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Chapter 2 Orthogonal Representation,

Fourier Series and Power Spectra Orthogonal Series Representation of Signals and Noise

• Orthogonal Functions• Orthogonal Series

Fourier Series.• Complex Fourier Series• Quadrature Fourier Series• Polar Fourier Series• Line Spectra for Periodic Waveforms• Power Spectral Density for Periodic Waveforms

Huseyin BilgekulEeng360 Communication Systems I

Department of Electrical and Electronic Engineering Eastern Mediterranean University

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Orthogonal FunctionsOrthogonal Functions

Definition: Functions ϕn(t) and ϕm(t) are said to be Orthogonal with respect to each other the interval a < t < b if they satisfy the condition,

where

• δnm is called the Kronecker delta function.

• If the constants Kn are all equal to 1 then the ϕn(t) are said to be orthonormal functions.

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Example 2.11 Orthogonal Complex Exponential Example 2.11 Orthogonal Complex Exponential FunctionsFunctions

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Orthogonal SeriesOrthogonal SeriesTheorem: Assume w(t) represents a waveform over the interval a < t <b. Then w(t) can be represented over the interval (a, b) by the series where, the coefficients an are given by following where n is an integer value :

• If w(t) can be represented without any errors in this way we call the set of functions {φn} as a “Complete Set”

• Examples for complete sets:• Harmonic Sinusoidal Sets {Sin(nw0t)}• Complex Expoents {ejnwt}• Bessel Functions• Legendare polynominals

*

( ) ( )

1( ) ( )

n nn

b

n nan

w t a t

a w t t dtK

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Orthogonal SeriesOrthogonal SeriesProof of theorem: Assume that the set {φn} is sufficient to represent the waveform w(t) over the interval a < t <b by the series

We operate the integral operator on both sides to get,

• Now, since we can find the coefficients an writing w(t) in series form is possible. Thus theorem is proved.

( ) ( )n nn

w t a t

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Application of Orthogonal SeriesApplication of Orthogonal Series It is also possible to generate w(t) from the ϕj(t) functions and the coefficients aj.

In this case, w(t) is approximated by using a reasonable number of the ϕj(t)

functions.

w(t) is realized by adding weighted versions of orthogonal functions

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Ex. Square Waves Using Sine Ex. Square Waves Using Sine Waves.Waves.

http://www.educatorscorner.com/index.cgi?CONTENT_ID=2487

n =1n =3n =5

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Fourier SeriesFourier SeriesComplex Fourier SeriesComplex Fourier Series

The frequency f0 = 1/T0 is said to be the fundamental frequency and the frequency nf0 is said to be the nth harmonic frequency, when n>1.

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Some Properties of Complex Fourier Some Properties of Complex Fourier SeriesSeries

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Some Properties of Complex Fourier SeriesSome Properties of Complex Fourier Series

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Quadrature Fourier SeriesQuadrature Fourier Series

The Quadrature Form of the Fourier series representing any physical waveform w(t) over the interval a < t < a+T0 is,

n

nn

n

nn tnbtnatw

00

00 )sin()cos()(

where the orthogonal functions are cos(nω0t) and sin(nω0t).

Using we can find the Fourier coefficients as:

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Quadrature Fourier SeriesQuadrature Fourier Series• Since these sinusoidal orthogonal functions are periodic, this series is periodic with the fundamental period T0.

• The Complex Fourier Series, and the Quadrature Fourier Series are equivalent representations.

• This can be shown by expressing the complex number cn as below

For all integer values of n

and

Thus we obtain the identities

and

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Polar Fourier SeriesPolar Fourier Series• The POLAR F Form is

where w(t) is real and

The above two equations may be inverted, and we obtain

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Polar Fourier Series CoefficientsPolar Fourier Series Coefficients

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Line Spetra for Periodic WaveformsLine Spetra for Periodic Waveforms

Theorem: If a waveform is periodic with period T0, the spectrum of the waveform w(t) is

where f0 = 1/T0 and cn are the phasor Fourier coefficients of the waveform

Proof:

Taking the Fourier transform of both sides, we obtain

Here the integral representation for a delta function was used.

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Line Spectra for Periodic WaveformsLine Spectra for Periodic Waveforms

Theorem: If w(t) is a periodic function with period T0 and is represented by

Where,

then the Fourier coefficients are given by:

The Fourier Series Coefficients can also be calculated from the periodic sample values of the Fourier Transform.

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Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Ex. 2.12 Fourier Coeff. for a Periodic Rectangular WaveWave

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T Sa(fT)

Now compare the spectrum for this periodic rectangular wave (solid lines) with the spectrum for the rectangular pulse.• Note that the spectrum for the periodic wave contains spectral lines, whereas the spectrum for the nonperiodic pulse is continuous.• Note that the envelope of the spectrum for both cases is the same |(sin x)/x| shape, where x=Tf.• Consequently, the Null Bandwidth (for the envelope) is 1/T for both cases, where T is the pulse width. • This is a basic property of digital signaling with rectangular pulse shapes. The null bandwidth is the reciprocal of the pulse width.

Now evaluate the coefficients from the Fourier Transform

Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Ex. 2.12 Fourier Coeff. for a Periodic Rectangular WaveWave

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Single PulseContinous Spectrum

Periodic Pulse Train Line Spectrum

Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Ex. 2.12 Fourier Coeff. for a Periodic Rectangular WaveWave

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Normalized PowerNormalized Power

Theorem: For a periodic waveform w(t), the normalized power is given by:

where the {cn} are the complex Fourier coefficients for the waveform.

Proof: For periodic w(t), the Fourier series representation is valid over all time and may be substituted into Eq.(2-12) to evaluate the normalized power:

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Power Spectral Density for Periodic WaveformsPower Spectral Density for Periodic Waveforms

Theorem: For a periodic waveform, the power spectral density (PSD) is given by

where T0 = 1/f0 is the period of the waveform and {cn} are the corresponding Fourier coefficients for the waveform.

PSD is the FT of the Autocorrelation function

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Power Spectral Density for a Square WavePower Spectral Density for a Square Wave• The PSD for the periodic square wave will be found.• Because the waveform is periodic, FS coefficients can be used to evaluate the PSD. Consequently this problem becomes one of evaluating the FS coefficients.