2.00 LTI System Properties and Time-Domain Convolution Compa

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Chapter 2: LTI System Properties & Time-Domain Convolution - Compact Review

This compact review card synthesizes all core concepts, mathematical definitions, final-line derivations, and architectural realization structures for continuous-time and discrete-time systems under Chapter 2: LTI System Properties & Time-Domain Convolution.

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1. Systems Classification & Mathematical Operators

1.1 Linearity: Superposition & Homogeneity

*(Target: Theory Descriptive / Linearity Verification)*

• Concept: A system is linear if and only if it simultaneously satisfies the principles of additivity (superposition) and homogeneity (scaling). A linear system has the property that a zero input strictly produces a zero output.
• Governing Equations:
  • Additivity: $\mathbb{T}[x_1(t)+x_2(t)]=y_1(t)+y_2(t)$
  • Homogeneity: $\mathbb{T}[a\cdot x_1(t)]=a\cdot y_1(t)$
  • Unified Linearity Condition: $\mathbb{T}[a\cdot x_1(t)+b\cdot x_2(t)]=a\cdot y_1(t)+b\cdot y_2(t)ext(derivation)$

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1.2 Time-Invariance (Shift-Invariance)

*(Target: Theory Descriptive / Time-Invariance Verification)*

• Concept: A system is time-invariant if its internal parameters do not change over time. A time shift applied to the input signal must produce an identical time shift in the output response.
• Governing Formula: $y(t-t_0)=\mathbb{T}[x(t-t_0)]ext(derivation)$
• The Two-Step Verification Checklist:
  1. Step 1: Delay the system output $y(t)$ by $t_0$ directly: $y(t,t_0)=y(t-t_0)$.
  2. Step 2: Pass a delayed input $x_1(t)=x(t-t_0)$ through the operator: $y_1(t)=\mathbb{T}[x(t-t_0)]$.
  3. Check: If $y_1(t)\equiv y(t-t_0)$, the system is Time-Invariant. If they are not equal, the system is Time-Variant.

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1.3 Causality

*(Target: Theory Descriptive / Causality Boundary Check)*

• Concept: A system is causal if its output at any instant depends strictly on current and past input values, but is completely independent of future input values. A continuous-time LTI system described by a differential equation is causal when it is initially relaxed {if input $x(t)=0$ for $t\le t_0$, then output $y(t)=0$ for $t\le t_0$}.
• s-Domain/z-Domain Causal Boundary: The impulse response must satisfy: $h(t)=0extfort<0$ $h[n]=0extforn<0$

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1.4 Memory (Dynamic) vs. Memoryless (Static)

*(Target: Theory Descriptive / Memoryless Bound)*

• Concept: A system is memoryless (static) if its output at any instant $t_0$ depends strictly on the input applied at that exact instant $t_0$. If the output depends on inputs at other times (past or future), the system has memory (dynamic).
• Memoryless LTI Bound: An LTI system is memoryless if and only if its impulse response is a scaled Dirac delta function: $h(t)=c\cdot \delta (t)extandh[n]=c\cdot \delta [n]$

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1.5 Bounded-Input Bounded-Output (BIBO) Stability

*(Target: Theory Descriptive / Stability Test)*

• Concept: A system is BIBO stable if every bounded input sequence produces a bounded output response.
• Input-Output Bounded Limits: $extIf|x(t)|\le M_x<\infty ⟹|y(t)|\le M_y<\infty extforallt$

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2. Continuous-Time Convolution Integral

2.1 The Representation of Signals in Terms of Impulses

*(Target: Mathematical Proof / 5-Mark derivation)*

• Concept: Any arbitrary continuous-time signal $x(t)$ can be represented as a weighted superposition of infinite, shifted continuous impulse functions.
• Governing Formula: $x(t)={\int }_{-\infty }^{\infty }x(au)\delta (t-au)dauext(derivation)$

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2.2 The Continuous Convolution Integral

*(Target: Numerical Solving / System Response)*

• Concept: The continuous-time convolution operates on an input $x(t)$ and system impulse response $h(t)$ to calculate the exact time-domain system response $y(t)$.
• Governing Formula: $y(t)=x(t)\ast h(t)={\int }_{-\infty }^{\infty }x(au)h(t-au)dau$

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2.3 Mathematical Properties of Continuous Convolution

*(Target: Theory Descriptive & Proofs)*

• Commutative Property: $x(t)\ast h(t)=h(t)\ast x(t)ext(derivation)$
• Distributive Property: $x(t)\ast [h_1(t)+h_2(t)]=x(t)\ast h_1(t)+x(t)\ast h_2(t)ext(derivation)$
• Associative Property: $[x(t)\ast h_1(t)]\ast h_2(t)=x(t)\ast [h_1(t)\ast h_2(t)]ext(derivation)$
• Convolution with a Unit Impulse: $x(t)\ast \delta (t)=x(t)ext(derivation)$
• Convolution with Shifted / Delayed Inputs: $extIfy(t)=x(t)\ast h(t)⟹x(t-t_1)\ast h(t-t_2)=y(t-t_1-t_2)ext(derivation)$
• Convolution Time-Scaling Identity:

ight] \quad ext{for } a eq 0$\ast \ast SpecialCase(DoubleScaling):\ast$y(2t) = 2 \left[ x(2t) * h(2t) ight] \quad ext{(derivation)}$$

• Differentiation Property of Convolution: rac{dy(t)}{dt} = rac{dx(t)}{dt} * h(t) = x(t) * rac{dh(t)}{dt} \quad ext{(derivation)}

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3. Discrete-Time Convolution Sum

3.1 The Discrete Convolution Sum

*(Target: Numerical Solving / Sequence Inversion)*

• Concept: The discrete counterpart of the convolution integral, evaluating the output sequence $y[n]$ by summing the weighted and shifted impulse response sequences triggered by each input sample.
• Governing Formula: $y[n]=x[n]\ast h[n]={\sum }_{k=-\infty }^{\infty }x[k]h[n-k]$

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3.2 Sequence Length Theorem

*(Target: Numerical Solving)*

• Concept: Convolving two finite-duration sequences of lengths $N_1$ and $N_2$ yields a resulting convolved sequence of a larger finite duration.
• Governing Formula: $L=N_1+N_2-1$

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3.3 Periodic (Circular) vs. Linear Convolution

*(Target: Theory Descriptive / 5-Mark comparison)*

• Concept: Linear convolution operates on infinite or finite aperiodic sequences. If signals $x_1(t)$ and $x_2(t)$ are periodic with common period $T$, linear convolution does not converge. Periodic convolution integrates/sums only over a single period $T$ (or $N$ samples), forcing the output response to be strictly periodic.
• Continuous Periodic Convolution: $y_p(t)={\int }_0^T{x}_{1,p}(au)x_{2,p}(t-au)dauext(derivation)$
• Discrete Periodic Convolution:

angle_N]$$

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4. Continuous & Discrete BIBO Stability Proofs

4.1 Continuous-Time Absolute Integrability Proof

*(Target: Mathematical Proof / 8-Mark Derivation)*

• Premise: A continuous-time LTI system is BIBO stable if and only if its impulse response is absolutely integrable.
• Governing Formula: ${\int }_{-\infty }^{\infty }|h(au)|dau<\infty ext(derivation)$

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4.2 Discrete-Time Absolute Summability Proof

*(Target: Mathematical Proof / 5-Mark Derivation)*

• Premise: A discrete-time LTI system is BIBO stable if and only if its impulse response sequence is absolutely summable.
• Governing Formula: ${\sum }_{n=-\infty }^{\infty }|h[n]|<\infty ext(derivation)$

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5. Systems Described by Differential & Difference Equations

5.1 Linear Constant-Coefficient Differential Equations (LCCDE)

*(Target: Numerical Solving / Continuous-Time System)*

• Concept: Represents a continuous-time system where the relationship between the output $y(t)$ and input $x(t)$ is governed by a linear combination of derivatives.
• Governing Equation: \sum_{n=0}^{N} a_n rac{d^n y(t)}{dt^n} = \sum_{m=0}^{M} b_m rac{d^m x(t)}{dt^m}

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5.2 Linear Constant-Coefficient Difference Equations

*(Target: Numerical Solving / Discrete-Time System)*

• Concept: Governs discrete-time LTI recursive (IIR) or non-recursive (FIR) systems.
• Governing Equation: ${\sum }_{k=0}^N{a}_{k}y[n-k]={\sum }_{m=0}^M{b}_{m}x[n-m](extwith{a}_0=1)$

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5.3 Complete Response Decompositions

*(Target: Theory Descriptive)*

• The Output Splitting Law: $y(t)=y_c(t)+y_p(t)extandy[n]=y_c[n]+y_p[n]$
• The Complementary (Homogeneous) Solution: Evaluated by setting the input terms to zero ($x(t)=0$ or $x[n]=0$).
• The Particular Solution: Represents the system’s steady-state output corresponding to the specific profile of the input signal.

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5.4 Characteristic Roots & Homogeneous Solution Forms

*(Target: Numerical Solving / Algebraic Rules)*

• Distinct Roots (${\lambda }_i$): y_c[n] = lpha_1 \lambda_1^n + lpha_2 \lambda_2^n + \dots + lpha_N \lambda_N^n
• Repeated Roots (Root ${\lambda }_1$ of multiplicity $m$):

ight) \lambda_1^n + eta_2 \lambda_2^n + \dots$$

• Complex Conjugate Roots (${\lambda }_{1,2}=a\pm jb$):

ight) \quad ext{where } r = \sqrt{a^2 + b^2}, ; heta = an^{-1}\left(rac{b}{a} ight)$$

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6. Block Diagram Realizations & Interconnections

6.1 Direct Form Realizations

*(Target: Block Diagram Realization / 5-to-8 Mark Draw)*

• Direct Form I Realization: Separately realizes input zeros (feedforward delays) and output poles (feedback delays). It is non-canonic because it uses a separate set of delay/integrator elements for both parts.
• Direct Form II Realization: Merges the delay paths for both input and output sections into a single, unified delay/integrator chain. It is canonic because it minimizes the total number of delay elements required ($N$ delays for an $N$-th order system).

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6.2 System Interconnections

*(Target: Theory Descriptive / Cascade vs. Parallel)*

  Cascade (Series Connection):      Parallel Connection:
                                        ┌───────┐
       ┌────────┐    ┌────────┐      ┌─>│ h₁(t) ├──┐
  x(t)─│  h₁(t) ├───>│  h₂(t) ─>y(t) │  └───────┘  v  ┌───┐
       └────────┘    └────────┘  x(t)┤             ├──>│ + ├──> y(t)
                                     │  ┌───────┐  ^  └───┘
                                     └─>│ h₂(t) ├──┘
                                        └───────┘

• Cascade (Series) Connection: The overall impulse response is the convolution of individual subsystem responses: $h_{extoverall}(t)=h_1(t)\ast h_2(t)extand{h}_{extoverall}[n]=h_1[n]\ast h_2[n]$
• Parallel Connection: The overall impulse response is the direct algebraic sum of subsystem responses: $h_{extoverall}(t)=h_1(t)+h_2(t)extand{h}_{extoverall}[n]=h_1[n]+h_2[n]$

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6.3 Inverse Systems & Deconvolution

*(Target: Theory Descriptive)*

• Concept: A system $h(t)$ is invertible if its input $x(t)$ can be completely recovered from its output response $y(t)$ using an inverse system $g(t)$.
• Governing Formula: $h(t)\ast g(t)=\delta (t)extandh[n]\ast g[n]=\delta [n]$

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7. Master Comparison Tables

Table 2.1: System Properties Test Matrix

*(Pre-Exam Quick Lookup)*

Property	Continuous Condition	Discrete Condition	Practical Checklist
Linearity	$\mathbb{T}[a{x}_1+b{x}_2]=a{y}_1+b{y}_2$	$\mathbb{T}[a{x}_1+b{x}_2]=a{y}_1+b{y}_2$	Zero input must produce zero output. No nonlinear terms (squares, logs, sines of signals).
Time-Invariance	$\mathbb{T}[x(t-t_0)]=y(t-t_0)$	$\mathbb{T}[x[n-n_0]]=y[n-n_0]$	Check independent variable coefficient. If scaled (e.g., $x(2t)$, $x[-n]$), it is Time-Variant.
Causality	$h(t)=0extfort<0$	$h[n]=0extforn<0$	Output at $t_0$ must not depend on inputs at $t>t_0$ (e.g., $x(t+3)$, $x[n+1]$ are noncausal).
Memory	$h(t)
eq c\delta(t)$	$h[n]
eq c\delta[n]$	If output depends only on the present input, it is memoryless; else, it is dynamic.
BIBO Stability	$\int_{-\infty}^{\infty} \lvert h(t)
vert dt < \infty$	$\sum_{-\infty}^{\infty} \lvert h[n]
vert < \infty$	Every bounded input must produce a bounded output (e.g., integrations, step responses are unstable).

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Table 2.2: Continuous vs. Discrete Convolution Properties

*(Symmetry and Sizing Mapping)*

Parameter	Continuous-Time Convolution	Discrete-Time Convolution
Governing Operator	Integral: ${\int }_{-\infty }^{\infty }x(au)h(t-au)dau$	Infinite Summation: ${\sum }_{k=-\infty }^{\infty }x[k]h[n-k]$
Aperiodic Sequence Bounds	Evaluated via piecewise continuous limits.	Sequence Length Theorem: $L=N_1+N_2-1$.
Unit Impulse Response	$x(t)\ast \delta (t)=x(t)$	$x[n]\ast \delta [n]=x[n]$
Integrability/Summability	Area: $A_y=A_x\cdot A_h$	Sum: $\sum y[n] = \left( \sum x[n]
ight) \cdot \left( \sum h[n]
ight)$

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Table 2.3: Recursive (IIR) vs. Non-Recursive (FIR) Difference Equations

*(Structural Properties)*

Attribute	Non-Recursive (FIR) Systems	Recursive (IIR) Systems
Feedback Paths	None. No output feedback loop is present.	Present. Output terms $y[n-k]$ are fed back.
Impulse Response Duration	Finite Duration (Finite Impulse Response).	Infinite Duration (Infinite Impulse Response).
Stability	Always stable (since $h[n]$ is finite and bounded).	Conditionally stable (depends on pole locations inside the unit circle).
Memory Elements	Fixed memory storage proportional to order $N$.	Requires infinite decay tracking of feedback states.

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8. Verbatim Chapter 2 PYQ Bank

2025/2022 Exam Section B Q. 1

If $y(t)=x(t)\ast h(t)$, then show that: $y(2t)=2x(2t)\ast h(2t)ext(03Marks)$ Answer Plan: Use variable substitutions inside the convolution integral of $y(at)$ with $a=2$, scaling both dummy variable $au$ and actual time $t$.

2023/2017/2015 Exam Section B

Show that the output response of an LTI system is the convolution sum of the input signal and the impulse response of the system. $ext(03to04Marks)$ Answer Plan: Represent the arbitrary input signal $x[n]$ as a weighted summation of discrete impulses: $x[n]=\sum x[k]\delta [n-k]$. Apply the system operator $\mathbb{T}[\cdot ]$, utilizing Linearity (superposition) to move the operator inside the sum, and Time-Invariance ($\mathbb{T}[\delta [n-k]]=h[n-k]$) to yield the convolution sum.

2024 Exam Section B Q. 2

Draw the block diagram of the system described by: rac{d^2 y(t)}{dt^2} + 3y(t) = rac{dx(t)}{dt} + 2rac{d^2 x(t)}{dt^2} \quad ext{(05 Marks)} Answer Plan: Integrate the second-order equation twice to avoid differentiators, isolate $y(t)$, and construct the Direct Form II canonic block diagram utilizing two integrators, multipliers, and adders.

2021 Exam Section B Q. 3

Derive the expression of convolution with delayed input and delayed impulse response. $ext(09Marks)$ Answer Plan: Starting from $y(t)=x(t)\ast h(t)=\int x(au)h(t-au)dau$, substitute delayed variables $x(t-t_1)$ and $h(t-t_2)$ into the integral, use variable substitution $v=au-t_1$, and match the resulting expression directly to $y(t-t_1-t_2)$.

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9. Common Mistakes That Cost Marks

Critical Exam Pitfalls

1. Direct Form II Feedback Coefficient Sign Swap: When sketching Direct Form II diagrams, failing to negate the feedback coefficients. Remember: feedback terms in difference/differential equations must be subtracted on summing junctions to represent negative feedback multipliers.
2. Evaluating Causal Overlap Boundaries incorrectly: Performing graphical convolution of piecewise signals without checking overlap boundaries. Always verify the active integration interval for each segment of time $t$ before computing areas.
3. Discrete Origin ($n=0$) Arrow Omission: When evaluating discrete tabular convolution, failing to locate and mark the $n=0$ index in convolved sequences.
4. Applying Value Theorems to Unstable LTI Systems: Evaluating the Final Value Theorem on unstable systems with poles outside or on the imaginary axis.

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10. Pre-Exam Self-Check Checklist

• Can you verify linearity and time-invariance for any continuous-time system equations?
• Can you prove that a continuous LTI system is BIBO stable if and only if ${\int }_{-\infty }^{\infty }|h(t)|dt<\infty$?
• Can you calculate the length $L=N_1+N_2-1$ of convolved finite discrete arrays?
• Can you draw canonic Direct Form II representations of second-order continuous differential and discrete difference equations?