communication system Chapter 3

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Communication System Communication System Ass. Prof. Ibrar Ullah

BSc (Electrical Engineering)

UET Peshawar

MSc (Communication & Electronics Engineering)

UET Peshawar

PhD (In Progress) Electronics Engineering

(Specialization in Wireless Communication)

MAJU Islamabad

E-Mail: [email protected]

Ph: 03339051548 (0830 to 1300 hrs)

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Chapter-3

• Aperiodic signal representation by Fourier integral (Fourier Transform)

• Transforms of some useful functions• Some properties of the Fourier transform• Signal transmission through a linear system• Ideal and practical filters• Signal; distortion over a communication channel• Signal energy and energy spectral density• Signal power and power spectral density• Numerical computation of Fourier transform

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Fourier Transform

Motivation • The motivation for the Fourier transform comes from the study of Fourier

series.

• In Fourier series complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines.

• Due to the properties of sine and cosine it is possible to recover the amount of each wave in the sum by an integral

• In many cases it is desirable to use Euler's formula, which states that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of the basic waves e2πiθ.

• From sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex valued. complex number gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave.

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Fourier Transform

• The Fourier series can only be used for periodic signals.

• We may use Fourier series to motivate the Fourier transform.

• How can the results be extended for Aperiodic signals such as g(t) of limited length T ?

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Fourier Transform

To is made long enough to avoid overlapping between the repeating pulses

To is made long enough to avoid overlapping between the repeating pulses

The pulses in the periodic signal repeat after an infinite interval

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Fourier Transform

Observe the nature of the spectrum changes as To increases. Let define G(w) a continuous function of w

Fourier coefficients Dn are 1/To times the samples of G(w) uniformly spaced at wo rad/sec

Fourier coefficients Dn are 1/To times the samples of G(w) uniformly spaced at wo rad/sec

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Fourier Transform

is the envelope for the coefficients Dn

Let To by doubling To repeatedly Doubling To halves the fundamental frequency wo and twice samples in the spectrum

Doubling To halves the fundamental frequency wo and twice samples in the spectrum

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Fourier Transform

If we continue doubling To repeatedly, the spectrum becomes denser while its magnitude becomes smaller, but the relative shape of the envelope will remain the same.

oT 0ow0nD Spectral components are spaced at zero (infinitesimal) interval

Spectral components are spaced at zero (infinitesimal) interval

Then Fourier series can be expressed as:

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Fourier Transform

As

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Fourier Transform

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Fourier Transform

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Fourier Transform

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Fourier Transform

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Example 3.1

Solution:

dtetgwG jwt

)()(

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Example 3.1

Fourier spectrum

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Compact Notation for some useful Functions

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2) Unit triangle function:

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3) Interpolation function sinc(x):

The function “sine over argument” is called sinc function given by

x

xsin

sinc function plays an important role in signal processing


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Fourier

Fourier series:

Fourier transform:

and

and

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Some useful Functions

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2) Unit triangle function:

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3) Interpolation function sinc(x):

The function “sine over argument” is called sinc function given by

x

xsin



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Example 3.2

Consider

Fourier transformFourier transform

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Example 3.2

Therefore

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Example 3.2

Spectrum:

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Example 3.3

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Example 3.4

Spectrum of a constant signal g(t) =1 is an impulse

Spectrum of a constant signal g(t) =1 is an impulse )(2 w

Fourier transform of g(t) is spectral representation of everlasting exponentials components of of the form . Here we need single exponential component with w = 0, results in a single spectrum at a single frequency

w = 0

Fourier transform of g(t) is spectral representation of everlasting exponentials components of of the form . Here we need single exponential component with w = 0, results in a single spectrum at a single frequency

w = 0

jwte jwte

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Example 3.5

Spectrum of the everlasting exponential is a single impulse atSpectrum of the everlasting exponential is a single impulse attjwoe oww

Similarly we can represent:

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Example 3.6

According to Euler formula:

and andAs

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Example 3.6

Spectrum:

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Some properties of Fourier transform

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Example 3.5

Spectrum of the everlasting exponential is a single impulse atSpectrum of the everlasting exponential is a single impulse attjwoe oww

Similarly we can represent:

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Example 3.6

According to Euler formula:

and andAs

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Example 3.6

Spectrum:

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Symmetry of Direct and Inverse Transform Operations—

1- Time frequency duality: •g(t) and G(w) are remarkable similar.

•Two minor changes, and opposite signs in the exponentials

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2- Symmetry property

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Symmetry property on pair of signals:

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3- Scaling property:

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The function g(at) represents the function g(t) compressed in time by a factor aThe scaling property states that:

Time compression spectral expansion

Time expansion spectral compression

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Reciprocity of the Signal Duration and its Bandwidth

As g(t) is wider, its spectrum is narrower and vice versa.

Doubling the signal duration halves its bandwidth.

Bandwidth of a signal is inversely proportional to the signal duration or width.

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4- Time-Shifting Property


Delaying a signal by does not change its spectrum.

Phase spectrum is changed by

Delaying a signal by does not change its spectrum.

Phase spectrum is changed by

ot

owt

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Physical explanation of time shifting property:

Time delay in a signal causes linear phase shift in its spectrum

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5- Frequency-Shifting Property:


Multiplication of a signal by a factor of shifts its spectrum byMultiplication of a signal by a factor of shifts its spectrum byte ojwoww

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is not a real function that can be generatedtjwoe

In practice frequency shift is achieved by multiplying g(t) by a sinusoid as:

Multiplying g(t) by a sinusoid of frequency

shift the spectrum G(w) by

Multiplying g(t) by a sinusoid of frequency

shift the spectrum G(w) by

owow

Multiplication of sinusoid by g(t) amounts to modulating the sinusoid amplitude. This type of modulation is called amplitude modulation.

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communication system Chapter 3