Continuous-time Signals
What is Continuous-time Signals?
A signal whose independent variable (time) is continuous, meaning its amplitude is defined for every real value of time within a given interval.
Key formula / rule: Energy of a Continuous-time Signal
Key points
- Define and identify continuous-time signals.
- Classify signals as periodic/aperiodic and energy/power signals.
- Perform basic operations on signals: time shifting, time scaling, time reversal, amplitude scaling.
- Calculate energy and average power of continuous-time signals.
Common exam trap
Confusing continuous-time with continuous-amplitude (analog vs. digital).
Definitions
- Term
Continuous-time Signal
- Meaning
A signal whose independent variable (time) is continuous, meaning its amplitude is defined for every real value of time within a given interval.
- Term
Periodic Signal
- Meaning
A signal x(t) for which there exists a positive constant T₀ (fundamental period) such that x(t) = x(t + T₀) for all t.
- Term
Aperiodic Signal
- Meaning
A signal that does not satisfy the condition for periodicity.
- Term
Energy Signal
- Meaning
A signal with finite total energy (E < ∞) and consequently zero average power (P = 0).
- Term
Power Signal
- Meaning
A signal with finite average power (P < ∞) and consequently infinite total energy (E = ∞).
Learning objectives
Define and identify continuous-time signals.
Classify signals as periodic/aperiodic and energy/power signals.
Perform basic operations on signals: time shifting, time scaling, time reversal, amplitude scaling.
Calculate energy and average power of continuous-time signals.
Recognize standard continuous-time signals (e.g., impulse, step, ramp, exponential, sinusoidal).
Formulae
- Name
Energy of a Continuous-time Signal
- Note
Applicable for energy signals.
- Expression
E = ∫_{-∞}^{∞} |x(t)|^2 dt
- Name
Average Power of a Continuous-time Signal
- Note
Applicable for power signals. For periodic signals with period T₀, P = (1/T₀) ∫_{0}^{T₀} |x(t)|^2 dt.
- Expression
P = limT→∞ (1/(2T)) ∫_{-T}^{T} |x(t)|^2 dt
- Name
Periodicity Condition
- Note
T₀ is the fundamental period.
- Expression
x(t) = x(t + T₀)
Prerequisites
Basic calculus (differentiation, integration).
Understanding of functions and their properties.
Familiarity with basic mathematical operations.
Common mistakes
Confusing continuous-time with continuous-amplitude (analog vs. digital).
Incorrectly applying energy/power formulas, especially for periodic signals.
Errors in time-shifting and time-scaling operations (e.g., x(at+b) vs x(a(t+b))).
Assuming all continuous-time signals are analog; they can be quantized in amplitude.
Keywords
Continuous
Time-domain
Analog
Periodic
Aperiodic
Energy
Power
Shifting
Scaling
Reversal
Impulse
Step
Ramp
Exponential
Sinusoidal
Practice preview
Which of the following continuous-time signals is periodic?…
easy
Consider the continuous-time signal x(t) = 5cos(2t). Is this an energy signal, a power signal, or neither?…
medium
Given a continuous-time signal x(t), what is the relationship between y(t) = x(2t + 4) and x(t)?…
medium