Signals and Systems is the theoretical core of Part B1 (Electrical) in the GATE Robotics and Automation (RA) 2027 paper, and it underpins Control Systems and Analog Circuits alike. The questions cluster around a few dependable skills: classifying signals, convolving them through LTI systems, and moving between time and transform domains with the Fourier, Laplace and Z transforms. This guide reviews each area with its working formulas and then solves five GATE-style problems in full.
TABLE OF CONTENTS
Signal classification
Every signal question starts with classification: continuous vs discrete, periodic vs aperiodic, energy vs power, and even vs odd. A continuous signal is periodic if x(t) = x(t+T) for some T; a discrete signal x[n] is periodic only if its normalised frequency is a rational multiple of 2π. An energy signal has finite total energy (and zero average power); a power signal has finite non-zero average power.
LTI systems & convolution
A linear time-invariant (LTI) system is completely described by its impulse response h(t) or h[n], and its output is the convolution of input with impulse response. For discrete signals the convolution sum slides one sequence past the other; the output length is (length of x) + (length of h) − 1. Convolution in time becomes multiplication in the transform domain, which is why transforms are so powerful.
Fourier analysis
The Fourier series represents a periodic signal as a sum of harmonics; the Fourier transform generalises this to aperiodic signals, giving the frequency spectrum X(jω). Key properties — linearity, time shift (a phase term), frequency shift, and the convolution/multiplication duality — solve most spectrum questions faster than direct integration.
Laplace & Z transforms
The Laplace transform handles continuous LTI systems (poles, stability, transfer functions), and the Z transform is its discrete counterpart. The pair to memorise is L{e−atu(t)} = 1/(s+a). A system is stable if its region of convergence includes the imaginary axis (Laplace) or the unit circle (Z). Partial fractions convert a transform back to a time-domain sum of exponentials.
Sampling theorem
The Nyquist–Shannon sampling theorem states that a band-limited signal with highest frequency fm can be reconstructed perfectly only if it is sampled at a rate fs ≥ 2fm. The minimum rate 2fm is the Nyquist rate; sampling below it causes aliasing, where high frequencies fold down and masquerade as low ones.
Worked examples (GATE-style)
Example 1 — discrete convolution
Compute the linear convolution of x[n] = {1, 2, 3} and h[n] = {1, 1}.
Solution. The output has length 3 + 2 − 1 = 4. y[0] = 1×1 = 1; y[1] = 1×1 + 2×1 = 3; y[2] = 2×1 + 3×1 = 5; y[3] = 3×1 = 3.
y[n] = {1, 3, 5, 3}.
Example 2 — Nyquist rate
A signal x(t) = 2cos(100πt) + 5sin(300πt) is to be sampled. Find the Nyquist rate.
Solution. The angular frequencies are 100π and 300π rad/s, i.e. frequencies 50 Hz and 150 Hz. The highest frequency is fm = 150 Hz.
Nyquist rate = 2fm = 300 Hz (300 samples/second).
Example 3 — fundamental period of a discrete signal
Find the fundamental period of x[n] = cos(πn/4).
Solution. A discrete sinusoid cos(Ωn) is periodic with period N = 2πm/Ω for the smallest integer N. Here Ω = π/4, so N = 2π/(π/4) = 8, which is already an integer (m = 1).
Fundamental period N = 8 samples.
Example 4 — energy of a signal
Find the total energy of x(t) = e−2tu(t).
Solution. E = ∫0∞(e−2t)² dt = ∫0∞e−4t dt = [−e−4t/4]0∞.
E = 0 − (−1/4) = 0.25 J. (Finite energy ⇒ it is an energy signal.)
Example 5 — LTI output via Laplace
An LTI system has impulse response h(t) = e−3tu(t). Find its response to x(t) = e−2tu(t).
Solution. In the s-domain, X(s) = 1/(s+2) and H(s) = 1/(s+3), so Y(s) = 1/[(s+2)(s+3)].
Partial fractions: 1/[(s+2)(s+3)] = 1/(s+2) − 1/(s+3).
Inverse transform: y(t) = (e−2t − e−3t)u(t).
Key formulas
FORMULA SHEET
Convolution: y[n] = ∑k x[k]h[n−k], length = Lx + Lh − 1
Energy / power: E = ∫|x|²dt, P = lim (1/T)∫|x|²dt
Laplace: e−atu(t) ↔ 1/(s+a)
Z-transform: anu[n] ↔ z/(z−a), |z|>|a|
Sampling: fs ≥ 2fm, Nyquist rate = 2fm
DT period: N = 2πm/Ω (smallest integer N)
Common mistakes to avoid
- Assuming every discrete sinusoid is periodic. cos(Ωn) is periodic only if Ω/2π is rational; otherwise it never repeats.
- Getting the convolution length wrong — it is Lx + Lh − 1, and the first output aligns the leading samples.
- Forgetting the region of convergence when using Laplace/Z transforms; the same algebraic expression can be stable or unstable depending on the ROC.
- Confusing Nyquist rate with Nyquist frequency: the rate is 2fm, the frequency is half the sampling rate.
- Dropping the unit step u(t), which sets the lower limit of energy and transform integrals to zero.
GATE ROBOTICS & AUTOMATION 2027
Master Signals & Systems for GATE RA
Get in-depth lectures on convolution, Fourier, Laplace and Z transforms and sampling, with PYQs and full-length tests, taught by Piyush Wairale (IIT Madras).
Frequently asked questions
What is the difference between an energy and a power signal?
An energy signal has finite total energy and zero average power — typically transient signals like e−2tu(t). A power signal has finite non-zero average power and infinite energy — typically periodic signals like sinusoids. A signal cannot be both.
Why is convolution so important for LTI systems?
Because an LTI system is fully characterised by its impulse response, the output for any input is the convolution of the input with that impulse response. In the transform domain this convolution becomes a simple multiplication, which is why transforms simplify system analysis.
What is aliasing?
Aliasing occurs when a signal is sampled below its Nyquist rate (less than twice its highest frequency). High-frequency components fold back and appear as lower frequencies, distorting the reconstructed signal irreversibly. An anti-aliasing filter prevents it.
This solved-problems guide is part of the complete GATE RA 2027 Syllabus overview. For the full topic breakdown, see the Signals and Systems syllabus guide.
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