cs lecture 3.ppt

8/14/2019 CS Lecture 3.ppt

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Analysis and Transmission of

Signals

Lecture 3


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Introduction

Integrals are used for taking care of continuous value

function sum

Summation is used for totaling discrete values function

For periodic signals we use Fourier series used indiscrete spectrum

For aperiodic signals, we use Fourier transform used

in continuous spectrum


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Aperiodic Signal Representation by Fourier Integral

Figure: 3.1

1. An aperiodic function never repeats, although technically an aperiodic function can be considered like

a periodic function with an infinite period.

2. A periodic waveform lasts for ever, it is an everlast ingsignal. An everlasting signal exists in theory, but

not in real world.

3. The Fourier Transform is used similarly to the Fourier Series, in that it converts a time-domain function

into a frequency domain representation


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Aperiodic Signal Representation by Fourier

Integral

Including the dc component having f=0

Limiting process


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Aperiodic Signal Representation by Fourier ..


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Fundamental frequency

and


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Everlasting signalstarts from t-infinity and runs towards t-infinity.

Causal signalstarts at t=0 and continues to run towards t-infinity.

Fig: 3.3

Fig: 3.2


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1. Fourier series is used for spectral analysis of periodic waves. FT is used for spectral

analysis of aperiodic signal.

2. This property is called Duality.


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i. Cos(x) = Cos(-x)

ii. Sin(-x) = -Sinx


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Sol.

FT of a signal g(t) is a spectral representation of g(t) in terms of everlasting exponential component of the form e(iwt)


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Load distribution is discrete


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Load distribution is continuous


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Discussion

Spectral distribution is discrete

Spectral distribution is continuous


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1.

The rectangular function(also known as the rectangle function, rect function, Pi function, gate

function, unit pulse, or the normalized boxcar function).

In mathematics, a boxcar functionis any function which is zero over the entire real line

except for a single interval where it is equal to a constant,
http://localhost/var/www/apps/conversion/tmp/scratch_3//upload.wikimedia.org/wikipedia/commons/d/d5/Boxcar_function.svghttp://localhost/var/www/apps/conversion/tmp/scratch_3//upload.wikimedia.org/wikipedia/commons/d/d5/Boxcar_function.svghttp://localhost/var/www/apps/conversion/tmp/scratch_3//upload.wikimedia.org/wikipedia/commons/d/d5/Boxcar_function.svg


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2.

The function is useful in signal processing and comm unicat ion systems engineer ingas a representation

of an idealized signal, and as a prototype or kernel from which more realistic signals can be derived.

It also has applications in pulse code modulation as a pulse shape for transmitting digital signals and as a

matched filter for receiving the signals.

Proof: The transform is easily determined using the convolution property of FT and the FT ofthe rect function:

Applications


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EXAMPLE 3.2

Solution

Fig: 3.10


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Example 3.4


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Solution


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cs lecture 3.ppt

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