[[1.01_Signals,Systems&Singularity_Functions|← 1.01 Signals, Systems & Singularity Functions]] | [[00_Chapter_Map-_Introduction_to_Signals|Chapter 1 Map]] | 2.01 Systems Classification, LTI Properties & Stability →
Chapter 1: Signal Fundamentals & Operations — Compact Review
This note compiles all core concepts, mathematical models, essential formulas, and comparison matrices for Chapter 1: Signal Fundamentals & Operations. Designed for rapid pre-exam review, all intermediate algebraic transitions are omitted, showing only the final boxable expressions.
1. Introduction to Signals & Systems
*(Target: Theory Descriptive / 2-Mark to 4-Mark Definitions)*
- Signal: Any physical quantity that carries information and varies with time, space, or any other independent variable. Mathematically modeled as a function of an independent variable (continuous-time) or index (discrete-time):
- System: A physical device, process, or mathematical algorithm that operates on an input signal to produce a transformed output signal :
- Chain of Signal Dependence: Event Sequence Time Frequency Phase Amplitude. Ultimately, all signal characteristics depend on time.
- Real-World Engineering Applications: Communication systems (transmitting/filtering voice/data), biomedical monitoring (ECG/EEG heart and brain signals), navigation (GPS satellite synchronization), geological tracking (seismic vibrations), and target detection (radar and sonar echoes).
2. Elementary Continuous-Time Singularity Functions
*(Target: Theory Descriptive / Graphical Sketching / 3-Mark Definitions)*
Singularity functions serve as highly idealized mathematical building blocks to model switching transitions, impulsive excitations, and boundary envelopes.
2.1 The Unit Step Function,
Models a DC switch closing instantaneously at : u(t) = egin{cases} 1, & t > 0 \ 0, & t < 0 \end{cases}
u(t)
│
1 ──┼──────────
│
────┴────────── t
0
2.2 The Unit Impulse Function (Dirac Delta),
An infinitely narrow, infinitely tall pulse centered at containing an area (weight) of exactly 1:
eq 0, \quad ext{and} \quad \int_{-\infty}^{\infty} \delta(t) \, dt = 1$$ ``` δ(t) (1) ↑ │ │ ────┼──┴─────── t 0 ``` ### 2.3 The Unit Ramp Function, $r(t)$ A linearly growing signal representing the time-integrated step function: $$r(t) = egin{cases} t, & t > 0 \ 0, & t \le 0 \end{cases} \quad \implies \quad r(t) = t \cdot u(t)$$ ``` r(t) │ / (Slope = 1) │ / │ / ────┼───┴────── t 0 ``` ### 2.4 The Signum Function, $ ext{sgn}(t)$ A polarity indicator extracting the algebraic sign of the time variable $t$: $$ ext{sgn}(t) = egin{cases} 1, & t > 0 \ 0, & t = 0 \ -1, & t < 0 \end{cases} \quad \implies \quad ext{sgn}(t) = u(t) - u(-t)$$ ``` sgn(t) │ 1 ──┼────────── ├─── 0 (at t=0) ────┼─── -1 │ ``` ### 2.5 The Rectangular Gate Pulse, $\Pi(t/ au)$ A symmetrical rectangular window centered at the origin with total duration (width) $ au$ and amplitude 1: $$\Pi\left( rac{t}{ au} ight) = egin{cases} 1, & - rac{ au}{2} < t < rac{ au}{2} \ 0, & ext{otherwise} \end{cases}$$ ``` Π(t/τ) ┌─────┐ 1 │ │ ─────┴──┬──┴───── t -τ/2 0 τ/2 ``` ### 2.6 The Sinc Function, $ ext{sinc}(t)$ Representing the frequency-domain spectrum of a continuous rectangular time pulse: $$ ext{sinc}(t) = rac{\sin(\pi t)}{\pi t}$$ ### 2.7 Key Calculus Relationships * **Integral relationship (unilateral step):** $$u(t) = \int_{-\infty}^{t} \delta( au) \, d au \quad ext{(derivation)}$$ * **Derivative relationship (impulse):** $$\delta(t) = rac{du(t)}{dt} \quad ext{(derivation)}$$ --- ## 3. Core Elementary Operations on the Independent Variable (Time) `*(Target: Numerical Solving / 4-Mark to 8-Mark Signal Sketching)*` Time-domain transformations alter when a signal occurs or how fast it unfolds without modifying its vertical amplitude. ### 3.1 Mathematical Definitions * **Time Shifting:** Delaying (right-shift) or advancing (left-shift) a signal along the time axis: $$y(t) = x(t - t_0) \quad ext{where} \quad egin{cases} t_0 > 0 \implies ext{Delay (Right-Shift)} \ t_0 < 0 \implies ext{Advance (Left-Shift)} \end{cases}$$ * **Time Scaling:** Compressing or expanding the signal along the time axis: $$y(t) = x(at) \quad ext{where} \quad egin{cases} |a| > 1 \implies ext{Time Compression (Squeezed)} \ |a| < 1 \implies ext{Time Expansion (Stretched)} \end{cases}$$ * **Time Reversal (Folding):** Reflecting the signal symmetrically across the vertical $y$-axis: $$y(t) = x(-t) \quad ext{or} \quad y[n] = x[-n]$$ ### 3.2 Standard Precedence Rule for Composite Operations When a signal contains simultaneous shifting, scaling, and reversal (e.g., evaluating $x(at - b)$): ``` [ x(t) ] ───( Step 1: Shift by b )───> [ x(t - b) ] │ ( Step 2: Scale/Reverse by a ) │ ▼ [ x(at - b) ] ``` > [!WARNING] > > **The Precedence Trap:** Shifting must **always** be performed before scaling/reversal when working directly on the un-factored expression $x(at - b)$. > Alternatively, if you scale first, you must factor the argument to scale the delay properly: $x\left(a(t - rac{b}{a}) ight)$, shifting by $ rac{b}{a}$ instead of $b$. --- ## 4. Signal Classifications (Structural & Predictability) `*(Target: Theory Descriptive / Comparison Tables)*` ### 4.1 Continuous-Time (CT) vs. Discrete-Time (DT) CT signals are defined over a continuous interval of time, whereas DT signals represent indexed sequences defined strictly at discrete integer instances. | Criterion | Continuous-Time (CT) Signals | Discrete-Time (DT) Signals | | :--- | :--- | :--- | | **Independent Variable** | Time $t \in \mathbb{R}$ (Continuous) | Sample Index $n \in \mathbb{Z}$ (Discrete Integer) | | **Mathematical Form** | $x(t)$ | $x[n]$ | | **Standard Plot Type** | Continuous curves | Stem (lollipop) discrete sequences | ### 4.2 Analog vs. Digital Signals Analog signals possess a continuous range of amplitude values, while digital signals restrict their amplitudes to a quantized finite set. | Criterion | Analog Signals | Digital Signals | | :--- | :--- | :--- | | **Amplitude Range** | Continuous (infinite possible values in a range) | Quantized (discrete, finite set of levels) | | **Axis Definition** | Usually continuous-time | Strictly discrete-time | | **Enabling Process** | Physical measurements (transducer outputs) | **Quantization** of discrete-time values | ### 4.3 Deterministic vs. Random (Nondeterministic) Deterministic signals can be perfectly described by a closed-form algebraic formula, whereas random signals are governed by statistical probability densities. | Criterion | Deterministic Signals | Random (Nondeterministic) Signals | | :--- | :--- | :--- | | **Predictability** | 100% predictable for any future time instant | Unpredictable; future values can only be described statistically | | **Mathematical Modeling** | Exact time-domain functions (e.g., $A\cos(\omega t)$) | Probability density functions, mean, and variance | | **Physical Example** | Pure sinusoidal carrier wave | Atmospheric static, thermal semiconductor **noise** | --- ## 5. Periodic vs. Aperiodic Signals & Composite Periodicity `*(Target: Theory Descriptive / 5-Mark to 6-Mark Numerical Solving)*` A signal is periodic if it repeats its amplitude profile exactly after a constant fundamental period $T_0$ (for CT) or $N$ (for DT). ### 5.1 Continuous-Time Periodicity * **Definition Condition:** $$x(t + T) = x(t) \quad ext{for all } t \in \mathbb{R}$$ * **Fundamental Period ($T_0$):** The smallest positive non-zero value of $T$ satisfying the definition condition. ### 5.2 Periodicity Test for Composite CT Signals (The Rationality Rule) For a composite signal $x(t) = x_1(t) + x_2(t) + \dots + x_k(t)$ with individual periods $T_1, T_2, \dots, T_k$: The composite signal is periodic if and only if the ratios of all individual periods represent rational numbers: $$ rac{T_i}{T_j} \in \mathbb{Q} \quad ( ext{ratio of two integers } rac{p}{q}) \quad ext{(derivation)}$$ If this criterion is met, the fundamental period $T_0$ is evaluated using the fractional Least Common Multiple (LCM) formula: $$ ext{LCM}\left( rac{a}{b}, rac{c}{d} ight) = rac{ ext{LCM}(a, c)}{ ext{HCF}(b, d)}$$ *(where $ ext{HCF}$ is the Highest Common Factor, or Greatest Common Divisor).* ### 5.3 Discrete-Time Periodicity * **Definition Condition:** $$x[n + N] = x[n] \quad ext{for all } n \in \mathbb{Z}$$ * **The DT Sinusoid Periodicity Constraint:** A discrete sinusoid $x[n] = A\cos(\omega_0 n + heta)$ is periodic **if and only if** its angular frequency $\omega_0$ is a rational multiple of $2\pi$: $$ rac{\omega_0}{2\pi} = rac{m}{N} \in \mathbb{Q} \quad \implies \quad N = m \left( rac{2\pi}{\omega_0} ight) \quad ( ext{for integer } N)$$ --- ## 6. Time Boundary Classifications (Causal, Anticausal, Non-causal) `*(Target: Theory Descriptive / 4-Mark to 5-Mark Signal Classification)*` This classification evaluates where a signal is defined on the time axis relative to the origin $t = 0$. ``` Anticausal Signals Causal Signals (Only negative time) (Only positive time) ───────────────────────┬────────────────────────> t 0 └─────────── Non-causal Signals ───────────────┘ ``` * **Causal Signal:** Exists only in positive time ($t \ge 0$): $$x(t) = 0 \quad ext{for } t < 0 \quad \implies \quad x(t) \cdot u(t) \equiv ext{Causal}$$ * **Anticausal Signal:** Exists only in negative time ($t \le 0$): $$x(t) = 0 \quad ext{for } t > 0 \quad \implies \quad x(t) \cdot u(-t) \equiv ext{Anticausal}$$ * **Non-causal Signal:** Exists on both sides of the origin (extends into both negative and positive time boundaries). --- ## 7. Waveform Symmetry (Even and Odd Decompositions) `*(Target: 7-Mark to 10-Mark Mathematical Proofs)*` ### 7.1 Symmetry Types * **Even Signal (Symmetrical across y-axis):** $$x(t) = x(-t)$$ * **Odd Signal (Anti-symmetrical across origin):** $$x(t) = -x(-t)$$ ### 7.2 Decomposition Formulas Any arbitrary signal $x(t)$ can be split into a sum of its even and odd parts: $$x(t) = x_e(t) + x_o(t)$$ * **Even Part:** $$x_e(t) = rac{x(t) + x(-t)}{2}$$ * **Odd Part:** $$x_o(t) = rac{x(t) - x(-t)}{2}$$ ### 7.3 Even-Odd Energy Orthogonality Proof [!theorem] Symmetrical Integral Identity Prove that the total energy of a signal equals the sum of the energies of its even and odd components: $$\int_{-\infty}^{\infty} x^2(t) \, dt = \int_{-\infty}^{\infty} x_e^2(t) \, dt + \int_{-\infty}^{\infty} x_o^2(t) \, dt \quad ext{(derivation)}$$ --- ## 8. Energy and Power Classifications `*(Target: Theory Descriptive / 5-Mark to 6-Mark Mathematical Classification)*` Measures a signal's physical dissipation properties over infinite boundaries to see if it decays (Energy) or continues indefinitely (Power). ### 8.1 Continuous-Time Definitions * **Total Energy ($E$):** $$E = \int_{-\infty}^{\infty} |x(t)|^2 \, dt$$ * **Average Power ($P$):** $$P = \lim_{T o \infty} rac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 \, dt$$ ### 8.2 Discrete-Time Definitions * **Total Energy ($E$):** $$E = \sum_{n=-\infty}^{\infty} |x[n]|^2$$ * **Average Power ($P$):** $$P = \lim_{N o \infty} rac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$$ ### 8.3 The Golden Classification Rules 1. **Energy Signal:** Finite energy and zero average power: $$0 < E < \infty \quad ext{and} \quad P = 0$$ *(Examples: Transient pulses, single square gates, exponentially decaying signals).* 2. **Power Signal:** Finite average power and infinite energy: $$0 < P < \infty \quad ext{and} \quad E = \infty$$ *(Examples: All periodic signals, infinite-duration step signals).* 3. **Mutual Exclusivity:** A signal cannot represent both classifications simultaneously. If a signal grows without bound (e.g., $e^{at}$ or $t \cdot u(t)$), it represents **neither** ($E = \infty, P = \infty$). --- ## 9. Typical Signal Processing Operations `*(Target: Theory Descriptive / 3-Mark to 4-Mark Short Notes)*` * **Correlation:** Measures the mathematical similarity between signals. Unlike convolution, correlation does not involve time-reversal (folding): $$R_{xy}( au) = \int_{-\infty}^{\infty} x(t) y(t + au) \, dt$$ * **Filtering:** Selective frequency suppression to reshape spectra. Categorized into Low-Pass Filter (LPF), High-Pass Filter (HPF), Band-Pass Filter (BPF), Band-Stop Filter (BSF), and sharp notch-cut suppressors. * **Modulation:** Shifting a low-frequency baseband message $m(t)$ up to a high-frequency carrier band for wireless transmission: $$y(t) = m(t) \cos(\omega_c t)$$ * **Transformation:** Coordinate-domain mapping to simplify analysis. Maps differential equations to simple algebraic polynomials (Laplace, Fourier, Z-transformations). * **Multiplexing:** Combining multiple data streams over a shared medium: * **Time Division Multiplexing (TDM):** Interleaving signals in non-overlapping time slots. * **Frequency Division Multiplexing (FDM):** Assigning distinct frequency bands separated by guard bands. * **Code Division Multiplexing (CDM):** Separating channels using unique orthogonal codes. --- ## 10. Signal Hold & Reconstruction Circuits `*(Target: Theory Descriptive / Circuit Modeling)*` [147, 4.03] Physical reconstruction of analog signals from their discrete samples requires signal hold configurations [147, 4.03]. * **Zero-Order Hold (ZOH):** Holds the amplitude of the last sample constant until the next sample is acquired [147, 4.03]: $$h_{ ext{ZOH}}(t) = u(t) - u(t - T_s)$$ Creates a staircase-like waveform with high-frequency spectral droop (the Aperture Effect) [4.03]. * **First-Order Hold (FOH):** Performs linear interpolation between consecutive samples [147, 4.03], generating a continuous piecewise linear ramp. --- ## 11. Common Mistakes That Cost Marks > [!CAUTION] **Critical Exam Pitfalls** > > 1. **Direct-Period Sinusoid Scaling Assumption:** Swapping continuous and discrete angular frequency properties. A continuous sinusoid $A\cos(\omega_0 t)$ is **always** periodic regardless of $\omega_0$. A discrete sinusoid $A\cos(\omega_0 n)$ is periodic **if and only if** $\omega_0/2\pi$ is a rational number. > 2. **Fractional LCM Swap:** Inverting the fractional LCM formula. Remember that the Least Common Multiple of fractional periods is $ rac{ ext{LCM of numerators}}{ ext{HCF of denominators}}$. Writing HCF/LCM is a guaranteed zero-marks error. > 3. **Even/Odd Cross-Product Omission:** Dropping the cross-product evaluation during the Even-Odd energy identity proof. You **must** explicitly state and prove that the integral of $x_e(t)x_o(t)$ is zero because the product is an odd function. > 4. **Signal Multiplication Energy Multiplying:** Assuming the energy of a combined signal is $E_1 + E_2$ for any arbitrary sum. This is **only** true if the signals are strictly orthogonal ($\int x_1(t)x_2(t) \, dt = 0$). --- ## 12. PYQ Bank — Verbatim Questions & Answer Plans ### Q1: [PYQ 2022 - 8 Marks] Define: (i) Unit step function, (ii) Unit impulse function, (iii) Ramp function, and (iv) Signum function. * *Answer Plan:* Provide the mathematical piecewise conditions and hand-sketch the continuous-time waveforms with labeled axes. ### Q2: [PYQ 2024, 2023, 2021 - 7 to 10 Marks] Show that: $\int_{-\infty}^{\infty} x^2(t) \, dt = \int_{-\infty}^{\infty} x_e^2(t) \, dt + \int_{-\infty}^{\infty} x_o^2(t) \, dt$. * *Answer Plan:* Substitute $x(t) = x_e(t) + x_o(t)$, square the term, and prove that the cross-product term integrates to zero due to odd symmetry over symmetrical limits. ### Q3: [PYQ 2018 - 6 Marks] Distinguish between: (i) deterministic and nondeterministic signals, (ii) even and odd signals. * *Answer Plan:* Use comparative Markdown tables detailing predictability, mathematical models, boundary definitions, and symmetrical axes. ### Q4: [PYQ 2017 - 6 Marks] Determine whether $x[t] = \sin(20\pi t) + \sin(5\pi t)$ is periodic and find its period. * *Answer Plan:* Evaluate individual periods $T_1 = 1/10$ s, $T_2 = 2/5$ s, verify the rational ratio (1/4), and compute LCM(1/10, 2/5) = 2/5 = 0.4 seconds. --- ## 13. Self-Check Before Moving On - [ ] Can you define and sketch $u(t)$, $\delta(t)$, $r(t)$, and $ ext{sgn}(t)$ with correct vertical limits? - [ ] Can you solve a composite periodicity problem using fractional LCM? - [ ] Can you write out the complete Even-Odd energy proof without skipping steps? - [ ] Do you know the exact conditions for a signal to be Energy, Power, or Neither? --- _Source: (k.Deergha Rao) signals and systems.pdf, 1.01 Signals, Systems & Singularity Functions.md, 1.03 Signal Classification I — Theory & Periodicity.md, 1.04 Signal Classification II — Symmetry, Energy & Power.md_