Download Spectrum Representation

lecture 3
MATLAB LABORATORY 3
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(2*pi*fk*t + øk)
k=1
=
N
Xo + Re{ ∑ Xk exp(j*2*pi*fk*t)}
k=1
Where Xo = Ao is a real constant,
and Xk = Ak exp(j*øk)
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.
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.
Formulae :
N
x(t)=Ao + ∑ Ak cos(2*pi*fk*t +øk)
k=1
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(j*2*pi*fk*t)}
K=1
Where Xo = Ao represents a real constant
component, and each phasor
Xk = Ak exp(j*øk)
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(j*2*pi*fk*t) + Xk*/2 exp
k=1
(-j*2*pi*fk*t)}
We define the two sided spectrum of a signal
composed of sinusoids as in above x(t) to be the
set of 2N+1 complex phasors and 2N+1
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 .
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(200*pi*t -pi/3) + 8cos(500*pi*t+pi/2)
Applying inverse Euler’s formula we get
x(t) = 10 + 7 exp(-j*pi/3) exp(j*2*pi*100*t)
+ 7 exp(j*pi/3) exp(- j*2*pi*100*t)
+ 4 exp(j*pi/2) exp(j*2*pi*250*t)
+ 4 exp(- j*pi/2) exp(- j*2*pi*250*t)
The spectrum of the signal is the set of five
rotating phasors represented by
{ (10,0) , (7 exp(- j*pi/3) ,100) ,
(7 exp( j*pi/3) ,-100) , (4 exp(j*pi/2) ,250)
(4exp(- j*pi/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(j*0*t) =10.
>> f =[-250,-100,0,100,250];
>> y =[4,7,10,7,4];
>> stem(f,y)
>> axis([-300 300 ,0 15 ])
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
1) The relative location of the frequencies
2) The relative amplitudes of the sinusoidal
components.
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
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.
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.
As the previous example suggests that the sum
of two sinusoids can also be written as a
product.
Let x(t)=cos(2*pi*f1*t) +cos(2*pi*f2*t)
The two frequencies are expressed as
f1=fc – fΔ
f2=fc + fΔ
Where fc is the center frequency and fΔ is the
deviation frequency.
fc = (f1+f2)/2
fΔ = (f1-f2)/2
After the use of Euler formula to the equation
we obtain the equation of x(t) as
x(t)=2*cos(2*pi*fΔ*t)*cos(2*pi*fc*t)
Let fc=200 and fΔ=20
>> t=0:1/2000:0.1;
>> x=2*cos(2*pi*20*t).*cos(2*pi*200*t);
>> plot(t,x)
>> sound(x,2000)
If we listen to such x(t) we can hear that the fΔ
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 fΔ is reduced to 9Hz we can see that the
envelope of the 200Hz tone changes slowly.
>> t=0:1/2000:0.1;
>> x=2*cos(2*pi*9*t).*cos(2*pi*200*t);
>> plot(t,x)
>> sound(x,2000)
The time interval between nulls of the
envelops is ½(1/fΔ),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.
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.
Example say v(t)=(5+2*cos(40*pi*t))
x(t)=v(t)*cos(400*pi*t)
>> t=0:1/2000:0.1;
>> x=(5+2*cos(40*pi*t)).*cos(2*pi*200*t);
>> 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.
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