Continuous-Time vs Discrete-Time Signals

Running speed of a player plotted against time


Classification of continuous-time and discrete-time deals with the type of independent variable. If the signal amplitude is defined for every possible value of time, the signal is called a continuous-time signal. However, if the signal takes values at specific instances of time but not anywhere else, it is called a discrete-time signal. Basically, a discrete-time signal is just a sequence of numbers.

Example


Consider a football (soccer) player participating in a 20-match tournament. Suppose that his running speed is recorded at each instant of time in the 90-minute duration of a particular match and plotted against time. The result shown in this Figure is clearly a continuous-time signal.

On the other hand, the Figure below shows the number of goals he scored during those 20 matches, which is defined only for each match and not in between. Hence, it is a discrete-time signal.

Number of goals scored by player 1 in each match

Discrete-time signals usually arise in two ways:

  1. By acquiring values of a continuous-time signal at fixed time instants. This process is called sampling and is discussed in detail later. For example, the actual temperature outside varies continuously throughout the day, but weather stations log the data after specific intervals, say every 30 minutes.
  2. By recording the number of events over finite time periods. For example, number of trees cut every year in a city for housing and development projects.

Mathematically, we represent a continuous-time signal as s(t) and a discrete-time signal as s[n], where t is a real number while n is an integer. So a discrete-time signal s[n] = 3n^2 can be plotted by finding s[n] for various values of n. Each member s[n] of a discrete-time signal is called a sample.

    \begin{align*}   n = -5 \quad \rightarrow \quad s[n] &= 75 \nonumber \\   n = -4 \quad \rightarrow \quad s[n] &= 48 \nonumber \\   n = -3 \quad \rightarrow \quad s[n] &= 27 \nonumber \\   n = -2 \quad \rightarrow \quad s[n] &= 12 \nonumber \\   n = -1 \quad \rightarrow \quad s[n] &= 3 \nonumber \\   n =  0 \quad \rightarrow \quad s[n] &= 0 \nonumber \\   n = +1 \quad \rightarrow \quad s[n] &= 3 \nonumber \\   n = +2 \quad \rightarrow \quad s[n] &= 12 \nonumber \\   n = +3 \quad \rightarrow \quad s[n] &= 27 \nonumber \\   n = +4 \quad \rightarrow \quad s[n] &= 48 \nonumber \\   n = +5 \quad \rightarrow \quad s[n] &= 75 \nonumber \end{align*}

Plotting each value of s[n] against every n is then straightforward as shown below.

Plot of a quadratic signal with time

Another way of representing a discrete-time signal is in the form of a sequence with an underline indicating the time origin (n=0), such as

    \begin{equation*}   s[n] = \{\cdots,75, 48, 27, 12, 3, \underline{0}, 3, 12, 27, 48, 75, \cdots\}. \nonumber \end{equation*}

Finally, it is incorrect to assume that a discrete-time signal is zero between two values of n. It is simply not defined for non-integer values.

Voltage as a signal


In electrical engineering, the time-varying quantity is usually voltage (or sometimes current). Therefore, when we work with a signal, just think of it as a voltage changing over time.

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