--- title: "Define LTI; properties of DT LTI systems" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:define-lti-dt-properties" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/define-lti-dt-properties site: BCT Engineering --- # Define LTI; properties of DT LTI systems Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2080 Chaitra, 6 marks - Video: [Linear Time-Invariant(LTI) system- concept, convolution, properties, deconvolution, identity system](https://www.youtube.com/watch?v=m_FgpCpGYNc) — Shrenik Jain - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/define-lti-dt-properties - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question Define LTI system. Explain the properties of discrete time LTI system. ## Short answer An LTI system satisfies superposition (linear) and shift-invariance. Output is convolution y[n]=x[n]*h[n]. Properties: commutativity, associativity, distributivity; memoryless if h[n]=Kδ[n]; causal if h[n]=0 for n<0; BIBO stable if sum|h[n]| finite. ## Answer A **linear time-invariant (LTI) system** is a system that is both **linear** (obeys superposition: $a x_1+b x_2 \rightarrow a y_1+b y_2$) and **time-invariant** (a shift of the input only shifts the output: $x[n-n_0]\rightarrow y[n-n_0]$). A discrete-time LTI system is completely described by its impulse response $h[n]$, and its output is the convolution sum $$ y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty}x[k]\,h[n-k] $$ #### Properties of discrete-time LTI systems 1. **Commutative:** $x[n]*h[n]=h[n]*x[n]$. Input and impulse response can be interchanged. 2. **Associative:** $\{x[n]*h_1[n]\}*h_2[n]=x[n]*\{h_1[n]*h_2[n]\}$. Two systems in **cascade** are equal to one system with $h=h_1*h_2$, and the order of the cascade does not matter. 3. **Distributive:** $x[n]*\{h_1[n]+h_2[n]\}=x*h_1+x*h_2$. Two systems in **parallel** equal one system with $h=h_1+h_2$. 4. **Memory:** the system is memoryless only if $h[n]=K\delta[n]$, i.e. $y[n]=Kx[n]$. Otherwise it has memory. 5. **Causality:** causal if and only if $h[n]=0$ for $n<0$. Then $y[n]=\sum_{k=0}^{\infty}h[k]x[n-k]$ uses only present and past inputs. 6. **Stability (BIBO):** stable if and only if $\sum_{n}|h[n]|<\infty$. 7. **Invertibility:** invertible if an inverse system $h_i[n]$ exists with $h[n]*h_i[n]=\delta[n]$. Example: accumulator $h=u[n]$ and first difference $h_i=\delta[n]-\delta[n-1]$. 8. **Unit step response:** $s[n]=\sum_{k=-\infty}^{n}h[k]$ and $h[n]=s[n]-s[n-1]$. 9. **Eigenfunction property:** complex exponentials pass through unchanged in shape: $z^n \rightarrow H(z)z^n$, $e^{j\omega n}\rightarrow H(e^{j\omega})e^{j\omega n}$. This is why Fourier and z-transforms are used for LTI systems. | Property | Condition on $h[n]$ | |---|---| | Memoryless | $h[n]=K\delta[n]$ | | Causal | $h[n]=0,\ n<0$ | | Stable | $\sum\lvert h[n]\rvert<\infty$ | | Invertible | $h*h_i=\delta[n]$ | Example: $h[n]=(0.5)^n u[n]$ is causal (zero for $n<0$), has memory, and is stable since $\sum (0.5)^n = 2<\infty$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. An LTI system is one that is: A. Linear and time-invariant B. Linear and causal C. Time-invariant and stable D. Memoryless and linear Answer: A. LTI means superposition holds and a shift of the input only shifts the output. (Key point: Definition) 2. The output of a DT LTI system with impulse response h[n] is: A. x[n] h[n] B. x[n] * h[n] = sum x[k]h[n-k] C. x[n] + h[n] D. h[n] / x[n] Answer: B. The system is fully described by h[n] through the convolution sum. (Key point: Output formula) 3. A DT LTI system is causal if and only if: A. h[n] = 0 for n > 0 B. h[n] = 0 for n < 0 C. h[n] is absolutely summable D. h[n] = K delta[n] Answer: B. Then the output uses only present and past inputs. (Key point: Causality) 4. How are the unit step response s[n] and h[n] related? A. h[n] = s[n] - s[n-1], s[n] = running sum of h B. s[n] = h[n] - h[n-1] C. s[n] = h[n] D. h[n] = s[n] + s[n-1] Answer: A. u[n] is the running sum of delta[n], so s[n] is the running sum of h[n]. (Key point: Step response) ## Close variants set in other exams - [Properties of DT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-lti-properties) — Signals and Systems, chapter 6, asked 3× ## Related in Signals and Systems - [Describe LTI system properties](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-describe) — Signals and Systems, chapter 5, asked 3× - [Properties of continuous time systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Act-system-properties) — Signals and Systems, chapter 5, asked 2× - [System properties; is y(t) = sin[x(t)] time-variant](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asystem-properties-sin-x) — Signals and Systems, chapter 5, asked 2× - [LTI system: linearity, stability, TI, causality](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-four) — Signals and Systems, chapter 5, asked 2× - [Commutative property of CT LTI systems](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Acommutative-property-ct) — Signals and Systems, chapter 5 - [LTI properties; is y[n] = e^(x[n]) linear](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Alti-properties-exponential-system) — Signals and Systems, chapter 6 ## Related in other subjects - [Hilbert transformation; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ahilbert-define-lti-properties) — Communication Systems, chapter 1 - [LTI system output in time domain](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Alti-output-time-domain) — Communication Systems, chapter 1 - [Plot sequence; properties of LTI system](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aplot-sequence-lti-properties) — Digital Signal Analysis and Processing, chapter 1 - [LTI system and closed loop in modern systems](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Alti-define-cl-pronounced) — Control Systems, chapter 1