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Lecture-9: Frequency Analysis of Signals and Systems

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Lecture-9: Frequency Analysis of Signals and Systems

Fourier Transform can convert time domain signal into frequency domain signal.

Mathematical Expression of coefficient of Fourier transform:

The periodic signal which is a combination of harmonically related complex exponentials,

where, = Fourier Coefficient

Now, dt = ()dt

dt =

So, finally = where, =

 

(2)

Dirichlet Conditions

Dirichlet Conditions: There are three conditions:

1. The signal x(t) has a finite number of discontinuities in any period

2. The signal x(t) contains a finite number of maxima and minima during any period.

3. The signal x(t) is absolutely integrable in any period. That is,

All periodic signals of practical interest satisfy these conditions.

 

(3)

Power Density Spectrum of Periodic Signals

A periodic signal has infinite energy and a finite average power, which

Px= = = So,

Px= = =

This is called Parseval’s relation for power signal.

 

(4)

Problems

Example-4.1.1: Determine the Fourier series and the power density spectrum of the rectangular pulse train signal illustrated in the following figure:

Figure: Continuous-time period train of rectangular pulses

Referensi

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