Signals
Important signals

In previous chapter was explained the meaning of the word signal. It was not mentioned if the signal is one or more dimensional. In this chapter the most important signal in digital signal processing and multimedia will be introduced.

## One-dimensional signals

A signal which is a function of single independent variable is called one-dimensional signal. Usually, the only independent variable is time t (for example f(t)=5t), in case of the discrete signal the independent variable is number n (for example f(n) = n+1).

In all following definitions and formulas x refers to a set of real numbers {R} and n to a set of natural numbers {N}.

Dirac delta function or ɗ function is generalized function on the real number line that is zero everywhere except at zero. The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike. In the area of signal processing it is often referred to as the unit impulse symbol.

Mathematical definition:

and which is also constrained to satisfy the identity

On the picture below is displayd ideal and approximated Dirac delta function. The appoximated Dirac is just for better explanation, how we can get in real word Dirac delta function.

Dirac delta function – ideal and approximated by sinc() function

In the discrete domain the equivalent of Dirac delta function is Kronecker delta function. In the area of digital signal processing, the function is referred to as an impulse, or unit impulse. And when it stimulates a signal processing element, the output is called the impulse response of the element.

Mathematical definition:

Kronecker delta function

Unit step function, usually denoted as u, is a discontinuous function whose values is zero for negative argument (negative number) and one for positive argument. The function is used in signal processing to represent a signal that switches on at a specified time and stays switched on indefinitely. The unit step function is the integral of the Dirac delta function.

Mathematical definition:

Unit step function (continuous)

Discrete form of unit step function:

Unit step function (discrete)

Other special group of signals is periodic signals. Periodic function (which describe periodic signal) is a function that repeats its values in regular intervals or periods. Here belong for example all trigonometric functions (sine, cosine, tangent, cotangent with period 2π). If the period is P, then mathematical definition of a periodic function is:

## Two-dimensional signals

A signal which is a function of two independent variables is called two-dimensional signal. A typical example of two – dimensional signal is a picture. Picture consists of a brightness and luminescence signal. A 2D image can have a continuous spatial domain, as in a traditional photograph or painting; or the image can be discretized in space, as in a raster scanned digital image.

All important signals listed for one-dimensional domain are defined also in two-dimensional domain. Only mathematical definitions will be listed.

Dirac delta function

2D Dirac delta function – ideal and approximated by sinc() function

Kronecker delta

2D Kronecker delta

2D Unit step function (continuous)

2D Unit step function (continuous)

Unit step function (discrete)

2D Unit step function (discrete)