1.00 Chapter 1: Signal Fundamentals & Operations — Compact Review Sheet

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💡 Syllabus & PYQ Weightage Dashboard

• Target Domain: Instructor 2 (Time-Domain Fundamentals) [1.297, 1.365].
• Total Exam Value: 15 to 25 Marks per paper [1.352, 1.518].
• High-Yield Targets (Must Master):
  1. Even-Odd Energy Orthogonality Proof [PYQ 2024, 2023, 2021] (7 to 10 Marks) [1.518].
  2. Composite Periodicity Tests & Fractional LCM [PYQ 2019, 2017, 2015] (5 to 6 Marks) [1.518].
  3. Composite Signal Sketching (The Precedence Rule) [PYQ 2024, 2019, 2015] (4 to 8 Marks) [1.518].
  4. Energy vs. Power Classification Integrals [PYQ 2025, 2023, 2022, 2019] (2 to 6 Marks) [1.518].
  5. ZOH Transfer Function s-Domain Derivation [Syllabus / Lecture Notes] (5 Marks) [1.113, 1.518].

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1. Introduction to Signals & Systems

*(Target: Theory Descriptive / 2-Mark to 6-Mark Vocabulary)* [1.106, 1.352]

• Signal: Any physical quantity that carries information and varies with time, space, or any other independent variable [1.10, 1.99]. Mathematically modeled as: $x(t)\text{(Continuous-Time)}\text{or}x[n]\text{(Discrete-Time)}[1.99]$
• System: A physical device, process, or mathematical algorithm that operates on an input signal $x(t)$ (excitation) to produce a transformed output signal $y(t)$ (response) [1.11, 1.99]: $y(t)=\mathbb{T}[x(t)]\text{or}y[n]=\mathbb{T}[x[n]][1.11,1.99]$ where $\mathbb{T}$ represents the transformation operator [1.11, 1.99].
• The Chain of Signal Dependence: Event $\to$ Sequence $\to$ Time $\to$ Frequency $\to$ Phase $\to$ Amplitude [1.99]. Ultimately, all signal characteristics depend on time [1.99].
• Real-World Applications: Biomedical monitoring (ECG/EEG heart and brain waves), target detection (radar and sonar echo delays), satellite navigation (GPS synchronization), geological tracking (seismic vibrations), and media processing (voice/TV broadcast codecs) [1.10, 1.99].

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2. Elementary Continuous-Time Singularity Functions

*(Target: Waveform Sketching / 3-Mark to 8-Mark Calculus Derivations)* [1.100, 1.368]

Singularity functions act as highly idealized building blocks to model instantaneous switching transitions, impulse excitations, and gating envelopes [1.100, 1.368].

2.1 Piecewise Mathematical Definitions

• Unit Step Function, $u(t)$: Models a DC switch closing instantaneously at $t=0$ [1.17, 1.100]: $u(t)=\{\begin{array}{ll}1,& t>0\\ 0,& t<0\\ \text{undefined},& t=0\end{array}[1.17,1.100]$
• Unit Impulse Function (Dirac Delta), $\delta (t)$: An infinitely narrow, infinitely tall spike centered at $t=0$ containing an area (weight) of exactly 1 [1.17, 1.100]: $\delta (t)=0\text{for }t\ne 0,\text{and}{\int }_{-\infty }^{\infty }\delta (t)dt=1[1.17,1.100]$
  • Sifting Property: ${\int }_{-\infty }^{\infty }x(t)\delta (t-t_0)dt=x(t_0)$ [1.18]
  • Scaling Property: $\delta (at+b)=\frac{1}{|a|}\delta \left(t+\frac{b}{a}\right)$ [1.19]
• Unit Ramp Function, $r(t)$: A linearly growing signal representing the time integral of the step function [1.19, 1.100]: $r(t)=\{\begin{array}{ll}t,& t>0\\ 0,& t\le 0\end{array}⟹r(t)=t\cdot u(t)[1.19]$
• Signum Function, $\text{sgn}(t)$: A polarity indicator extracting the algebraic sign of time $t$ [1.8, 1.100]: $\text{sgn}(t)=\{\begin{array}{ll}1,& t>0\\ 0,& t=0\\ -1,& t<0\end{array}⟹\text{sgn}(t)=u(t)-u(-t)[1.8]$
• Rectangular Gate Pulse, $\Pi (t/\tau )$: A symmetrical window centered at the origin of width $\tau$ [1.8]: $\Pi \left(\frac{t}{\tau }\right)=\{\begin{array}{ll}1,& -\frac{\tau }{2}<t<\frac{\tau }{2}\\ 0,& \text{otherwise}\end{array}[1.8]$

2.2 Sinc Function & Calculus Limits

The continuous-time normalized sinc function represents the frequency spectrum of a rectangular time pulse [1.8, 1.113]: $\text{sinc}(t)=\frac{\sin (\pi t)}{\pi t}[1.8,1.113]$

• Proof of Peak Value at the Origin ($t=0$): Direct substitution yields $0/0$. Applying L’Hôpital’s Rule: ${\lim }_{t\to 0}\text{sinc}(t)={\lim }_{t\to 0}\frac{\frac{d}{dt}[\sin (\pi t)]}{\frac{d}{dt}[\pi t]}={\lim }_{t\to 0}\frac{\pi \cos (\pi t)}{\pi }=\cos (0)=1\text{(proved)}[1.113]$
• Zero-Crossing Points: $\sin (\pi t)=0⟹\pi t=k\pi ⟹t=k\text{for }k\in \mathbb{Z},k\ne 0[1.113]$
  • Zero-crossings occur at all non-zero integer values: $t=\pm 1,\pm 2,\pm 3,\dots$ [1.113]

2.3 Rigorous Step-Impulse Derivations [1.19, 1.101]

• Derivative relationship (impulse): $\delta (t)=\frac{du(t)}{dt}\text{(derivation)}[1.19,1.101]$
  • Mathematical Proof: Model step $u(t)$ as the limit of a continuous ramping function $g_{\Delta }(t)$ as the rise time $\Delta \to 0$. Its derivative $\frac{d{g}_{\Delta }(t)}{dt}$ is a rectangular pulse $p_{\Delta }(t)$ of width $\Delta$ and height $1/\Delta$. As $\Delta \to 0$, pulse width $\to 0$, height $\to \infty$, while the area remains constant: $\int p_{\Delta }(t)dt=\frac{1}{\Delta }\cdot \Delta =1$. This matches the definition of Dirac delta $\delta (t)$.
• Integral relationship (step): $u(t)={\int }_{-\infty }^t\delta (\tau )d\tau \text{(derivation)}[1.19,1.101]$

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3. Core Elementary Operations on the Independent Variable (Time)

*(Target: Numerical Solving / 4-Mark to 8-Mark Signal Sketching)* [1.101, 1.369]

3.1 Core Definitions

• Time Shifting: Delaying (shifting right) or advancing (shifting left) a waveform [1.12, 1.101]: $y(t)=x(t-t_0)\text{where}\{\begin{array}{l}{t}_0>0⟹\text{Delay (Right-Shift)}\\ t_0<0⟹\text{Advance (Left-Shift)}\end{array}[1.12]$
• Time Scaling: Compressing or expanding a waveform along the time axis [1.12, 1.101]: $y(t)=x(at)\text{where}\{\begin{array}{l}|a|>1⟹\text{Time Compression (Squeezed)}\\ |a|<1⟹\text{Time Expansion (Stretched)}\end{array}[1.12]$
• Time Reversal (Folding): Symmetric reflection across the vertical axis [1.13]: $y(t)=x(-t)\text{or}y[n]=x[-n][1.13]$

3.2 The Unbending Precedence Rule

When drawing simultaneous shifting, scaling, and reversal (e.g., plotting $x(at-b)$) [1.369]:

$\text{Step 1: Shift First }[x(t-b)]\stackrel{\text{Scale/Reverse by }a}{\to }\text{Step 2: Plot }[x(at-b)][1.369]$

The Precedence Trap

Shifting must always be performed before scaling/reversal when working on the un-factored argument $x(at-b)$ [1.101]. Alternatively, if you scale first, you must factor the scaling term to scale the delay properly: $x\left(a(t-\frac{b}{a})\right)$, shifting the scaled waveform by $\frac{b}{a}$ instead of $b$ [1.101].

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4. Signal Classifications (Structural & Boundaries)

*(Target: Theory Descriptive / Comparison Matrices)* [1.108, 1.518]

4.1 Continuous-Time (CT) vs. Discrete-Time (DT)

CT signals are defined over a continuous timeline $t\in \mathbb{R}$, while DT signals are indexed sequences defined strictly at integer samples $n\in \mathbb{Z}$ [1.7, 1.108].

• Continuous-Time: $x(t)$ (Continuous curve) [1.7, 1.108]
• Discrete-Time: $x[n]$ (Discrete stem plot) [1.7, 1.108]

4.2 Analog vs. Digital Signals

• Analog: Amplitudes are continuous (can take infinite possible values over a range) [1.7].
• Digital: Amplitudes are quantized (restricted to a pre-defined, finite set of levels) [1.7].
  • The Interface: Quantization is the process of mapping continuous amplitudes to a digital finite set [1.7].

4.3 Deterministic vs. Random (Nondeterministic)

• Deterministic: Signal behavior is completely predictable and described by a closed-form algebraic formula (e.g., $A\cos (\omega t)$) [1.7].
• Random: Signal values are unpredictable and can only be characterized statistically using probability densities (e.g., thermal noise) [1.7].

4.4 Time Boundary Classifications

• Causal Signal: Exists strictly in positive time [1.7]: $x(t)=0\text{for }t<0⟹x(t)=x(t)\cdot u(t)[1.7]$
• Anticausal Signal: Exists strictly in negative time [1.7]: $x(t)=0\text{for }t>0⟹x(t)=x(t)\cdot u(-t)[1.7]$
• Non-causal Signal: Exists on both sides of the origin (extends into both positive and negative time boundaries) [1.7].

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5. Periodic vs. Aperiodic Signals & Composite Periodicity

*(Target: 5-Mark to 6-Mark Periodicity Test Calculations)* [1.108, 1.518]

A signal is periodic if it repeats its amplitude profile exactly after a constant fundamental period [1.7]: $x(t+T_0)=x(t)\text{(for CT)}\text{and}x[n+N]=x[n]\text{(for DT)}[1.7]$

5.1 Continuous-Time Periodicity Rules

• A continuous sinusoid $A\cos ({\omega }_{0}t+\theta )$ is always periodic regardless of the value of ${\omega }_0$ [1.102].
• The Rationality Rule: A composite signal $x(t)=x_1(t)+x_2(t)+\cdots +x_k(t)$ with individual periods $T_1,T_2,\dots ,T_k$ is periodic if and only if the ratios of all individual periods are rational numbers [1.14, 1.101]: $\frac{T_i}{T_j}=\frac{p}{q}\in \mathbb{Q}(\text{ratio of two integers }p,q)\text{(derivation)}[1.14,1.101]$
• If rational, the overall fundamental period $T_0$ is evaluated using the fractional LCM formula [1.14]: ${\mathbf{T}}_0=\text{LCM}({\mathbf{T}}_1,{\mathbf{T}}_2,\dots ,{\mathbf{T}}_{\mathbf{k}})⟹\text{LCM}\left(\frac{a}{b},\frac{c}{d}\right)=\frac{\text{LCM}(a,c)}{\text{HCF}(b,d)}[1.14,1.101]$

5.2 Discrete-Time Periodicity Constraints

Unlike continuous sines, a discrete sinusoid $x[n]=A\cos ({\omega }_{0}n+\theta )$ is periodic if and only if its angular frequency ${\omega }_0$ is a rational multiple of $2\pi$ [1.102]: $\frac{{\omega }_0}{2\pi }=\frac{m}{N}\in \mathbb{Q}⟹\text{Fundamental Period: }N=m\left(\frac{2\pi }{{\omega }_0}\right)(\text{for the smallest integers }m,N)[1.102]$

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6. Waveform Parity & Decompositions

*(Target: 7-Mark to 10-Mark Symmetrical Proofs)* [1.101, 1.518]

6.1 Symmetry Decompositions

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 [1.101]: $x(t)=x_e(t)+x_o(t)[1.101]$

• Even Part (Symmetric across y-axis): $x_e(t)=\frac{x(t)+x(-t)}{2}⟹x_e(-t)=x_e(t)[1.101]$
• Odd Part (Anti-symmetric across origin): $x_o(t)=\frac{x(t)-x(-t)}{2}⟹x_o(-t)=-x_o(t)[1.101]$

6.2 Even-Odd Energy Orthogonality Proof

[!theorem] Symmetrical Power Conservation Proof [PYQ 2024, 2023, 2021] Prove that the total energy of a signal equals the sum of the energies of its even and odd components [1.101, 1.518]: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt\text{(derivation)}[1.101,1.518]$

Step-by-Step Proof:

1. Express the signal as the sum of its decomposed parts: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }[x_e(t)+x_o(t){]}^{2}dt$
2. Expand the squared binomial integrand: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt+2{\int }_{-\infty }^{\infty }{x}_e(t)x_o(t)dt$
3. Analyze the cross-product integrand $g(t)=x_e(t)x_o(t)$. Test its parity under time reversal: $g(-t)=x_e(-t)x_o(-t)=x_e(t)[-x_o(t)]=-x_e(t)x_o(t)=-g(t)$ The cross-product $g(t)$ is a purely odd function.
4. By definite integration laws, the integral of any odd function over symmetrical limits is strictly zero: $2{\int }_{-\infty }^{\infty }{x}_e(t)x_o(t)dt=0[1.102]$
5. The cross-product vanishes, completing the proof: ${\int }_{-\infty }^{\infty }{x}^2(t)dt={\int }_{-\infty }^{\infty }{x}_e^2(t)dt+{\int }_{-\infty }^{\infty }{x}_o^2(t)dt■$

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7. Energy and Power Classifications

*(Target: 2-Mark to 6-Mark Numerical Integrals)* [1.101, 1.518]

Measures a signal’s physical dissipation properties over infinite boundaries to classify it [1.101].

7.1 Continuous-Time Definition Integrals

• Total Energy ($E$): $E={\int }_{-\infty }^{\infty }|x(t){|}^{2}dt[1.101]$
• Average Power ($P$): $P={\lim }_{T\to \infty }\frac{1}{T}{\int }_{-T/2}^{T/2}|x(t){|}^{2}dt[1.101]$
  • Periodic Signal power shortcut: $P=\frac{1}{T_0}{\int }_0^{T_0}|x(t){|}^{2}dt$ (integrated over exactly one period $T_0$).

7.2 Discrete-Time Definition Sums

• Total Energy ($E$): $E={\sum }_{n=-\infty }^{\infty }|x[n]{|}^2[1.101]$
• Average Power ($P$): $P={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=-N}^N|x[n]{|}^2[1.101]$

7.3 Bounded Classification Rules

1. Energy Signal: Bounded non-zero energy and zero average power ($0<E<\infty$ and $P=0$) [1.102].
   • Examples: Transient pulses, decay signals, finite rectangular gates.
2. Power Signal: Bounded non-zero average power and infinite total energy ($0<P<\infty$ and $E=\infty$) [1.102].
   • Examples: All periodic signals, infinite-duration step signals ($u(t)$).
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 $P=\infty$) [1.102].

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8. Typical Signal Processing Operations

*(Target: Theory Descriptive / 3-Mark to 4-Mark Short Notes)* [1.101, 1.518]

• Correlation: Measures mathematical similarity between waveforms [1.13, 1.101]. Unlike convolution, correlation does not fold the sliding signal [1.110]: $R_{xy}(\tau )={\int }_{-\infty }^{\infty }x(t)y(t+\tau )dt[1.13,1.101]$
  • Auto-correlation ($R_{xx}$): Signal compared with a shifted version of itself to find periodicities.
  • Cross-correlation ($R_{xy}$): Compares two different signals.
• Filtering: Frequency-selective operation to suppress unwanted spectral components [1.101]. Categorized as Low-Pass (LPF), High-Pass (HPF), Band-Pass (BPF), Band-Stop (BSF), and sharp Notch filters (replaces a single frequency, e.g., 50 Hz power hum) [1.110].
• Modulation: Shifting a low-frequency baseband message $m(t)$ up to a high-frequency carrier band: $s(t)=m(t)\cos ({\omega }_{c}t)[1.101]$
  • Physical Importance: Antenna size must be proportional to wavelength ($\approx \lambda /4$). Modulation to high-frequencies decreases the wavelength, enabling practical, compact antennas and allowing channel sharing (FDM) [1.111].
• Transformation: Coordinate-domain mapping (Laplace, Fourier, Z-transform) to simplify calculus differential equations into algebraic equations [1.101, 1.110].
• Multiplexing: Combining multiple data streams over a single shared channel [1.101]:
  • Time-Division (TDM): Splits users into non-overlapping time slots.
  • Frequency-Division (FDM): Assigns distinct frequency bands separated by guard bands.

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9. DAC Hold & Reconstruction Circuits

*(Target: s-Domain Transfer Function Derivations)* [1.112, 1.113]

Reconstructing continuous analog waveforms $x_a(t)$ from discrete samples requires a signal hold circuit to bridge sample intervals [1.112, 1.113].

9.1 Zero-Order Hold (ZOH)

The ZOH holds the amplitude of the last sample constant until the next sample is acquired, generating a staircase-like waveform [1.113].

• Impulse Response: A rectangular gate pulse of width $T_s$ [1.113]: $h_0(t)=u(t)-u(t-T_s)[1.113]$

s-Domain Transfer Function Derivation

Take the Laplace transform of the impulse response $h_0(t)$ [1.113]: $H_0(s)=\mathcal{L}\{u(t)-u(t-T_s)\}=\mathcal{L}\{u(t)\}-\mathcal{L}\{u(t-T_s)\}$ Using the Laplace time-shifting property: ${\mathbf{H}}_0(\mathbf{s})=\frac{1-{\mathbf{e}}^{-\mathbf{s}{\mathbf{T}}_{\mathbf{s}}}}{\mathbf{s}}[1.113]$

Frequency Response Derivation

Substitute $s=j\Omega$ [1.113]: $H_0(j\Omega )=\frac{1-e^{-j\Omega T_s}}{j\Omega }$ Factor out $e^{-j\Omega T_s/2}$ using Euler’s identity: $H_0(j\Omega )=\frac{e^{-j\Omega T_s/2}\left(e^{j\Omega T_s/2}-e^{-j\Omega T_s/2}\right)}{j\Omega }$ Since $e^{j\theta }-e^{-j\theta }=2j\sin \theta$, substitute: $H_0(j\Omega )=e^{-j\Omega T_s/2}\left[\frac{2j\sin \left(\frac{\Omega T_s}{2}\right)}{j\Omega }\right]=2\frac{\sin \left(\frac{\Omega T_s}{2}\right)}{\Omega }{e}^{-j\Omega T_s/2}$ Multiply numerator and denominator by $T_s$: ${\mathbf{H}}_0(\mathbf{j\Omega })={\mathbf{T}}_{\mathbf{s}}\text{sinc}\left(\frac{\mathbf{\Omega }{\mathbf{T}}_{\mathbf{s}}}{2\pi }\right){\mathbf{e}}^{-\mathbf{j\Omega }{\mathbf{T}}_{\mathbf{s}}/2}[1.113]$

Physical Implications

1. Aperture Distortion: Sinc envelope rolls off before the Nyquist frequency, dampening high frequencies (aperture effect) [1.113].
2. Phase Delay: Phase term $e^{-j\Omega T_s/2}$ introduces a constant delay of half a sampling interval ($T_s/2$) [1.113].

9.2 First-Order Hold (FOH)

The FOH performs linear interpolation, joining sample points with straight lines to avoid staircase discontinuities [1.113].

• Triangular Impulse Response: $h_1(t)=\left(1-\frac{|t|}{T_s}\right)\left[u(t+T_s)-u(t-T_s)\right]$ [1.113]
• Frequency Response: $H_1(j\Omega )=T_s{\left(\text{sinc}\left(\frac{\Omega T_s}{2\pi }\right)\right)}^2$ [1.113]
• Trade-off: FOH suppresses high-frequency harmonics much faster than ZOH (${\text{sinc}}^2(f)$ vs. $\text{sinc}(f)$ decay) but requires a delay of $T_s$ for physical causality [1.113].

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10. 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 ${\omega }_0$ [1.102]. A discrete sinusoid $A\cos ({\omega }_{0}n)$ is periodic if and only if ${\omega }_0/2\pi$ is a rational number [1.102].
2. Fractional LCM Swap: Inverting the fractional LCM formula. Remember that the Least Common Multiple of fractional periods is $\frac{\text{LCM of numerators}}{\text{HCF of denominators}}$ [1.102]. Writing HCF/LCM is a guaranteed zero-marks error [1.102].
3. Even/Odd Cross-Product Omission: Dropping the cross-product evaluation during the Even-Odd energy identity proof [1.102]. 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 [1.102].
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 ($\int x_1(t)x_2(t)dt=0$) [1.102].

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11. Complete Chapter 1 PYQ Answer Plans

Q1: [PYQ 2024 - 6 Marks]

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

• Answer Plan: Define Signal and System mathematically (Section 1) [1.10, 1.11, 1.99]. Detail how they allow engineers to transmit data, suppress noise, and filter out-of-band interference [1.99]. List the real-world communication applications (biomedical, radar/sonar, GPS) [1.10, 1.99].

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 and hand-draw the 2D waveforms with labeled axes for all four functions (Section 2.1) [1.100].

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

Prove that the total energy of a signal is the sum of the energies of its even and odd parts.

• Answer Plan: State $x(t)=x_e(t)+x_o(t)$. square the term, integrate, and write out the complete proof showing why the cross-product vanishes over symmetric limits due to odd parity (Section 6.2) [1.102].

Q4: [PYQ 2024, 2019 - 4 Marks]

Sketch the continuous signal: $x(t)=-4r(3t-1)$.

• Answer Plan: Apply the Precedence Rule (Section 3.2): Shift $r(t)$ right by 1, scale amplitude by $-4$ (slope is now $-4$), then compress time by 3. The new ramp starts at $t=1/3$ and has a compressed slope of $-12$ [1.107].

Q5: [PYQ 2017 - 6 Marks]

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

• Answer Plan: Find individual periods: $T_1=2\pi /20\pi =0.1\text{ s}$ and $T_2=2\pi /5\pi =0.4\text{ s}$. Ratio is $1/4$ (rational $⟹$ Periodic). Compute LCM of fractions: $\text{LCM}(1/10,2/5)=\frac{\text{LCM}(1,2)}{\text{HCF}(10,5)}=2/5=0.4$ seconds [1.103].

Q6: [PYQ 2023, 2019 - 5 to 6 Marks]

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

• Answer Plan: Since $n$ is an integer index, $\cos (\pi n)=(-1{)}^n$. Square and sum for energy: $E={\sum }_{n=0}^{\infty }|(-1{)}^n{|}^2={\sum }_{n=0}^{\infty }1=\infty$. Calculate average power limit: $P={\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=0}^N(1)={\lim }_{N\to \infty }\frac{N+1}{2N+1}=0.5$ Watts. Classify as a Power Signal [1.135].

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12. Interactive Revision Checklist

• Can you write and sketch the four basic singularity functions [1.100]?
• Do you know how to apply the Precedence Rule without scaling the shift parameter [1.101]?
• Can you prove why the sum of sines is periodic using the fractional LCM formula [1.101]?
• Can you complete the Even-Odd orthogonality proof without omitting the cross-product explanation [1.102]?
• Can you derive the s-domain transfer function of a ZOH from its impulse response [1.113]?

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Source: (k.Deergha Rao) signals and systems.pdf, Chapter 1 - Signal Fundamentals & Operations.md, 1.00_Signal_Fundamentals_and_Operations_Ultimate_Mega_Note.md, 1.06 Typical Signal Processing Operations.md, 1.07 Supplementary Signal Functions, Hold Circuits & DAC Reconstruction.md