Discrete-time signals are sequences of values defined at specific, discrete instants in time. Unlike continuous-time signals, which exist at every point in time, discrete-time signals are defined only at integer values of time. They are especially relevant in the era of digital systems, where signals are processed in a digital format.

Characteristics and Properties:

1. Amplitude: The value or strength of the signal at any given discrete point in time.
2. Frequency: The rate at which patterns in the signal repeat.
3. Phase: The offset or delay of the signal relative to a reference point.

Mathematical Representation:
Discrete-time signals are typically represented by a sequence ( x[n] ), where ( n ) is an integer variable denoting the discrete-time index.

Examples:

1. Sinusoidal Sequence: ( x[n] = A \sin(2\pi f n + \phi) ).
2. Exponential Sequence: ( x[n] = A \alpha^n ), where ( \alpha ) is a constant.
3. Unit Step Sequence: ( x[n] = \begin{cases} 0, & \text{for } n < 0 \ 1, & \text{for } n \geq 0 \end{cases} )

Transformations:
Several mathematical transformations can be applied to discrete-time signals:

1. Discrete Fourier Transform (DFT): Converts a finite-length discrete-time signal to its frequency domain representation.
2. Fast Fourier Transform (FFT): An algorithm to compute the DFT efficiently.
3. Z-Transform: Provides a complex frequency domain representation for discrete-time signals and systems.
4. Time Reversal: Reflects the sequence about the vertical axis.
5. Time Scaling: Expands or compresses the sequence.
6. Time Shifting: Delays or advances the sequence.

Applications:
Discrete-time signals are fundamental in:

1. Digital signal processing (DSP) applications such as filtering, spectrum analysis, and convolution.
2. Digital audio and video systems.
3. Control systems that use digital controllers.
4. Communications systems where signals are processed digitally.

Conversion from Continuous-Time Signals:
Continuous-time signals can be transformed into discrete-time signals via a process called sampling. When sampling, values of the continuous signal are captured at regular intervals. The resulting set of values forms the discrete-time signal. To accurately reconstruct the original continuous-time signal from its samples, the sampling rate must adhere to the Nyquist-Shannon sampling theorem.