1.00 Chapter 1: Signal Fundamentals & Operations — Ultimate Master Note

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Chapter 1: Signal Fundamentals & Operations — Ultimate Master Note

This master note serves as your exhaustive, concepts-complete, and exam-ready study guide for Chapter 1: Signal Fundamentals & Operations [171, 565]. It meticulously compiles and merges all core definitions, mathematical models, step-by-step derivations, comparison matrices, and past year question (PYQ) solutions from your lecture notes, syllabus, and K. Deergha Rao’s textbook [1, 516, 566]. This is the only document you need to study to secure full marks on Chapter 1 in your Class Tests and Final Examinations [547].

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1. Introduction to Signals & Systems

*(Target: Theory Descriptive / 2-Mark to 6-Mark Vocabulary Questions)* [198, 552]

1.1 Formal Mathematical Definitions

• Signal: A signal is defined as any physical quantity that carries information and varies with time, space, or any other independent variable or variables [4, 174, 194]. Mathematically, it is modeled as a function of an independent variable $t$ (for continuous-time) or a sample index $n$ (for discrete-time) [180, 195]: $x(t)\text{or}x[n]$
  • Physical Examples: Voltage or current waveforms in circuits, sound/pressure waves, physiological traces (ECG/EEG), or temperature measurements [194, 195].
• System: A system is a physical device, process, or mathematical algorithm that manipulates or operates on an input signal $x(t)$ (excitation) to produce a transformed output signal $y(t)$ (response) [5, 174, 196]: $y(t)=\mathbb{T}[x(t)]\text{or}y[n]=\mathbb{T}[x[n]]$ where $\mathbb{T}[\cdot ]$ represents the transformation operator acting on the input [196, 280].

1.2 The Chain of Signal Dependence

In physical propagation, a signal undergoes a strict sequence of dependencies [180, 195]: $\text{Event}\to \text{Sequence}\to \text{Time}\to \text{Frequency}\to \text{Phase}\to \text{Amplitude}$ Ultimately, all signal characteristics depend on time [180, 195].

1.3 Importance in Communication Engineering

In communication systems, raw physical signals (like human speech) are rarely usable in their original form. Signals and systems allow us to transmit data over vast distances, filter out-of-band interference, and extract extremely weak signals from background noise [197, 232].

• Key Real-World Applications:
  • Biomedical Monitoring: ECG (electrocardiogram) and EEG (electroencephalogram) for tracking heart and brain electrical activity [4, 174, 197].
  • Target Detection: Radar and sonar echoes to calculate object range and velocity [4, 174, 197].
  • Navigation: Global Positioning System (GPS) satellite time synchronization [4, 174, 197].
  • Geological Tracking: Seismic vibrations for earthquake detection and mineral mapping [4, 174, 197].
  • Speech & TV Processing: Broadcast transmissions, voice codecs, and multimedia streaming [4, 174, 197].

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2. Elementary Continuous-Time Singularity & Basic Functions

*(Target: Theory Descriptive / Waveform Sketching / 3-Mark to 8-Mark Derivations)* [190, 203]

Singularity functions are mathematical idealizations that serve as the fundamental building blocks to construct complex waveforms, model switching transitions, and represent impulsive excitations [181, 199].

2.1 The Unit Step Function, $u(t)$

Models a DC switch closing instantaneously at $t=0$ [181, 386]: $u(t)=\{\begin{array}{ll}1,& t>0\\ 0,& t<0\\ \text{undefined},& t=0\end{array}$

    u(t) 
     │
 1 ──┼──────────
     │
 ────┴────────── t
     0

2.2 The Unit Impulse Function (Dirac Delta), $\delta (t)$

An infinitely narrow, infinitely tall pulse centered at $t=0$ containing an area (weight) of exactly 1 [182, 238]: $\delta (t)=0\text{for }t\ne 0,\text{and}{\int }_{-\infty }^{\infty }\delta (t)dt=1$

    δ(t) 
    (1) ↑
     │  │
 ────┼──┴─────── t
     0

• Sifting (Shifting) Property: If an arbitrary continuous signal $x(t)$ is integrated against a shifted impulse, it isolates the value of $x(t)$ at the impulse’s location [34]: ${\int }_{-\infty }^{\infty }x(t)\delta (t-t_0)dt=x(t_0)$
• Scaling Property: Scaling the independent variable inside the delta function scales its area [35]: $\delta (at+b)=\frac{1}{|a|}\delta \left(t+\frac{b}{a}\right)$

2.3 The Unit Ramp Function, $r(t)$

A linearly growing signal representing the time-integrated step function [183, 241]: $r(t)=\{\begin{array}{ll}t,& t>0\\ 0,& t\le 0\end{array}⟹r(t)=t\cdot u(t)$

    r(t)
     │      / (Slope = 1)
     │     /
     │    /
 ────┼───┴────── t
     0

2.4 The Signum Function, $\text{sgn}(t)$

A polarity indicator extracting the algebraic sign of the time variable $t$ [183]: $\text{sgn}(t)=\{\begin{array}{ll}1,& t>0\\ 0,& t=0\\ -1,& t<0\end{array}⟹\text{sgn}(t)=u(t)-u(-t)$

   sgn(t)
     │
 1 ──┼──────────
     ├─── 0 (at t=0)
 ────┼─── -1
     │

2.5 The Rectangular Gate Pulse, $\Pi (t/\tau )$

A symmetrical rectangular window centered at the origin with total duration (width) $\tau$ and amplitude 1 [184]: $\Pi \left(\frac{t}{\tau }\right)=\{\begin{array}{ll}1,& -\frac{\tau }{2}<t<\frac{\tau }{2}\\ 0,& \text{otherwise}\end{array}$

    Π(t/τ)
      ┌─────┐  1
      │     │
 ─────┴──┬──┴───── t
       -τ/2 0 τ/2

2.6 The Normalized Sinc Function, $\text{sinc}(t)$

Representing the frequency-domain spectrum of a continuous rectangular time pulse [374]: $\text{sinc}(t)=\frac{\sin (\pi t)}{\pi t}$

[!tip] Sinc Notation Alert In pure mathematics, the Unnormalized Sinc Function (often written as $\text{Sa}(t)$, sine-arg) is defined as $\text{Sa}(t)=\frac{\sin (t)}{t}$. To transition from unnormalized to normalized: $\text{sinc}(t)=\text{Sa}(\pi t)$ [241]. Always read the axis labels in exam sheets!

Proof of Peak Value at the Origin ($t=0$) — L’Hôpital’s Derivation

Direct substitution of $t=0$ into $\text{sinc}(t)$ yields an indeterminate form ($0/0$). Applying L’Hôpital’s Rule {differentiating the numerator and denominator separately with respect to t} [241]: ${\lim }_{t\to 0}\text{sinc}(t)={\lim }_{t\to 0}\frac{\frac{d}{dt}[\sin (\pi t)]}{\frac{d}{dt}[\pi t]}={\lim }_{t\to 0}\frac{\pi \cos (\pi t)}{\pi }=\cos (0)=1\text{(proved)}$

Proof of Zero-Crossing Points

The sinc function crosses the horizontal axis at values of $t$ where the numerator is zero but the denominator is non-zero [241]: $\sin (\pi t)=0⟹\pi t=k\pi ⟹t=k\text{for }k\in \mathbb{Z},k\ne 0$ Zero-crossings occur at all non-zero integer values: $t=\pm 1,\pm 2,\pm 3,\dots$ [241]

2.7 Advanced Elementary Signals

Real Exponential Function, $A{e}^{\sigma t}$

Changes its amplitude monotonically {unidirectionally, either always growing or always shrinking} with time, governed by a real exponent [239]:

• Exponential Decay ($\sigma <0$): $x(t)=A{e}^{-|\sigma |t}$ [239]. Amplitude decays asymptotically toward zero as $t\to \infty$. Models passive circuit element discharges (e.g., source-free RC circuits) [239].
• Exponential Growth ($\sigma >0$): $x(t)=A{e}^{|\sigma |t}$ [239]. Amplitude grows boundlessly as $t\to \infty$, representing unstable positive-feedback systems [239].

Complex Exponential Function, $A{e}^{st}$

Unifies real exponential damping and sinusoidal oscillations using the complex frequency variable $s=\sigma +j\Omega$ [239, 461]: $x(t)=A{e}^{(\sigma +j\Omega )t}=A{e}^{\sigma t}{e}^{j\Omega t}$ Applying Euler’s Identity ($e^{j\theta }=\cos \theta +j\sin \theta$) [239]: $x(t)=A{e}^{\sigma t}\cos (\Omega t)+jA{e}^{\sigma t}\sin (\Omega t)$

• $\sigma =0$ (Undamped / Pure Sinusoid): Constant amplitude oscillation [239].
• $\sigma <0$ (Damped Sinusoid): Oscillation enclosed inside an exponentially decaying envelope [239].
• $\sigma >0$ (Growing Sinusoid): Oscillation whose amplitude grows exponentially [239].

2.8 Rigorous Step-Impulse Calculus Derivations

[!theorem] The Step-to-Impulse Calculus Identity

1. Prove that $\delta (t)=\frac{du(t)}{dt}$: The unit step function $u(t)$ can be modeled as the limit of a continuous ramping transition function $g_{\Delta }(t)$ as its rise duration $\Delta \to 0$: $g_{\Delta }(t)=\{\begin{array}{ll}0,& t<-\Delta /2\\ \frac{t}{\Delta }+\frac{1}{2},& -\Delta /2\le t\le \Delta /2\\ 1,& t>\Delta /2\end{array}$ Taking the derivative of this transition function: $\frac{d{g}_{\Delta }(t)}{dt}=p_{\Delta }(t)=\{\begin{array}{ll}\frac{1}{\Delta },& -\Delta /2\le t\le \Delta /2\\ 0,& \text{otherwise}\end{array}$ This derivative $p_{\Delta }(t)$ is a rectangular pulse of height $1/\Delta$ and width $\Delta$. As $\Delta \to 0$:

   • The pulse height $1/\Delta \to \infty$.
   • The pulse width $\Delta \to 0$.
   • The total area remains constant: ${\int }_{-\infty }^{\infty }{p}_{\Delta }(t)dt=\frac{1}{\Delta }\cdot \Delta =1$. By definition, this limit is the Dirac delta function $\delta (t)$. Therefore: $\delta (t)={\lim }_{\Delta \to 0}{p}_{\Delta }(t)={\lim }_{\Delta \to 0}\frac{d{g}_{\Delta }(t)}{dt}=\frac{du(t)}{dt}\text{(proved)}$

2. Prove that $u(t)={\int }_{-\infty }^t\delta (\tau )d\tau$: Since $\delta (t)=\frac{du(t)}{dt}$, we integrate both sides from $-\infty$ to $t$: ${\int }_{-\infty }^t\delta (\tau )d\tau ={\int }_{-\infty }^t\frac{du(\tau )}{d\tau }d\tau =u(t)-u(-\infty )$ Since $u(-\infty )=0$ by definition: $u(t)={\int }_{-\infty }^t\delta (\tau )d\tau \text{(proved)}$

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3. Elementary Operations on the Time Variable

*(Target: Numerical Solving / 4-Mark to 8-Mark Graphical Sketching Problems)* [203, 569]

Time-domain operations shift or rescale the independent time variable $t$, altering when a signal occurs or how fast it unfolds, without changing its vertical amplitude [187, 204].

3.1 Mathematical Definitions

• Time Shifting: Delaying or advancing a signal along the time axis [204]. $y(t)=x(t-t_0)\text{where}\{\begin{array}{l}{t}_0>0⟹\text{Time Delay (Shift Right)}\\ t_0<0⟹\text{Time Advance (Shift Left)}\end{array}$
• Time Scaling: Compressing or expanding a signal along the time axis [205]. $y(t)=x(at)\text{where}\{\begin{array}{l}|a|>1⟹\text{Time Compression (Squeezed)}\\ |a|<1⟹\text{Time Expansion (Stretched)}\end{array}$
• Time Reversal (Folding): Mirroring the signal across the vertical $y$-axis [206]. $y(t)=x(-t)$

3.2 The Unbending Precedence Rule for Composite Operations

When a signal is subjected to multiple simultaneous time operations (e.g., plotting $x(at-b)$), the operations must be performed in a precise sequence to avoid boundary errors [180, 209]:

               [ x(t) ]  ───( Step 1: Shift by b )───>  [ x(t - b) ]
                                                            │
                                                     ( Step 2: Scale/Reverse by a )
                                                            │
                                                            ▼
                                                       [ x(at - b) ]

The Precedence Trap

You must always apply time shifting first ($x(t-b)$) before applying time scaling or reversal ($x(at-b)$) on the un-factored argument [180, 209]. Alternatively, if you choose to scale first, you must factor the argument to scale the delay properly: $x\left(a(t-\frac{b}{a})\right)$, meaning you shift the scaled waveform by $\frac{b}{a}$ instead of $b$ [180, 210].

3.3 Core Solved Numerical Sketching Problems (The “Exam Killers”)

Problem 1: Sketch $x(t)=4r(3t-1)$ [KUET 2024, 2019 - 4 Marks] [210]

• Step 1: Draw the base function: Start with the unit ramp $r(t)$ which has a slope of 1 starting at $t=0$.
• Step 2: Scale the amplitude: Multiply by 4 to get $4r(t)$, which has a slope of 4 starting at $t=0$.
• Step 3: Shift the time origin: Apply the shift of $1$ unit to the right to get $4r(t-1)$ [210]. The ramp now starts at $t=1$ with a slope of 4.
• Step 4: Scale the time variable: Apply the compression factor of 3 to get $4r(3t-1)$ [210].
  • Find the new starting boundary: $3t-1=0⟹t=1/3$.
  • The slope scales up proportionally to the compression: $\text{New Slope}=\text{Amplitude Slope}\times \text{Compression Factor}=4\times 3=12$ [210].
  • Mathematical Verification: $x(t)=\{\begin{array}{ll}12t-4,& t\ge 1/3\\ 0,& t<1/3\end{array}$

Problem 2: Sketch $x(t)=r(-0.5t+2)$ [KUET 2015 - 8 Marks] [211]

• Step 1: Factor the internal argument: $x(t)=r\left(-0.5(t-4)\right)$
• Step 2: Draw the base function: Start with the unit ramp $r(t)$.
• Step 3: Apply time reversal: Flip the ramp across the $y$-axis to get $r(-t)$ [211]. The ramp now exists for negative time ($t\le 0$), growing as time goes left.
• Step 4: Apply time scaling (expansion): Stretch the ramp by a factor of $0.5$ to get $r(-0.5t)$ [211]. The slope becomes shallower (slope = $0.5$).
• Step 5: Apply time shifting: Shift the entire expanded, reversed waveform to the right by $4$ units to obtain $r(-0.5(t-4))$ [211].
  • Find the starting boundary: $-0.5t+2=0⟹t=4$.
  • Due to time reversal, the signal exists strictly for $t\le 4$.
  • Evaluate key coordinate points to draw:
    • At $t=4⟹x(4)=r(0)=0$.
    • At $t=0⟹x(0)=r(2)=2$.
    • At $t=-2⟹x(-2)=r(3)=3$.

Problem 3: Sketch $x(t)=u(t-5)-u(t-7)$ [KUET 2021 - 5 Marks] [212]

• Step 1: Plot $u(t-5)$: A unit step function turning on at $t=5$ with an amplitude of $+1$.
• Step 2: Plot $-u(t-7)$: A negative unit step function turning off (subtracting 1) at $t=7$.
• Step 3: Sum the waveforms: The addition of these two step functions results in a flat rectangular gate window of amplitude 1 starting at $t=5$ and ending at $t=7$ (total width of 2).

Problem 4: Sketch $x(t)=t^2[u(t-1)-u(t-2)]$ [KUET 2021 - 5 Marks] [602]

• Step 1: Identify the window interval: The term $[u(t-1)-u(t-2)]$ represents a rectangular window pulse of amplitude 1 that exists strictly between $t=1$ and $t=2$.
• Step 2: Multiply the quadratic growth by the window: The function behaves as $t^2$ inside the active window interval and is 0 everywhere else: $x(t)=\{\begin{array}{ll}{t}^2,& 1\le t<2\\ 0,& \text{otherwise}\end{array}$
• Step 3: Sketch key coordinates:
  • At $t=1⟹x(1)=1^2=1$.
  • At $t=1.5⟹x(1.5)=1.5^2=2.25$.
  • At $t\to 2⟹x(2)\to 2^2=4$ (discontinuous jump back to 0 at $t=2$).

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4. Mathematical Representation of Piecewise Signals using Unit Steps

*(Target: 8-Mark Graphical Analysis & Synthesis Problems)* [602]

In university exam questions, you are often given a piecewise waveform and asked to express it algebraically [602].

4.1 Addition / Subtraction Method

Any arbitrary piecewise signal can be built by summing (adding or subtracting) shifted unit step and unit ramp functions that starting at the jump locations.

• Rule for Steps: A vertical jump of height $h$ at $t=t_0$ is represented by $h\cdot u(t-t_0)$.
• Rule for Ramps: A change in slope of $\Delta m$ at $t=t_0$ is represented by $\Delta m\cdot r(t-t_0)$.
  • Calculating slope changes: $\Delta m=m_{\text{new}}-m_{\text{old}}$.

Worked Example [KUET 2015 Q1(e) - 8 Marks]:

Express the staircase pulse shown below using a sum of unit step functions.

         x(t)
          │
      2 ──┼─────┐ (from t=1 to t=2)
          │     │
      1 ──┼──┐  │ (from t=0 to t=1)
          │  │  │
    ──────┴──┴──┴───── t
          0  1  2

• Step 1: Trace the jumps step-by-step:
  • At $t=0$: Signal jumps from $0$ up to $1$ (change of $+1$). $⟹+1\cdot u(t)$.
  • At $t=1$: Signal jumps from $1$ up to $2$ (change of $+1$). $⟹+1\cdot u(t-1)$.
  • At $t=2$: Signal jumps from $2$ down to $0$ (change of $-2$). $⟹-2\cdot u(t-2)$.
• Step 2: Combine the terms: $x(t)=u(t)+u(t-1)-2u(t-2)\text{(proved)}$

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4.2 Multiplication Method

Alternatively, any piecewise segment can be mathematically isolated by multiplying the analytical function of that segment by a rectangular gating step function.

• Gating step function: $[u(t-t_{\text{start}})-u(t-t_{\text{end}})]$ which is exactly 1 between $t_{\text{start}}$ and $t_{\text{end}}$ and 0 elsewhere.

Worked Example [KUET 2015 Q1(e) - 8 Marks]:

Express the same staircase pulse above using multiplication of unit step functions.

• Step 1: Isolate each constant interval:
  • Segment 1: Amplitude is $1$ between $t=0$ and $t=1$. $⟹1\cdot [u(t)-u(t-1)]$.
  • Segment 2: Amplitude is $2$ between $t=1$ and $t=2$. $⟹2\cdot [u(t-1)-u(t-2)]$.
• Step 2: Sum the gated segments: $x(t)=1\cdot [u(t)-u(t-1)]+2\cdot [u(t-1)-u(t-2)]$
• Step 3: Algebraic verification (expanding and grouping terms): $x(t)=u(t)-u(t-1)+2u(t-1)-2u(t-2)=u(t)+u(t-1)-2u(t-2)$ Both addition and multiplication representations are mathematically equivalent, but writing both in your exam script secures the full 8 marks [602]!

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5. Signal Classifications

*(Target: Theory Descriptive / 4-Mark to 6-Mark Comparative Tables)* [198, 552]

5.1 Continuous-Time (CT) vs. Discrete-Time (DT) [9, 10, 193]

• Continuous-Time Signals: Defined over a continuous interval of time $t\in \mathbb{R}$.
• Discrete-Time Signals: Defined strictly at discrete integer instances of time $n\in \mathbb{Z}$.

5.2 Analog vs. Digital Signals [9, 10, 194]

• Analog Signals: Continuous-time signals whose amplitudes can take on any infinite value within a continuous physical range.
• Digital Signals: Discrete-time signals whose amplitudes are restricted to a quantized, finite set of pre-defined levels.
  • The Bridge: The physical process of converting a continuous-amplitude discrete-time signal into a digital signal is called quantization [9, 10, 194].

5.3 Deterministic vs. Random (Nondeterministic)

Criterion	Deterministic Signals [36, 195]	Random (Nondeterministic) Signals [36, 195]
Predictability	100% predictable; values are completely specified for any future instant [195].	Unpredictable; future values can only be characterized statistically [195].
Mathematical Model	Described by closed-form equations (e.g., $A\cos (\omega t)$) [195].	Described by probability density functions, mean, and variance [195].
Physical Example	Pure sinusoidal carrier wave, DC voltage source.	Atmospheric static, thermal semiconductor noise [36, 195].

5.4 Time Boundary Classifications (Causal, Anticausal, Non-causal)

• Causal Signal: A signal that contains zero values for all negative time [22, 206]. $x(t)=0\text{for }t<0⟹x(t)=x(t)\cdot u(t)$
• Anticausal Signal: A signal that contains zero values for all positive time [22, 206]. $x(t)=0\text{for }t>0⟹x(t)=x(t)\cdot u(-t)$
• Non-causal Signal: A signal that contains non-zero values extending into both positive and negative time boundaries [22, 206].

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6. Periodic vs. Aperiodic Signals & Composite Periodicity

*(Target: 5-Mark to 6-Mark Periodicity Test Calculations)* [363]

6.1 Continuous-Time Periodicity

A continuous-time signal $x(t)$ is periodic if it satisfies [11, 196]: $x(t+T)=x(t)\text{for all }t\in \mathbb{R}$ The smallest positive, non-zero constant $T_0$ that satisfies this condition is the fundamental period [11, 196]. If no such $T$ exists, the signal is aperiodic [11].

6.2 Periodicity Test for Composite CT Signals (The Rationality Rule)

For a composite signal $x(t)=x_1(t)+x_2(t)+\cdots +x_k(t)$ with individual fundamental periods $T_1,T_2,\dots ,T_k$: The overall signal $x(t)$ is periodic if and only if the ratios of all individual periods are rational numbers [14, 197]: $\frac{T_i}{T_j}=\frac{p}{q}\in \mathbb{Q}\text{for integers }p,q$ If this condition is satisfied, the fundamental period $T_0$ of the composite wave is computed using the fractional Least Common Multiple (LCM) formula [198]: $T_0=\text{LCM}(T_1,T_2,\dots ,T_k)⟹\text{LCM}\left(\frac{a}{b},\frac{c}{d}\right)=\frac{\text{LCM}(a,c)}{\text{HCF}(b,d)}$ where $\text{HCF}$ is the Highest Common Factor (or Greatest Common Divisor).

6.3 Discrete-Time Periodicity Constraint

A discrete-time sinusoidal sequence $x[n]=A\cos ({\omega }_{0}n+\theta )$ is periodic if and only if its angular frequency ${\omega }_0$ is a rational multiple of $2\pi$ [116, 184]: $\frac{{\omega }_0}{2\pi }=\frac{m}{N}\in \mathbb{Q}⟹\text{Fundamental Period: }N=m\left(\frac{2\pi }{{\omega }_0}\right)\text{for the smallest integers }m,N$

6.4 High-Yield Solved Periodicity Problems

Problem 1: Determine the periodicity and period of $x(t)=\sin (\sqrt{2}\pi t)+\sin (15\pi t)$ [KUET 2019, 2015 - 5 Marks] [218]

• Step 1: Calculate individual periods:
  • ${\omega }_1=\sqrt{2}\pi ⟹T_1=\frac{2\pi }{\sqrt{2}\pi }=\sqrt{2}$ seconds.
  • ${\omega }_2=15\pi ⟹T_2=\frac{2\pi }{15\pi }=\frac{2}{15}$ seconds.
• Step 2: Evaluate the ratio: $\frac{T_1}{T_2}=\frac{\sqrt{2}}{2/15}=\frac{15\sqrt{2}}{2}$
• Step 3: Apply the Rationality Rule: Since $\sqrt{2}$ is irrational, the ratio $\frac{T_1}{T_2}$ is an irrational number ($\notin \mathbb{Q}$ ).
• Conclusion: The composite signal is Aperiodic.

Problem 2: Determine the periodicity and period of $x(t)=\sin (20\pi t)+\sin (5\pi t)$ [KUET 2017 - 6 Marks] [219]

• Step 1: Calculate individual periods:
  • ${\omega }_1=20\pi ⟹T_1=\frac{2\pi }{20\pi }=\frac{1}{10}$ seconds.
  • ${\omega }_2=5\pi ⟹T_2=\frac{2\pi }{5\pi }=\frac{2}{5}$ seconds.
• Step 2: Evaluate the ratio: $\frac{T_1}{T_2}=\frac{1/10}{2/5}=\frac{5}{20}=\frac{1}{4}$ Since $\frac{1}{4}$ is a rational number ($\in \mathbb{Q}$ ), the signal is Periodic.
• Step 3: Calculate the fundamental period ($T_0$): $T_0=\text{LCM}\left(\frac{1}{10},\frac{2}{5}\right)=\frac{\text{LCM}(1,2)}{\text{HCF}(10,5)}=\frac{2}{5}=0.4\text{ seconds}$

Problem 3: Determine if $x(t)=2\cos (4\pi t)+3\sin (3\pi t)$ is periodic or aperiodic. If periodic, find the period [KUET 2025 - 5 Marks] [210]

• Step 1: Calculate individual periods:
  • ${\omega }_1=4\pi ⟹T_1=\frac{2\pi }{4\pi }=\frac{1}{2}$ seconds.
  • ${\omega }_2=3\pi ⟹T_2=\frac{2\pi }{3\pi }=\frac{2}{3}$ seconds.
• Step 2: Evaluate the ratio: $\frac{T_1}{T_2}=\frac{1/2}{2/3}=\frac{3}{4}$ Since $\frac{3}{4}$ is a rational fraction, the composite signal is Periodic.
• Step 3: Compute the fundamental period ($T_0$): $T_0=\text{LCM}\left(\frac{1}{2},\frac{2}{3}\right)=\frac{\text{LCM}(1,2)}{\text{HCF}(2,3)}=\frac{2}{1}=2\text{ seconds}$

Problem 4: Consider a signal: $x(t)=10\sin \sqrt{5}\pi t$. Is it periodic? Find its fundamental period [KUET 2016 - 4 Marks] [209]

• Step 1: Analyze the composite rule boundary: The rationality check only applies to multi-tone additions. A single sinusoidal term is always periodic, regardless of whether its frequency is rational or irrational.
• Step 2: Find the fundamental period ($T_0$): $\omega =\sqrt{5}\pi ⟹T_0=\frac{2\pi }{\sqrt{5}\pi }=\frac{2}{\sqrt{5}}\text{ seconds}$

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7. Waveform Symmetry (Even & Odd Decompositions)

*(Target: 7-Mark to 10-Mark Symmetrical Proofs & Decompositions)* [209, 363]

7.1 Symmetrical axes and Decomposition

Any arbitrary signal $x(t)$ can be mathematically split into a sum of a purely symmetric (even) part and a purely anti-symmetric (odd) part [18, 186, 223]: $x(t)=x_e(t)+x_o(t)\text{where}{x}_e(t)=\frac{x(t)+x(-t)}{2}\text{and}{x}_o(t)=\frac{x(t)-x(-t)}{2}$

7.2 High-Yield Worked Decomposition Problems

Problem 1: Find even/odd parts of $x(t)=u(t)$ [KUET 2020 - 3 Marks] [213, 602]

• Even part: $x_e(t)=\frac{u(t)+u(-t)}{2}=\frac{1}{2}$ (since $u(t)+u(-t)=1$ for $t\ne 0$).
• Odd part: $x_o(t)=\frac{u(t)-u(-t)}{2}=\frac{1}{2}\text{sgn}(\mathbf{t})$ (since $u(t)-u(-t)=\text{sgn}(t)$).

Problem 2: Find even/odd parts of $x(t)=e^{-\alpha t}u(t)$ [KUET 2020 - 3 Marks] [602]

• Even part: $x_e(t)=\frac{e^{-\alpha t}u(t)+e^{\alpha t}u(-t)}{2}=\frac{1}{2}{\mathbf{e}}^{-\alpha |t|}$
• Odd part: $x_o(t)=\frac{e^{-\alpha t}u(t)-e^{\alpha t}u(-t)}{2}=\frac{1}{2}{\mathbf{e}}^{-\alpha \mathbf{t}}\text{sgn}(\mathbf{t})$

Problem 3: Find even/odd parts of $x(t)=\cos (20\pi t-5\pi )$ [KUET 2018, 2015 - 1 Mark] [602]

• Step 1: Simplify the expression algebraically: We know that $\cos (\theta -5\pi )=\cos (\theta -\pi )=-\cos (\theta )$. $x(t)=-\cos (20\pi t)$
• Step 2: Evaluate symmetry properties: Since $\cos (-\theta )=\cos (\theta )$, the signal is symmetric across the vertical axis: $x(-t)=-\cos (-20\pi t)=-\cos (20\pi t)=x(t)⟹\text{Purely Even}$
• Step 3: State decomposed components: $x_e(t)=-\cos (20\pi \mathbf{t})\text{and}{x}_o(t)=0$

7.3 Even-Odd Energy Orthogonality Proof [KUET 2024, 2023, 2021 - 7 to 10 Marks] [190, 224]

Theorem: Prove that the total energy of a signal is the sum of the energies of its even and odd components: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt$

Step-by-Step Mathematical Proof [21, 225]:

1. Express the signal as the sum of its decomposed parts: $x(t)=x_e(t)+x_o(t)$
2. Square both sides of the equation: $x^2(t)={\left[x_e(t)+x_o(t)\right]}^2=x_e^2(t)+x_o^2(t)+2x_e(t)x_o(t)$
3. Integrate both sides across the entire real timeline: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt+2{\int }_{-\infty }^{\infty }{x}_e(t)x_o(t)dt$
4. Analyze the cross-product integrand $g(t)=x_e(t)x_o(t)$:
   • Apply time reversal to the product: $g(-t)=x_e(-t)x_o(-t)$
   • Since $x_e(t)$ is even ($x_e(-t)=x_e(t)$) and $x_o(t)$ is odd ($x_o(-t)=-x_o(t)$): $g(-t)=[x_e(t)][-x_o(t)]=-x_e(t)x_o(t)=-g(t)$
   • The cross-product $g(t)$ is a purely odd function.
5. By the properties of definite integrals, the integral of any odd function over symmetrical limits is strictly zero [21, 225]: $2{\int }_{-\infty }^{\infty }{x}_e(t)x_o(t)dt=0$
6. The energy equation simplifies directly to the sum of the component energies [21, 225]: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt■$

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8. Energy and Power Classifications

*(Target: 2-Mark to 6-Mark Numerical Integrations & Classification Problems)* [211, 363]

8.1 Continuous-Time (CT) vs. Discrete-Time (DT) Metrics

• Continuous-Time: [24, 211, 226] $E={\int }_{-\infty }^{\infty }|x(t){|}^{2}dt|P={\lim }_{T\to \infty }\frac{1}{T}{\int }_{-T/2}^{T/2}|x(t){|}^{2}dt$
• Discrete-Time: [108, 109, 226] $E={\sum }_{n=-\infty }^{\infty }|x[n]{|}^2|P={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=-N}^N|x[n]{|}^2$

8.2 The Golden Classification Rules

1. Energy Signal: Finite, non-zero energy and zero average power ($0<E<\infty$ and $P=0$) [25, 212, 227].
2. Power Signal: Finite, non-zero average power and infinite total energy ($0<P<\infty$ and $E=\infty$) [25, 212, 227].
3. Neither: If a signal grows infinitely with time (such as $e^{at}$ or $t\cdot u(t)$), both its energy and power diverge to infinity ($E=\infty$ and $P=\infty$) [212, 227].

8.3 High-Yield Worked Classification Problems

Problem 1: Classify and find the metrics of $x(t)=u(t)$ [KUET 2022 - 2 Marks] [229]

• Step 1: Integrate for Energy ($E$): $E={\int }_{-\infty }^{\infty }|u(t){|}^{2}dt={\int }_0^{\infty }(1{)}^{2}dt=[t{]}_0^{\infty }=\infty ⟹\text{Not an Energy Signal}$
• Step 2: Integrate for Power ($P$): $P={\lim }_{T\to \infty }\frac{1}{T}{\int }_{-T/2}^{T/2}|u(t){|}^{2}dt={\lim }_{T\to \infty }\frac{1}{T}{\int }_0^{T/2}(1{)}^{2}dt={\lim }_{T\to \infty }\frac{1}{T}\left(\frac{T}{2}\right)=0.5\text{ Watts}$
• Conclusion: Since $E=\infty$ and $P=0.5\text{ W}$, the unit step function is a Power Signal.

Problem 2: Classify and find the metrics of $x[n]=\cos (\pi n)u[n]$ [KUET 2023, 2019 - 5 to 6 Marks] [228]

• Step 1: Simplify the sequence: Since $n$ is an integer sample index, $\cos (\pi n)=(-1{)}^n$ for $n\ge 0$.
• Step 2: Sum for Energy ($E$): $E={\sum }_{n=-\infty }^{\infty }|x[n]{|}^2={\sum }_{n=0}^{\infty }|(-1{)}^n{|}^2={\sum }_{n=0}^{\infty }(1)=\infty ⟹\text{Not an Energy Signal}$
• Step 3: Evaluate limit for Power ($P$): $P={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=-N}^N|x[n]{|}^2={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=0}^N(1)={\lim }_{N\to \infty }\frac{N+1}{2N+1}$ Dividing numerator and denominator by $N$: $P={\lim }_{N\to \infty }\frac{1+1/N}{2+1/N}=0.5\text{ Watts}$
• Conclusion: Since $E=\infty$ and $P=0.5\text{ W}$, the causal sequence is a Power Signal.

Problem 3: Classify and find the metrics of $x[n]=\sin (\pi n)u[n]$ [KUET 2025 - 5 Marks] [227]

• Step 1: Evaluate the sequence: Since $n$ is strictly an integer index, $\sin (\pi n)=0$ for all $n\in \mathbb{Z}$ [228].
• Step 2: Find Energy and Power: $E=0\text{and}P=0$
• Conclusion: This is a trivial zero-signal, representing neither a power nor an energy signal [228].

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9. Typical Signal Processing Operations

*(Target: Theory Descriptive / 3-Mark to 4-Mark Short Notes)* [283, 363]

• Correlation: A mathematical measure of similarity between two waveforms as a function of a relative time shift [13, 233]. It is used to compare a reference signal with a target signal [13].
  • Auto-correlation: Compares a signal against a shifted version of itself to detect hidden periodicities [233]: $R_{xx}(au)={\int }_{-\infty }^{\infty }x(t)x(t+\tau )dt$
  • Cross-correlation: Compares two different signals [233]: $R_{xy}(au)={\int }_{-\infty }^{\infty }x(t)y(t+\tau )dt$
  • Physical Application: Radar/sonar echo delay measurements, fingerprint matching, and channel noise suppression [13, 233].
• Filtering: A frequency-selective operation that passes desired spectral components of a signal while suppressing unwanted ones [234]. Standard designs include Low-Pass (LPF), High-Pass (HPF), Band-Pass (BPF), Band-Stop (BSF), and Notch filters [234].
  • Notch Filter: A specialized, extremely narrow band-stop filter designed to reject a single frequency (e.g., suppressing 50 Hz/60 Hz power-line interference in ECG devices) [234].
• Modulation: The process of varying parameters (amplitude, phase, or frequency) of a high-frequency carrier wave in proportion to a low-frequency message signal [234].
  • Amplitude Modulation (AM) Model: $s(t)=m(t)\cos ({\omega }_{c}t)$
  • Physical Importance: Human voice waves (20 Hz - 20 kHz) have extremely long wavelengths. Antennas must be proportional to wavelength ($\approx \lambda /4$). Multiplying by a high-frequency carrier (e.g., 100 MHz) shrinks the wavelength to meters, allowing compact, practical antennas and enabling frequency multiplexing [234].
• Transformation: Domain-mapping (Fourier, Laplace, Z-transforms) to simplify system equations [235]. It translates complex calculus differential equations into simple, solvable algebraic polynomials [235].
• Multiplexing: Combining multiple data streams over a single shared physical channel [236]:
  • Time-Division Multiplexing (TDM): Interleaves signals in non-overlapping time slots [236].
  • Frequency-Division Multiplexing (FDM): Assigns distinct frequency bands separated by unshaded guard bands to users [236].

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10. DAC Hold & Reconstruction Circuits

*(Target: Theory Descriptive / 5-Mark s-Domain Transfer Function Derivations)* [147, 4.03]

Digital-to-Analog Converters (DACs) reconstruct continuous-time signals from discrete samples [238]. Physical reconstruction requires a hold circuit to bridge sample intervals [147, 4.03].

+------------+      +--------------+      +---------------------+      +---------+
|  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)
+------------+      +--------------+      +---------------------+      +---------+

10.1 Zero-Order Hold (ZOH) Reconstruction

The ZOH holds the amplitude of the last acquired sample completely constant over the entire sampling period $T_s$ until the next sample is acquired, generating a staircase-like waveform [238].

• Impulse Response: A rectangular gate pulse of width $T_s$ [244]: $h_0(t)=u(t)-u(t-T_s)$

Derivation of ZOH s-Domain Transfer Function

To find $H_0(s)$, we take the unilateral Laplace transform of its impulse response: $H_0(s)=\mathcal{L}\{u(t)-u(t-T_s)\}=\mathcal{L}\{u(t)\}-\mathcal{L}\{u(t-T_s)\}$ Applying the s-domain time-shifting property: $H_0(s)=\frac{1}{s}-\frac{1}{s}{e}^{-s{T}_s}=\frac{1-{\mathbf{e}}^{-\mathbf{s}{\mathbf{T}}_{\mathbf{s}}}}{\mathbf{s}}$

Derivation of ZOH Frequency Response

Substitute $s=j\Omega$ into the transfer function: $H_0(j\Omega )=\frac{1-e^{-j\Omega T_s}}{j\Omega }$ Factor out $e^{-j\Omega T_s/2}$ using Euler’s identity to establish the phase delay: $H_0(j\Omega )=\frac{e^{-j\Omega T_s/2}\left(e^{j\Omega T_s/2}-e^{-j\Omega T_s/2}\right)}{j\Omega }$ Since $e^{j\theta }-e^{-j\theta }=2j\sin \theta$: $H_0(j\Omega )=e^{-j\Omega T_s/2}\left[\frac{2j\sin \left(\frac{\Omega T_s}{2}\right)}{j\Omega }\right]=2\frac{\sin \left(\frac{\Omega T_s}{2}\right)}{\Omega }{e}^{-j\Omega T_s/2}$ Multiply numerator and denominator by $T_s$: $H_0(j\Omega )=T_s\left[\frac{\sin \left(\frac{\Omega T_s}{2}\right)}{\frac{\Omega T_s}{2}}\right]e^{-j\Omega T_s/2}={\mathbf{T}}_{\mathbf{s}}\text{sinc}\left(\frac{\mathbf{\Omega }{\mathbf{T}}_{\mathbf{s}}}{2\pi }\right){\mathbf{e}}^{-\mathbf{j\Omega }{\mathbf{T}}_{\mathbf{s}}/2}$

Physical Implications

1. Aperture Distortion (Sinc Droop): The magnitude $|H_0(j\Omega )|$ acts as a low-pass filter, but rolls off before the Nyquist frequency, dampening high-frequency components [244].
2. Phase Delay: The phase term $e^{-j\Omega T_s/2}$ introduces a constant linear phase delay of exactly half a sampling interval ($T_s/2$) [244].

10.2 First-Order Hold (FOH) Reconstruction

The FOH reconstructs continuous waveforms using linear interpolation, joining sample points with straight lines (polygonal waveforms) [242].

• Symmetric Impulse Response: A triangular pulse centered at the origin [242]: $h_1(t)=\left(1-\frac{|t|}{T_s}\right)\left[u(t+T_s)-u(t-T_s)\right]$
• Frequency Response: The squared sinc function [242]: $H_1(j\Omega )=T_s{\left(\text{sinc}\left(\frac{\Omega T_s}{2\pi }\right)\right)}^2$
• Comparison: FOH decays as ${\text{sinc}}^2(f)$, drastically reducing high-frequency staircase harmonic leakage compared to ZOH’s $\text{sinc}(f)$ decay [242]. However, FOH requires a delay of $T_s$ for physical causality, while ZOH only delays by $T_s/2$ [242].

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11. Common Mistakes That Cost Marks

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 ${\omega }_0$ [201]. A discrete sinusoid $A\cos ({\omega }_{0}n)$ is periodic if and only if ${\omega }_0/2\pi$ is a rational number [106, 300].
2. Fractional LCM Swap: Inverting the fractional LCM formula. Remember that the Least Common Multiple of fractional periods is $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$ [198]. Writing HCF/LCM is a guaranteed zero-marks error [198].
3. Even/Odd Cross-Product Omission: Dropping the cross-product evaluation during the Even-Odd energy identity proof [210]. 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 [210].
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 ($\int x_1(t)x_2(t)dt=0$) [50].

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12. PYQ Bank — Verbatim Questions & Answer Plans

Q1: [PYQ 2024 - 7 Marks]

What are signal and system? What are the importance of these in communication engineering?

• Answer Plan: Define Signal and System mathematically (Section 1.1) [4, 5, 174]. Provide the real-world communication applications (ECG, radar, sonar, GPS) and explain the importance of domain transformation and noise suppression (Section 1.3) [197, 235].

Q2: [PYQ 2022 - 8 Marks]

Define: (i) Unit step function, (ii) Unit impulse function, (iii) Ramp function, and (iv) Signum function.

• Answer Plan: Write down the mathematical piecewise equations for each function and sketch their continuous-time waveforms with labeled axes (Section 2) [181, 182, 183, 184].

Q3: [PYQ 2024, 2023, 2021 - 7 to 10 Marks]

If $x_e(t)$ and $x_o(t)$ are even and odd parts of $x(t)$, 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: Provide the complete 6-step algebraic and calculus proof showing the cancellation of the cross-product integral due to its odd parity over symmetrical boundaries (Section 7.3) [21, 225].

Q4: [PYQ 2024, 2019 - 4 Marks]

Sketch the following signals: (i) $x(t)=-4r(3t-1)$, (ii) $x(t)=\Pi (2t+5)$.

• Answer Plan:
  • (i) Apply the Precedence Rule: Shift right by 1, scale amplitude by 4, compress time by 3 to start the ramp at $t=1/3$ with a slope of 12 (Section 3.3, Problem 1) [210].
  • (ii) Shift left by 5 units, compress width by 2 to center a gate pulse of width 0.5 at $t=-2.5$ [209].

Q5: [PYQ 2017 - 6 Marks]

Determine whether $x(t)=\sin (20\pi t)+\sin (5\pi t)$ is periodic or not. If it is periodic, then find the fundamental period.

• Answer Plan: Compute individual periods $T_1=1/10$ s, $T_2=2/5$ s. Verify the rational ratio (1/4) and compute the fractional LCM to find the fundamental period of $T_0=0.4$ seconds (Section 6.4, Problem 2) [219].

Q6: [PYQ 2023 - 5 Marks]

Determine whether $x[n]=\cos (\pi n)u[n]$ is energy or power signal and find its value.

• Answer Plan: Prove $E=\infty$ and $P=0.5\text{ W}$, classifying it as a Power Signal (Section 8.3, Problem 2) [121, 229].

Q7: [PYQ 2015 - 8 Marks]

Write down the corresponding equation for the given signal in figure 1(e). i) Represent through addition of unit step functions. ii) Represent through multiplication of unit step functions.

• Answer Plan:
  • (i) Trace vertical jumps step-by-step to sum shifted unit steps (Section 4.1) [602].
  • (ii) Gated rectangular segments multiplied by steps of the form $[u(t-t_s)-u(t-t_e)]$ (Section 4.2) [602].

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13. Self-Check Before Moving On

• Can you define and sketch $u(t)$, $\delta (t)$, $r(t)$, and $\text{sgn}(t)$ with correct vertical limits [181, 182, 183, 184]?
• Can you solve a composite periodicity problem using fractional LCM [198, 200]?
• Can you write out the complete Even-Odd energy proof without skipping steps [210]?
• Do you know the exact conditions for a signal to be Energy, Power, or Neither [212]?
• Can you derive the s-domain transfer function of a ZOH from its impulse response step-by-step?

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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, 1.06 Typical Signal Processing Operations.md, 1.07_Supplementary_Signal_Functions_Hold_Circuits_and_DAC_Reconstruction.md