01 Chapter Map - Signal Fundamentals & Operations
Chapter 1 Overview & Map of Content (MOC)
Foundational continuous-time signal classifications, singularity functions, elementary operations, parity decompositions, and hold-reconstruction circuits.
📚 Study Notes Index
Read in order — each note assumes the previous one.
| # | Note | What it covers |
|---|---|---|
| 1.00 | 1.00 Signal Fundamentals and Operations Compact Review | Signal Operations Compact Review, Chapter 1 Formula Sheet |
| 1.01 | 1.01 Signals, Systems & Singularity Functions | Signal Definition, System Definition, Unit Step Function, Unit Impulse Function, Ramp Function, Signum Function, Rectangular Pulse |
| 1.02 | 1.02 Elementary Operations on Signals | Time Shifting, Time Scaling, Time Reversal, Signal Folding, Discrete Signal Operations |
| 1.03 | 1.03 Signal Classification I — Theory & Periodicity | Signal Classification, Continuous-Time Signal, Discrete-Time Signal, Analog Signal, Digital Signal, Deterministic Signal, Random Signal, Periodic Signal, Aperiodic Signal, Periodicity Test |
| 1.04 | 1.04 Signal Classification II — Symmetry, Energy & Power | Causal Signal, Anti-causal Signal, Even Signal, Odd Signal, Even-Odd Decomposition, Energy Signal, Power Signal |
| 1.05 | 1.05 The Sampling Theorem Bridge & Multi-Rate Processing | Sampling Theorem, Nyquist Rate, Aliasing, Natural Sampling, Flat-top Sampling, Down-sampling, Up-sampling, Decimation, Interpolation |
| 1.06 | 1.06 Typical Signal Processing Operations | Typical Operations, Signal Operations, Correlation, Modulation, Filtering, Multiplexing |
| 1.07 | 1.07 Supplementary Signal Functions Hold Circuits and DAC Re | Sinc and Exponential Signals, ZOH and FOH, DAC Reconstruction |
🎯 Exam Weight
ECE 2107 Exam Relevance
Master the core derivations, mathematical definitions, and problem-solving techniques. Refer to ECE 2107 - Signals and Systems for syllabus boundaries and past year questions.
🔗 Related Resources
- Course Teaching Plan: ECE 2107 - Signals and Systems
- Previous chapter: None
- Next chapter: 02 Chapter Map - LTI System Properties & Time-Domain Convolution
[[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_ --- **Related Concepts:** [[1.02 Elementary Operations on Signals]] | [[1.03 Signal Classification I — Theory & Periodicity]] | [[1.04 Signal Classification II — Symmetry, Energy & Power]] # 1.01 Signals, Systems & Singularity Functions > [!abstract] Course Focus & Fundamentals > > These concepts form the absolute prerequisite for all time-domain and frequency-domain transformations. Master these definitions, properties, and standard shapes for guaranteed exam marks. --- ## 1. Introduction to Signals and Systems ### 1.1 What is a Signal? > [!abstract] Definition: Signal > > A **signal** is defined as any physical quantity that carries information and varies with time, space, or any other independent variable or variables. > > $$x(t) \quad \text{or} \quad x[n]$$ - **Mathematical Context:** A signal is represented as a function of one or more independent variables. For example, $x(t)$ represents a continuous-time signal varying with time $t$. - **Physical Examples:** voltage or current waveforms, sound/pressure waves, or temperature measurements. - **Event mapping:** Event $\to$ Sequence $\to$ Time $\to$ Phase $\to$ Amplitude. (Frequency matters first, then phase, then amplitude, but ultimately everything depends on time). --- ### 1.2 What is a System? > [!abstract] Definition: System > > A **system** is a physical device, process, or mathematical algorithm that manipulates or operates on an input signal to produce a desired output signal. > > $$y(t) = \mathbb{T}[x(t)]$$ - **Mathematical Representation:** - $x(t)$ = Input (Excitation / Independent variable) - $y(t)$ = Output (Response / Dependent variable) - $\mathbb{T}[\cdot]$ = The mathematical operator or process acting on the input. ```mermaid graph LR x["Input Signal x(t)"] --> System["System Operator T[•]"] System --> y["Output Signal y(t) = T[x(t)]"] ``` --- ### 1.3 Importance and Real-World Applications Signals and systems are the backbone of communication engineering, allowing us to transmit, filter, and extract data from noise. - **Key Applications:** Speech processing, television broadcast transmission, image/video compression, physiological signals (ECG/EEG), radar/sonar echoes, seismic vibration detection, GPS satellite tracking, and genetic sequence analysis. --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2025]:** What is signal & system? (04 Marks) > * **[PYQ 2024]:** What are signal and system? What are the importance of these in communication engineering? (06 Marks) - #missed-in-note > * **[PYQ 2023]:** What are meant by signal and system? (04 Marks) > * **[PYQ 2020]:** Define: (i) Signal, (ii) System. > * **[PYQ 2024, 2023, 2019, 2018, 2016]:** Define system. What are the properties of a system? (08 Marks) *(Note: Properties are covered in Chapter 2)* > * **[PYQ 2021]:** Define signal. Classify signals based on major criteria. (06 Marks) *(Note: Classifications are covered in Note 1.03 & 1.04)* --- ## 2. Basic Continuous-Time Singularity Functions Singularity functions are mathematical idealizations used as basic building blocks to construct or analyze more complex signals. ### 2.1 The Unit Step Function, $u(t)$ > [!abstract] Unit Step Function > > The unit step function represents a signal that turns "on" at $t=0$ and stays at a constant amplitude of 1. It models a DC switch closing. > > $$u(t) = \begin{cases} 1, & t > 0 \\ 0, & t < 0 \end{cases}$$ ```desmos-graph left = -4; right = 4 top = 2; bottom = -1 --- y = 1 | x > 0 | BLUE | SOLID y = 0 | x < 0 | BLUE | SOLID (0, 1) | label:u(t) | BLUE (0, 0) | open | BLUE ``` --- ### 2.2 The Unit Impulse Function (Dirac Delta), $\delta(t)$ > [!abstract] Unit Impulse Function > > An idealized pulse that is infinitely narrow, infinitely tall, but contains a total area (weight) of exactly 1. It is the derivative of the unit step function. > > $$\delta(t) = 0 \quad \text{for } t \neq 0$$ > $$\int_{-\infty}^{\infty} \delta(t) dt = 1$$ ```desmos-graph left = -2; right = 2 top = 2; bottom = -0.5 --- x = 0 | 0 <= y <= 1 | BLUE | SOLID (0, 1) | label:δ(t) (Area = 1) | BLUE ``` *Note: In physical drawings, the impulse $\delta(t)$ is represented by an upward-pointing arrow at the origin with parentheses denoting its area/weight.* --- ### 2.3 The Ramp Function, $r(t)$ > [!abstract] Ramp Function > > A signal that grows linearly with time for $t \ge 0$. It is the integral of the unit step function. > > $$r(t) = \begin{cases} t, & t > 0 \\ 0, & t \le 0 \end{cases} \quad \implies r(t) = t \cdot u(t)$$ ```desmos-graph left = -2; right = 4 top = 4; bottom = -1 --- y = x | x > 0 | BLUE | SOLID y = 0 | x <= 0 | BLUE | SOLID (0, 0) | BLUE ``` --- ### 2.4 The Signum Function, $\text{sgn}(t)$ > [!abstract] Signum Function > > A function that extracts the mathematical sign of the time variable $t$. > > $$\text{sgn}(t) = \begin{cases} 1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases} \quad \implies \text{sgn}(t) = u(t) - u(-t)$$ ```desmos-graph left = -4; right = 4 top = 2; bottom = -2 --- y = 1 | x > 0 | BLUE | SOLID y = -1 | x < 0 | BLUE | SOLID (0, 0) | label:sgn(t) = 0 | BLUE (0, 1) | open | BLUE (0, -1) | open | BLUE ``` --- ### 2.5 The Rectangular Pulse Function, $\Pi(t/\tau)$ > [!abstract] Rectangular Pulse Function > > A symmetrical gate pulse centered at the origin with a total duration (width) of $\tau$ and an amplitude of 1. > > $$\Pi\left(\frac{t}{\tau}\right) = \begin{cases} 1, & -\frac{\tau}{2} < t < \frac{\tau}{2} \\ 0, & \text{otherwise} \end{cases}$$ For example, a unit rectangular pulse with width $\tau = 2$ is plotted below: ```desmos-graph left = -3; right = 3 top = 2; bottom = -1 --- y = 1 | -1 < x < 1 | BLUE | SOLID y = 0 | x < -1 | BLUE | SOLID y = 0 | x > 1 | BLUE | SOLID (-1, 0) | open | BLUE (-1, 1) | open | BLUE (1, 0) | open | BLUE (1, 1) | open | BLUE ``` --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2022]:** Define: (i) Unit step function, (ii) Unit impulse function, (iii) Ramp function, and (iv) Signum function. (08 Marks) --- # 1.02 Elementary Operations on Signals > [!abstract] Focus & Objective > > Continuous and discrete signals undergo mathematical transformations along their horizontal (independent variable) axis. Shifting, scaling, and reversal are heavily tested, both individually and in combination. --- ## 1. Core Operations on the Independent Variable (Time) Manipulating the independent variable (time $t$ for continuous, or index $n$ for discrete) alters *when* and *how fast* a signal occurs without changing its amplitude. ### 1.1 Time Shifting Time shifting delays or advances the signal along the horizontal axis. It does not change the shape or width of the signal. * **Time Delay (Right Shift):** Mathematically represented as $x(t - t_0)$ where $t_0 > 0$. The signal occurs *later*. * **Time Advance (Left Shift):** Mathematically represented as $x(t + t_0)$ where $t_0 > 0$. The signal occurs *earlier*. ```desmos-graph left = -6; right = 6 top = 2; bottom = -0.5 --- y = 1 | -1 < x < 1 | BLUE | SOLID y = 1 | 2 < x < 4 | RED | DASHED y = 1 | -4 < x < -2 | GREEN | DASHED (0, 1) | label:x(t) | BLUE (3, 1) | label:x(t-3) [Delay] | RED (-3, 1) | label:x(t+3) [Advance] | GREEN ``` #self-question why right shift is called delay when its like in advance in x axis? --- ### 1.2 Time Scaling Time scaling speeds up or slows down the signal by multiplying the time variable. * **Compression:** Mathematically represented as $x(at)$ where $a > 1$. The signal occurs faster, shrinking horizontally. (e.g., $x(2t)$ is twice as fast, taking half the time). * **Expansion:** Mathematically represented as $x(at)$ where $0 < a < 1$. The signal occurs slower, stretching horizontally. (e.g., $x(0.5t)$ takes twice as long). Below is the original triangle $x(t)$ (blue), compressed $x(2t)$ (red dashed), and expanded $x(t/2)$ (green dotted): ```desmos-graph left = -5; right = 5 top = 1.5; bottom = -0.5 --- y = 1 - 0.5*abs(x) | -2 < x < 2 | BLUE | SOLID y = 1 - abs(x) | -1 < x < 1 | RED | DASHED y = 1 - 0.25*abs(x) | -4 < x < 4 | GREEN | DOTTED (0, 1.1) | label:x(t) | BLUE (0, 0.7) | label:x(2t) | RED (0, 0.3) | label:x(t/2) | GREEN ``` --- ### 1.3 Time Reversal (Folding) Time reversal flips the signal across the vertical axis ($t=0$). * **Mathematical Representation:** $x(-t)$. * Whatever happened at $t = t_0$ now happens at $t = -t_0$. If a signal is symmetric about $t=0$, then $x(t) = x(-t)$ (an even signal), and reversal has no visible effect. Below is an asymmetrical original signal $x(t)$ (blue) and its reversed version $x(-t)$ (red): ```desmos-graph left = -3; right = 3 top = 1.5; bottom = -0.5 --- y = 0.5*x | 0 < x < 2 | BLUE | SOLID y = -0.5*x | -2 < x < 0 | RED | SOLID (1.5, 0.8) | label:x(t) | BLUE (-1.5, 0.8) | label:x(-t) | RED ``` --- ## 2. Processing of Discrete Signals > [!question] High-Yield PYQ > > * **[PYQ 2019, 2018, 2017, 2015]:** What is signal? Explain the operations performed on the independent variables in the processing of discrete signals mathematically and graphically. (12 Marks) When answering for discrete signals $x[n]$, the operations must use integer indices $n$ and sample-by-sample representations. ### 2.1 Discrete Time Shifting * **Mathematical:** $y[n] = x[n - k]$. If $k > 0$, the sequence is shifted right (delayed) by $k$ samples. If $k < 0$, it is shifted left (advanced) by $|k|$ samples. * **Graphical (ASCII Representation):** ```text x[n]: 3 | o 2 | o | 1 | o | | -------+-+-+-+-+--> n 0 1 2 3 y[n] = x[n-2] (Delayed by 2): 3 | o 2 | o | 1 | o | | -------+-+-+-+-+-+--> n 0 1 2 3 4 ``` --- ### 2.2 Discrete Time Reversal (Folding) * **Mathematical:** $y[n] = x[-n]$. The sequence is mirrored around the origin $n = 0$. * **Graphical (ASCII Representation):** ```text x[n]: 2 | o 1 | o | -------+-+-+-+--> n 0 1 2 y[n] = x[-n] (Folded): 2 | o 1 | | o -------+-+-+-+--> n -2 -1 0 ``` --- ### 2.3 Discrete Time Scaling Unlike continuous time scaling, discrete time scaling is non-trivial because the independent variable must remain an integer. * **Down-sampling (Decimation / Compression):** $y[n] = x[Mn]$ where $M$ is an integer $> 1$. The system keeps only every $M$-th sample and discards the rest. **This results in data loss.** * **Up-sampling (Interpolation / Expansion):** $y[n] = x[n/M]$ only for indices where $n/M$ is an integer. For all other values, $y[n] = 0$. This inserts $M-1$ zeros between every sample. ```text x[n] = {0, 1, 2, 3, 4} for n=0..4: 4 | o 3 | o | 2 | o | | 1 | o | | | -------+-+-+-+-+-+--> n 0 1 2 3 4 y[n] = x[2n] (Decimated by M=2; sample 1 and 3 are lost): 4 | o 2 | o | -------+-+-+-+--> n 0 1 2 y[n] = x[n/2] (Interpolated by M=2; zeros inserted): 4 | o 3 | o | 2 | o | | 1 | o | | | -------+-+-+-+-+-+-+-+-+-+--> n 0 1 2 3 4 5 6 7 8 ``` --- ## 3. Combined Operations & Sketching > [!warning] Order of Operations > > To avoid boundary errors when sketching $x(at - b)$, **always** follow this sequence: > > 1. **Shift First:** Plot the delayed or advanced signal $x(t - b)$. > 2. **Scale Second:** Replace $t$ with $at$ to compress or expand. > 3. **Reverse Last (if $a < 0$):** Flip the entire scaled result across the vertical $y$-axis. --- > [!example] Sketching PYQ 1 > > **Question:** Sketch the following signal: $x(t) = 4r(3t - 1)$ (2024, 2019) > > **Step-by-Step Solution:** > 1. **Base Signal:** The ramp function $r(t) = t \cdot u(t)$, starting at origin with slope = 1. > 2. **Amplitude Scaling:** $4r(t)$. Slope is now 4. > 3. **Time Shifting:** $4r(t - 1)$. Shift the starting point right to $t = 1$. The formula is $4(t-1)$ for $t \ge 1$. > 4. **Time Scaling (Compression):** $4r(3t - 1)$. > - Starting point: $3t - 1 = 0 \implies t = 1/3$. > - The new slope: Amplitude factor $\times$ Compression factor $= 4 \times 3 = 12$. > - **Mathematical formula:** $x(t) = 12t - 4$ for $t \ge 1/3$. ```desmos-graph left = -1; right = 2 top = 10; bottom = -2 --- y = 12*x - 4 | x > 1/3 | RED | SOLID y = 0 | x <= 1/3 | RED | SOLID (0.333, 0) | label:t=1/3 | RED (1, 8) | label:Slope = 12 | RED ``` --- > [!example] Sketching PYQ 2 > > **Question:** Sketch the following signal: $x(t) = r(-0.5t + 2)$ (2015) > > **Step-by-Step Solution:** > 1. **Factor terms inside:** $r(-0.5(t - 4))$ to prevent scaling-shifting confusion. > 2. **Base Signal:** $r(t)$, starting at 0, slope 1. > 3. **Time Reversal:** $r(-t)$. The ramp now goes to the left (exists for $t \le 0$). > 4. **Time Scaling (Expansion):** $r(-0.5t)$. The ramp expands, slope becomes shallower ($0.5$). > 5. **Time Shifting:** Shift right by 4 units $\implies r(-0.5(t - 4))$. > - Boundary: $-0.5t + 2 = 0 \implies t = 4$. > - Due to reversal, signal exists for $t \le 4$. > - Value at $t = 4$ is $0$. > - Value at $t = 0$ is $r(2) = 2$. > - Value at $t = -2$ is $r(3) = 3$. ```desmos-graph left = -2; right = 6 top = 4; bottom = -1 --- y = -0.5*x + 2 | x < 4 | GREEN | SOLID y = 0 | x >= 4 | GREEN | SOLID (4, 0) | label:t=4 | GREEN (0, 2) | label:(0, 2) | GREEN ``` --- > [!example] Sketching PYQ 3 > > **Question:** Sketch the signal: $x(t) = u(t-5) - u(t-7)$ (2021) > > **Step-by-Step Solution:** > 1. $u(t-5)$ is a step function turning ON (+1) at $t = 5$. > 2. $-u(t-7)$ is a negative step turning OFF (-1 subtraction) at $t = 7$. > 3. Summing them creates a rectangular window pulse of width 2 between $t=5$ and $t=7$. ```desmos-graph left = 3; right = 9 top = 2; bottom = -1 --- y = 1 | 5 < x < 7 | PURPLE | SOLID y = 0 | x < 5 | PURPLE | SOLID y = 0 | x > 7 | PURPLE | SOLID (5, 0) | open | PURPLE (5, 1) | open | PURPLE (7, 0) | open | PURPLE (7, 1) | open | PURPLE ``` --- **Related Concepts:** [[1.01 Signals, Systems & Singularity Functions]] | [[1.02 Elementary Operations on Signals]] | [[1.04 Signal Classification II — Symmetry, Energy & Power]] # 1.03 Signal Classification I — Theory & Periodicity > [!abstract] Overview > > Signals are classified based on properties of their independent variables (time), dependent variables (amplitude), predictability, and repetition. Periodicity calculations are highly repeated numericals in Section A of examinations. ```mermaid graph TD S["Signal Classifications"] --> IV["By Independent Variable"] S --> DV["By Dependent Variable"] S --> PR["By Predictability"] S --> PER["By Repetition"] IV --> CT["Continuous-Time x(t)"] IV --> DT["Discrete-Time x[n]"] DV --> AN["Analog"] DV --> DIG["Digital"] PR --> DET["Deterministic"] PR --> RAN["Random / Noise"] PER --> PERIODIC["Periodic"] PER --> APERIODIC["Aperiodic"] ``` --- ## 1. Fundamental Classifications ### 1.1 Continuous-Time vs. Discrete-Time Signals This classification is based strictly on the **independent variable** (domain of definition). * **Continuous-Time (CT) Signal:** Defined for a continuous, unbroken range of time. The independent variable $t$ is continuous. - *Example:* $x(t) = \cos(\omega t)$ * **Discrete-Time (DT) Signal:** Defined only at specific, isolated integer time instances. The independent variable is an integer index $n$. - *Example:* $x[n] = \{1, -2, 3, 0\}$ ```text Continuous-Time x(t) [Smooth Curve]: Discrete-Time x[n] [Lollipop/Stem Plot]: ^ ^ 1 | _--_ 1 | o o | _- -_ | o | o | ----+-----------> t ----+----+---+-+-> n | -1 | 0 1 2 ``` --- ### 1.2 Analog vs. Digital Signals This classification is based strictly on the **dependent variable** (range of amplitude values). * **Analog Signal:** The amplitude can take on an infinite number of continuous values within a given range. * **Digital Signal:** The amplitude is restricted to a finite, discrete set of quantized values. * *Concept Note:* The process of mapping a continuous amplitude discrete-time signal into a quantized digital signal is called **quantization**. --- ### 1.3 Deterministic vs. Random (Nondeterministic) Signals * **Deterministic Signal:** A signal whose future values are completely specified and predictable. It is defined by a known mathematical formula. - *Example:* $x(t) = 5\sin(100\pi t)$ * **Random Signal:** A signal that is unpredictable, containing random, statistically characterized values. It lacks a specific predictable frequency, amplitude, or phase. **Noise** is the most common example. #update-note add a difference table since pyq asked for it. ```desmos-graph left = -2; right = 10 top = 1.5; bottom = -1.5 --- y = sin(2*x) | BLUE | SOLID (2.5, 1.2) | label:Deterministic Sine wave | BLUE ``` --- ### 1.4 Periodic vs. Aperiodic Signals * **Periodic Signal:** A signal $x(t)$ is periodic if it exactly repeats its pattern after a specific, constant time interval $T$. - **Mathematical Condition:** $$x(t + T) = x(t) \quad \text{for all } t$$ where the smallest positive non-zero value $T_0$ satisfying this is the **Fundamental Period**. * **Aperiodic Signal:** Any signal that does not satisfy the periodicity condition (does not repeat). Below is a periodic sine wave (blue) and an aperiodic decaying sine wave (red): ```desmos-graph left = -1; right = 10 top = 1.5; bottom = -1.5 --- y = sin(2*x) | BLUE | SOLID y = exp(-0.4*x)*sin(2*x) | x >= 0 | RED | SOLID (4, 1.2) | label:Periodic | BLUE (4.5, 0.4) | label:Aperiodic Decaying | RED ``` --- ## 2. Periodicity Test for Composite Signals Examiners frequently ask whether a sum of two or more continuous-time signals is periodic, and if so, what its fundamental period is. ### 2.1 The Rationality Rule Let a continuous-time signal be $x(t) = x_1(t) + x_2(t)$, with fundamental periods $T_1$ and $T_2$. The composite signal $x(t)$ is periodic **if and only if** the ratio of their individual periods is a **Rational Number** (a ratio of two integers). $$\frac{T_1}{T_2} = \frac{p}{q} \quad (p, q \in \mathbb{Z})$$ If the ratio is rational, the fundamental period $T_0$ is the **Least Common Multiple (LCM)** of $T_1$ and $T_2$. > [!tip] Sir's Math Shortcut for Fractional LCM > > To find the LCM of fractional values: > > $$\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$$ > > where $\text{HCF}$ is the Highest Common Factor (Greatest Common Divisor). --- ## 3. PYQ Step-by-Step Solutions > [!example] Trap 1: The Irrational Ratio > > **Question:** Determine whether the signal $x(t) = \sin(\sqrt{2}\pi t) + \sin(15\pi t)$ is periodic or not. Find its fundamental period if periodic. (2019, 2015) > > **Solution:** > 1. Find individual frequencies and periods: > - Term 1: $\omega_1 = \sqrt{2}\pi \implies T_1 = \frac{2\pi}{\sqrt{2}\pi} = \sqrt{2}\text{ s}$ > - Term 2: $\omega_2 = 15\pi \implies T_2 = \frac{2\pi}{15\pi} = \frac{2}{15}\text{ s}$ > 2. Determine the ratio: > $$\frac{T_1}{T_2} = \frac{\sqrt{2}}{\frac{2}{15}} = \frac{15\sqrt{2}}{2}$$ > 3. Because $\sqrt{2}$ is an irrational number, this ratio is irrational. > 4. **Conclusion:** The signal is **Aperiodic** (does not have a fundamental period). --- > [!example] Trap 2: LCM of Fractions > > **Question:** Determine whether the signal $x(t) = \sin(20\pi t) + \sin(5\pi t)$ is periodic. If so, find its fundamental period. (2017) > > **Solution:** > 1. Find individual frequencies and periods: > - Term 1: $\omega_1 = 20\pi \implies T_1 = \frac{2\pi}{20\pi} = \frac{1}{10}\text{ s}$ > - Term 2: $\omega_2 = 5\pi \implies T_2 = \frac{2\pi}{5\pi} = \frac{2}{5}\text{ s}$ > 2. Determine the ratio: > $$\frac{T_1}{T_2} = \frac{\frac{1}{10}}{\frac{2}{5}} = \frac{5}{20} = \frac{1}{4}$$ > Since $\frac{1}{4}$ is a rational fraction, the signal is **Periodic**. > 3. Find the fundamental period $T_0$: > $$T_0 = \text{LCM}\left(\frac{1}{10}, \frac{2}{5}\right) = \frac{\text{LCM}(1, 2)}{\text{HCF}(10, 5)} = \frac{2}{5} = \mathbf{0.4\text{ seconds}}$$ --- > [!example] Trap 3: Single Term Irrational Frequency > > **Question:** Consider a signal $x(t) = 10\sin(\sqrt{5}\pi t)$. Is it periodic? Find its fundamental period. (2016) > > **Solution:** > * **Exam Warning:** The rational ratio rule only applies when adding *multiple* signals together. A single sinusoidal term is **always periodic**, even if its frequency is irrational. > * Fundamental frequency: $\omega = \sqrt{5}\pi\text{ rad/s}$ > * Fundamental period: > $$T_0 = \frac{2\pi}{\omega} = \frac{2\pi}{\sqrt{5}\pi} = \mathbf{\frac{2}{\sqrt{5}}\text{ seconds}}$$ --- > [!example] Trap 4: Mixed Sines & Cosines > > **Question:** Determine if $x(t) = 2\cos(4\pi t) + 3\sin(3\pi t)$ is periodic or aperiodic. If periodic, find the period. (2025) > > **Solution:** > 1. Term 1: $\omega_1 = 4\pi \implies T_1 = \frac{2\pi}{4\pi} = \frac{1}{2}\text{ s}$ > 2. Term 2: $\omega_2 = 3\pi \implies T_2 = \frac{2\pi}{3\pi} = \frac{2}{3}\text{ s}$ > 3. Ratio: > $$\frac{T_1}{T_2} = \frac{\frac{1}{2}}{\frac{2}{3}} = \frac{3}{4} \quad (\text{Rational } \implies \text{Periodic})$$ > 4. Fundamental period: > $$T_0 = \text{LCM}\left(\frac{1}{2}, \frac{2}{3}\right) = \frac{\text{LCM}(1, 2)}{\text{HCF}(2, 3)} = \frac{2}{1} = \mathbf{2\text{ seconds}}$$ --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2021, 2019, 2018, 2016, 2015]:** What is signal? What are the major classifications of signals? (06 to 10 Marks) > * **[PYQ 2020]:** Classify discrete time signals. > * **[PYQ 2018]:** Distinguish between deterministic and nondeterministic signals. --- **Related Concepts:** [[1.01 Signals, Systems & Singularity Functions]] | [[1.02 Elementary Operations on Signals]] | [[1.03 Signal Classification I — Theory & Periodicity]] # 1.04 Signal Classification II — Symmetry, Energy & Power > [!abstract] Overview > > Signals are further classified mathematically based on time boundaries, symmetry across axes, and physical energy/power content. --- ## 1. Causal, Anti-causal, and Non-causal Signals This classification is determined entirely by where the signal exists relative to the $t=0$ boundary on the horizontal axis. * **Causal Signal:** Only exists in positive time. It starts at or after $t=0$. - **Mathematical Condition:** $x(t) = 0$ for $t < 0$. - *Tip:* Any signal multiplied by $u(t)$ is automatically forced to be causal. * **Anti-causal Signal:** Only exists in negative time. It ends at or before $t=0$. - **Mathematical Condition:** $x(t) = 0$ for $t > 0$. - *Tip:* Any signal multiplied by $u(-t)$ is anti-causal. * **Non-causal Signal:** Exists on both sides of the $t=0$ boundary (extends into both positive and negative time). --- ### Graphical Examples #### Causal Signal ($e^{-2t}u(t)$) ```desmos-graph left = -3; right = 4 top = 1.5; bottom = -0.5 --- y = exp(-2*x) | x >= 0 | BLUE | SOLID y = 0 | x < 0 | BLUE | SOLID (0, 1) | label:Causal e^(-2t)u(t) | BLUE ``` --- #### Anti-causal Signal ($e^{2t}u(-t)$) ```desmos-graph left = -4; right = 3 top = 1.5; bottom = -0.5 --- y = exp(2*x) | x <= 0 | RED | SOLID y = 0 | x > 0 | RED | SOLID (0, 1) | label:Anti-causal e^(2t)u(-t) | RED ``` --- #### Non-causal Signal ($e^{-t^2}$) ```desmos-graph left = -3; right = 3 top = 1.5; bottom = -0.5 --- y = exp(-x^2) | BLUE | SOLID (0, 1) | label:Non-causal e^(-t^2) | BLUE ``` --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2025, 2020]:** State whether the following signals are causal, anticausal or noncausal. (i) $x(t) = e^{-2t}u(t)$; (ii) $x(t) = t u(t)$. (04 Marks) > - **Solution:** Both signals are multiplied by $u(t)$. Since $u(t) = 0$ for $t < 0$, both $x(t) = 0$ for $t < 0$, making them strictly **Causal** signals. --- ## 2. Even and Odd Signals (Symmetry) Any arbitrary signal $x(t)$ can be decomposed into a sum of an even component $x_e(t)$ and an odd component $x_o(t)$. * **Even Signal:** Symmetrical across the vertical $y$-axis. $$x(t) = x(-t)$$ * **Odd Signal:** Anti-symmetrical across the origin. $$x(t) = -x(-t)$$ * **Decomposition Formulas:** $$x_e(t) = \frac{x(t) + x(-t)}{2}$$ $$x_o(t) = \frac{x(t) - x(-t)}{2}$$ $$x(t) = x_e(t) + x_o(t)$$ --- ### Graphical Decomposition of $u(t)$ #### Even Component of Step Function: $u_e(t) = 1/2$ ```desmos-graph left = -4; right = 4 top = 1; bottom = -1 --- y = 0.5 | BLUE | SOLID (0, 0.5) | label:ue(t) = 1/2 | BLUE ``` --- #### Odd Component of Step Function: $u_o(t) = \frac{1}{2}\text{sgn}(t)$ ```desmos-graph left = -4; right = 4 top = 1; bottom = -1 --- y = 0.5 | x > 0 | RED | SOLID y = -0.5 | x < 0 | RED | SOLID (0, 0) | label:uo(t) = 1/2 sgn(t) | RED (0, 0.5) | open | RED (0, -0.5) | open | RED ``` --- ### Even/Odd Energy Component Proof > [!theorem] Symmetrical Definite Integral Energy Identity > > **Question:** If $x_e(t)$ and $x_o(t)$ are the even and odd components of $x(t)$ respectively, prove that: > > $$\int_{-\infty}^{\infty} x^2(t) dt = \int_{-\infty}^{\infty} x_e^2(t) dt + \int_{-\infty}^{\infty} x_o^2(t) dt$$ > *(PYQ 2024, 2023, 2021)* **Proof:** 1. Express $x(t)$ as the sum of its components: $$x(t) = x_e(t) + x_o(t)$$ 2. Square both sides and integrate over all time: $$\int_{-\infty}^{\infty} x^2(t) dt = \int_{-\infty}^{\infty} \left[x_e(t) + x_o(t)\right]^2 dt$$ $$\int_{-\infty}^{\infty} x^2(t) dt = \int_{-\infty}^{\infty} x_e^2(t) dt + \int_{-\infty}^{\infty} x_o^2(t) dt + 2\int_{-\infty}^{\infty} x_e(t)x_o(t) dt$$ 3. Analyze the cross-product term: $x_e(t) \cdot x_o(t)$. - The product of an even function and an odd function is always an **odd function**. - The definite integral of any odd function over symmetrical limits ($-\infty$ to $\infty$) is mathematically equal to **zero**. $$2\int_{-\infty}^{\infty} x_e(t)x_o(t) dt = 0$$ 4. Therefore, the expression simplifies to: $$\int_{-\infty}^{\infty} x^2(t) dt = \int_{-\infty}^{\infty} x_e^2(t) dt + \int_{-\infty}^{\infty} x_o^2(t) dt \quad \blacksquare$$ --- ## 3. Energy and Power Signals This classification measures whether a signal eventually decays to zero (Energy) or continues indefinitely with steady amplitude (Power). ### 3.1 Mathematical Definitions * **Total Energy ($E$):** - **Continuous-Time:** $E = \int_{-\infty}^{\infty} |x(t)|^2 dt$ - **Discrete-Time:** $E = \sum_{n=-\infty}^{\infty} |x[n]|^2$ * **Average Power ($P$):** - **Continuous-Time:** $P = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 dt$ - **Discrete-Time:** $P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^{N} |x[n]|^2$ ```mermaid graph TD Start["Calculate E and P"] --> CheckE{"Is E finite?"} CheckE -->| "Yes: 0 < E < ∞" | Energy["Energy Signal: P = 0"] CheckE -->| "No: E = ∞" | CheckP{"Is P finite?"} CheckP -->| "Yes: 0 < P < ∞" | Power["Power Signal: E = ∞"] CheckP -->| "No: P = ∞" | Neither["Neither Energy nor Power"] ``` > [!tip] The Golden Classification Rules > > 1. **Energy Signal:** $0 < E < \infty$ and $P = 0$. (Most transient, finite-duration, or exponentially decaying pulses). > 2. **Power Signal:** $0 < P < \infty$ and $E = \infty$. (All periodic signals, and infinite-duration steps). > 3. A signal **cannot** be both. > 4. If a signal grows infinitely (e.g., $e^{at}$, $t \cdot u(t)$), then $E = \infty$ and $P = \infty$, so it is **neither**. --- ### 3.2 Numerical Trap Solutions > [!example] Trap 1: The Discrete Zero-Energy Identity > > **Question:** Determine whether the discrete signal $x[n]$ is an energy or power signal, and find its value: > $$x[n] = \begin{cases} \sin(\pi n), & n \ge 0 \\ 0, & \text{otherwise} \end{cases}$$ > *(PYQ 2025)* > > **Solution:** > 1. Note that $n$ is strictly an integer index. > 2. For any integer $n$, $\sin(\pi n) = 0$ (e.g., $\sin(0) = 0, \sin(\pi) = 0, \sin(2\pi) = 0$). > 3. Therefore, $x[n] = 0$ for all $n$. > 4. $$E = 0 \quad \text{and} \quad P = 0$$ > 5. **Conclusion:** It is technically **neither** an energy nor a power signal (it is a trivial zero signal). --- > [!example] Trap 2: Discrete Cosine Power Signal > > **Question:** Determine whether the signal $x[n]$ is an energy or power signal, and find its value: > $$x[n] = \begin{cases} \cos(\pi n), & n \ge 0 \\ 0, & \text{otherwise} \end{cases}$$ > *(PYQ 2023, 2019)* > > **Solution:** > 1. For integers $n$, $\cos(\pi n) = (-1)^n$. > 2. Calculate Energy: > $$E = \sum_{n=0}^{\infty} |(-1)^n|^2 = \sum_{n=0}^{\infty} 1 = \infty \quad \implies \text{Not an Energy Signal}$$ > 3. Calculate Power: > $$P = \lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=0}^{N} |(-1)^n|^2 = \lim_{N \to \infty} \frac{N+1}{2N+1}$$ > Using L'Hôpital's Rule or dividing by $N$: > $$P = \mathbf{\frac{1}{2}\text{ Watts}}$$ > 4. **Conclusion:** Since $E = \infty$ and $P = 0.5\text{ W}$, it is a **Power Signal**. --- > [!example] Trap 3: The Continuous-Time Step > > **Question:** Determine the power and energy of the continuous-time step function $x(t) = u(t)$. (2022) > > **Solution:** > 1. Calculate Energy: > $$E = \int_{-\infty}^{\infty} |u(t)|^2 dt = \int_{0}^{\infty} 1 dt = \infty$$ > 2. Calculate Power: > $$P = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} |u(t)|^2 dt = \lim_{T \to \infty} \frac{1}{T} \int_{0}^{T/2} 1 dt = \lim_{T \to \infty} \frac{T/2}{T} = \mathbf{\frac{1}{2}\text{ Watts}}$$ > 3. **Conclusion:** It is a **Power Signal** with $P = 0.5\text{ Watts}$. --- > [!example] Trap 4: Bounded Discrete Range > > **Question:** Determine whether the signal $x[n]$ is an energy or power signal, and find its value: > $$x[n] = \begin{cases} \cos(\pi n), & -4 \le n \le 4 \\ 0, & \text{otherwise} \end{cases}$$ > *(PYQ 2018)* > > **Solution:** > * **Exam Warning:** Any bounded, finite-duration signal is always an **Energy Signal**. > * Calculate Energy: > $$E = \sum_{n=-4}^{4} |\cos(\pi n)|^2 = \sum_{n=-4}^{4} |(-1)^n|^2 = \sum_{n=-4}^{4} 1 = \mathbf{9\text{ Joules}}$$ > * Average Power $P = 0$. > * **Conclusion:** It is an **Energy Signal** with $E = 9\text{ Joules}$. --- > [!question] Additional Past Year Questions (PYQs) > > * **[PYQ 2020]:** Find the odd and even components for each of the following signals: (i) $x(t) = u(t)$; (ii) $x(t) = e^{-\alpha t}u(t)$. (08 Marks) > - *Solution (i):* $u_e(t) = \frac{u(t) + u(-t)}{2} = \frac{1}{2}$ (for $t \neq 0$); $u_o(t) = \frac{u(t) - u(-t)}{2} = \frac{1}{2}\text{sgn}(t)$. > - *Solution (ii):* $x_e(t) = \frac{e^{-\alpha t}u(t) + e^{\alpha t}u(-t)}{2}$; $x_o(t) = \frac{e^{-\alpha t}u(t) - e^{\alpha t}u(-t)}{2}$. > * **[PYQ 2018, 2015]:** Find the even and odd components of the given signal $x(t) = \cos(20\pi t - 5\pi)$ and sketch the signal. (10 Marks) > - *Hint:* $\cos(20\pi t - 5\pi) = -\cos(20\pi t)$. Since $\cos(-20\pi t) = \cos(20\pi t)$, this is an even function. Even component is $-\cos(20\pi t)$ and odd component is $0$. > * **[PYQ 2019]:** Determine the even and odd components of the signal $x(t) = \sin(\sqrt{2}\pi t) + \sin(15\pi t)$. (06 Marks) > - *Hint:* Since $\sin(-\theta) = -\sin(\theta)$, this is a purely odd function. Even component is $0$ and odd component is $x(t)$. > * **[PYQ 2017]:** Determine whether $x[n] = \begin{cases} n, & 0 \le n < 5 \\ 10-n, & 5 \le n \le 10 \\ 0, & \text{otherwise} \end{cases}$ is energy or power. > - *Answer:* Bounded finite duration $\implies$ Energy signal. $E = 0^2+1^2+2^2+3^2+4^2+5^2+4^2+3^2+2^2+1^2+0^2 = 85\text{ Joules}$. > * **[PYQ 2016]:** Consider a signal $x(t) = 5\cos(\pi t) + \sin(5\pi t)$. Determine if it is an energy or power signal and find its value. > - *Answer:* Power signal (sum of periodic components is periodic). $P = \frac{5^2}{2} + \frac{1^2}{2} = 13\text{ Watts}$. > * **[PYQ 2015]:** Compute the energy of $x(t) = \sin(\sqrt{2}\pi t) + \sin(15\pi t)$. > - *Answer:* Since sines run infinitely, $E = \infty$. It is a power signal with $P = \frac{1^2}{2} + \frac{1^2}{2} = 1\text{ Watt}$. --- **Related Concepts:** [[1.01 Signals, Systems & Singularity Functions]] | [[1.02 Elementary Operations on Signals]] | [[1.03 Signal Classification I — Theory & Periodicity]] | [[1.04 Signal Classification II — Symmetry, Energy & Power]] # 1.05 The Sampling Theorem Bridge & Multi-Rate Processing > [!abstract] Overview > > This note serves as the critical bridge from continuous-time (analog) to discrete-time (digital) signals. Understanding how to sample an analog waveform without data loss is a fundamental requirement of signal processing. --- ## 1. The Shannon-Nyquist Sampling Theorem ### 1.1 The Core Theorem To convert a continuous-time analog signal into a discrete-time digital signal without losing any information, the analog wave must be sampled at a sufficiently high rate. > [!theorem] Shannon-Nyquist Sampling Theorem > > A continuous-time, band-limited signal containing no frequency components higher than $f_{max}$ Hz can be uniquely and perfectly reconstructed from its samples if it is sampled at a rate $F_s$ samples per second that is at least twice the maximum frequency component. > > $$F_s \ge 2f_{max}$$ > > * **Nyquist Rate:** The absolute minimum theoretical sampling rate that avoids information loss: > $$F_N = 2f_{max}$$ > * **Nyquist Interval:** The maximum allowable time duration between consecutive samples: > $$T_N = \frac{1}{F_N} = \frac{1}{2f_{max}}$$ --- ### 1.2 The Three Sampling Conditions The relation between the sampling frequency $F_s$ and the Nyquist rate $2f_{max}$ determines the frequency-domain reconstruction behavior. #### 1. Over-sampling ($F_s > 2f_{max}$) The replicated spectral copies of the signal are spaced far apart. A guard band exists between them, allowing reconstruction with a cheap, non-ideal low-pass filter. ```text Replicated Spectrums (Spaced apart with Guard Bands): ^ ^ ^ / \ / \ / \ / \ / \ / \ ----+-----+----+-------+----+-----+----+-------+----+-----+----+--> Frequency (Ω) -2f_max 0 2f_max F_s |<----->| Guard Band ``` #### 2. Nyquist-rate sampling ($F_s = 2f_{max}$) The replicated spectral copies are exactly touching at their edges. An ideal "brick-wall" low-pass filter is required to reconstruct the original signal. ```text Nyquist-rate sampling (Spectrums exactly touching): ^ ^ ^ / \ / \ / \ / \ / \ / \ ----+-----+----+-----------+----+-----+----+-----------+----+-----+----+--> Frequency (Ω) -f_max 0 f_max F_s |<-------->| F_s = 2f_max ``` #### 3. Under-sampling ($F_s < 2f_{max}$) The replicated spectral copies overlap. This causes **Aliasing**, preventing accurate reconstruction of the original signal. ```text Under-sampling (Spectrums overlapping): ^ ^ ^ / \ / \ / \ / \XXXXX / \XXXXX / \ ----+-----+----+----+-----+----+----+-----+----+--> Frequency (Ω) |XXXXXXXXX| Overlapping Region (Aliasing Distortion) ``` --- ### 1.3 Aliasing Effect & Anti-Aliasing Filter * **What is Aliasing?** A destructive frequency-folding distortion that occurs when a signal is under-sampled ($F_s < 2f_{max}$). High-frequency components fold back and masquerade as false low-frequency components, permanently corrupting the signal. * **How to avoid it:** Place an **Anti-aliasing Filter** (a strict analog low-pass filter) *before* the Analog-to-Digital Converter (ADC). This filter strips away any frequencies higher than $F_s/2$ before sampling begins, ensuring the Nyquist criteria is never violated. --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2024, 2023, 2022, 2016, 2015]:** What is sampling theorem? (04 Marks) > * **[PYQ 2019, 2018, 2017]:** State and prove the Shannon-Nyquist sampling theorem. (08 to 10 Marks) > - *Note: Ensure you memorize the formal mathematical proof of $x_s(t) = x(t) \cdot \delta_T(t)$ mapping to frequency-domain convolution.* > * **[PYQ 2021, 2019, 2018, 2016, 2015]:** Explain the aliasing effect and the means to avoid it. (06 Marks) > * **[PYQ 2018, 2016]:** Write short notes on: (i) Over-sampling, (ii) Nyquist rate sampling, (iii) Aliasing. (06 Marks) --- ## 2. Classification of Sampling Techniques Physical implementation of the sampling switch is categorized into three types: 1. **Ideal Sampling (Impulse Sampling):** The analog signal is multiplied by a train of perfect, zero-width Dirac delta impulses. This is physically impossible to build and is used only for theoretical mathematical modeling. 2. **Natural Sampling:** The analog signal is multiplied by a train of finite-width rectangular pulses. The top of each sampled pulse naturally curves to follow the exact contour of the analog wave. 3. **Flat-top Sampling:** The analog signal is sampled and its voltage level is held strictly constant for the duration of the pulse width (using a Sample-and-Hold circuit). This creates a staircase-like waveform and is the standard method in modern ADCs. --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2018]:** Classify the sampling techniques. (04 Marks) > * **[PYQ 2017]:** Distinguish between Ideal, Natural, and Flat-top sampling techniques. (06 Marks) --- ## 3. Multi-Rate Signal Processing (Up/Down Sampling) Multi-rate processing manipulates the sampling rate of a discrete-time sequence. ### 3.1 Discrete-Time Down-sampler (Compressor) Used to reduce the sampling rate. * **Definition:** A down-sampler by an integer factor $M$ keeps every $M$-th sample of the input sequence and discards the $(M-1)$ intermediate samples. * **Mathematical Operation:** $$x_d[n] = x[nM]$$ * **Warning:** Because this operation discards data, down-sampling will cause digital aliasing unless the original signal is sufficiently over-sampled. --- ### 3.2 Discrete-Time Up-sampler (Interpolator) Used to increase the sampling rate. * **Definition:** An up-sampler by an integer factor $L$ inserts exactly $(L-1)$ zero-valued samples between each consecutive pair of original samples. * **Mathematical Operation:** $$x_e[n] = \begin{cases} x[n/L], & n = 0, \pm L, \pm 2L, \dots \\ 0, & \text{otherwise} \end{cases}$$ --- ### 3.3 Array Numerical Worked Example > [!example] High-Yield Array Numerical > > **Question:** If $x[n] = [2, 6, \underline{0}, 3, 1, 2]$, find the down-sampled sequence by a factor of 3, and the up-sampled sequence by a factor of 3. (2022) > > **Solution:** > 1. **Index Mapping:** > Identify the origin $n=0$ (underlined): > $$x[-2] = 2, \quad x[-1] = 6, \quad x[\underline{0}] = 0, \quad x[1] = 3, \quad x[2] = 1, \quad x[3] = 2$$ > > 2. **Down-sampling ($M=3$):** > Keep only indices that are multiples of 3 ($x_d[n] = x[3n]$): > - $n = 0 \implies x_d[0] = x[0] = \underline{0}$ > - $n = 1 \implies x_d[1] = x[3] = 2$ > - $n = -1 \implies x_d[-1] = x[-3]$ (Out of bounds, assume 0) > > $$\mathbf{x_d[n] = [\underline{0}, 2]}$$ > > 3. **Up-sampling ($L=3$):** > Insert $L-1 = 2$ zeros between each element of the original sequence: > > $$\mathbf{x_e[n] = [2, 0, 0, 6, 0, 0, \underline{0}, 0, 0, 3, 0, 0, 1, 0, 0, 2]}$$ --- > [!question] Past Year Questions (PYQs) > > * **[PYQ 2021]:** Define: (i) Discrete time up-sampler; (ii) Discrete time down-sampler. (04 Marks) > * **[PYQ 2024]:** Find out the up-sampling sequence $x_u[n]$ and down-sampling sequence $x_d[n]$ from the input signal sequence $x[n] = \dots$ for the discrete time system. (10 Marks) --- # 1.06 Typical Signal Processing Operations In engineering practice, raw physical signals are rarely usable in their original form. To extract information, transmit data over long distances, or suppress noise, we must perform **Typical Signal Processing Operations** *{standard mathematical manipulations applied to waveforms within a communication system}*. These standard operations form the physical building blocks of transmitters, channels, and receivers. --- ### 1.7.1 Correlation **Correlation** *{a mathematical measure of similarity between two waveforms as a function of a relative time shift}* is used to compare a reference signal with one or more target signals to determine how closely they match. > [!note] **Classifications of Correlation** > > 1. **Cross-correlation ($R_{xy}$ or $r_{xy}$):** Measures the similarity between two *different* signals, $x(t)$ and $y(t)$, as one is shifted in time relative to the other. > * **Continuous-Time Formula:** > $$R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t) y(t+\tau) \, dt$$ > * **Discrete-Time Formula:** > $$r_{xy}[l] = \sum_{n=-\infty}^{\infty} x[n] y[n-l]$$ > 2. **Auto-correlation ($R_{xx}$ or $r_{xx}$):** Measures the similarity of a signal with a *time-shifted version of itself*. It is highly effective at extracting periodic signals heavily corrupted by noise. > * **Continuous-Time Formula:** > $$R_{xx}(\tau) = \int_{-\infty}^{\infty} x(t) x(t+\tau) \, dt$$ > * **Discrete-Time Formula:** > $$r_{xx}[l] = \sum_{n=-\infty}^{\infty} x[n] x[n-l]$$ > [!info] **High-Yield Applications of Correlation** > > * **Radar & Sonar Echo Detection:** Measuring the time-of-flight delay ($\tau$) of a reflected echo to calculate the exact distance to an object. > * **Signal-in-Noise Extraction:** Pulling extremely weak communication signals out of channel noise. > * **Pattern Matching:** Biometric fingerprint matching and voice/image recognition algorithms. > * **Cross-spectral analysis:** Mapping similarities across frequency bands. --- ### 1.7.2 Filtering **Filtering** *{a frequency-selective operation that passes desired spectral components of a signal while suppressing unwanted ones}* is used to clean up signals and block out-of-band interference. #### The 5 Core Filter Types: 1. **Low-Pass Filter (LPF):** Passes all frequencies below a specified cutoff frequency ($\Omega_c$) and attenuates those above it. `[GRAPH: Ideal LPF frequency response |H(j\Omega)| showing a flat passband of height 1 from -\Omega_c to +\Omega_c, and a stopband of 0 for |\Omega| > \Omega_c]` 2. **High-Pass Filter (HPF):** Passes all frequencies above a specified cutoff frequency ($\Omega_c$) and blocks those below it. `[GRAPH: Ideal HPF frequency response |H(j\Omega)| showing a stopband of 0 for |\Omega| < \Omega_c, and a flat passband of height 1 for |\Omega| > \Omega_c]` 3. **Band-Pass Filter (BPF):** Passes frequencies within a specific frequency band ($\Omega_{c1}$ to $\Omega_{c2}$) and attenuates all frequencies outside this band. `[GRAPH: Ideal BPF frequency response |H(j\Omega)| showing a flat passband of height 1 in the intervals [\Omega_{c1}, \Omega_{c2}] and [-\Omega_{c2}, -\Omega_{c1}], and 0 elsewhere]` 4. **Band-Stop Filter (BSF):** Blocks frequencies within a specific frequency band ($\Omega_{c1}$ to $\Omega_{c2}$) and passes all frequencies outside it. `[GRAPH: Ideal BSF frequency response |H(j\Omega)| showing a stopband notch of 0 in the intervals [\Omega_{c1}, \Omega_{c2}] and [-\Omega_{c2}, -\Omega_{c1}], and a passband of height 1 elsewhere]` 5. **Notch Filter:** A specialized, extremely narrow band-stop filter designed to reject a single, highly specific frequency component. * *Example:* Used in biomedical systems (ECG/EEG machines) to suppress the 50 Hz/60 Hz power-line hum without distorting sensitive physiological signals. > [!tip] **Filtering in LTI Systems** > > In the time domain, filtering is modeled by **convolving** the input signal $x(t)$ with the filter's impulse response $h(t)$: > $$y(t) = x(t) * h(t)$$ > In the frequency domain, this complex convolution simplifies to simple algebraic **multiplication** with the transfer function $H(j\omega)$: > $$Y(j\omega) = X(j\omega) H(j\omega)$$ --- ### 1.7.3 Modulation and Demodulation * **Modulation** *{the process of varying one or more parameters of a high-frequency carrier wave in proportion to a low-frequency message signal}* shifts baseband frequencies to higher bands [1.7.3]. * **Demodulation** *{the reverse process of extracting the original baseband message signal from the received modulated wave}* restores the signal at the receiver. > [!warning] **The Physical Importance of Modulation** > > Baseband signals (like human speech, 20 Hz to 20 kHz) have extremely long wavelengths ($\lambda = c/f$). Because an antenna's physical size must be proportional to the wavelength ($\approx \lambda/4$), transmitting raw baseband signals would require antennas kilometers tall! > > Modulation translates the message to a high-frequency carrier (e.g., 100 MHz), reducing the wavelength to meters and allowing compact, practical antennas. It also enables **Frequency-Division Multiplexing** to prevent multiple transmitters from interfering. #### Mathematical Model of Amplitude Modulation (AM): $$s(t) = m(t) \cos(\omega_c t)$$ where $m(t)$ is the low-frequency modulating message signal, $\cos(\omega_c t)$ is the high-frequency carrier wave, and $s(t)$ is the resulting modulated signal. `[DIAGRAM: Modulation block diagram showing input message m(t) and carrier source \cos(\omega_c t) multiplying in a mixer block to produce s(t) = m(t)\cos(\omega_c t)]` --- ### 1.7.4 Transformation **Transformation** *{a mathematical mapping that shifts a signal from its original time-domain representation to an alternative frequency-domain coordinate system}* simplifies system analysis by turning complex differential or difference equations into easy-to-solve algebraic equations. ```mermaid graph LR Time[Time Domain] -- Fourier Transform --> Freq[Frequency Domain] Time -- Laplace Transform --> sDomain[s-Domain s = σ + jω] Time -- Z-Transform --> zDomain[z-Plane] ``` * **Fourier Transform:** Analyzes the steady-state frequency spectra of continuous-time signals. $$F(j\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} \, dt$$ * **Laplace Transform:** Generalizes the Fourier transform by introducing a damping envelope $e^{-\sigma t}$. Essential for analyzing transient responses in analog circuits and unstable LTI systems. $$F(s) = \int_{0^{-}}^{\infty} f(t) e^{-st} \, dt$$ * **Z-Transform:** The discrete-time equivalent of the Laplace transform [10.01]. Maps discrete sequences to the complex z-plane. $$X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}$$ --- ### 1.7.5 Multiplexing and Demultiplexing * **Multiplexing** *{the simultaneous transmission of multiple independent message signals over a single shared physical channel}* maximizes medium efficiency [1.7.5]. * **Demultiplexing** *{separating the combined multiplexed signals back into their independent original forms at the receiver}* routes signals to their proper destinations. > [!info] **The Three Major Multiplexing Paradigms** > > 1. **Time-Division Multiplexing (TDM):** Allocates different, non-overlapping time slots to each user over the entire channel bandwidth. > `[DIAGRAM: TDM frame structure showing sequential time slots t_1, t_2, t_3 occupied by User 1, User 2, User 3 repeating in time]` > 2. **Frequency-Division Multiplexing (FDM):** Allocates different, non-overlapping frequency bands (separated by guard bands to prevent adjacent-channel interference) to each user simultaneously. > `[DIAGRAM: FDM spectrum showing Channel 1, Channel 2, Channel 3 separated by unshaded guard bands in the frequency domain]` > 3. **Code-Division Multiplexing (CDM):** Assigns a unique mathematical code to each user, allowing multiple users to transmit simultaneously over the entire common frequency band without interference. --- ### 1. Common Mistakes That Cost Marks * **Convolving vs. Correlating:** Students often confuse these two operations. Convolution requires time-reversing (folding) the sliding signal, whereas correlation does *not* fold the signal. $$x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) \, d\tau \quad \text{vs.} \quad R_{xy}(\tau) = \int_{-\infty}^{\infty} x(t) y(t+\tau) \, dt$$ * **Notch vs. Band-Stop Filter:** A Notch filter is *not* a standard band-stop filter. A Notch filter has an extremely narrow, sharp rejection band designed to suppress one specific frequency (such as power line noise) while leaving the rest of the spectrum untouched. * **Multiplexing vs. Modulation:** Modulation is frequency translation for a *single* message, whereas Multiplexing is combining *multiple* distinct user messages to share a single channel. --- ### 2. Verbatim PYQ Bank > [!question] **Potential Exam Questions (Syllabus/Lecture Alignment)** > > > **Q1. Define: (i) Correlation, (ii) Filtering, (iii) Modulation. State the physical importance of modulation.** > * **Grounded Answer Guidelines:** > * Define **Correlation** as a mathematical tool to compare signals and determine similarity. > * Define **Filtering** as a frequency-selective operation designed to suppress unwanted spectral components. > * Define **Modulation** as translating baseband message frequencies using a high-frequency carrier wave [1.7.3]. > * State that modulation is physically crucial because baseband signal wavelengths are too long, requiring impractically large antennas. Modulation reduces the wavelength, allowing compact antennas. > > **Q2. Distinguish between Time-Division Multiplexing (TDM) and Frequency-Division Multiplexing (FDM).** > * **Grounded Answer Guidelines:** > * TDM splits the shared medium by allocating non-overlapping **time slots** to each user over the entire channel bandwidth [1.7.5]. > * FDM splits the shared medium by allocating non-overlapping **frequency bands** separated by guard bands, allowing users to transmit simultaneously [1.7.5]. --- ### 3. Self-Check Before Moving On - [ ] Can you write down the exact continuous-time integrals for both cross-correlation and convolution, highlighting their algebraic differences? - [ ] Do you know why a Notch filter is used instead of a standard Low-Pass Filter to remove power hum from an ECG signal? (To prevent removing high-frequency bio-potential data!) - [ ] Can you explain the relationship between wavelength and antenna size, proving why modulation is necessary for wireless communication? --- ### 4. Source Citations * **Textbook:** K. Deergha Rao, *Signals and Systems*, Section 1.7 (Typical Signal Processing Operations), pp. 30-32. * **Class Handnotes:** Rabiul Sir Class Lectures, Week 3 (Lec 3, pg. 30) [Lec 3]. * **Laboratory Guide:** ECE 2108 Signals & Systems Lab Experiment 3 (Correlation & Convolution), pg. 30. --- # 1.07 Supplementary Signal Functions, Hold Circuits & DAC Reconstruction Related Concepts: [[1.01 Signals, Systems & Singularity Functions.md]] | [[1.05 The Sampling Theorem Bridge & Multi-Rate Processing.md]] | [[1.06 Typical Signal Processing Operations.md]] --- > [!abstract] Overview > > This note plugs critical curriculum gaps in Chapter 1. **Part I** expands on advanced continuous-time and discrete-time signals—**Real Exponentials**, **Complex Exponentials**, and the **Sinc Function**. **Part II** details physical signal reconstruction in the Digital-to-Analog (DAC) pipeline, focusing on the mathematical transfer functions of **Zero-Order Hold (ZOH)** and **First-Order Hold (FOH)** circuits. --- ### Part I: Advanced Elementary Signals #### 1. Real Exponential Function A real exponential signal changes its amplitude monotonically *{unidirectionally, either always growing or always shrinking}* with time. It is governed by a real constant exponent. ##### A. Continuous-Time Real Exponential Mathematically represented as: $$x(t) = A e^{\sigma t}$$ Where: * $A \in \mathbb{R}$ is the amplitude scaling factor at $t = 0$. * $\sigma \in \mathbb{R}$ is the exponential parameter *{damping factor or growth rate}*. Depending on the sign of $\sigma$, the signal exhibits two distinct physical behaviors: 1. **Exponential Decay ($\sigma < 0$):** The signal's amplitude decays asymptotically toward zero as $t o \infty$. This models physical systems returning to equilibrium, such as charge dissipating across a resistor. 2. **Exponential Growth ($\sigma > 0$):** The signal's amplitude grows boundlessly as $t o \infty$. This represents unstable positive-feedback systems. [GRAPH: Continuous Exponential Decay (y = e^{-0.5t}) vs. Growth (y = e^{0.5t}) — source: Textbook Fig 1.16] ##### B. Discrete-Time Real Exponential For discrete index $n$, the real exponential is defined as: $$x[n] = A a^n$$ Where $a \in \mathbb{R}$ is the base of the exponential. * **Decaying Sequence ($|a| < 1$):** If $a$ is positive ($0 < a < 1$), the sequence decays monotonically toward zero. If $a$ is negative ($-1 < a < 0$), the sequence oscillates back and forth between positive and negative values while its envelope decays toward zero. * **Growing Sequence ($|a| > 1$):** The sequence grows boundlessly toward infinity. [GRAPH: Discrete stems showing monotonic decay for a = 0.8 and oscillating decay for a = -0.8 — source: Textbook Ch 1] --- #### 2. Complex Exponential Function The complex exponential function is the mathematical bridge that unifies real exponential dampening and sinusoidal oscillations. ##### A. Continuous-Time Complex Exponential Mathematically represented as: $$x(t) = A e^{s t}$$ Where $s$ is a complex variable of frequency: $$s = \sigma + j\Omega$$ * $\sigma$ represents the real part *{the damping coefficient, measured in Nepers/sec}*. * $\Omega$ represents the imaginary part *{the angular oscillation frequency, measured in radians/sec}*. ##### B. Euler's Expansion Substituting $s = \sigma + j\Omega$ into the complex exponential equation: $$x(t) = A e^{(\sigma + j\Omega)t} = A e^{\sigma t} e^{j\Omega t}$$ Applying **Euler's Identity** ($e^{j heta} = \cos heta + j\sin heta$): $$x(t) = A e^{\sigma t} \left[ \cos(\Omega t) + j \sin(\Omega t) ight] = A e^{\sigma t}\cos(\Omega t) + j A e^{\sigma t}\sin(\Omega t)$$ ##### C. Physical Interpretations * **Case 1: $\sigma = 0 \implies x(t) = A e^{j\Omega t}$ (Undamped / Pure Sinusoid):** Represents a pure complex sinusoidal wave of constant amplitude. * **Case 2: $\sigma < 0 \implies x(t) = A e^{\sigma t} e^{j\Omega t}$ (Damped Sinusoid):** Represents a sinusoidal oscillation enclosed inside an exponentially decaying envelope. * **Case 3: $\sigma > 0 \implies x(t) = A e^{\sigma t} e^{j\Omega t}$ (Growing Sinusoid):** Represents a sinusoidal oscillation whose amplitude grows exponentially. [GRAPH: Exponentially damped sinusoid 5*sin(2*pi*1*t).*exp(-0.4*t) showing sinusoidal peaks bounded by an exponential envelope — source: Rabiul Sir Notes, Fig 1.20] --- #### 3. Sinc Function The **Sinc Function** (abbreviation of *sine cardinal*) is highly prominent in signal processing and Fourier analysis because it represents the time-domain impulse response of an ideal brick-wall low-pass filter. ##### A. Definition There are two common definitions of the sinc function. ECE 2107 utilizes the **Normalized Sinc Function** (DSP standard): $$ ext{sinc}(t) = rac{\sin(\pi t)}{\pi t}$$ > [!tip] **Sinc Notation Alert** > > In pure mathematics, the **Unnormalized Sinc Function** (often written as $ ext{Sa}(t)$, sine-arg) is defined as: > $$ ext{Sa}(t) = rac{\sin(t)}{t}$$ > To transition from unnormalized to normalized: $ ext{sinc}(t) = ext{Sa}(\pi t)$. Always read the axis labels in exam sheets! ##### B. Mathematical Evaluation at the Origin ($t = 0$) Direct substitution of $t = 0$ yields an indeterminate form ($0/0$). Applying **L'Hôpital's Rule** *{differentiating the numerator and denominator separately with respect to t}*: $$\lim_{t o 0} ext{sinc}(t) = \lim_{t o 0} rac{ rac{d}{dt}[\sin(\pi t)]}{ rac{d}{dt}[\pi t]} = \lim_{t o 0} rac{\pi \cos(\pi t)}{\pi} = \cos(0) = 1$$ Therefore, the function has a peak amplitude of **1** at $t = 0$. ##### C. Zero Crossings The sinc function crosses the horizontal axis at values of $t$ where the numerator is zero but the denominator is non-zero: $$\sin(\pi t) = 0 \implies \pi t = k\pi \implies t = k \quad ext{for } k \in \mathbb{Z}, \ k eq 0$$ Thus, zero crossings occur at all non-zero integer values: $t = \pm 1, \pm 2, \pm 3, \dots$ [GRAPH: Sinc(t) = sin(\pi t)/(\pi t) showing a central lobe of height 1 at t=0, crossing zero at all non-zero integers, and decaying side lobes — source: Senior Notes, Fig 1.19] --- ### Part II: Practical Signal Reconstruction (Hold Circuits) #### 1. The Digital-to-Analog (DAC) Reconstruction Pipeline In **Note 1.05**, we explored sampling—discretizing a continuous signal $x_a(t)$ into impulses. The reverse process, converting a discrete sequence $x[n]$ back into a continuous-time signal $x_a(t)$, takes place in a DAC. ``` +------------+ +--------------+ +---------------------+ +---------+ | Discrete | ---> | Impulse | ---> | Hold Circuit | ---> | Smoothing| ---> Analog | Sequence | | Generator | | (ZOH or FOH LPF) | | Filter | Signal | x[n] | | x_p(t) | | "Staircase/Polygon" | | (Ideal) | x_a(t) +------------+ +--------------+ +---------------------+ +---------+ ``` --- #### 2. Zero-Order Hold (ZOH) Reconstruction The **Zero-Order Hold** is the simplest and most common hardware reconstruction filter. ##### A. Operating Concept The ZOH takes a discrete sample value at $t = nT_s$ and holds its amplitude completely constant over the entire sampling interval until the next sample arrives at $t = (n+1)T_s$. This produces a jagged, staircase-like continuous waveform. ##### B. Impulse Response The impulse response $h_0(t)$ of a ZOH is a single rectangular gating pulse of width $T_s$ starting at the origin: $$h_0(t) = u(t) - u(t - T_s)$$ [GRAPH: Rectangular pulse of height 1 spanning from t = 0 to t = T_s — source: Textbook Ch 6] ##### C. Derivation of the ZOH s-Domain Transfer Function To find the transfer function $H_0(s)$, we take the Laplace Transform of its impulse response $h_0(t)$: $$H_0(s) = \mathcal{L}\{u(t) - u(t - T_s)\} = \mathcal{L}\{u(t)\} - \mathcal{L}\{u(t - T_s)\}$$ Using the Laplace time-shifting property: $$H_0(s) = rac{1}{s} - rac{1}{s} e^{-s T_s} = rac{1 - e^{-s T_s}}{s}$$ ##### D. Derivation of the ZOH Frequency Response To analyze its filtering characteristics, substitute $s = j\Omega$: $$H_0(j\Omega) = rac{1 - e^{-j\Omega T_s}}{j\Omega}$$ Using Euler's identity to factor out a phase delay of half a sampling period ($e^{-j\Omega T_s/2}$): $$H_0(j\Omega) = rac{e^{-j\Omega T_s/2} \left( e^{j\Omega T_s/2} - e^{-j\Omega T_s/2} ight)}{j\Omega}$$ We know that $e^{j heta} - e^{-j heta} = 2j\sin heta$. Substituting this in: $$H_0(j\Omega) = e^{-j\Omega T_s/2} \left[ rac{2j \sin\left( rac{\Omega T_s}{2} ight)}{j\Omega} ight] = e^{-j\Omega T_s/2} \cdot 2 rac{\sin\left( rac{\Omega T_s}{2} ight)}{\Omega}$$ Multiply numerator and denominator by $T_s$: $$H_0(j\Omega) = T_s \left[ rac{\sin\left( rac{\Omega T_s}{2} ight)}{ rac{\Omega T_s}{2}} ight] e^{-j\Omega T_s/2}$$ Using the unnormalized sinc notation ($ ext{Sa}( heta) = rac{\sin heta}{ heta}$): $$H_0(j\Omega) = T_s \, ext{Sa}\left( rac{\Omega T_s}{2} ight) e^{-j\Omega T_s/2} \quad \iff \quad T_s \, ext{sinc}\left( rac{\Omega T_s}{2\pi} ight) e^{-j\Omega T_s/2}$$ ##### E. Physical Implications of the Frequency Response 1. **Aperture Distortion (Sinc Roll-off):** The magnitude spectrum $|H_0(j\Omega)| = T_s \left| ext{sinc}\left( rac{\Omega T_s}{2\pi} ight) ight|$ acts as a low-pass filter, but its main lobe rolls off significantly before reaching the Nyquist frequency ($\Omega = \pi/T_s$). This dampens high-frequency signal components (aperture effect). 2. **Phase Delay:** The phase term $e^{-j\Omega T_s/2}$ introduces a constant linear phase delay of exactly half a sampling interval ($T_s/2$), lagging the reconstructed output behind the actual digital values. 3. **Spectral Leakage:** Sinc side lobes exist outside the Nyquist band. These lobes leak unwanted high-frequency staircase-harmonics into the analog output, requiring an active low-pass smoothing filter to suppress. --- #### 3. First-Order Hold (FOH) Reconstruction The **First-Order Hold** is a more sophisticated reconstruction filter that utilizes linear interpolation. ##### A. Operating Concept Instead of holding a constant value, the FOH reconstructs the analog signal by joining the sample points with straight lines (linear interpolation). It estimates the slope between consecutive samples, producing a continuous polygonal line waveform with no sharp staircase discontinuities. ##### B. Impulse Response (Symmetric Model) To perform linear interpolation, the impulse response $h_1(t)$ must be a symmetric triangular pulse centered at $t = 0$ with a total base width of $2T_s$: $$h_1(t) = \left( 1 - rac{|t|}{T_s} ight) \left[ u(t + T_s) - u(t - T_s) ight]$$ [GRAPH: Triangular pulse centered at origin of height 1 spanning from -T_s to +T_s — source: Textbook Ch 6] ##### C. Frequency Response of FOH The Fourier transform of a symmetrical triangular pulse is the squared sinc function: $$H_1(j\Omega) = T_s \left( ext{sinc}\left( rac{\Omega T_s}{2\pi} ight) ight)^2$$ ##### D. Comparison: ZOH vs. FOH * **Harmonic Suppression:** The FOH magnitude response drops off as the square of the sinc function, causing its side lobes to be significantly smaller than ZOH. This dramatically reduces high-frequency staircase harmonic leakage and simplifies the design of the post-DAC smoothing filter. * **Causality & Delay:** The symmetric FOH is non-causal *{requires future samples to calculate the connecting slope}*. A causal physical implementation of FOH requires delaying the output by exactly one sampling interval ($T_s$), whereas the ZOH only introduces a delay of $T_s/2$. --- ### Common Mistakes That Cost Marks > [!warning] **Exam Pitfall Checklist** > > * **The Sinc Peak Value:** Never state $ ext{sinc}(0) = 0$ or undefined. Differentiate using L'Hôpital's Rule to prove that $ ext{sinc}(0) = 1$. > * **Complex vs. Real Exponentials:** Do not mix up the real damping factor $\sigma$ with the complex frequency $s$. Remember that $e^{\sigma t}$ represents simple decay/growth, while $e^{j\Omega t}$ represents oscillatory movement. > * **ZOH Phase Delay:** In derivation questions, do not omit the phase shift term $e^{-j\Omega T_s/2}$ when stating the transfer function. This phase shift is a physical delay of $T_s/2$ and is highly critical for stability analysis. > * **ZOH vs. FOH Spectral Roll-off:** Remember that ZOH magnitude decays as $ ext{sinc}(f)$, while FOH magnitude decays much faster as $ ext{sinc}^2(f)$. --- ### Potential Exam Questions (PYQ Prep) > [!question] **Unilateral vs. Bilateral Exponentials** > > **Question:** Define a real exponential signal. Sketch the continuous-time signal $x(t) = e^{-3t}u(t)$ and explain why multiplying by $u(t)$ is necessary in practical systems. > **Answer Plan:** > 1. Write the general definition of a real exponential $x(t) = Ae^{\sigma t}$. > 2. Detail that $\sigma = -3 < 0 \implies$ decay. > 3. Explain that multiplying by $u(t)$ forces the signal to be causal ($x(t) = 0$ for $t < 0$). This matches physically realizable systems that are turned on at a starting time $t = 0$. > [!question] **Sinc Function Peak and Zero Crossings** > > **Question:** Mathematically define the normalized sinc function. Prove its value at the origin $t = 0$ and find all of its zero-crossing points. > **Answer Plan:** > 1. State the formula: $ ext{sinc}(t) = rac{\sin(\pi t)}{\pi t}$. > 2. Use L'Hôpital's rule to derive the limit as $t o 0$, showing the peak is $1$. > 3. Solve $\sin(\pi t) = 0$ for $t eq 0$ to prove that zero crossings occur at all non-zero integers $t = \pm 1, \pm 2, \dots$ > [!question] **ZOH Transfer Function Derivation** > > **Question:** Draw the impulse response of a Zero-Order Hold (ZOH) circuit with a sampling period $T_s$. Derive its transfer function in both the s-domain and the frequency domain. > **Answer Plan:** > 1. Draw a rectangular pulse of height 1 from $t = 0$ to $T_s$. > 2. Write $h_0(t) = u(t) - u(t - T_s)$ and take the Laplace transform to find $H_0(s) = rac{1 - e^{-s T_s}}{s}$. > 3. Substitute $s = j\Omega$ and factor out $e^{-j\Omega T_s/2}$ to arrive at the final sinc form: $H_0(j\Omega) = T_s ext{sinc}\left( rac{\Omega T_s}{2\pi} ight) e^{-j\Omega T_s/2}$. --- ### Self-Check Before Moving On - [ ] Can you write Euler's identity and expand a complex exponential into its real and imaginary parts? - [ ] Do you understand why the sinc function represents the impulse response of an ideal low-pass filter? - [ ] Can you derive the s-domain transfer function of a ZOH from its impulse response step-by-step? - [ ] Do you know the physical trade-off between ZOH and FOH in terms of causal delay versus harmonic leakage? --- ### Source Citations * *K. Deergha Rao, Signals and Systems, Chapter 1 (Section 1.5 - Singularity Functions).* * *Senior Lecture Notes, "1 intro to Signal & systems.pdf" (Sampling Process & Hold Circuits).* * *Syllabus ECE 2107, Weeks 10-11 (DAC & Hold Circuits).*