--- title: "Time shifting, scaling and inversion" subject: "Signals and Systems" chapter: 1 key: "semester_4/signals_and_systems:time-shift-scale-inversion" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/time-shift-scale-inversion site: BCT Engineering --- # Time shifting, scaling and inversion Signals and Systems · Chapter 1: Signals - Asked in: 2081 Asoj, 3 marks; 2076 Baisakh, 3 marks (asked 2 times) - Video: [Operations on Time of Signals](https://www.youtube.com/watch?v=613MVjAoAHg) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/time-shift-scale-inversion - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1 ## Question Explain time shifting, time scaling and time inversion of a continuous time signal with example. ## Short answer Time shifting: y(t)=x(t−t0) delays (t0>0) or advances (t0<0) the signal. Time scaling: y(t)=x(at) compresses (a>1) or expands (0 0$: the signal is **delayed** (shifted right) by $t_0$. - $t_0 < 0$: the signal is **advanced** (shifted left). Example: if $x(t)$ is a pulse from 0 to 1, then $x(t-2)$ is the same pulse from 2 to 3. A radar echo is a delayed copy of the transmitted signal. #### Time scaling $y(t) = x(at)$, $a > 0$. - $a > 1$: signal is **compressed** in time (plays faster). - $0 < a < 1$: signal is **expanded** (plays slower). Example: for the pulse $x(t)$ on $0 \le t \le 2$, $x(2t)$ lies on $0 \le t \le 1$ and $x(t/2)$ on $0 \le t \le 4$. Playing an audio tape at double speed gives $x(2t)$. #### Time inversion (reflection/folding) $y(t) = x(-t)$: the signal is mirrored about $t = 0$. A value at $t = 2$ in $x(t)$ appears at $t = -2$ in $x(-t)$. Example: playing a recording backwards. ```text x(t) x(t-2) x(2t) x(-t) 1 +--+ 1 +--+ 1 +-+ 1 +--+ | | | | | | | | --+--+---t --+----+--+-t ----+-+---t ---+--+-+--t 0 1 0 2 3 0 .5 -1 0 ``` **Combined operation** $x(at - b)$: first shift by $b$ to get $x(t-b)$, then scale by $a$ (replace $t$ by $at$). Example: $x(2t-2)$ for the pulse on $[0,1]$ lies on $1 \le t \le 1.5$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. x(t - 2) compared with x(t) is: A. Advanced (shifted left) by 2 B. Delayed (shifted right) by 2 C. Compressed by 2 D. Reflected about t = 2 Answer: B. For t0 > 0, x(t - t0) is a delayed copy shifted right by t0. (Key point: Time shifting direction) 2. A pulse x(t) on 0 <= t <= 2 becomes what for x(2t)? A. A pulse on 0 to 4 B. A pulse on 0 to 1 C. A pulse on 2 to 4 D. A pulse on -2 to 0 Answer: B. a = 2 > 1 compresses the signal in time, so the width halves to 0 to 1. (Key point: Time scaling effect) 3. What does x(-t) do to the value of x at t = 2? A. Places it at t = -2 B. Keeps it at t = 2 C. Places it at t = 1 D. Negates its amplitude only Answer: A. x(-t) mirrors the signal about t = 0, so the value at t = 2 appears at t = -2. 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