Representation of a Discrete Time Signal

Discrete Time Signals

The signals which are defined only at discrete instants of time are known as discrete time signals. The discrete time signals are represented by x(n) where n is the independent variable in time domain.

Representation of Discrete Time Signals

A discrete time signal may be represented by any one of the following four ways −

• Graphical Representation
• Functional Representation
• Tabular Representation
• Sequence Representation

Graphical Representation of Discrete Time Signals

Consider a discrete time signal x(n) with the values,

• x(−3) = −2,
• x(−2) = 3,
• x(−1) = 0,
• x(0) = −1,
• x(1) = 2,
• x(2) = 3,
• x(3) = 1

This discrete time signal can be represented graphically as shown in the figure below.

Functional Representation of Discrete Time Signal

In the functional representation of discrete time signals, the magnitude of the signal is written against the values of n. Therefore, the above discrete time signal x(n) can be represented using functional representation as given below.

$$\mathrm{x(n) \:=\: \left\{\begin{matrix} -2\: for \: n\:=\:-3\ 3\: for \: n\:=\:-2\ 0 \: for \: n\:=\:-1\ -1 \: for\:n\:=\:0\ 2\:for\:n\:=\:1\ 3\:for\:n\:=\:2\ 1\:for\:n\:=\:3\ \end{matrix}\right.}$$

Tabular Representation of Discrete Time Signal

In the tabular representation of discrete time signals, the sampling instant n and the magnitude of the discrete time signal at the corresponding sampling instant are represented in the form of a table. The above discrete time signal x(n) can be represented in the tabular form as given below.

Sequence Representation of Discrete Time Signal

The discrete time signal x(n) can be represented in the sequence representation as follows −

$$\mathrm{x(n) \:=\: \begin{Bmatrix} -2,3,0,-1,2,3,1\ \uparrow \ \end{Bmatrix}}$$

Here, the arrow mark (↑) denotes the term corresponding to n = 0. When no arrow is indicated in the sequence representation of a discrete time signal, then the first term of the sequence corresponds to n = 0.

Sum and Products of Discrete Time Sequences −

pThe sum of two discrete time sequences is obtained by adding the corresponding elements of sequences, i.e.,

$$\mathrm{\left \{ C_{n} \right \}\:=\:\left \{ a_{n} \right \}\:+\:\left \{ b_{n} \right \}\; \rightarrow \; C_{n}\:=\:a_{n}\:+\:b_{n}}$$

The product of two discrete time sequences is obtained by multiplying the corresponding elements of the sequences, i.e.,

$$\mathrm{\left \{ C_{n} \right \}\:=\:\left \{ a_{n} \right \}\left \{ b_{n} \right \}\; \rightarrow \; C_{n}\:=\:a_{n}b_{n}}$$

The product of a sequence and a constant k is obtained by multiplying each element of the sequence by that constant, i.e.,

$$\mathrm{\left \{ C_{n} \right \}\:=\:k\left \{ a_{n} \right \} \rightarrow \; C_{n}\:=\:ka_{n}}$$