--- title: "LTI properties; is y[n] = e^(x[n]) linear" subject: "Signals and Systems" chapter: 6 key: "semester_4/signals_and_systems:lti-properties-exponential-system" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-properties-exponential-system site: BCT Engineering --- # LTI properties; is y[n] = e^(x[n]) linear Signals and Systems · Chapter 6: Discrete-Time Systems - Asked in: 2072 Magh, 4+5 marks - Video: [Properties Of Systems | Example 1](https://www.youtube.com/watch?v=NYtQ542JRw0) — EnggClasses - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/lti-properties-exponential-system - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6 ## Question What are the properties of LTI systems? Determine whether the given system is linear or not? y[n] = e^(x[n]) ## Short answer LTI systems are linear and time-invariant; convolution is commutative, associative, and distributive, and causality, stability, invertibility, and eigenfunction behavior follow from h[n]. The system y[n]=e^(x[n]) is nonlinear because e^(a1x1+a2x2) does not equal a1 e^(x1)+a2 e^(x2), and zero input gives y=1 instead of 0. ## Answer #### Properties of LTI systems An LTI system is linear and time-invariant; its output is $y[n]=x[n]*h[n]$. Its main properties: 1. **Commutative:** $x*h=h*x$. 2. **Associative:** $(x*h_1)*h_2=x*(h_1*h_2)$; a cascade equals one system $h_1*h_2$, in any order. 3. **Distributive:** $x*(h_1+h_2)=x*h_1+x*h_2$; parallel systems add their impulse responses. 4. **Memoryless** only if $h[n]=K\delta[n]$. 5. **Causal** if and only if $h[n]=0$ for $n<0$. 6. **BIBO stable** if and only if $\sum_n|h[n]|<\infty$. 7. **Invertible** if some $h_i[n]$ satisfies $h*h_i=\delta[n]$. 8. **Step response** $s[n]=\sum_{k\le n}h[k]$, and $h[n]=s[n]-s[n-1]$. 9. **Eigenfunction:** $e^{j\omega n}\rightarrow H(e^{j\omega})e^{j\omega n}$. #### Is $y[n]=e^{x[n]}$ linear? A system is linear if $a_1x_1+a_2x_2\rightarrow a_1y_1+a_2y_2$. Let $y_1[n]=e^{x_1[n]}$ and $y_2[n]=e^{x_2[n]}$. For $x_3=a_1x_1+a_2x_2$: $$ y_3[n]=e^{a_1x_1[n]+a_2x_2[n]}=\left(e^{x_1[n]}\right)^{a_1}\left(e^{x_2[n]}\right)^{a_2} $$ while $$ a_1y_1[n]+a_2y_2[n]=a_1e^{x_1[n]}+a_2e^{x_2[n]} $$ These are not equal in general (one is a product, the other a sum). Also, zero input gives $y[n]=e^0=1\ne0$, which a linear system cannot do. Numerical check: $x=1\rightarrow y=e\approx2.718$; $x=2\rightarrow y=e^2\approx7.389\ne2(2.718)=5.437$. **Answer: $y[n]=e^{x[n]}$ is non-linear.** (It is, however, time-invariant, memoryless, causal, and BIBO stable, since $|x|\le B$ gives $|y|\le e^{B}$.) ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the associative property say about a cascade of LTI systems? A. It equals one system with h = h1*h2, in any order B. It equals one system with h = h1+h2 C. It equals one system with h = h1/h2 D. Order matters, only h1 then h2 works Answer: A. A cascade convolves impulse responses; commutativity makes the order irrelevant. (Key point: Associative property) 2. Which pair of conditions for LTI systems is correct? A. Causal iff h[n]=0 for n>0; stable iff sum h[n]=0 B. Causal iff h[n]=0 for n<0; BIBO stable iff sum|h[n]| is finite C. Causal iff h[n]=delta[n]; stable iff h[n]>0 D. Causal iff sum|h|