--- title: "Plot sequence; properties of LTI system" subject: "Digital Signal Analysis and Processing" chapter: 1 key: "semester_7/digital_signal_analysis_and_processing:plot-sequence-lti-properties" source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/plot-sequence-lti-properties site: BCT Engineering --- # Plot sequence; properties of LTI system Digital Signal Analysis and Processing · Chapter 1: Discrete Time Signals and Systems - Asked in: 2074 Chaitra, 3+2 marks - Video: [Linear Time-Invariant (LTI) Systems](https://www.youtube.com/watch?v=Y8iFJVmSQIk) — Neso Academy - Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/plot-sequence-lti-properties - Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-1 ## Question Plot the sequence x[n] = u[n] − u[n−3] + 5δ[n−4] + n u[n−6]. List out the properties of LTI system. ## Short answer The sequence values are x[0]=x[1]=x[2]=1, x[3]=0, x[4]=5, x[5]=0, and x[n]=n for n≥6. LTI properties include linearity, time invariance, convolution description, commutativity, associativity, distributivity, causality iff h[n]=0 for n<0, BIBO stability iff sum |h[n]| is finite, memoryless iff h[n]=K delta[n], invertibility, and eigenfunction property. ## Answer #### Plot of x[n] = u[n] − u[n−3] + 5δ[n−4] + n u[n−6] Term by term: - u[n] − u[n − 3]: a rectangle, value 1 at n = 0, 1, 2. - 5δ[n − 4]: a single sample of height 5 at n = 4. - n u[n − 6]: a ramp, value n for n ≥ 6 (6, 7, 8, …). So x[n] = 1 (n = 0, 1, 2); 0 (n = 3); 5 (n = 4); 0 (n = 5); n (n ≥ 6); 0 for n < 0. | n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |---|---|---|---|---|---|---|---|---|---| | x[n] | 1 | 1 | 1 | 0 | 5 | 0 | 6 | 7 | 8 | ```text x[n] 8 | | 7 | | 6 | | 5 | | 1 | | | | 0 +-+--+--+--+--+--+--+--+--+-- n 0 1 2 3 4 5 6 7 8 ... ``` #### Properties of LTI systems 1. **Linearity and time-invariance:** superposition holds and a shift in input shifts the output equally. The system is described by h[n]: y[n] = x[n] * h[n]. 2. **Commutative:** x * h = h * x. 3. **Associative:** (x * h₁) * h₂ = x * (h₁ * h₂), so cascaded systems can be interchanged (h = h₁ * h₂). 4. **Distributive:** x * (h₁ + h₂) = x * h₁ + x * h₂, giving parallel combination. 5. **Causality:** causal if and only if h[n] = 0 for n < 0. 6. **Stability (BIBO):** stable if and only if Σ |h[n]| < ∞. 7. **Memoryless:** only if h[n] = 0 for n ≠ 0 (h[n] = Kδ[n]). 8. **Invertibility:** an inverse system exists if h * h_inv = δ[n]. 9. Identity: x * δ[n] = x[n]; shift: x * δ[n − n₀] = x[n − n₀]. 10. Complex exponentials are eigenfunctions: e^(jωn) → H(e^(jω)) e^(jωn). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. u[n] - u[n-3] is A. 1 for n = 0..3 B. 1 for n = 3 C. 1 for n = 0, 1, 2 D. 1 for n >= 3 Answer: C. Step minus delayed step is a three-sample rectangle. (Key point: Rectangle term) 2. n u[n-6] is A. n for n >= 5 B. n for n >= 6 C. 6 for n >= 6 D. n-6 for n >= 6 Answer: B. The ramp is switched on at n = 6 with value n. (Key point: Ramp term) 3. x[4] equals A. 5 B. 1 C. 6 D. 0 Answer: A. Only 5 delta[n-4] is non-zero at n = 4. (Key point: Value at n = 4) 4. An LTI system is causal iff and BIBO stable iff A. h[n]=0 for n>0; sum h < infinity B. h[n]=delta[n]; sum|h|=0 C. h[n] finite; sum h^2 finite D. h[n]=0 for n<0; sum|h|