Discrete-time LTI system : the convolution sum representation

1. LTI system 

LTI system is the abbreviation of linear ,time-invariant system . Linearity and time invariance are the two distinguishing features of this kind of system . 

 

An example of a LTI system :

if 

the response of x1(1) is y1(t) 

the response of x2(1) is y2(t)

then 

the response of ax(t-t0) + bx2(t-t0) is ay1(t) + by2(t)  , where ' a ' and  ' b ' are any complex constants 

2. Convolution 

Convolution is a mathematical  operation .  The convolutions of two functions in continuous time and discrete time are given by equation (1) and (2)

3. The Representation of Discrete-Time Signals in Terms of Unit Impulse 

Any discrete signals can be represented as the  sum of  scaled and time-shifted unit impulses  . This is apparent when we think of a discrete-time signal as the combination of a sequence of individual impulses (figure 1 ). The mathematical representation is given by equation (3) .

4.The Representation of the Discrete-Time Signal Responses in Terms of Unit Impulse Response

Due to the linearity and time-invariance properties of LTI system , we can construct any signal responses by using the response of the unit impulse. 

if the response of unit impulse δ[n] is h [n ] and x[n] is defined by eq.(4) 

then the response of x[n] (i.e. y[n]) can be constructed as eq.(5) 

More simple form is 

5. Two important convolution sum representations 

From the dissusions above , we finally got two useful representions of the discrete-time LTI system .