Computation of linear convolution

The different method used for the computation of linear convolution is as follows :

• Graphical method
• Using the mathematical equation of convolution
• Tabulation method
• Multiplication method

This article gives information about the graphical method to how to find computation of linear convolution.

Graphical method :

                   
              ∞

y (n)  =  ∑  x (k) h(n-k)

           k= -∞

    

for n = 0              
              ∞

y (0)  =  ∑  x (k) h(-k)

           k= -∞        

    

for n =1

              ∞

y (1)  =  ∑  x (k) h(1-k)

           k= -∞       

Here the term (1-k) can be written as h(-k+1).

              ∞

y (1)  =  ∑  x (k) h(-k+1)

           k= -∞ 

Here h(-k+1) indicates of folded signal h(-k). It indicates that h(-k) is delayed by 1 sample. Similarly for other values of n output y (n) is calculated.

Thus different operation involved in the calculation of linear convolution are as follow :

• Folding operation: It indicates folding of sequence h (k)
• Shifting operation: It indicates time shifting of h (-k) e.g h (-k+1)
• Multiplication: It indicates multiplication of x (k) and h (n-k)
• Summation: It indicates the addition of all product terms obtained because of multiplication of x (k) and h (n-k)

Properties of even and odd signals

• The sum of two even signals is even signal
• The sum of two odd signals are odd
• The sum of an even signal and an odd signal is neither even nor odd signal
• The product of two even signals is even
• The product of two odd signals is  even
• The product of two odd signals and even the signal is odd

Show in  :

Addition / Subtraction :

In addition :

Even + Even = Even

Odd + Odd = Odd

Even + Odd = we can't say anything

In Subtraction :

Even - Even = Even

Odd - Odd = Odd

Even - Odd = we can't say anything

In Multiplication :

Odd * Odd = Even

Even * Odd = Odd

Even * Even = Even

What is LTI system

LTI is also called an LSI system. LTI stands for the linear time-invariant system also stands for a linear shift-invariant system. LTI is a class of system used in signal and system in digital communication for a linear and time-invariant system.

Linear system is the system whose outputs for a linear combination of input are the same as a linear combination of input is as a linear combination of individual response to those inputs and also have the time-invariant system are a system where the output does not depend on when the input was applied.

Linear convolutions in DSP

Consider relaxed LTI system. A relaxed system means if input x (n) is zero then output y (n) = 0 is zero. Let us say unit impulse 𝛿 (n) is applied to this system then its output is denoted by h (n). h (n) we called as the impulse response of the system.

Step 1 :

         T
𝛿 (n) → y (n) = h (n)

Step 2 :

             T
𝛿 (n-k)  → y (n) = h (n-k)

Step 3 :

                       T

x (k) 𝛿 (n-k)    →  y (n) = x (k) h(n-k) 

Step 4 :

 ∞                                       T   ∞  

∑        = x (k) 𝛿 (n-k) =   y (n) →  ∑      x (k) h(n-k)

k= -∞                                             k= -∞

y (n) = 

 ∞                                        

∑        = x (k) h (n-k)  

k= -∞           

                   
                          ∞

x (n) * h (n)  =   ∑      x (k) h(n-k)

                          k= -∞                                            

                          

Power of energy signal

Let x (t) be an energy signal. x(t) has a finite non zero energy. Let us calculate the power of x(t). By definition, stated equation the power of x(t) is given by :

Power P = 

P = 0 * E = 0 

Symmetrical and Anti symmetrical signal

Symmetrical signal or even signal :

A signal x (t) is said to be symmetrical or even if it satisfies the following condition.

Condition for symmetry : x(t) = x(-t)

Where,  x(t) = Value of the signal for positive "t" and x(-t) = Value of the signal for negative "t".

An example of symmetrical signal is a cosine wave shown in figure : This article also gives the some properties of even and odd signals 

Anti symmetrical signal or odd signal :

A signal x(t) is said to be Anti symmetrical or odd if it satisfies the following condition,

Condition for anti-symmetry : x (t) = -x(-t)

An example of odd signal is a sine wave shown in figure :

Energy of power signal

Let x(t) be a power signal. The normalized energy of this signal is given by :

Energy E = 

               

 

              E = ∞

Basic signals

In signals and system, we need to use some standard or elementary signals. In this section, we will show some important standard signal graphically and express them mathematically.

Some of the standard continuous and discrete time signals are :

• A DC signal
• Unit step signal
• Delta or unit impulse function
• Sinusoidal signal
• Exponential function
• Signum function
• Sinc function

A DC signal :

A DC signal is shown in the figure. As seen from the figure or waveform the amplitude A of a direct current signal remains constant independent of time.

A DC signal is x (t) = A  - ∞ < t <  ∞

Sinusoidal signal :

The sinusoidal signal includes sine and cosine signals.

Mathematically the can be represented as follows :

A sine signal   x(t) = A sin ωt = A sin (2∏ft )
Same way in a cosine signal   x(t) = A cos ωt = A sin (2∏ft )

Here = A = Amplitude 
ω = Angular frequency = 2∏f

Unit step signal :

The unit step signal is as shown in the figure. It has a constant amplitude of unity(1) for the zero of the positive value of time "t" . Whereas it has zero value for a negative value of t.

The unit step signal is mathematically represented as, 

Unit step signal called as    u (t) = 1  for t > 0

                                                   = 0  for t < 0

Signum function :

The signum function is as shown in figure. It is represented mathematically as follows :

sgn(t) =  1 for t > 0 
          = - 1  for t < 0 

Delta or unit impulse function :

The delta function is an extremely function used for the analysis of the communication system. The impulse response of a system is its response to a delta function applied at the input signal.

Delat function : = 0 for t ≠ 0 

 Unit ramp function :

A continuous time unit ramp signal is denoted by ramp called r (t). Mathematically it is expressed as, 

r (t) = t for t >= 0
         = 0 for t< 0

What is signal

A signal is basically an electrical or electromagnetic current that is used for carrying data from one device or network to another. It is a physical quantity. It varies with some dependent or independent variables. 

So term of Signal can be defines as "A physical quantity which contain some information and which is function of one or more independent variables."

A signal can be analog type or digital type. Signal basically one dimensional and two dimensional. In one dimensional signal the function depends on a signal variable, i.e. speech signal whose amplitude varies with time while in multi dimensional signal depends on two or more variables, i.e an image because it is horizontal and vertical co-ordinates.