Define discrete time signal

A discrete-time signal is a type of signal where the amplitude of the signal is defined only at discrete instances of time. Unlike continuous-time signals, which are defined for all real values of time ttt, discrete-time signals are defined only at integer values of time nnn. These signals are widely used in digital signal processing, communication systems, control systems, and many other fields. Here are the key characteristics and definitions of discrete-time signals:

  1. Definition: A discrete-time signal x[n]x[n]x[n] is a sequence of numbers that represent the amplitude of the signal at each discrete time index nnn, where nnn is an integer.

  2. Mathematical Representation: Formally, a discrete-time signal can be represented as:

    x[n]=x(nT),x[n] = x(nT),x[n]=x(nT),

    where x(nT)x(nT)x(nT) is the value of the continuous-time signal x(t)x(t)x(t) sampled at discrete time instants nTnTnT, with TTT being the sampling interval.

  3. Sampling: Discrete-time signals are typically obtained by sampling a continuous-time signal at regular intervals. The sampling process converts the continuous-time signal x(t)x(t)x(t) into a sequence of discrete values x[n]=x(nT)x[n] = x(nT)x[n]=x(nT).

  4. Discrete Values: The values of x[n]x[n]x[n] are typically real numbers, complex numbers, or integers, depending on the nature of the signal being represented.

  5. Representation: Discrete-time signals can be represented graphically as a sequence of points or as a plot where the amplitude x[n]x[n]x[n] is plotted against the discrete time index nnn.

  6. Applications: Discrete-time signals are fundamental in digital signal processing (DSP), where operations such as filtering, modulation, and sampling theorem are applied to process and manipulate signals efficiently in computer systems.

In summary, a discrete-time signal x[n]x[n]x[n] represents a sequence of discrete values of a signal, where the values are defined at specific discrete time instances nnn. These signals are essential in digital systems for their ability to be processed and analyzed using computational methods and algorithms.

 

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