ASSIGNMENT IN DC

ASSIGNMENT IN DIGITAL                     COMMUNICATION

                               HARISH PANDIAN.R

                                                                                             10EC03

1)    THE DIFFERENCE BETWEEN THE FOURIER SERIES AND FOURIER TRANSFORM

ANS.

Fourier series :

                       Fourier series is an expansion of a periodic function using infinite sum of sines and cosines. Fourier series was initially developed when solving heat equations but later it was found out that the same technique can be used to solve a large set of mathematical problems specially the problems that involve linear differential equations with constant coefficients

Fourier transform:

                     Fourier transform defines a relationship between a signal in the time domain and its representation in the frequency domain. The Fourier transform decomposes a function into oscillatory functions. Since this is a transformation, the original signal can be obtained from knowing the transformation, thus no information is created or lost in the process.

Difference:

1)      Fourier series is an expansion of periodic signal as a linear combination of sines and cosines while Fourier transform is the process or function used to convert signals from time domain in to frequency domain

2)      Fourier series is defined for periodic signals and the Fourier transform can be applied to aperiodic  signals

3)      The study of Fourier series actually provides motivation for the Fourier transform.

2)    FOURIER TRANSFORM TIME SHIFTING PROPERTY

Ans.

              If 

            

             then 

           

      In other words, a shift in time corresponds to a change in phase in the Fourier       

      transform.

3)      TYPES OF SIGNALS (OR) FUNTIONS:

Ans.

      Step signal:

         The step signal is zero for negative input and one for positive. Thus, it is defined    

          by 

      

         

      Ramp signal:

            The Step signal can be interpreted as a ramp of width zero, when the ramp is    

       defined as follows 
       

Impulse signal:          The impulse function, also known as Dirac delta function or Dirac impulse, is               defined by

      Sinc function :

           The Sinc function is defined in the following manner:

       Rect  function :

        The Rect Function is a function which produces a rectangular-shaped pulse with a     

        width of 1 centered at t = 0. The Rect function pulse also has a height of 1. The

         Sinc function and the rectangular function form a Fourier transform pair.

          A Rect function can be written in the form:

                                    

4)    POWER AND ENERGY SIGNALS :

ENERGY SIGNALS:

     Some signals qualify to be classified as energy signals, whereas some other signals qualify to be classified as power signals. Given a continuous-time signal f(t), the energy contained over a finite time interval is defined as follows.
                                   
      POWER SIGNAL:
           When a reference to power in a signal is made, it points to the average power. Power is defined as energy per second. For a continuous-time signal, we can obtain an expression for power from equation