GATE Solved Questions

Introduction

GATE (Graduate Aptitude Test in Engineering) frequently tests signals and systems concepts, particularly Fourier transforms, Laplace transforms, convolution, system properties, and sampling theory. This section presents carefully solved problems representative of actual GATE difficulty levels, covering the most commonly tested topics.

The key to GATE success in this subject is: (1) memorize fundamental transform pairs, (2) master property-based problem solving (avoid direct integration when possible), and (3) practice systematic approaches for system analysis.

Question 1: Fourier Transform (GATE-style)

Q: The Fourier transform of $x(t) = e^{-|t|}$ is $X(\omega) = \frac{2}{1+\omega^2}$. Find the Fourier transform of $y(t) = te^{-|t|}$.

Solution: We need the FT of $tx(t)$. Using the property:

$$\mathcal{F}\{tx(t)\} = j\frac{dX(\omega)}{d\omega}$$

$$\frac{dX}{d\omega} = \frac{d}{d\omega}\left(\frac{2}{1+\omega^2}\right) = \frac{-4\omega}{(1+\omega^2)^2}$$

$$Y(\omega) = j \cdot \frac{-4\omega}{(1+\omega^2)^2} = \frac{-4j\omega}{(1+\omega^2)^2}$$

Answer: $Y(\omega) = \frac{-4j\omega}{(1+\omega^2)^2}$ (purely imaginary — expected since $te^{-|t|}$ is odd).

Question 2: System Stability

Q: A causal LTI system has transfer function $H(s) = \frac{s+2}{s^2+5s+6}$. Is the system stable?

Solution: Factor denominator: $s^2+5s+6 = (s+2)(s+3)$

Poles at $s = -2$ and $s = -3$ (both in left half-plane).

For a causal system, stability requires ALL poles in the left half s-plane. ✓

Answer: YES, the system is BIBO stable.

Question 3: Convolution

Q: Find $y(t) = e^{-t}u(t) * e^{-2t}u(t)$.

Solution: Using frequency domain: $Y(s) = \frac{1}{s+1} \cdot \frac{1}{s+2} = \frac{1}{s+1} - \frac{1}{s+2}$

$$y(t) = (e^{-t} - e^{-2t})u(t)$$

Verification: At $t=0$: $y(0) = 0$ ✓ (convolution of two causal signals starts at 0). As $t\to\infty$: $y\to 0$ ✓.

Question 4: Sampling Theorem

Q: A signal $x(t) = \cos(200\pi t) + 2\cos(600\pi t)$ is sampled at $f_s = 500$ Hz. What frequencies appear in the reconstructed signal?

Solution: Signal frequencies: $f_1 = 100$ Hz, $f_2 = 300$ Hz. Nyquist rate = $2 \times 300 = 600$ Hz.

Since $f_s = 500 < 600$ Hz, aliasing occurs for $f_2 = 300$ Hz.

Aliased frequency of 300 Hz: $|f_2 - f_s| = |300 - 500| = 200$ Hz.

Answer: Reconstructed signal contains 100 Hz and 200 Hz (the 300 Hz component aliases to 200 Hz).

Question 5: Z-Transform

Q: Find the inverse z-transform of $X(z) = \frac{z}{(z-0.5)(z-0.25)}$, ROC: $|z| > 0.5$.

Solution: Partial fractions in $X(z)/z$:

$$\frac{X(z)}{z} = \frac{1}{(z-0.5)(z-0.25)} = \frac{A}{z-0.5} + \frac{B}{z-0.25}$$

$A = 1/(0.5-0.25) = 4$, $B = 1/(0.25-0.5) = -4$

$$X(z) = \frac{4z}{z-0.5} - \frac{4z}{z-0.25}$$

Since ROC is $|z| > 0.5$ (outside the outermost pole), both sequences are causal:

$$x[n] = 4(0.5)^n u[n] - 4(0.25)^n u[n] = 4[(0.5)^n - (0.25)^n]u[n]$$

Question 6: Laplace Transform Initial/Final Value

Q: Given $X(s) = \frac{5(s+2)}{s(s+1)(s+3)}$, find $x(0^+)$ and $x(\infty)$.

Solution:

Initial value: $x(0^+) = \lim_{s\to\infty} sX(s) = \lim_{s\to\infty} \frac{5s(s+2)}{s(s+1)(s+3)} = \lim_{s\to\infty}\frac{5(s+2)}{(s+1)(s+3)} = 5/1 = 0$

Wait: $\lim_{s\to\infty} \frac{5(s+2)}{(s+1)(s+3)} = \lim_{s\to\infty}\frac{5s}{s^2} = 0$

Final value: $x(\infty) = \lim_{s\to 0} sX(s) = \frac{5(0+2)}{(0+1)(0+3)} = 10/3$

Answer: $x(0^+) = 0$, $x(\infty) = 10/3$

Question 7: DFT Properties

Q: The 4-point DFT of $x[n] = \{1, 0, 2, 0\}$ is $X[k] = \{3, -1, 3, -1\}$. Find the DFT of $y[n] = \{0, 1, 0, 2\}$.

Solution: $y[n] = x[(n-1)_4]$ — circular shift by 1.

By circular shift property: $Y[k] = X[k] \cdot W_4^k = X[k] \cdot e^{-j\pi k/2}$

$Y[0] = 3 \cdot 1 = 3$, $Y[1] = -1 \cdot (-j) = j$, $Y[2] = 3 \cdot (-1) = -3$, $Y[3] = -1 \cdot (j) = -j$

Answer: $Y[k] = \{3, j, -3, -j\}$

Question 8: System Properties

Q: Determine if $y[n] = x[n^2]$ is (a) linear, (b) time-invariant, (c) causal.

Solution:

(a) Linear: $T\{ax_1 + bx_2\} = ax_1[n^2] + bx_2[n^2] = aT\{x_1\} + bT\{x_2\}$ ✓ YES

(b) Time-invariant: Input $x_1[n] = x[n-n_0]$ gives output $y_1[n] = x[(n-n_0)^2] \neq x[n^2-n_0] = y[n-n_0]$. NOT time-invariant ✗

(c) Causal: At $n=2$, output depends on $x[4]$ (future value). NOT causal ✗

Tips for GATE Preparation

1. Transform pairs: Memorize 10-15 key pairs for each transform
2. Properties: Master shift, scale, differentiation, convolution theorem
3. Partial fractions: Essential for inverse transforms
4. System properties: Practice checking linearity, TI, causality, stability systematically
5. Sampling: Know Nyquist rate calculation and aliasing frequency formula
6. Typical traps: ROC specification, circular vs linear convolution, discrete periodicity conditions

Key Takeaways

• GATE problems often test property application rather than direct integration
• For inverse transforms: always use partial fractions first
• System stability for causal systems: all poles in left half s-plane (or inside unit circle for z)
• Aliased frequency = $|f_{signal} - k \cdot f_s|$ where $k$ gives the closest integer multiple
• Practice is essential — these problem types repeat with variations every year