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