Signals in Natural Domain
Chapter 1 : Introduction
 
Elementary Signals

There are several elementary signals that occur prominently in the study of digital signals and digital signal processing.
 
(a) UNIT SAMPLE SEQUENCE: 

Defined by  
$\displaystyle \delta[n]=\left \{ \begin{aligned}1,\, \, n=0\\ 0,\,\,n\neq 0 \end{aligned} \right. $

Graphically this is as shown below.
$ \indent \indent \indent \indent {\delta[n]}$

Unit sample sequence is also known as impulse sequence.

This plays role akin to the impulse function $ {\delta(t)}$ of continous time. The continues time impulse $ {\delta(t)}$ is purely a mathematical construct while in discrete time we can actually generate the impulse sequence.

(b) UNIT STEP SEQUENCE:  $ {u[n]}$

Defined by :  
$\displaystyle u[n]=\left \{ \begin{aligned}1,\, \, n\geq0\\ 0,\,\,n< 0 \end{aligned} \right. $

Graphically this is as shown below
$ \indent \indent \indent \indent {u[n]}$

(c) EXPONENTIALSEQUENCE:
The complex exponential signal or sequence {x[n]}   is defined by  x[n] = C αn

where C  and α are, in general, complex numbers.
Note that by writing  α = eβ , we can write the exponential sequence as x[n] = c eβn
Real exponential signals:
: If C and $ \alpha$ are real, we can have one of the several type of behavior illustrated below
               
                               
              
              

For   |α| > 1          magnitude of the signals grows exponentially,
         |α| < 1          It is decaying exponential.
For   α > 1            all terms of {x[n]} have same sign,
         α < 1            sign of terms in {x[n]}  alternates.

(d)SINUSOIDAL SIGNAL:

The sinusoidal signal {x[n]}   is defined by      
                                              $\displaystyle x[n]=A\, \cos (w_0 n + \phi)$

Euler's relation allows us to relate complex exponentials and sinusoids as

                                            $\displaystyle e^{j\,w_0 n}=\cos w_0 n +j\, \sin w_0 n$    

      and               $\displaystyle A\,\cos (w_0 n+ \phi)=\frac{A}{2}e^{j \phi}e^{j\,w_0 n}+\frac{A}{2}e^{-j\,\phi}e^{-jw_0 n}$
The general discrete time complex exponential can be written in terms of real exponential and sinusiodal signals.
Specifically if we write C and α in polar form $ C=\vert C\vert e^{j\,\theta}$   and $ \alpha=\vert\alpha\vert e^{j\,w_0}$ then
$\displaystyle C\, \alpha^n=\vert C\vert\vert\alpha\vert^n\, \cos(w_0 n +\theta)+j\vert C\vert\vert\alpha\vert^n \sin (w_0n+\theta)$

Thus for |α| = 1 , the real and imaginary parts of a complex exponential sequence are sinusoidal.
                |α| < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential,
                |α| > 1 , they correspond to sinusiodal sequence multiplied by a growing exponential.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
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