Name :- Smit Shah -140410109096

S.Y electrical 2 Sem 4

Subject:-SS

Topic :- Frequency spectrum of periodic signal..

Representation of periodic signals: Frequency spectrum of periodic signal..Spectrum : The entire range over which some

measurable property of a physical system can vary, such as a frequency of sound , the wavelength etc. Its also called as a graphic or photographic representation of such measurable range..

Frequency Domain RepresentationAll the signal till now were drawn with

respect to time. That representation of signal with respect to time is called as its time domain representation.

The time domain representation of the signal is not sufficient for its analysis. Hence we have to use the frequency domain representation of the signal for the sake of analysis.

In the frequency domain representation , the variable plotted on the X-axis is frequency “f” rather than “t”.

The signal represented in the frequency domain is called as the line spectrum. The line spectrum consists of two graphs mainly.

1. Amplitude spectrum: A graph of amplitude on Y-axis versus frequency on X-axis.

2. Phase Spectrum : A graph of phase on Y-axis versus frequency on X-axis.

The signal x(t) and its line spectrum is show in above fig.

The graph of Instantaneous signal voltage plotted on y-axis versus time plotted on X-axis is called as time domain representation.

The time domain representation gives us the following information:

1. Shape of the signal2. Its frequency3. Type of the signal (periodic or

nonperiodic )4. One cycle peroid

But we do not know anything about what frequency components are present and in what proportion they have been mixed in order to obtain the particular shape of signal.

All this information can be obtained from the line spectrum of a signal.

Line spectrum is the representation of the same signal x(t), now in the frequency domain.

It can be obtained by using either Fourier series or Fourier Transform. It consists of the amplitude and phase spectrum of the signal.

The line spectrum indicates the amplitude and phase of various frequency components present in the given signal. The line spectrum enables us to analyze and synthesize a signal….

Frequency Domain Representation of Sinusoids: Continuous Time

Consider a sinusoid in continuous time:

tFjjtFjj eeAeeA

tFAtx

00 22

0

22

)2cos()(

Frequency Domain Representation:

0F

2A

0F

2A

magnitude

phaseradians

)(HzF

)(HzF

Example

Consider a sinusoid in continuous time:

tjjtjj eeee

ttx

200015.0200015.0 55

)15.02000cos(10)(

Represent it graphically as:

000,1

5

15.0

magnitude

phase

15.0

radians

5

000,1 )(HzF

)(HzF

Frequency Domain Representation of Sinusoids: Discrete Time

Same for a sinusoid in discrete time:

njjnjj eeAeeA

nAnx

00

22

)cos(][ 0

2A

magnitude

phase

0

2A

0 )(rad

)(rad

Frequency Domain Representation:

Example

Consider a sinusoid in discrete time:

njjnjj eeee

nnx

45.035.045.035.0 44

)35.045.0cos(8][

Represent it graphically as:

45.0

4

35.0

magnitude

phase 35.0radians

4

45.0 )(rad

)(rad

So as we show in above example for real periodic signal the amplitude spectrum is even symmetric & phase spectrum is odd symmetric…