MATLAB LABORATORY 3

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lecture 3

• MATLAB LABORATORY 3

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Spectrum Representation

• Definition A spectrum is a graphical
  representation of the frequency content of a
  signal .
• Formulae Sum Of Sinusoidal Signals
• N
• x(t) Ao ? Ak cos(2pifkt
  øk)
  
• k1
• N
• Xo Re ? Xk exp(j2pifkt)
• k1

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Where Xo Ao is a real constant, and Xk Ak
exp(jøk) is the complex amplitude (i.e. the
phasor) for the complex exponential frequency
fk. The spectrum is also a graphical
presentation of the individual sinusoidal
components that make up the signal.
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Spectrum of Sinusoids

• Sinusoids are very important as they are basic
  building blocks for making more complicated
  signals.
• The most important and powerful method of
  producing new signals from sinusoids is the
  additive linear combination.
• Where a signal is created by adding together a
  constant and N sinusoids of different
  frequencies, amplitude and phase.

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Formulae N x(t)Ao ?
Ak cos(2pifkt øk)
k1 Where each amplitude, phase and frequency
can be chosen independently. In Phasor
representation x(t) can also be represented as
N x(t) Xo ? Re Xk
exp(j2pifkt)
K1 Example 3.1
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Where Xo Ao represents a real constant
component, and each phasor Xk Ak
exp(jøk) represents the magnitude and phase of
a rotating phasor whose frequency is fk. Using
Inverse Euler formula we can write x(t) as
N x(t)Xo ? Xk/2
exp(j2pifkt) Xk/2 exp k1

(-j2pifkt)
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We define the two sided spectrum of a signal
composed of sinusoids as in above x(t) to be the
set of 2N1 complex phasors and 2N1 frequencies
that specify the signal representation of
x(t). The definition of spectrum is just set of
pairs (Xo,0) , (1/2 X1,f1) , (1/2 X1,- f1)
Each pair (1/2Xk,fk) indicates the size
and relative phase of the sinusoidal component
contributing at frequency fk .
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It is common to refer spectrum as the frequency
domain representation of the signal. The
frequency domain representation simply gives the
information required to synthesize the
signal. Example x(t) 10 14cos(200pit
-pi/3) 8cos(500pitpi/2) Applying inverse
Eulers formula we get x(t) 10 7 exp(-jpi/3)
exp(j2pi100t) 7 exp(jpi/3)
exp(- j2pi100t) 4 exp(jpi/2)
exp(j2pi250t) 4 exp(- jpi/2)
exp(- j2pi250t)
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The spectrum of the signal is the set of five
rotating phasors represented by (10,0) , (7
exp(- jpi/3) ,100) ,
(7 exp( jpi/3) ,-100) , (4
exp(jpi/2) ,250) (4exp(- jpi/2 , -250) The
constant component of the signal, often called as
the DC component can be expressed as a complex
exponential signal with zero frequency i.e. 10
exp(j0t) 10. gtgt f -250,-100,0,100,250 gtgt
y 4,7,10,7,4 gtgt stem(f,y) gtgt axis(-300 300
,0 15 )

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Graphical plot of the spectrum

• A plot of the spectrum contains frequency
  component represented by a straight line at that
  frequency. And its length is given by the
  magnitude of the corresponding phasor.
• This simple but effective plot makes easy two
  things
• The relative location of the frequencies
• 2) The relative amplitudes of the sinusoidal
  components.

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A general procedure to compute and plot the
spectrum for any given signal. It is necessary to
express the signals as complex exponentials (by
using inverse Euler relation) And then plot the
complex amplitude of each of the positive and
negative frequency components at the
corresponding frequency. In other words the
process of analyzing the signal to find its
spectral components involves simply looking at an
equation and picking off the amplitude, phase and
frequency
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Beat Notes

• When two sinusoidal signals of different
  frequencies are multiplied ,then it creates an
  interesting audio effect called a Beat note.
• This interesting sound can be best heard by
  picking one of the frequencies to be very small.
• Another use of multiplying sinusoids is for
  Modulation.

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To plot spectrum of a signal the signal should be
expressed as additive linear combination of
complex exponential signal. And a product of two
sinusoids can be written as sum of complex
exponential signal by using inverse Euler
formula. Beat notes are also produced by adding
two sinusoids with nearly identical
frequencies, e.g., by playing two neighboring
piano keys.
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As the previous example suggests that the sum of
two sinusoids can also be written as a
product. Let x(t)cos(2pif1t)
cos(2pif2t) The two frequencies are
expressed as f1fc f? f2fc f? Where fc is
the center frequency and f? is the deviation
frequency. fc (f1f2)/2 f?
(f1-f2)/2 .
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After the use of Euler formula to the equation
we obtain the equation of x(t)
as x(t)2cos(2pif?t)cos(2pifct) Let
fc200 and f?20 gtgt t01/20000.1 gtgt
x2cos(2pi20t).cos(2pi200t) gtgt plot(t,x)
gtgt sound(x,2000)
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If we listen to such x(t) we can hear that the
f? Variation causes the signal to fade in and
fade out because of the signal envelope falling
and rising. This is the phenomenon called Beating
of tones in music. If f? is reduced to 9Hz we
can see that the envelope of the 200Hz tone
changes slowly. gtgt t01/20000.1 gtgt
x2cos(2pi9t).cos(2pi200t) gtgt plot(t,x)
gtgt sound(x,2000)
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The time interval between nulls of the envelops
is ½(1/f?),so the more closely spaced the
sinusoids ,the slower the envelope
variation. Musicians use this phenomenon as an
aid in tuning two instruments to the same pitch.
When two notes are close but not identical in
frequency, the beating phenomenon is heard. As on
pitch is changed to become closer and closer to
the other ,this effect disappears ,and the two
instruments are then in tune.
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Amplitude Modulation

• Multiplying two sinusoids is also useful in
  modulation for communication systems.
• Amplitude Modulation is the process of
  multiplying a low frequency signal with a
  frequency sinusoid.
• In other words changing the amplitude of the high
  frequency signal according to the change in the
  message signal to be transmitted is called as
  amplitude modulation.
• Where the high frequency signal is called as the
  carrier signal and its frequency is called as
  carrier frequency.

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Example say v(t)(52cos(40pit)) x(t)v(t)cos
(400pit) gtgt t01/20000.1 gtgt
x(52cos(40pit)).cos(2pi200t) gtgt
plot(t,x) The primary difference between this
Amplitude modulated signal and the beat signal
is that the envelope never goes to zero as the
case of beat signal.
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In the frequency domain the Am signal spectrum is
nearly same as the beat signal ,the only
difference being a large term at f fc Where fc
is the center frequency. HW Problem 3.1,
Exercise 3.1