Week_III

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University of Malaya

Dr.Harikrishnan

Department of Electrical Engineering

e-mail: [email protected]

KEET 1101/ KEEE 2142Communication

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Spectral Analysis

The Fourier Series

Essentially any repetitive waveform that is comprised of more than one harmonically related sine or

cosine wave is nonsinusiodal, complex wave. A complex wave is any periodic (repetitive) waveform

that is not a sinusoid, such as square waves, rectangular waves and triangular waves.

To analyze a complex periodic wave, it is necessary to use a mathematical series, called Fourier

Series.

University of Malaya KEEE 2142/KEET 1101 HRK 2/16

1 2 3 n

1 2 3 n

cos cos cos .... cos n

B sin B sin 2 B sin 3 ....B sin n

=

+ + + +

where =

Any periodic waveform is comprised of an average dc component and a series of harmonically

related sine or cosine waves. A harmonic is an integral multiple of fundamental frequency.

The second multiple of the fundamental frequency is called the second harmonic, the third multiple is

called the third harmonic.

( )t dc fundamental 2nd harmonic 3rd harmonic .... nth harmonicf = + + + + +

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Spectral Analysis (contd)

The coefficients Ao, B1, Bn and A1 to An can be valuated:

( )T

0

1A t dt

Tf =

( )T

n

0

2A t cos n t dt

Tf=


( )n 0B t sin n t dtT f= Example 1Determine Fourier series to represent the function f(t)=2t in the range of - to +. Use the expression

( ) n nn 1

f t A A cos nt B sin nt

=

= + +

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Fourier Series Summary

Waveform Fourier Series

Odd

V+

tT 2 T

( )

( )( )

( )

N

2N 2

V V 2V 2Vv t sin t cos 2 t cos 4 t ...

2 3 15

V 1 1V Vv t sin t cos N t

2 1 N

=

= + +

+ = + +



Odd

( )

( )( )

( )

N

2N 1

2V 4V 4Vv t cos t cos 2 t ....3 15

4V 12Vv t cos N t

1 2N

=

= + +

= +

( )

( )N odd

4V 4Vv t sin t sin 3 t ....

3

4Vv t sin N t

N

=

= + +

=

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Fourier Series Summary

Waveform Fourier Series

Even

( )

( )N Odd

4V 4V 4V

v t cos t cos 3 t cos 5 t ...3 5

Vsin N 2v t cos N t

N 2

=

= + +

=



Even

( )N 1

V 2V sin N t Tv t cos N t

T T N t T

=

= +

( )( ) ( )

( )( )

2 22

2N odd

8V 8V 8Vv t cos t cos 3 t cos 5 t ...

3 5

8Vv t cos N t

N

=

= + + +

=

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Example 2For the train of square waves shown below.a. Determine the peak amplitudes and frequencies of the first five odd harmonicsb. Draw the frequency spectrumc. Calculate the total instantaneous voltage for several times and sketch the time-domain harmonics



Example 3Determine the first three harmonics of the waveform shown below.

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Frequency Spectrum and Bandwidth

The frequency spectrum of a waveform consists of all frequencies contained in the waveform and

their respective amplitudes in the frequency domain.

Frequency spectrums can show absolute values of frequency versus voltage or frequency versus

power level, or they can plot frequency versus some relative unit of measurement such as dB.

The bandwidth of an information signal is simply the difference between the highest and lowest



.

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Frequency Spectrum and Bandwidth


Example 4Determine the bandwidth for the frequency spectrum shown below.


Example 5Determine the channel bandwidth for the frequency envelope below.

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Frequency Series for a Rectangular Waveform

The duty cycle (DC) for the rectangular waveform is the ratio of the active time of the pulse to the

period of the waveform



Duty Cycle, DC T

=

( )DC % 100T

=

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Fourier Transform a Rectangular Waveform

Regardless of the duty cycle, a rectangular waveform is made up of a series of harmonically related

sine waves. Rectangular Waveforms are similar to the square wave waveform above, the difference

being that the two pulse widths of the waveform are of an unequal time period. Rectangular

waveforms are therefore classed as "Non-symmetrical" waveforms. However, the amplitude of the

spectral components depends on the duty cycle.



( ) ( ) ( ) ( )V 2V sin x sin 2x sin nxv t cos t cos 2 t ... cos n tT T x 2x nx = + + + +

where v(t) = Time varying voltage wave

= Pulse width of the rectangular wave (seconds)

T = Period of the rectangular wave (seconds)

X = (/T)

n = nth harmonic and can be any positive integer value

V = Peak pulse amplitude (volts)

(1)

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From equation (1) it can be seen that a rectangular waveform has 0-Hz (dc) component equal to:

V V or V DCT

=


where V0 = dc voltage (volts)

DC = Duty cycle as a decimal


= Pulse width of rectangular wave (seconds)

T = Period of rectangular wave (seconds)

The narrower the pulse width is, the smaller the dc component will be. Also from equation (1), the

amplitude of the nth harmonic is:

( )

( )

n

n

2V sin nx

V T nx

or

sin n T2VV

T n T

=

=

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where Vn = Peak amplitude of the nth harmonic (volts)

n = nth harmonic (any positive integer)

= 3.14159 radians

V = Peak amplitude of the rectangular wave (seconds)

= Pulse width of the rectangular wave (seconds)


T = Period of the rectangular wave (seconds)

The , function is used to describe repetitive pulse waveforms. Sin x is simply a sinusoidal

waveform whose instantaneous amplitude depends on x and varies both positively and negatively

between its peak amplitudes at a sinusoidal rate as x increases. With only x in the denominator, the denominator increases with x. Therefore, a , function is

simply a damped sine wave in which each successive peak is smaller than the preceding one.

( )sinx x

( )sinx x

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Figure shows the frequency spectrum for a rectangular pulse with a pulse width-to-period ratio of 0.1.

It can be seen that the amplitudes of the harmonics follow a damped sinusoidal shape.

At the frequency whose period equals to 1/ (at the frequency 10f Hz), there is a 0V component. A

second null occurs at 20f and so on.

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All spectrum components between 0 Hz and the first null frequency are considered in the first lobe of

the frequency spectrum and are positive.

All spectrum components between the first and second null frequency are in second lobe and arenegative.


Envelope

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Example 6For the pulse waveform shown below.a. Determine the dc componentb. Determine the peak amplitude of the first 10 harmonicsc. Plot the (sin x)/x functiond. Sketch the frequency spectrum


Example 7

If T= 1ms, = 0.1 ms and Vpeak= +2 V in Example 4.a. Determine the dc componentb. Determine the peak amplitude of the first 5 harmonicsc. Plot the (sin x)/x functiond. Sketch the frequency spectrum

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Power Spectra


Electrical power is the rate at which energy is dissipated, delivered or used and is a function of the

square of the voltage or current. For power relationship, in the Fourier equation, f(t) is replaced by

[f(t)]2

. The envelope resembles voltage-versus-frequency spectrum except it has more lobes a muchlarger primary lobe.


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