GATE ECE Signals and Systems Formula Sheet 2027, with Traps

Networks, Signals and Systems is the largest technical section of the GATE EC paper, with a mean of 17.0 marks across the five papers counted for the ECE book. Its questions rarely need a new formula. They need the right one, applied under the right convention.

This sheet gives the most-used formulas topic by topic, each with a short worked example and the trap that costs marks. Use it to check recall, not to learn from scratch.

Key takeaways

• Phasor analysis, the z-transform and two-port networks appeared in every one of the five counted papers: 2010, 2019, 2020, 2025 and 2026.
• Most errors in this section are convention errors, such as peak against rms, $\omega$ against $f$, or a missing minus sign in the h-parameters.
• In the z-domain, the region of convergence decides which sequence a formula describes, and the unit circle decides stability.
• The final value theorem is valid only when every pole of $sF(s)$ is in the open left half-plane.
• FIR and IIR filter design is newly named in 2027, but 2019 Q39 and Q54 had already asked it through zero placement and least squares.

The notation this sheet uses

Angular frequency $\omega$ is in rad/s and frequency $f$ is in Hz, so $\omega = 2\pi f$. A voltage or current is rms unless the word peak is written. A phasor is the complex number that carries a sinusoid's amplitude and phase. The passive sign convention holds throughout.

An LTI system is linear and time-invariant, and $h(t)$ or $h[n]$ is its impulse response. The ROC is the region of convergence of a Laplace or z-transform. The DFT is unnormalised in the forward direction. For the section's topic ranking, read the GATE ECE important topics guide.

Phasors, impedance and AC power

Turn each element into an impedance and solve as if the circuit were resistive.

Trap: Ideal voltmeters and ammeters read rms, while a source written as $V_m\cos\omega t$ gives the peak. GATE 2025 Q39 framed a phasor problem through ideal meter readings, so the rms convention was the whole difficulty.

Thevenin, Norton and superposition in dc networks

The Thevenin voltage is the open-circuit voltage at the terminals. The Thevenin resistance is what the terminals see with the independent sources switched off.

DC network analysis appeared in four of the five counted papers (2010 Q34, 2020 Q19, 2025 Q19, 2026 Q30). The 2019 paper had none.

First-order RL and RC transients

Every first-order response moves from its initial value to its final value along one exponential:

Here $x$ is any voltage or current, and $R$ is the Thevenin resistance seen by the capacitor or inductor. Take $R = 1\text{ k}\Omega$ and $C = 1\,\mu\text{F}$ charging from $0$ to $10\text{ V}$. Then $\tau = 1\text{ ms}$ and $v_C(1\text{ ms}) = 10(1 - e^{-1}) = 6.32\text{ V}$.

The trap is continuity: $v_C$ and $i_L$ do not jump, unless switching forms a capacitor loop or an inductor cut-set. Then conserve charge or flux.

Series and parallel resonance

Two-port parameters, reciprocity and symmetry

Each parameter set opens or shorts one port. For example, $\displaystyle z_{11} = \frac{V_1}{I_1}\Big\vert_{I_2 = 0}$ and $\displaystyle y_{11} = \frac{I_1}{V_1}\Big\vert_{V_2 = 0}$.

A single series impedance $Z$ has ABCD matrix $\begin{bmatrix} 1 & Z \\ 0 & 1 \end{bmatrix}$. Then $AD - BC = 1$ and $A = D$, so it is reciprocal and symmetric.

Remember: A two-port question appeared exactly once in each of the five counted papers, but the set kept changing. 2019 Q14 named no matrix and stated reciprocity as excitation and response interchanged.

LTI properties and convolution

Test linearity with two checks, additivity and scaling. Test time invariance by delaying the input and comparing.

An interconnection is LTI only if every block in it is LTI.

Fourier series symmetry rules

A zero-mean square wave centred on $t = 0$ is even and half-wave symmetric, so it holds only cosine terms at odd harmonics.

This sheet keeps to the most-used rows. The GATE ECE 2027 book has a last-minute revision sheet covering all nine sections, ranks 78 recurring concepts, and includes 925 questions with worked solutions and 10 full mock tests.

Fourier, Laplace and Parseval

Apply the final value theorem to $\displaystyle F(s) = \frac{1}{s^{2} + 1}$ and you get $0$, yet $f(t) = \sin t$ never settles.

Sampling and aliasing

Sample faster than twice the highest frequency: $f_s > 2f_{\max}$. The Nyquist rate is $2f_{\max}$; the Nyquist frequency is $\displaystyle \frac{f_s}{2}$.

A tone at $f$ appears at $\vert f - kf_s\vert$ for the integer $k$ that brings it below $\displaystyle \frac{f_s}{2}$. So $7\text{ kHz}$ sampled at $10\text{ kHz}$ appears at $3\text{ kHz}$. Squaring a signal doubles its bandwidth: a $4\text{ kHz}$ signal squared needs a Nyquist rate of $16\text{ kHz}$.

z-transform: ROC, causality and stability

In one line: One $H(z)$ can describe several sequences; the ROC picks the sequence, and the unit circle inside it means stability.

DTFT, DFT and all-pass systems

The DTFT is $X(e^{j\omega}) = \sum_n x[n]e^{-j\omega n}$, periodic in $\omega$ with period $2\pi$. It exists when $\sum \vert x[n]\vert < \infty$, that is, when the ROC contains the unit circle. $2^{n}u[n]$ has a z-transform but no DTFT.

The $N$-point DFT is:

For $x = [1, 1, 1, 1]$, $X = [4, 0, 0, 0]$, and both energy sums give $4$. A cosine $\cos(2\pi k_0 n/N)$ fills only bins $k_0$ and $N - k_0$. That holds only for a whole number of periods; otherwise leakage spreads into every bin.

An all-pass system has $\vert H\vert = 1$ at every frequency. In continuous time, $\displaystyle H(s) = \frac{s - a}{s + a}$ mirrors its zero and pole about the $j\omega$ axis. In discrete time, with real $a$, $\displaystyle H(z) = \frac{z^{-1} - a}{1 - az^{-1}}$ puts the zero at $\displaystyle \frac{1}{a}$. Its phase is not linear, so an all-pass system still distorts. Group delay is $\displaystyle \tau_g = -\frac{d\theta}{d\omega}$.

FIR and IIR filter design

The 2027 syllabus names FIR and IIR filter design for the first time. GATE 2019 Q39 and Q54, set by IIT Madras, were both filter-design questions, 4 of that paper's 14 Networks marks. They asked design by zero placement and by least-squares approximation. One paper is a precedent, not a rate; the 2027 syllabus changes guide covers the full comparison.

An FIR filter has poles only at the origin, so it is always stable and can have exact linear phase. An IIR filter needs a lower order, but a causal, stable one cannot have exactly linear phase. The last two rows have no counted instance; learn them from the syllabus wording.

Trap: To reject a frequency, a filter needs a zero on the unit circle at that frequency, never a pole. With real taps, nulling one frequency costs a conjugate pair of zeros and three taps.

On exam day the marking scheme, which is common to every GATE paper, puts negative marks only on MCQs.

More formula sheets: all of GATE ECE · Analog Circuits · Communications · Electromagnetics

Quick revision

1. $\omega = 2\pi f$, meters read rms, and $\displaystyle V_{\text{rms}} = \frac{V_m}{\sqrt{2}}$.
2. Never deactivate a dependent source; use a test source, and never superpose power.
3. Series $\displaystyle Q = \frac{\omega_0 L}{R}$; parallel $\displaystyle Q = \frac{R}{\omega_0 L}$; $\displaystyle \text{BW} = \frac{\omega_0}{Q}$.
4. Reciprocity: $z_{12} = z_{21}$, $y_{12} = y_{21}$, $AD - BC = 1$, and $h_{12} = -h_{21}$.
5. Even kills sine terms, odd kills cosines and dc, half-wave kills even harmonics.
6. Final value theorem only with every pole of $sF(s)$ in the open left half-plane.
7. Stable iff the ROC contains $\vert z\vert = 1$; causal and stable iff every pole is inside it.
8. To null $\omega_0$ with real taps, place zeros at $e^{\pm j\omega_0}$ on the unit circle.