Transcript Chapter2_Lect5.ppt

Chapter 2
Orthogonal Representation,
Fourier Series and Power Spectra
 Orthogonal Series Representation of Signals and Noise
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Orthogonal Functions
Orthogonal Series
 Fourier Series.
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Complex Fourier Series
Quadrature Fourier Series
Polar Fourier Series
Line Spectra for Periodic Waveforms
Power Spectral Density for Periodic Waveforms
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Orthogonal 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 Functions
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Orthogonal Series
Theorem: 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 :
w(t )   ann (t )
n
1
an 
Kn

b
a
w(t ) (t )dt
*
n
• 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
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Orthogonal Series
Proof of theorem: Assume that the set {φn} is sufficient to represent
the waveform w(t) over the interval a < t <b by the series
w(t )   an n (t )
n
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.
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Application 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 Waves.
n =1
n =3
n =5
http://www.educatorscorner.com/index.cgi?CONTENT_ID=2487
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Fourier Series
Complex 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 Series
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Some Properties of Complex Fourier Series
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Quadrature Fourier Series
 The Quadrature Form of the Fourier series representing any
physical waveform w(t) over the interval a < t < a+T0 is,
n 
n 
n 0
n 0
w(t )   an cos( n0t )   bn sin( n0t )
where the orthogonal functions are cos(nω0t) and sin(nω0t).
Using
we can find the Fourier coefficients as:
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Quadrature 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 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 Coefficients
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Line 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 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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Line Spectra for Periodic Waveforms
w(t ) 
n 
h(t  nT )

Line Spectra
for Periodic Waveforms
o
n 
cn  fo H (nfo )
h(t)
W( f ) 
h(t )  H ( f )
The Fourier Series Coefficients
of the periodic signal can be
calculated from the Fourier
Transform of the similar
nonperiodic signal.
n 
 c  ( f  nf
n 
n 
n
o
)
=   f o H (nf o )   ( f  nf o )
n 
The sample values for the
Fourier transform gives the
Fourier series coefficients.
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Line Spectra for Periodic Waveforms
Single Pulse
Continous Spectrum
Periodic Pulse Train Line Spectrum
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Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave
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Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave
 Now evaluate the coefficients from the Fourier Transform
 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.
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Ex. 2.12 Fourier Coeff. for a Periodic Rectangular Wave
Single Pulse
Continous Spectrum
Periodic Pulse Train Line Spectrum
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Normalized 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 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 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.
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