UNIT 1: SIGNALS AND SYSTEMS FUNDAMENTALS
I. CLASSIFICATION OF SIGNALS
| Basis of Classification | Types | Key Definition & Examples |
|---|---|---|
| Time Domain | Continuous-Time (CT) | Defined for every instant of time, $x(t)$. <br> Example: Analog audio signal. |
| Discrete-Time (DT) | Defined only at discrete instants, $$\displaystyle x[n] = x(nT_s) $$. <br> Example: Sampled signal. | |
| Determinism | Deterministic | Completely predictable for all $t$. Can be expressed by a mathematical formula. <br> Example: $$\displaystyle x(t) = \sin(2\pi t) $$. |
| Random (Non-deterministic) | Unpredictable; described by statistical properties (e.g., mean, variance). <br> Example: Thermal noise voltage. | |
| Periodicity | Periodic | Repeats after a fixed interval $$\displaystyle T_0 $$: $$\displaystyle x(t) = x(t + nT_0) $$. <br> Example: $$\displaystyle x(t) = \cos(2\pi f_0 t) $$ with $$\displaystyle T_0 = 1/f_0 $$. |
| Aperiodic (Non-periodic) | Does not repeat. <br> Example: Rectangular pulse of finite duration. | |
| Energy & Power | Energy Signal | Finite total energy, zero average power. <br> Energy: $$\displaystyle E = \int_{-\infty}^{\infty} |x(t)|^2 dt $$ <br> Example: Pulse, finite-duration signal. |
| Power Signal | Finite average power, infinite total energy. <br> Average Power: $$\displaystyle P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt $$ <br> Example: Periodic signals (sinusoid), infinite-duration signals. | |
| Causality | Causal | $$\displaystyle x(t) = 0 $$ for $$\displaystyle t < 0 $$. Depends only on present/past. <br> Example: $$\displaystyle x(t) = u(t) $$. |
| Non-causal | Non-zero for $$\displaystyle t < 0 $$. <br> Example: $$\displaystyle x(t) = \sin(\omega_0 t) $$. |
\[!TIP\] Common Pitfall: A finite-duration rectangular pulse is an Energy Signal (finite area under $$\displaystyle |x(t)|^2 $$), not a Power Signal. A sinusoid of infinite duration is a Power Signal (finite average power), not an Energy Signal.
II. STANDARD SIGNALS (EXPRESSIONS & WAVEFORMS)
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Unit Impulse (Delta) Function, $\delta(t)$
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Definition: $$\displaystyle \delta(t) = 0 $$ for $t \neq 0$, $$\displaystyle \int_{-\infty}^{\infty} \delta(t) dt = 1 $$.
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Sifting Property: $$\displaystyle \int_{-\infty}^{\infty} x(t) \delta(t - t_0) dt = x(t_0) $$.
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Key: It "picks out" the value of $x(t)$ at $$\displaystyle t = t_0 $$.
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Unit Step Function, $u(t)$
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Definition: $$\displaystyle u(t) = 1 $$ for $t \geq 0$, $$\displaystyle u(t) = 0 $$ for $$\displaystyle t < 0 $$.
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Relation to Delta: $$\displaystyle \frac{d}{dt}u(t) = \delta(t) $$.
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FT: $$\displaystyle \mathcal{F}\{u(t)\} = \pi \delta(\omega) + \frac{1}{j\omega} $$ (requires careful handling).
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Rectangular Pulse (Gate Function), $\text{rect}(t/\tau)$
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Definition: $$\displaystyle \text{rect}(t/\tau) = 1 $$ for $|t| \leq \tau/2$, $0$ otherwise.
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FT: $$\displaystyle \mathcal{F}\{\text{rect}(t/\tau)\} = \tau \text{sinc}(\omega \tau / 2\pi) $$.
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Key: Its FT is a $\text{sinc}$ function.
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Triangular Pulse
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Convolution of two rectangular pulses: $$\displaystyle \text{tri}(t/\tau) = \text{rect}(t/\tau) * \text{rect}(t/\tau) $$.
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FT: Square of the sinc function.
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Signum (Sign) Function, $\text{sgn}(t)$
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Definition: $$\displaystyle \text{sgn}(t) = 1 $$ for $$\displaystyle t > 0 $$, $-1$ for $$\displaystyle t < 0 $$, $0$ at $$\displaystyle t=0 $$.
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Relation: $$\displaystyle \text{sgn}(t) = 2u(t) - 1 $$.
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III. SYSTEM PROPERTIES
| Property | Definition | Test/Condition |
|---|---|---|
| Linearity | Superposition holds: $$\displaystyle T\{a x_1(t) + b x_2(t)\} = a y_1(t) + b y_2(t) $$. | Check homogeneity & additivity. |
| Time-Invariance | A time shift in input causes identical time shift in output. | If $$\displaystyle y(t) = T\{x(t)\} $$, then $$\displaystyle T\{x(t-t_0)\} = y(t-t_0) $$. |
| Causality | Output depends only on present/past input. | $$\displaystyle h(t) = 0 $$ for $$\displaystyle t < 0 $$ (for LTI). |
| Stability (BIBO) | Bounded Input produces Bounded Output. | $$\displaystyle \int_{-\infty}^{\infty} |h(t)| dt < \infty $$ (for LTI). |
| LTI System | Linear & Time-Invariant. Characterized by impulse response $h(t)$. | Output: $$\displaystyle y(t) = x(t) * h(t) $$. Frequency response: $$\displaystyle H(\omega) = \mathcal{F}\{h(t)\} $$. |
\[!TIP\] For LTI systems, causality $\iff$ $$\displaystyle h(t)=0 $$ for $$\displaystyle t<0 $$. BIBO stability $\iff$ $$\displaystyle \int |h(t)| dt < \infty $$.
IV. FOURIER TRANSFORM (FT)
1. Definition & Significance
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Definition: $$\displaystyle X(\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt $$.
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Inverse FT: $$\displaystyle x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} d\omega $$.
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Physical Significance: Represents a signal in the frequency domain. $|X(\omega)|$ shows amplitude spectrum, $\angle X(\omega)$ shows phase spectrum.
2. Key Properties (with Statement)
| Property | Time Domain | Frequency Domain |
|---|---|---|
| Linearity | $$\displaystyle a x_1(t) + b x_2(t) $$ | $$\displaystyle a X_1(\omega) + b X_2(\omega) $$ |
| Time-Shifting | $$\displaystyle x(t - t_0) $$ | $$\displaystyle X(\omega) e^{-j\omega t_0} $$ |
| Frequency-Shifting | $$\displaystyle x(t) e^{j\omega_0 t} $$ | $$\displaystyle X(\omega - \omega_0) $$ |
| Time-Scaling | $x(at)$ | $$\displaystyle \frac{1}{|a|} X(\omega/a) $$ |
| Duality | $X(t)$ | $2\pi x(-\omega)$ |
| Convolution | $$\displaystyle x_1(t) * x_2(t) $$ | $$\displaystyle X_1(\omega) X_2(\omega) $$ |
| Parseval's Theorem | $$\displaystyle \int |x(t)|^2 dt = \frac{1}{2\pi} \int |X(\omega)|^2 d\omega $$ | Relates energy in time & frequency domains. |
\[!TIP\] Time-Scaling Proof (Jun 2022):
$$Y(\omega) = \int_{-\infty}^{\infty} x(at) e^{-j\omega t} dt$$
Substitute $$\displaystyle \tau = at \Rightarrow t = \tau/a, dt = d\tau/a $$:
$$Y(\omega) = \int_{-\infty}^{\infty} x(\tau) e^{-j\omega (\tau/a)} \frac{d\tau}{a} = \frac{1}{a} \int_{-\infty}^{\infty} x(\tau) e^{-j(\omega/a)\tau} d\tau = \frac{1}{a} X\left(\frac{\omega}{a}\right)$$
For $$\displaystyle a<0 $$, the absolute value ensures positive scaling: $$\displaystyle \frac{1}{|a|} X(\omega/a) $$.
3. FT of Standard Signals
| Signal $x(t)$ | Fourier Transform $X(\omega)$ |
|---|---|
| Unit Impulse $\delta(t)$ | $1$ |
| Unit Step $u(t)$ | $$\displaystyle \pi \delta(\omega) + \frac{1}{j\omega} $$ |
| Rectangular Pulse $\text{rect}(t/\tau)$ | $\tau \text{sinc}(\omega \tau / 2\pi)$ |
| Exponential $$\displaystyle e^{-at}u(t) $$ ($$\displaystyle a>0 $$) | $$\displaystyle \frac{1}{a + j\omega} $$ |
| Complex Exponential $$\displaystyle e^{j\omega_0 t} $$ | $$\displaystyle 2\pi \delta(\omega - \omega_0) $$ |
4. Frequency Response of LTI Systems
- For an LTI system with impulse response $h(t)$, the frequency response is:
$$H(\omega) = \mathcal{F}\{h(t)\} = \int_{-\infty}^{\infty} h(t) e^{-j\omega t} dt$$
- For input $x(t)$ with FT $X(\omega)$, output FT is: $$\displaystyle Y(\omega) = H(\omega) X(\omega) $$.
\[!EXAM TIP\] For Nov 2023-type question (causal LTI $$\displaystyle h(t)=e^{-2t}u(t) $$):
$$H(\omega) = \int_{0}^{\infty} e^{-2t} e^{-j\omega t} dt = \int_{0}^{\infty} e^{-(2+j\omega)t} dt = \left[ \frac{e^{-(2+j\omega)t}}{-(2+j\omega)} \right]_{0}^{\infty} = \frac{1}{2 + j\omega}$$
Magnitude: $$\displaystyle |H(\omega)| = \frac{1}{\sqrt{4 + \omega^2}} $$; Phase: $$\displaystyle \angle H(\omega) = -\tan^{-1}(\omega/2) $$.