GATE Electrical Engineering Signals and Systems Syllabus

State the impulse-response condition for an LTI system to be BIBO stable.

The impulse response must be absolutely integrable/summable: $\int_{-\infty}^{\infty}|h(t)|\,dt<\infty$ (CT) or $\sum_{n=-\infty}^{\infty}|h[n]|<\infty$ (DT).

Is the system $y(t)=x(t+2)$ causal? Why?

No. The output at time $t$ depends on a future input value $x(t+2)$, which violates causality (it is non-causal / anticipatory).

Define a memoryless (static) system and give an LTI example.

A system whose output at time $t$ depends only on the input at the same time $t$. For an LTI system this means $h(t)=K\delta(t)$, i.e. $y(t)=Kx(t)$.

What is the definition of a continuous-time periodic signal?

$x(t)$ is periodic if there exists $T>0$ such that $x(t)=x(t+T)$ for all $t$. The smallest such $T$ is the fundamental period $T_0$.

For the CT sinusoid $x(t)=A\cos(\omega_0 t+\phi)$, what are the fundamental period and frequency?

Fundamental angular frequency $\omega_0$, fundamental period $T_0=\dfrac{2\pi}{\omega_0}$, and frequency $f_0=\dfrac{\omega_0}{2\pi}$ Hz.

Are all continuous-time sinusoids $\cos(\omega_0 t)$ periodic? Explain.

Yes. Every CT sinusoid is periodic for any real $\omega_0\neq 0$, with period $T_0=\frac{2\pi}{\omega_0}$, because $t$ is continuous.

What is the definition of a discrete-time periodic signal?

$x[n]$ is periodic if there exists a positive integer $N$ such that $x[n]=x[n+N]$ for all $n$. The smallest such integer is the fundamental period $N_0$.

When is the DT sinusoid $x[n]=\cos(\Omega_0 n)$ periodic?

Only when $\dfrac{\Omega_0}{2\pi}$ is a rational number, i.e. $\Omega_0=2\pi\dfrac{k}{N}$ for integers $k,N$. Then the fundamental period is $N=2\pi k/\Omega_0$ (smallest integer $N$).

Find the fundamental period of $x[n]=\cos\!\left(\dfrac{3\pi}{4}n\right)$.

Set $\Omega_0=\frac{3\pi}{4}=2\pi\frac{k}{N}\Rightarrow \frac{N}{k}=\frac{2\pi}{\Omega_0}=\frac{8}{3}$. Smallest integers give $N=8$ (with $k=3$).

Why is the DT frequency $\Omega$ only unique over an interval of length $2\pi$?

Because $e^{j(\Omega+2\pi)n}=e^{j\Omega n}e^{j2\pi n}=e^{j\Omega n}$ for integer $n$. DT exponentials are periodic in frequency with period $2\pi$, so $\Omega$ and $\Omega+2\pi$ are indistinguishable.

If $x_1[n]$ has period $N_1$ and $x_2[n]$ has period $N_2$, what is the period of their sum?

$N=\operatorname{lcm}(N_1,N_2)$, the least common multiple of the individual periods.

If two CT periodic signals have periods $T_1$ and $T_2$, when is their sum periodic and with what period?

Their sum is periodic iff $T_1/T_2$ is rational. Then the period is the least value $T=k_1T_1=k_2T_2$, found as the LCM of the periods (smallest integers $k_1,k_2$).

Define even and odd parts of a CT signal $x(t)$.

Even part: $x_e(t)=\frac{1}{2}[x(t)+x(-t)]$ (satisfies $x_e(t)=x_e(-t)$). Odd part: $x_o(t)=\frac{1}{2}[x(t)-x(-t)]$ (satisfies $x_o(-t)=-x_o(t)$). Then $x(t)=x_e(t)+x_o(t)$.

Define the energy and power of a CT signal.

Energy: $E=\int_{-\infty}^{\infty}|x(t)|^{2}\,dt$. Power: $P=\lim_{T\to\infty}\dfrac{1}{2T}\int_{-T}^{T}|x(t)|^{2}\,dt$.

Define the energy and power of a DT signal.

Energy: $E=\sum_{n=-\infty}^{\infty}|x[n]|^{2}$. Power: $P=\lim_{N\to\infty}\dfrac{1}{2N+1}\sum_{n=-N}^{N}|x[n]|^{2}$.

How do you classify a signal as an energy signal vs a power signal?

Energy signal: $0<E<\infty$ (then $P=0$). Power signal: $0<P<\infty$ (then $E=\infty$). A signal cannot be both; periodic signals are power signals.

What is the average power of the periodic signal $x(t)=A\cos(\omega_0 t+\phi)$?

$P=\dfrac{A^{2}}{2}$ (independent of frequency and phase), and its energy is infinite, so it is a power signal.

For any general periodic waveform of period $T_0$, how is its average power computed over one period?

$P=\dfrac{1}{T_0}\int_{T_0}|x(t)|^{2}\,dt$ — integration over any one full period suffices because the signal repeats.

State the exponential (complex) Fourier series for any general periodic waveform $x(t)$ of period $T_0$.

$x(t)=\sum_{k=-\infty}^{\infty}c_k\,e^{jk\omega_0 t}$ with $\omega_0=\frac{2\pi}{T_0}$ and coefficients $c_k=\dfrac{1}{T_0}\int_{T_0}x(t)\,e^{-jk\omega_0 t}\,dt$.

State Parseval's relation (power) for a general periodic waveform in terms of its Fourier coefficients $c_k$.

$P=\dfrac{1}{T_0}\int_{T_0}|x(t)|^{2}\,dt=\sum_{k=-\infty}^{\infty}|c_k|^{2}$ — total power equals the sum of the powers in all harmonics.

What is the trigonometric Fourier series form for a general periodic waveform $x(t)$?

$x(t)=a_0+\sum_{k=1}^{\infty}\big[a_k\cos(k\omega_0 t)+b_k\sin(k\omega_0 t)\big]$, where $a_0$ is the DC (average) value and $\omega_0=\frac{2\pi}{T_0}$.

For a general periodic waveform, what symmetry makes all sine terms ($b_k$) vanish, and what makes all cosine terms ($a_k$) and DC vanish?

Even symmetry ($x(t)=x(-t)$) eliminates all $b_k$ (only cosines/DC remain). Odd symmetry ($x(-t)=-x(t)$) eliminates DC and all $a_k$ (only sines remain).

What does half-wave symmetry $x(t)=-x(t-T_0/2)$ imply about a periodic waveform's Fourier series?

Only odd harmonics are present; all even-harmonic coefficients (and the DC term) are zero.

What are the Dirichlet conditions for the Fourier series of a periodic waveform to converge?

Over one period: (1) $x(t)$ is absolutely integrable, $\int_{T_0}|x(t)|\,dt<\infty$; (2) finite number of maxima and minima; (3) finite number of finite discontinuities.

What is the difference between deterministic and random signals?

A deterministic signal can be described by an explicit mathematical expression (its future is exactly known), whereas a random (stochastic) signal can only be characterized statistically (probabilistically).

What is an anti-causal (non-causal) system in terms of its impulse response?

A system whose impulse response is nonzero only for negative time: $h(t)=0$ for $t>0$. Its output depends solely on future input values.

Distinguish an energy signal from a power signal regarding boundedness and time extent.

Energy signals have finite energy and are typically time-limited or decaying (e.g. $e^{-t}u(t)$). Power signals have finite nonzero average power and exist over all time (e.g. periodic signals, $u(t)$).

Is the unit step $u(t)$ an energy or a power signal? Give its power.

It is a power signal: $E=\infty$, and $P=\lim_{T\to\infty}\frac{1}{2T}\int_{0}^{T}1\,dt=\frac{1}{2}$.

How does the fundamental period of $x[n]=e^{j\Omega_0 n}$ compare to that of the CT signal $e^{j\omega_0 t}$ as frequency increases?

For CT, period $T_0=2\pi/\omega_0$ always decreases as $\omega_0$ increases. For DT it is not monotonic: the rate of oscillation increases only as $\Omega_0$ goes from $0$ to $\pi$, then decreases from $\pi$ to $2\pi$, and the signal may not even be periodic.

State the commutative and shift relationship: how does convolving a signal with a shifted impulse $\delta(t-t_0)$ act?

$x(t)*\delta(t-t_0)=x(t-t_0)$. Convolution with a shifted impulse simply shifts the signal; this combines the sifting and time-shift properties used throughout LTI analysis.