Signals & Systems
LTI systems, convolution, Fourier and Laplace transforms, sampling and the Z-transform - the mathematical spine of the whole paper.
About this subject
Signals and Systems is the mathematics behind the rest of the paper. It describes signals in time and in frequency, and systems by what they do to those signals. The core ideas are linearity and time invariance, convolution, the Fourier, Laplace and Z-transforms, and sampling.
Most questions come down to choosing the right domain. A convolution in time is a multiplication in frequency; a differential equation in time is an algebraic equation in $s$. Picking the domain where the work is easy is half the answer.
Solved questions
Each question is a practice problem written for this portal at GATE standard. Try it first, then open the full derivation, and read the note on the idea behind it.
- NAT Value of the convolution of two rectangular pulses at t = 2 Convolution · 2 marks
- NAT Nyquist rate for a product of two cosines Sampling · 2 marks
- NAT Final value of a signal from its Laplace transform Laplace transform · 1 mark
- MCQ Steady-state output of a first-order LTI system Frequency response · 2 marks
Key formulas
The results these questions lean on most. Know what each symbol means, and when each one applies.
| What | Formula |
|---|---|
| Convolution | $y(t) = \int_{-\infty}^{\infty} x(\tau)\,h(t-\tau)\,d\tau$ |
| Convolution theorem | $x(t) * h(t) \;\leftrightarrow\; X(f)\,H(f)$ |
| Product of cosines | $\cos a \cos b = \tfrac{1}{2}[\cos(a-b) + \cos(a+b)]$ |
| Nyquist rate | $f_s = 2 f_{max}$ |
| Final value theorem | $\lim_{t\to\infty} y(t) = \lim_{s\to 0} s\,Y(s)$, if all poles of $sY(s)$ are in the left half-plane |
| Initial value theorem | $y(0^+) = \lim_{s\to\infty} s\,Y(s)$ |
| Sinusoidal steady state | $A\cos(\omega t) \to A\,|H(j\omega)|\cos(\omega t + \angle H(j\omega))$ |
| First-order system | $h(t) = e^{-at}u(t) \;\leftrightarrow\; H(s) = 1/(s+a)$ |
How to prepare
- Learn a short table of transform pairs by heart; most questions are built from them.
- Before sampling a signal, expand any products into sums so you can see its highest frequency.
- Always check the conditions of a theorem - region of convergence, pole positions - before using it.
- Sketch signals. A quick picture of two pulses sliding past each other makes convolution almost automatic.
Common traps
Each of these costs marks every year. The questions above are built to catch them.
- Reading the Nyquist rate off one factor of a product, instead of the highest frequency in the expanded signal.
- Using the final value theorem on a signal that oscillates or grows.
- Dropping the negative phase of a lagging system.
- Confusing the height of a convolution with the product of the heights of its inputs.