_2)$. Distributive: $x*(h_1 + h_2) = x*h_1 + x*h_2$. Also, convolution with $\delta(t)$ is identity: $x(t)*\delta(t) = x(t)$.

Define the bilateral Laplace transform and state its relationship to the Fourier transform.

$X(s) = \int_{-\infty}^{\infty} x(t)\,e^{-st}\,dt$, with $s = \sigma + j\omega$. The Fourier transform is the Laplace transform evaluated on the imaginary axis, $X(j\omega) = X(s)\big|_{s=j\omega}$, valid when the ROC includes the $j\omega$-axis.

Give the Laplace transforms of $\delta(t)$, $u(t)$, $e^{-at}u(t)$, and $\cos(\omega_0 t)u(t)$.

$\mathcal{L}\{\delta(t)\} = 1$; $\mathcal{L}\{u(t)\} = \dfrac{1}{s}$; $\mathcal{L}\{e^{-at}u(t)\} = \dfrac{1}{s+a}$; $\mathcal{L}\{\cos(\omega_0 t)u(t)\} = \dfrac{s}{s^{2}+\omega_0^{2}}$.

State the Initial Value Theorem and Final Value Theorem for Laplace transforms.

IVT: $x(0^{+}) = \lim_{s\to\infty} sX(s)$. FVT: $\lim_{t\to\infty} x(t) = \lim_{s\to 0} sX(s)$, valid only if all poles of $sX(s)$ lie in the left half-plane (system is stable).

Define the continuous-time Fourier transform (CTFT) and its inverse.

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

State Parseval's theorem for the Fourier transform.

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

State the time-convolution and time-shift properties of the Fourier transform.

Convolution: $x(t)*h(t) \leftrightarrow X(j\omega)H(j\omega)$. Time shift: $x(t-t_0) \leftrightarrow X(j\omega)e^{-j\omega t_0}$ (magnitude unchanged, linear phase added).

Give the Fourier transform pair for a rectangular pulse of width $T$ and the corresponding duality result.

A rectangular pulse $\text{rect}(t/T)$ transforms to a sinc: $T\,\text{sinc}\!\left(\dfrac{\omega T}{2}\right)$. By duality, a sinc in time gives a rectangle (ideal low-pass) in frequency.

Define the bilateral Z-transform and the unit-circle relationship to the DTFT.

$X(z) = \sum_{n=-\infty}^{\infty} x[n]z^{-n}$. The DTFT is the Z-transform on the unit circle: $X(e^{j\Omega}) = X(z)\big|_{z=e^{j\Omega}}$, valid when the ROC includes $|z| = 1$.

Give the Z-transforms of $\delta[n]$, $u[n]$, and $a^{n}u[n]$, including ROC.

$\mathcal{Z}\{\delta[n]\} = 1$ (all $z$); $\mathcal{Z}\{u[n]\} = \dfrac{1}{1-z^{-1}}$, $|z|>1$; $\mathcal{Z}\{a^{n}u[n]\} = \dfrac{1}{1-az^{-1}}$, $|z|>|a|$.

State the ROC-based stability and causality conditions for a discrete-time LTI system in the Z-domain.

Causal: ROC is the exterior of the outermost pole and includes $z=\infty$. Stable: ROC includes the unit circle $|z|=1$. A causal and stable system has all poles strictly inside the unit circle.

Define the N-point Discrete Fourier Transform (DFT) and its inverse.

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

How many complex multiplications does a direct $N$-point DFT need versus the radix-2 FFT?

Direct DFT: $N^{2}$ complex multiplications. Radix-2 FFT: about $\dfrac{N}{2}\log_2 N$ complex multiplications — a major reduction for large $N$.

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 $f_s > 2f_m$. The minimum rate $f_{s,min} = 2f_m$ is the Nyquist rate, and $2f_m$ is the Nyquist frequency.

Define aliasing and how it is avoided.

Aliasing is the overlap/folding of spectral replicas when $f_s < 2f_m$, causing high frequencies to masquerade as lower ones. It is prevented by band-limiting the signal with an anti-aliasing (low-pass) filter before sampling and/or increasing $f_s$.

Write the intrinsic carrier concentration relation and the mass-action law for a semiconductor.

Mass-action law: $np = n_i^{2}$ in thermal equilibrium. Intrinsic concentration $n_i = \sqrt{N_C N_V}\,e^{-E_g/2kT}$, so $n_i$ increases strongly with temperature and decreases with band gap $E_g$.

Write the built-in potential of a PN junction and the diode (Shockley) equation.

Built-in potential: $V_{bi} = \dfrac{kT}{q}\ln\!\left(\dfrac{N_A N_D}{n_i^{2}}\right)$. Diode current: $I = I_S\left(e^{V/\eta V_T} - 1\right)$, where thermal voltage $V_T = \dfrac{kT}{q} \approx 26\text{ mV}$ at 300 K.

Distinguish avalanche breakdown from Zener breakdown in a reverse-biased PN junction.

Avalanche: occurs in lightly doped (wide depletion) junctions at higher voltages via impact ionization; breakdown voltage increases with temperature (positive temperature coefficient). Zener: occurs in heavily doped (narrow depletion) junctions at low voltages via quantum tunneling; breakdown voltage decreases with temperature (negative coefficient).

Name the four regions of BJT operation and the junction bias conditions for each.

Active: emitter-base forward, collector-base reverse (amplification). Saturation: both junctions forward. Cutoff: both junctions reverse. Reverse-active: emitter-base reverse, collector-base forward.

Relate the BJT current gains $\alpha$ and $\beta$.

$\beta = \dfrac{\alpha}{1-\alpha}$ and $\alpha = \dfrac{\beta}{1+\beta}$. Also $I_C = \beta I_B$ and $I_E = I_C + I_B = (\beta+1)I_B$. Here $\alpha$ is common-base gain (<1) and $\beta$ is common-emitter gain.

Write the drain-current equations for an n-channel MOSFET in the triode and saturation regions.

Triode ($V_{DS} < V_{GS}-V_{th}$): $I_D = \mu_n C_{ox}\dfrac{W}{L}\left[(V_{GS}-V_{th})V_{DS} - \dfrac{V_{DS}^{2}}{2}\right]$. Saturation ($V_{DS} \geq V_{GS}-V_{th}$): $I_D = \dfrac{1}{2}\mu_n C_{ox}\dfrac{W}{L}(V_{GS}-V_{th})^{2}(1+\lambda V_{DS})$.

Give the small-signal transconductance $g_m$ of a BJT and of a MOSFET in saturation.

BJT: $g_m = \dfrac{I_C}{V_T}$. MOSFET: $g_m = \mu_n C_{ox}\dfrac{W}{L}(V_{GS}-V_{th}) = \sqrt{2\mu_n C_{ox}\dfrac{W}{L}I_D} = \dfrac{2I_D}{V_{GS}-V_{th}}$.

Why is the voltage-divider (self-bias) configuration preferred for BJT biasing?

It gives the most stable Q-point against variations in $\beta$ and temperature because the base voltage is set by the divider and the emitter resistor $R_E$ provides negative feedback (stabilization factor is low). The collector current is nearly independent of $\beta$.

State the four feedback topologies and the effect of negative feedback on gain and bandwidth.

Topologies: voltage-series (series-shunt), voltage-shunt (shunt-shunt), current-series (series-series), current-shunt (shunt-series). Negative feedback reduces gain by factor $(1+A\beta)$ but increases bandwidth by the same factor, so the gain-bandwidth product is roughly constant; it also reduces distortion and stabilizes gain.

What causes the low-frequency and high-frequency roll-off in an amplifier's frequency response?

Low-frequency roll-off: coupling and bypass capacitors (their reactance rises as $f$ falls). High-frequency roll-off: internal device junction/parasitic capacitances and the Miller effect. Mid-band gain is flat between these.

State the Miller effect for a capacitance $C$ bridging input and output of an inverting amplifier with gain $-A_v$.

The effective input capacitance is multiplied: $C_{in} = C(1 + A_v)$, and the effective output capacitance is $C\!\left(1 + \dfrac{1}{A_v}\right) \approx C$. This Miller multiplication dominates the high-frequency response of CE/CS stages.

List the ideal op-amp assumptions and the closed-loop gains of inverting and non-inverting configurations.

Ideal op-amp: infinite open-loop gain, infinite input impedance ($I_{in}=0$), zero output impedance, infinite bandwidth, and a virtual short ($V_+ = V_-$). Inverting gain: $A_v = -\dfrac{R_f}{R_1}$. Non-inverting gain: $A_v = 1 + \dfrac{R_f}{R_1}$.

Give the output expressions for an op-amp integrator and differentiator.

Integrator (R at input, C feedback): $V_{out} = -\dfrac{1}{RC}\int V_{in}\,dt$. Differentiator (C at input, R feedback): $V_{out} = -RC\dfrac{dV_{in}}{dt}$.

State the Barkhausen criterion for sustained oscillations.

For sustained sinusoidal oscillation the loop gain magnitude must be unity, $|A\beta| = 1$, and the total loop phase shift must be $0^{\circ}$ (or a multiple of $360^{\circ}$). For start-up, $|A\beta| > 1$ initially.

Compare the frequency-determining elements of Wien-bridge, RC phase-shift, Hartley, and Colpitts oscillators.

Wien-bridge: RC network, $f = \dfrac{1}{2\pi RC}$. RC phase-shift: three RC sections giving $180^{\circ}$. Hartley: LC tank with tapped inductors ($L_1, L_2$). Colpitts: LC tank with tapped capacitors ($C_1, C_2$), $f = \dfrac{1}{2\pi\sqrt{L\,C_{eq}}}$ with $C_{eq} = \dfrac{C_1 C_2}{C_1+C_2}$.

Define the quality factor $Q$ of a second-order active filter and give the cutoff/center frequency for a basic Sallen-Key filter.

$Q$ measures selectivity: $Q = \dfrac{f_0}{BW}$, where $BW$ is the $-3\text{ dB}$ bandwidth. For a Sallen-Key (or similar RC active filter) the characteristic frequency is $f_0 = \dfrac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}}$.