Continuous-time signals are signals that are defined for every instant of time. They are often represented graphically as a continuous curve, where the horizontal axis represents time and the vertical axis represents the amplitude of the signal. Unlike discrete-time signals, which are defined only at specific intervals, continuous-time signals have values at every point in time.
Characteristics and Properties:
- Amplitude: The value or strength of the signal at any given point in time.
- Frequency: The rate at which the signal repeats its values.
- Phase: The offset or delay of the signal relative to a reference point.
Mathematical Representation:
Continuous-time signals are typically represented by a function ( x(t) ), where ( t ) is a continuous variable denoting time.
Examples:
- Sinusoidal Signal: ( x(t) = A \sin(2\pi f t + \phi) ), where ( A ) is the amplitude, ( f ) is the frequency, and ( \phi ) is the phase.
- Exponential Signal: ( x(t) = Ae^{at} ), where ( a ) is a constant.
- Unit Step Signal: ( x(t) = \begin{cases} 0, & \text{for } t < 0 \ 1, & \text{for } t \geq 0 \end{cases} )
Transformations:
Several mathematical transformations can be applied to continuous-time signals to analyze them in different domains:
- Fourier Transform: Converts a signal from the time domain to the frequency domain.
- Laplace Transform: Provides a complex frequency domain representation, which is particularly useful for solving differential equations.
- Time Reversal: Reflects the signal about the vertical axis.
- Time Scaling: Expands or compresses the signal in time.
- Time Shifting: Delays or advances the signal in time.
Applications:
Continuous-time signals have various applications:
- In analog electronics, like audio amplifiers and radio receivers.
- In control systems, to understand the behavior of systems over time.
- In communications, for modulating and demodulating analog signals.
- In biomedical engineering, for signals like ECG and EEG.
Conversion to Discrete-Time Signals:
Continuous-time signals can be converted to discrete-time signals using a process called sampling. During sampling, the continuous signal is measured at regular intervals. The Nyquist-Shannon sampling theorem defines the conditions under which a continuous-time signal can be fully reconstructed from its samples.
In practice, while many real-world signals are continuous, they often get converted to discrete-time signals for processing in digital systems. However, the concepts and tools developed for continuous-time signals play a foundational role in understanding and analyzing the behavior of signals and systems.