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.