---
title: "Energy/power signals; shifting and scaling"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:energy-power-shifting-scaling"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-shifting-scaling
site: BCT Engineering
---

# Energy/power signals; shifting and scaling

Signals and Systems · Chapter 1: Signals

- Asked in: 2071 Bhadra, 3+4 marks
- Video: [Energy and Power of Continuous Time Signals](https://www.youtube.com/watch?v=0Kmqp-XX_F0) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/energy-power-shifting-scaling
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Define energy and power signal with examples. Describe time shifting, time scaling of a signal.

## Short answer

Energy signal: finite energy, zero power (e.g., e^(-2t)u(t)). Power signal: finite power, infinite energy (e.g., A cos(ω0 t)). Time shifting y(t)=x(t-t0) delays or advances the signal; time scaling y(t)=x(at) compresses (a>1) or expands (0<a<1).

## Answer

#### Energy and power signals
For a signal $x(t)$, the total energy and average power are

$$
E = \int_{-\infty}^{\infty} |x(t)|^2\,dt, \qquad P = \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|x(t)|^2\,dt
$$

(for discrete time, replace the integrals with sums over $n$ and $1/2T$ with $1/(2N+1)$).

- **Energy signal:** $0 < E < \infty$, so $P = 0$. These are usually time-limited or decaying signals. Example: $x(t) = e^{-2t}u(t)$ has $E = 1/4$ J, $P = 0$; a single rectangular pulse.
- **Power signal:** $0 < P < \infty$, so $E = \infty$. These are usually periodic or lasting forever. Example: $x(t) = A\cos(\omega_0 t)$ has $P = A^2/2$, $E = \infty$; the unit step $u(t)$ has $P = 1/2$.
- Some signals are neither, e.g. the ramp $r(t) = t\,u(t)$ (both $E$ and $P$ are infinite).

| Point | Energy signal | Power signal |
|---|---|---|
| Energy | finite | infinite |
| Average power | zero | finite |
| Typical form | pulse, decaying | periodic, step |
| Example | $e^{-at}u(t)$ | $\sin\omega_0 t$ |

#### Time shifting
Time shifting moves a signal along the time axis without changing its shape: $y(t) = x(t - t_0)$.

- $t_0 > 0$: **delay**, signal moves right.
- $t_0 < 0$: **advance**, signal moves left.
- Discrete time: $y[n] = x[n - n_0]$ with integer $n_0$.

Example: if $x(t)$ is a pulse on $0 \le t \le 2$, then $x(t-3)$ is the same pulse on $3 \le t \le 5$. Delays appear in echoes and transmission lines.

#### Time scaling
Time scaling compresses or expands a signal in time: $y(t) = x(at)$, $a > 0$.

- $a > 1$: **compression**, the signal is squeezed (plays faster). $x(2t)$ on $0 \le t \le 1$ if $x(t)$ was on $0 \le t \le 2$.
- $0 < a < 1$: **expansion**, the signal is stretched (plays slower). $x(t/2)$ lies on $0 \le t \le 4$.
- $a < 0$ also reflects the signal (time reversal).
- Discrete time: $x[2n]$ keeps only even samples (decimation, information can be lost); $x[n/2]$ needs interpolation.

```text
x(t):    ____          x(2t):  __        x(t/2): ________
        |    |                |  |              |        |
   -----+----+---       ------+--+---     ------+--------+--
        0    2                0  1              0        4
```

When shifting and scaling are combined, $y(t) = x(at - b)$: first shift by $b$, then scale by $a$ (or scale first and shift by $b/a$).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For an energy signal, what is the average power?
   A. Infinite
   B. Equal to E
   C. Equal to 1/2
   D. Zero
   Answer: D. Finite E divided by infinite time gives zero power. (Key point: Energy signal power)

2. Which signal is neither an energy nor a power signal?
   A. Ramp t u(t)
   B. A cos ω0 t
   C. e^(-2t)u(t)
   D. u(t)
   Answer: A. Both E and P are infinite for the ramp. (Key point: Neither type)

3. Which describes x(t - 3) for a pulse on 0 ≤ t ≤ 2?
   A. A pulse on -3 ≤ t ≤ -1
   B. The same pulse on 3 ≤ t ≤ 5
   C. A pulse on 0 ≤ t ≤ 1
   D. A pulse on 0 ≤ t ≤ 6
   Answer: B. t0 > 0 delays the signal, moving it right. (Key point: Delay)

4. If x(t) lies on 0 ≤ t ≤ 2, where does x(t/2) lie?
   A. 0 ≤ t ≤ 1
   B. 0 ≤ t ≤ 2
   C. 0 ≤ t ≤ 4
   D. -2 ≤ t ≤ 0
   Answer: C. a = 1/2 < 1 expands (stretches) the signal. (Key point: Expansion)

## Related in Signals and Systems

- [Time shifting, scaling and inversion](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Atime-shift-scale-inversion) — Signals and Systems, chapter 1, asked 2×
- [CTFS time shifting and time scaling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-properties) — Signals and Systems, chapter 2, asked 2×
- [CTFS shifting, scaling and conjugation](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-shift-scale-conjugation) — Signals and Systems, chapter 2
- [Even/odd signals; energy or power check](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aeven-odd-energy-power-check) — Signals and Systems, chapter 1
- [Power of energy signal is zero](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aenergy-vs-power-signal-example) — Signals and Systems, chapter 1
- [CTFS time reversal, scaling, frequency shift](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Actfs-reversal-scaling-shift) — Signals and Systems, chapter 2

## Related in other subjects

- [Energy/power signals; periodicity of complex exponential](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-1) — Digital Signal Analysis and Processing, chapter 1
- [Energy versus power signals](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aenergy-power-distinguish) — Data Communication, chapter 2, asked 6×
- [Causality, stability; energy and power signals](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Acausality-stability-energy-power) — Data Communication, chapter 2
- [Energy/power signals; periodicity of cos + sin](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Aenergy-power-periodic-2) — Digital Signal Analysis and Processing, chapter 1