What are the elementary discrete time signals

In discrete-time signal processing, elementary signals are basic building blocks that are often used as reference signals or to construct more complex signals. Here are some elementary discrete-time signals commonly encountered:

1. **Unit Impulse Signal (\(\delta[n]\))**:
   - Defined as:
     \[
     \delta[n] = 
     \begin{cases}
     1, & n = 0 \\
     0, & n \neq 0
     \end{cases}
     \]
   - Represents a signal that is zero everywhere except at \( n = 0 \), where it takes the value 1. It is used as a fundamental signal in defining other signals and in system analysis.

2. **Unit Step Signal (u[n])**:
   - Defined as:
     \[
     u[n] = 
     \begin{cases}
     1, & n \geq 0 \\
     0, & n < 0
     \end{cases}
     \]
   - Represents a signal that is 0 for all negative values of \( n \) and 1 for all non-negative values of \( n \). It is used to model signals that begin at a specific time or to define the starting point of other signals.

3. **Exponential Signal (a^n)**:
   - Defined as:
     \[
     x[n] = a^n
     \]
     where \( a \) is a constant.
   - Represents a signal whose amplitude varies exponentially with time \( n \). Depending on whether \( a \) is greater than 1, less than 1, or equal to 1, the signal can grow, decay, or remain constant over time.

4. **Sinusoidal Signal (\(\sin(\omega n + \phi)\)) and Cosinusoidal Signal (\(\cos(\omega n + \phi)\))**:
   - Defined as:
     \[
     x[n] = A \cos(\omega n + \phi) \quad \text{or} \quad x[n] = A \sin(\omega n + \phi)
     \]
   - Represents signals that oscillate sinusoidally over discrete time \( n \). \( A \) represents the amplitude, \( \omega \) the angular frequency, and \( \phi \) the phase shift. These signals are fundamental in many applications, including signal analysis and communication systems.

5. **Rectangular Pulse Signal (rect[n])**:
   - Defined as:
     \[
     \text{rect}[n] = 
     \begin{cases}
     1, & 0 \leq n < N \\
     0, & \text{otherwise}
     \end{cases}
     \]
   - Represents a signal that is 1 over a finite interval \( 0 \leq n < N \) and 0 otherwise. It is used in modeling pulses and in shaping signals for specific time durations.

These elementary discrete-time signals serve as fundamental components in the analysis, synthesis, and processing of discrete-time signals and systems. They are essential for understanding signal properties, system response, and designing digital signal processing algorithms.