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2 “Signals”
Transcript of analise sinais ppt02e (Signals) - UBIwebx.ubi.pt/~felippe/texts2/analise_sinais_ppt02e.pdf · 2.2...
2
“Signals”
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.1 – Introduction to Signals
The notion of signals is intuitive and comes from a large variety of contexts.
Any measurement that is done: in numbers for example, any register that is made
o of the performance of a machine
o of the performance of an engine
o of the consumption of a vehicle a long of a trip
any measurement done: by using any measurement device or tool; or
any recording done, of a sound, or of an image (photo) or even of a video, can easily become in a signal.
There is a ‘language’ used to describe signals, as well as there is also a very powerful set of ‘tools’ to analyse them.
In this present chapter we deal with the ‘language’ that describe the signals. In the others chapters ahead we deal with the ‘tools’ to analyse them.
2.2 – Examples of Signals
The signals are used to describe a large variety of physical phenomenon and can be defined by many different ways: by number, or by graphics, or by a sequence of digits
(bits) to be introduced in the computer, etc.
Circuito RCThe voltage signal vs(t) at the source or
the voltage signal vc(t) in the capacitor, as
well as the current signal i(t) that crosses the only loop of this circuit can be measured by tools
(voltmeter / amperemeter)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Car The cars run when we accelerate them.
the force is equal to mass x acceleration
[ f(t) = m⋅a(t) ] ,
where m = car mass.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
But this is equivalent to inflict a force f(t)that will pull the car, since by Newton
Second Law,
Signals of force f(t), of displacement x(t)and of velocity v(t) of a car.
Voice / human speech
The human vocal mechanism produce speech by creating fluctuations in the acoustic pressure.
The voice signal is obtained through the use of a microphone that receives the variations of acoustic pressure and convert in electric signals.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
The air is expelled from the lungs by the diaphragm and in its way produce vibrations.
These vibrations are modified, or moulded, when passing through the vocals cords, such as by the mouth, lips and the tongue in order to produce the sounds that on wishes.
These signals can be used to record the voice sound or to be transmitted (telephone or mobile for example).
Radio transmissions (AM & FM)
A radio transmission is also composed of electric signals that transport the sound(voice, music, etc.)
A carrier wave (signal of much higher frequency) carries the modulated signal
(sound), either modulated in amplitude (AM) or in frequency (FM).
signal of the carrierModulator signal
signal modulated in amplitude (AM) signal modulated in frequency (FM)
Signal Analysis ______________________________________________________________________________________________________________________________________________________________________________________
The music recorded in a CD or stored in the computer (format wav, wma or mp3, for example) is done through a series of numbers, a digital sequence of “zeros” and “ones”, that represent the electric voltages (in Volts) of the audio signal a long of the time.
So, the analogic signal of audio is converted in a digital signal, that is, binary data, at a rate that is measured in “bps” (bits per second).
It is obvious that the bigger is the number of bits per second the better will be the quality of the reproduction of the sound.
Some usual values of the rate in music recording are:
96 thousands bits per second [96kbps], or
128 thousands bits per second [128 kbps], or
192 thousands bits per second [192 kbps], or
256 thousands bits per second [256 kbps].
There are electronic devices that transform the analogic signal in digital (A/D converters) as well as electronic devices that transform a digital signal in analogic (D/Aconverters).
Digital Music (mp3 and others)
Electrocardiogram (ECG)
The electrocardiograph is a device that measures electrical signals from the heart to produce an electrocardiogram (ECG).
Electrocardiography studies the electrical activity of the heart from electrodes placed in certain points of the human body.
Typical ECG signal
Electrocardiogram (ECG) recording is a common practice in today's medicine, since it is of recognized value for the identification and prognosis of cardiovascular diseases such as myocardial infarction, arrhythmia, among other pathological conditions.
Signal Analysis ______________________________________________________________________________________________________________________________________________________________________________________
Electroencephalogram (EEG)
The electroencephalograph is a machine that records the graph of electrical brain signals developed in the brain producing electroencephalogram (EEG).
Therefore the electrodes are placed in predefined positions on the scalp of the patient and an amplifier increases the intensity of the electrical potentials to be built an graphic (EEG) that can be analog or digital (depending on the equipment).
This is done through electrodes that are applied to the scalp, the brain surface, or even (in some cases) within the brain.
These observed brain signals are very weak.
Signal Analysis ______________________________________________________________________________________________________________________________________________________________________________________
Typical EEG signal
Monochrome image (black-white)
A monochromatic (black-white) image consists of a pattern of variations in brightness across it.
A monochrome photo (black-white)the brightness intensity signal.
That is, the image signal is a function of the brightness intensity at all points of the image (two-dimensional).
Therefore, the signal of a monochromatic photo is a function f(x,y), that is,
it must have intensity (x) and brightness (y) information at each point on the surface of the photo.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Coloured images
If the image is colored, the signal obviously becomes more complex. In this case the colors will have a tonal representation of 3 basic elements / colors.
Sometimes the image is broken down into 3 basic colors, which are commonly
“red”, “green” and “blue”
which is called as the color-coding RGB.
But other color codes are also used, such as the “magenta”, the “cyan” and the “yellow”, which is common in color printers and in computer systems in general.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Alternatively, the tonal representation can be made by supplying the 3 elementsbelow:
o hue (tonality),o saturation eo luminosity.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Therefore, each point on the photo surface of a color photo will have to have 3 colour information or 3 elements (not just one as in the monochromeor black-white photos).
For example, this color seen here below has a tonal representation given by:
(255, 128,64) in the RGB form, or by
(13, 240,150) in the hue, saturation and luminosity form.
On the other hand, this other color (green), for example, is represented by:
(0, 128,0) in the RGB form, or by
(83, 240,60) in the hue, saturation and luminosity form.
The transmission of images such as on “television broadcasting”, for example, requires even more sophisticated signals.
TV broadcasts
While a photograph is a “static” signal, fixed in time, the transmissions of images via TV are “dynamic” signals because they change with time.
Since colour TV has
emerged, many broadcast
systems have already been
created, such as the PALsystem (European), the
NTSC system (American),
or more recently, the
HDTV.
Figure shows an examples of a signal RGB (red,green and blue) from a TV broadcast.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Besides that, in TV broadcast the sound information also has to follow along with the
image.
Economic and demographic indexes
Economic indexes (or indicators) (which normally only go out once a month) such as:
inflation (monthly);
unemployment rate (monthly);
give rise to discrete signals (i.e., non-continuous signal).
The stock exchange index is also an example of a discrete signal, although this is not monthly but rather daily.
There are many other examples of economic indices or indicators such as exchange rates or growth rates of the Gross Domestic Product (GDP), etc.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Any of these indices, if taken over a long period of time and the points are connected, one gets the impression that the signal is continuous.
The exchange rates of one currency against the other are examples of discrete signals although they can be taken daily, hourly or even minute by minute, if we wish.
This is similar to the case of music, or images, or videos scanned and saved in our computer (digital audio or video systems) or digital transmission of images, cases already mentioned in previous examples.
Other cases of discrete signals:
o birth rates of a nation (year over year over a period);
o consumption of a vehicle [ l /100 km ] (measured each time it is supplied);
o profit of a commercial establishment (month by month, over the years);
etc.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
For example, the Euro exchange rate (€) against the US dollar (US $) over several years.
2.3 – Continuous and discrete signals
To distinguish continuous and discrete signals in time we will use
“t” to denote time as a continuous independent variable and
“n” to denote time as a discrete independent variable.
Moreover, in continuous signals we use normal parentheses ( ),
x(t), y(t), v(t), etc.
while in discrete signals we use brackets [ ], x[n], y[n], v[n], etc.
This is a notation commonly adopted in the Signal Analysis.
A discrete signal may be the representation of an inherently discrete phenomenon (system), such as the case of demographic indices or stock market indices.
On the other hand there are also discrete time signals that come from the sampling of continuous signals.
digital audio or video systems,
the digital audio or video systems, already mentioned above, or, to mention another example:
digital autopilot;
These systems require the use of time-discrete sequences which are representations (discretizations) of time-continuous signals.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Thus, signals that are naturally continuous in time are made discrete signals (by
sampling) for this purpose, for example:
the voice;
the music;
the sound in general;
the photos appearing in newspapers and books;
the images from a recorded movie video;
etc.
the position of the aircraft;
the velocity of the aircraft;
the direction of the aircraft.
digital autopilot
digital imaging systems
digital audio systems
Note that this scan is done with a very large number of points. In the case of digital music, as we have seen, it can have more than 250,000 points in each second [256 kbps]
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.4 – Dynamic and static signals
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Signals are represented mathematically as functions of one or more independent variables.
In several signals the time ‘t’ is the independent variable (or one of the independent variables), for example, in the case of:
dynamic physical systems.
However, there are signals that ‘time’ does not appear as an independent variable.These signals are called
static signals, or non-dynamic signals,
because they do not evolve in time, and therefore represent static physical systems.
RC circuitcarradio broadcasts
dynamic signals,
Therefore, these signals are of the type x(t), y(t), f(t) or f(x,t), etc. and are called
because they vary with time (or evolve in time, or propagate in time, etc.), and therefore represent
human voice / speechradio broadcastsdigital music (mp3)
stock Exchange
TV broadcastsECG
EEG
Some signals that are static:
the monochrome imagethe colour image
the weather signals
the geophysical signals
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.5 – Energy and Power of a Signal
In many, but not all, applications, the signals are directly related to physical quantities that capture or absorb energy and power in the physical system.
For example, in the case of the RC circuit the instantaneous power in the resistor R is:
)t(vR
1)t(i)t(v)t(p 2=⋅=
where:
v(t) = voltage in the resistor R;
i(t) = current in the resistor R.
and the total energy spent in the time interval t1 ≤ t ≤ t2 is
== 2
1
2
1
t
t
2t
tTotal dt)t(v
R
1dt)t(pE
and the average power in this interval [t1, t2] is:
( ) ( ) ⋅=⋅=−−
2
1
2
1
t
t
2t
tmédia dt)t(v
R
11dt)t(p
1P
1212 tttt
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Similarly in the case of the above example of the car, the power dissipated by the friction is:
)t(v)t(p 2⋅ρ=where ρ = friction coefficient of the surface.
And in this case the total energy and average power in the range [t1, t2] are respectively:
⋅ρ== 2
1
2
1
t
t
2t
tTotal dt)t(vdt)t(pE
( ) ( ) ⋅ρ⋅=⋅=−−
2
1
2
1
t
t
2t
tmédia dt)t(v
1dt)t(p
1P
1212 tttt
Motivated by examples like these above, power and energy are defined for any
continuous signal x(t) and any discrete signal x[n] by the following way:
The instantaneous power of a continuous signal x(t) or of a discrete signal x[n]:
2)t(x)t(p = or
2]n[x]n[p =
Note: x(t) or x[t] can be real or complex; and | x(t) | or | x[t] | is the absolute value
of the number x(t) or x[t], as we have seen in chapter 1.
eq. (2.1)
The average power of this signal in this interval [t1 , t2] is defined as:
( ) ⋅⋅=−
2
1
t
t
2dt)t(x
1P
12 tt
The total energy in the interval [t1 , t2] of a continuous signal x(t) is defined as:
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
⋅=⋅= 2
1
2
1
t
t
2t
tdt)t(xdt)t(pE
The total energy in the interval [t1 , t2] of a discrete signal x[n] is defined as:
[ ]==
==2
1
2
1
n
nn
2n
nn
nx]n[pE
( ) [ ]=
⋅=+−
2
1
n
nn
2
12
nx1
P1nn
eq. (2.2)
eq. (2.3)
eq. (2.4)
eq. (2.5)
The average power in the interval [t1 , t2] of a discrete signal x[n] is defined as:
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
In the case of an infinite time interval:
–∞ < t < ∞ or –∞ < n < ∞The definitions of total energy and average power, in the case of a continuous signal
in time, we have:
∞
∞−−∞→⋅=⋅=∞ dt)t(xdt)t(xlimE
2T
T
2
T
−→∞⋅⋅=∞
T
T
2
Tdt)t(x
T2
1limP
and, for a discrete signal in time, we have:
[ ] [ ] −=
∞
−∞=∞→==∞
N
Nn n
22
NnxnxlimE
( ) [ ]−=∞→
⋅=+∞
N
Nn
2
NnxlimP
1N2
1
eq. (2.6)
eq. (2.7)
eq. (2.8)
eq. (2.9)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Note that for some signals E∞ and/or P∞ may not converge.
If a signal has energy E∞ < ∞ (finite total energy), then:
P∞ = 0
That is because
0T2
ElimPT
== ∞∞∞ →In the continuous case
( ) 0E
limP1N2N
==+∞
∞∞ →in the discrete case
On the other hand, for the same reason, that is, using eq. (2.10) and eq. (2.11), we
conclude that: if a signal has finite power ≠ 0 (0 < P∞ < ∞), then:
E∞ = ∞.
Finally, there are signals that both have: E∞ = ∞ and P∞ = ∞.
eq. (2.10)
eq. (2.11)
For example, if x(t) or x[n] = constant ≠ 0 for all t, then this signal has infinite energy
(E∞ = ∞).
Example 2.1:
Consider the signal x(t)
[ ]
∉<<
=2,0tse0
2t0se1)t(x
2
020
dt0dt1dt0
dt)t(xE
2
22
0
20 2
2
=++=
⋅+⋅+⋅=
⋅=
∞
∞−
∞
∞−∞
P∞ = 0
thus,
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Exemplo 2.2:
Consider the signal n,2]n[x ∀=
( ) [ ]
( )
( )4
4)1N2(lim
)4444(lim
nxlimP
1N2
1
1N2
1
1N2
1
N
N
N
Nn
2
N
=
=⋅+⋅=
=+++++⋅=
=⋅=
+
+
+
∞→
∞→
−=∞→∞
LL E∞ = ∞.
and,
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Example 2.3:
Consider the signal ,2,1,0,1,2n,2]n[x −−== ,2,1,0,1,2n,0]n[x −−≠∀=and
−−≠−−=
=2,1,0,1,2nse0
2,1,0,1,2nse2]n[x
For this signal x[n]:P∞ = 0.[ ] 202nxlimE
N
Nn
2
2n
22
N===
−= −=∞→∞
Example 2.4
Consider the signal x(t) = 0,25 t, ∀t It is easily observed that for this signal
x(t) both E∞ and P∞ are infinite.
E∞ = ∞,
P∞ = ∞.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.6 – Transformations of the independent variable
Time shifting (translation):
The “time shifting”, or just “shift”, is the lateral sliding (translation) to the right or
left of the signal x[n] (in the discrete case) or x(t) (in the continuous case).
This is achieved by changing the independent variable, the time ‘n’ or ‘t’:
n → n ± no
t → t ± to.
or
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
continuous signal: x(t) x(t – to), to > 0.
Shift to the right (retard):
discrete: signal x[n] x[n – no], no > 0.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
continuous signal: x(t) x(t + to), to > 0.
Shift to the left (advance):
discrete signal: x[n] x[n + no] , no > 0.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Time reversal (“reflected signal”) around t = 0:
discrete signal: x[n] x[–n]
continuous signal: x(t) x(–t)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Time scaling:
Time scaling is actually a change in the time scale ‘n’ (in the discrete case) or ‘t’ (in the continuous case).
n → a n or t → a t.
This is achieved by changing the independent variable, the time ‘n’ or ‘t’:
to a constant a > 0.
Compression or shrinkage
discrete signal: x[n] x[an] , a > 1.
continuous signal: x(t) x(at), a > 1.
Expansion or stretching
discrete signal: x[n] x[an] , 0 < a < 1.
continuous signal: x(t) x(at), 0 < a < 1.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
time scaling illustrative example:
Compression
Expansion
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
General case:
discrete signal: x[n] x[αn + β]
continuous signal: x(t) x(αt + β)
If | α | < 1 → signal is stretched ( ← → );
If | α | > 1 → signal is compressed ( → ← );
If α < 0 → signal is inverted;
If β < 0 → shift to the right ( → );
If β > 0 → shift to the left ( ← ).
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Example 2.5: Consider the signal x(t) given by the expression:
∉≤<≤≤
=]2,0[t0
2t15,0
1t01
)t(x
shift to the left of one unit of time
signal x(t) reflected
(“time reversal”)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
“time scaling”
“time scaling”
expansion of the scale of 1.5
(that is, α = 1/1.5 = 2/3 )
compression of the scale of 0.666
(that is, α = 1/0.666 = 3/2)
Example 2.5 (continued):
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
First a shift of one unit to the left, and then a compression of the scale of 0.666.
First a compression of the scale of 0.666,
and then a shift of one unit to the left.
Firs a shift of 0.5 to the right, and then a
compression of the scale of 0.5 (that is, ½).
Example 2.5 (continued):
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.7 – Periodic signals
A continuous signal x(t) is periodic if ∃ T > 0 such that
x(t) = x(t + T) , ∀ t eq. (2.12)
T is called the ‘period’ of x(t).
That is, a periodic signal x(t) stays unchangeable if we shift of T (the period) units of time.
If a signal x(t) is periodic with period T then x(t) is also periodic with period 2T,
3T, 4T, …
The fundamental period To of x(t), is the least positive value of T for which the eq. (2.12) above is valid.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
This definition has an exception, which is the case of
x(t) = C (constant) , ∀ t
which is also periodic since any value T > 0 is a period of this signal, but nevertheless
there is no fundamental period To for it.
Exemplo 2.6:
It is easy to verify that To = (2π/a) is the ‘fundamental period’ of the periodic signal:
x1(t) = b ⋅ cos (at + c)
and that To = (π/a) is the ‘fundamental period’ of the periodic signal:
x2(t) = b ⋅ | cos (at) |
A non periodic signal is called “aperiodic”.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x1(t) = b ⋅ cos (at + c)
x2(t) = b ⋅ | cos (at) |
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Example 2.7:
A discrete signal with fundamental period No = 3.
Similarly, a discrete signal x[n] is periodic if ∃ N such that
x[n] = x[n + N] , ∀ n eq. (2.13)
N is called period of x[n].
The fundamental period of x[n], No, is the least value of N for which eq. (2.13) is valid.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.8 – Odd and Even Signals
A continuous signal x(t) is even if: x(–t) = x(t)
A discrete signal x[n] is even if: x[–n] = x[n]
A continuous signal x(t) is odd if: x(–t) = –x(t)
A discrete signal x[n] is odd if: x[–n] = –x[n]
Example 2.8:
x(t) = sin (t) is an odd signal; and
x(t) = cos (t) is an even signal.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Example 2.9:
an odd signal
an even signal
Note that for an odd signal x(t) (continuous),
or x[n] (discrete), it satisfies respectively :
x(0) = 0, or
x[0] = 0.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Any signal can be decomposed into the sum of 2 signals being one of them odd and the other even.
In the case of a continuous signal:
{ } { })t(xOdx(t)Evx(t) +=where:
{ } ( ))t(x)t(x2
1x(t)Ev −+=
{ } ( ))t(x)t(x2
1x(t)Od −−=
In the case of a discrete signal:
[ ] [ ]{ } [ ]{ }nxOdnxEvnx +=where:
[ ]{ } [ ] [ ]( )nxnx2
1nxEv −+=
[ ]{ } [ ] [ ]( )nxnx2
1nxOd −−=
(even signal)
(even signal)
(odd signal)
(odd signal)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Example 2.10:
the signal x[n] below is called of unit step.
This signal can easily be decomposed in the two signals
xev[n] = Ev{x[n]} e xod[n] = Od{[n]}
[ ] [ ]{ }
>
=
<
==
0nse,2
1
0nse,1
0nse,2
1
nxEvnxev [ ] [ ]{ }od
1 , se n 02
x n Od x n 0, se n 0
1 , se n 02
− <= = =
>
given below:
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Signal xev[n], the even component
of x[n].
Signal xod[n], the odd component of
x[n].
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.9 – Continuous Exponential and sinusoidal signals
The continuous sinusoidal signal
The continuous exponential signal
The continuous sinusoidal signal:
o
o
2 T
ωπ=
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
This signal describes the characteristics of many physical processes, in particular: systems in which energy is conserved, such as LC circuits; simple harmonic motion (SHM); the variation of the acoustic pressure corresponding to the tone of a musical note; etc.
The above signal x(t) = A cos(ωot + φ), ωo ≠ 0 is periodic with
o
o
2 T
ωπ=
ωo is called of fundamental frequency.
fundamental period
The above equation shows that the fundamental frequency ωo and
the fundamental period To are inversely proportional.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
If we have 3 signals:
xo(t) = A cos(ωot + φ),
x1(t) = A cos(ω1t + φ), and
x2(t) = A cos(ω2t + φ),
with ω2 < ωo < ω1 (which is equivalent to T1 < To < T2), then x1(t) oscillates more
than xo(t) and on the other hand x2(t) oscillates less than xo(t)
That is, for the signal xo(t) = A cos(ωot + φ), the higher the frequency ωo, the
more it oscillates, and the lower frequency ωo, the less it oscillates.
Note the units of
T, To, T1, T2 [seconds]and of
ω, ωo, ω1, ω2 [radians / second]
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Three periodic signals of
the type x(t) = cos ωtwith different frequencies.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
The units of x(t) = A cos(ωot + ϕ)t [ seconds ]
φ [ radians ]
ωo [ radians/second]
Sometimes the natural frequency ωo is written as ωo = 2πfo where fo is the
Note also (the particular cases). For x(t) = A cos(ωot + ϕ)
frequency of the signal x(t) = A cos(2πfot + φ) and has as unit fo [Hertz ].
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
If ϕ = 0 or ϕ = ±2π, ±4π, … x(t) = A cos (ωot)
or 2 , 4 ,2 2 2
π π πϕ = ϕ = ± π ± π L
or 2 , 4 ,2 2 2
π π πϕ = − ϕ = − ± π − ± π L
or , 3 , 5 , 7 ,ϕ = −π ϕ = −π ± π ± π ± π L
if
if
if
x(t) = − A sin (ωot)
x(t) = A sin (ωot)
x(t) = − A cos (ωot)
Besides that: if ωo = 0 ==> x(t) = C (constant)
The signal x(t) = C (constant), ∀t is also a periodic signal, with
period T for any T > 0.
However, this signal x(t) = C (constant) does not have a fundamental period To.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Another detail: the signal x(t) written in the form of a linear combination of a
sine and a co-sine with the same frequency ωot and without phase shift, that is,
where:
A cosα = ⋅ ϕ A sinβ = ⋅ ϕ
22A β+α=
and
and arctgβ ϕ = α
eq. (2.14)
eq. (2.15)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x(t) = α⋅sin(ωot) + β⋅cos(ωot)
= A sin(ωot + ϕ)
On the other hand, the signal x(t) which we have seen above, expressed in the form
of a co-sine of frequency ωot and phase shift ϕ, that is,
can be written as a linear combination of a sine and a co-sine with the same
frequency ωot (and vice versa) as follows:
where α, β, A and ϕ are again given by eq. (2.14) and eq. (2.15), repeated here below:
A cosα = ⋅ ϕ A sinβ = ⋅ ϕ
22A β+α=
and
and arctgβ ϕ = α
eq. (2.14)
eq. (2.15)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x(t) = A cos(ωot + ϕ)
= α⋅cos(ωot) – β⋅sin(ωot)
x(t) = A cos(ωot + ϕ)
The continuous exponential signal:
In this case x(t) is called as a real exponential signal and can be
growth (if a > 0)
decay (if a < 0).
exponential growing (a > 0) exponential decaying (a < 0)
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Case 1:
R = set of real numbers.
x(t) = C eat ,= C = constant ∈ R e a ∈ R
The growing exponential is used to describe many physical phenomena such as the chain reaction in atomic explosions and certain complex chemical reactions.
The decaying exponential appears in the description of many physical processes
such as: the radioactive decay, the output vc(t) of the RC circuit and damped
mechanical systems.
Obviously if a = 0, then we have again
x(t) = C eat = C = constant
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The continuous exponential signal:
a = j⋅ωo (purely imaginary)
In this case x(t)is a complexexponential
signal for
each t.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Case 2: x(t) = C eat ,= C = 1 and a is a pure imaginary number ∈ C
C = set of complex numbers
x(t) = C e jω tothat is,
The continuous exponential signal:
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
a = j⋅ωo (purely imaginary)
Case 2: x(t) = C eat ,= C = 1 and a is a pure imaginary number ∈ C
C = set of complex numbers
In this case x(t)is a complexexponential
signal for
each t.
x(t) = C e jω tothat is,
We can interpret this signal x(t) as a point that moves in the circumference of radius 1
in the complex plane with angular velocity | ωo | rad/s.
Note that this signal
Is always periodic since:
)t(x
)Tt(x Tjtj)Tt(j ooo
====+ ωω+ω
eee
for many values of T (period) for which 1Toj =ωe
Actually, if
...,2,1k,k2
To
±±=ω
π=
1Tj o =ωe
Then
and T is a period of x(t).
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Observe that since | ejθ | = 1,=then: | x(t) | = 1, ∀t
x(t) = e jω to
In the particular case of
0,2
T o
o
o ≠ωω
π=
To is the fundamental period of x(t) and
ωo is called fundamental frequency of x(t).
then
The family of complex exponential signals
...,2,1,0k,)t( tkjk
o ±±=ω=φ e
is known as ‘harmonically related’ signals.
),t(kφ k ≠ 0, is
ook k ω⋅=ω
and the fundamental period is
k
T
k
2T o
o
ok =ω⋅π=
This signals are periodic and the fundamental frequency for each
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
and there is no fundamental frequency nor any fundamental period either.
The term “harmonic” comes from music and refers to tones resulting from variations in acoustic pressure at frequencies that are multiple of the fundamental frequency.
For example, the string vibration pattern of a musical instrument (such as a violin) can be described as the overlap (or weighted average) of harmonically
related periodic exponential signals.
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In the case of k = 0, then φo(t) = constant
Example 2.11:
( )t5,1jt5,1jt5,3j
t5jt2j)t(x
⋅⋅−⋅
⋅⋅
+=
+=
eee
ee
now, using the ‘Euler Equation’,
)t5,1cos(e2)t(xt5,3j ⋅= ⋅
and, since 1j =θe ∀θ, we have that )t5,1cos(2)t(x ⋅=
which is the complete rectified sinusoidal wave signal
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
The continuous exponential signal:
If C = |C| e jθ
a = σ + jωo
(‘C’ is written in the polar form)
(‘a’ is written in the Cartesian form)
then the continuous exponential signal
Thus:
Re{ x(t) } and Im{ x(t) }
sinusoidal signalsσ = 0
σ > 0
σ < 0
sinusoidal signals multiplied by growing exponentials
sinusoidal signals multiplied by decaying exponentials
)t(senCj)tcos(C
C
C
C)t(x
oo
j(
)j(j
at
tt
)tt
t
o
o
θ+ω⋅⋅+θ+ω⋅=
=⋅=
=⋅=
=
σσ
ωσ
ω+σθ
θ+
ee
ee
ee
e
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Case 3: x(t) = C eat ,= C ∈ C and a ∈ C
C = set of complex numbers
Re{x(t)} = C eσt⋅ cos(ωot + θ) , σ > 0 Re{x(t)} = C eσt⋅ cos(ωot + θ) , σ < 0
Two sinusoidal signals multiplied by exponentials.
Examples of physical systems where these signals appear are: RLC circuits; mechanical systems with damping and restorative forces (mass-spring, suspension of automobiles, etc.).
These systems have mechanisms that dissipate energy (such as resistors, damping
forces and friction) with oscillations that decay in time.
with σ > 0, the signal grows; with σ < 0, the signal decays.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
2.10 – Discrete Exponential and sinusoidal signals
The discrete sinusoidal signal
The discrete exponential signal
The discrete sinusoidal signal :
x[n] = A cos (ωon + φ)
Where the units of x[n] are:
n [dimensionless, no dimension
ωo [radians]
φ [radians]
fo = ωo / 2π [radians]
x1[n] = A cos (ωon), for ωo = 0,2π ≅ 0,628.
the fundamental
period is No = 10
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x2[n] = A cos (ωon), for ωo = 0,3π ≅ 0,944
The
fundamental
period is
No = 20
x3[n] = A cos (ωon), for ωo = 1
This signal is not periodic as we shall see later on.
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Using the Euler equations, a discrete sinusoidal signal x[n] can be written as:
o
o oj n j nj j
x[n] A cos ( n )
A A
2 2
−−ω ωϕ ϕ
= ω + ϕ =
= ⋅ ⋅ + ⋅ ⋅e e e e
12j =φ
e
and, sinceand
12
o nj =ωe
For this signal we have that the energy E∞and the total power P∞ are:
E∞ = ∞, and P∞ = 1.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
The discrete exponential signal:
where α = eβ which is an analogous form
to the continuous exponential signal.
C ∈ R and α∈R
R = set of real numbers.
In this case x[n] can bea growing signal (if α > 1) or
a decaying signal (if 0 < α < 1).
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x[n] = C αn = C eβnCase 1:
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Obviously, if α = 0, then
[ ] nCnx α= is the signal
Similarly, if α = ±1, then [ ] nCnx α= is one of the signals below
if α = 1 and C > 0, then
if α = –1 and C < 0, then
if α = –1 and C > 0, then
if α = 1 and C < 0, then
if α = 0, then [ ] 0 =Cnx nα=
[ ] , | C | =Cnx nα=
[ ] , | C | =Cnx nα=
[ ] , | C | =Cnx -nα=
[ ] , | C | =Cnx -nα=
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
C = 1 and β is the purely imaginary number
( | α | = 1 )
The discrete exponential signal:
The complex exponential signal [ ] nn CCnx α== βe ( )β=α e
for C = 1 and β = j ωo (purely imaginary), we have that |α| = 1, and x[n]becomes: [ ] nj onx ω= e
Using the Euler equations we have that:
[ ] nsenjncosnx oonj o ω⋅+ω== ω
e
Note that, since
,n,12
nj o ∀=ωe
then, for this signal we have again
E∞ = ∞, e P∞ = 1.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x[n] = C αn = C eβnCase 2:
Note that the exponential signal
satisfies the following property:
[ ]...,2,1,0m,
nx
n)mo(j
n)2o(jnoj
±±==
===π±ω
π+ωω
e
ee
that is, the signal x[n] is the same for frequency ωo and (ωo + 2π).
Actually it is the same for any frequency (ωo ± mπ), m = 0, ±1, ±2, …
That is, it is repeated for every 2π as the frequency ωo varies.
This situation is different from its continuous analogue signal x(t), where for
each ωo, x(t) was a different signal.
The continuous signal x(t) never repeated for values other than ωo.
In fact, the higher the frequency ωo, the higher the oscillation rate of x(t).
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x[n] = e jω no
In the discrete case that we analyze here
what happens is that as ωo increases from 0 to π, we get signals x[n] that oscillate faster and faster.
Then, continuing to increase ωo from π to 2π, the signals x[n] oscillate more and more
slowly until it is the same as it was in ωo = 0 for ωo = 2π.
To get an idea of how this occurs, the evolution of the real part of x[n], that is
{ } { }j
oon[n] Re x[n] Re cos( n) ,ωσ = = = ωe
from 0 (no oscillation) to π (maximum number of oscillation) and then
continuing up to 2π (no oscillation again).
0o =ωσ[n] = cos (ωon),
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x[n] = e jω no
No = 1
8oπ=ωσ[n] = cos (ωon),
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No = 12
σ[n] = cos (ωon),
4oπ=ω
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No = 8
2oπ=ωσ[n] = cos (ωon),
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No = 4
π=ωoσ[n] = cos (ωon),
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No = 2
23
oπ=ωσ[n] = cos (ωon),
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No = 4
47
oπ=ωσ[n] = cos (ωon),
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No = 8
σ[n] = cos (ωon),
815
oπ=ω
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
No = 12
σ[n] = cos (ωon),
If ωo = π, or ωo = ±nπ for a value of n odd, the oscitation is maximum since
[ ] j
j
j n no
n n
for n oddx n ,
( ) ( 1) .
ω π
π
= =
= = −
e e
e
that is, the signal x[n] jumps from +1 to –1 at each point n in time.
On the other hand, if ωo = 0, or ωo = ±nπ for each m even, there is no oscillation since [ ] n,1
0noj jnx ∀=
⋅== ⋅ω⋅ee
that is, the signal x[n] is constant for all values ‘n’ in time.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
ωo = 2π
No = 1
Therefore, the low oscillations (or slow variations) of the signal x[n] have values
ωo close to 0, 2π, etc. (even multiples of π), while the high oscillations (or quick
variations) of the signal x[n] are located close to ±π and odd multiples of π.
Another important property is the “periodicity”.
[ ] [ ]nxNnxnojNojNn(o nj)j
o ==ωω+ω ⋅==+ ω
eeee
Is only valid when
1Noj =ω
e
that is, if
...,2,1,0m,m2No ±±=π=ω eq. (2.16)
While the signal x(t) is always periodic, for the signal x[n] this does not always happen. Note that the equation
This situation here in x[n] is also different than in its continuous analogous x(t).
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
Therefore, the discrete signal
it is only periodical when ωo /2π is a rational number.
In other words x[n] will be periodic if ωo is multiple of π by a rational number.
what is equivalent to saying
πω2
o∈ Q = set of the rational numbers eq. (2.17)
ωo = α·π, α∈Q
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x[n] = e jω no
x1[n] = A cos (ωon), for ωo = 0,2π ,
x2[n] = A cos (ωon), for ωo = 0,3π ,
eq. (2.18)
eq. (2.19)
These two signals,
x1[n] from eq. (2.18)
and x2[n] fromeq. (2.19), are periodic because they have multiple
frequencies of π by a rational number.
Returning to discrete sinusoidal signals:
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x3[n] = A cos (ωon), for ωo = 1. eq. (2.20) This signal,
x3[n] from eq. (2.20) is not periodic since its frequency it is
not multiple of π by a rational number.
Note that in x1[n] an in x2[n] that the points have again the same value of x[n]periodically.
However, with the signal x3[n] this does not happen because ωo = 1 is not multiple of
π by a rational number and therefore it is not a periodic signal.
Observe that x2[n] and x3[n] are very close signals since
x2[n] = A cos (3πn) = A cos (0.9425n) and x3[n] = A cos (1⋅n).
However, for the signal x3[n] the points never have the same value again, since it is not periodic.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x3[–2] = –0.4161 x3[–1] = 0.5403 x3[0] = 1,0 x3[1] = 0,5403 x3[2] = –0,4161 x3[3] = –0,9899 x3[4] = –0,6536 x3[5] = 0,2837 x3[6] = 0,9602 x3[7] = 0,7539 x3[8] = –0,1455 x3[9] = –0,9111 x3[10] = –0,8391 x3[11] = 0,0044 x3[12] = 0,8439 x3[13] = 0,9074 x3[14] = 0,1367 x3[15] = –0,7597 x3[16] = –0,9577 x3[17] = –0,2752 x3[18] = 0,6603 x3[19] = 0,9887
x3[n] = A cos (ωon), for ωo = 1.
That is, x3[n] oscillates infinitely, but the sequences of values never repeat themselves.
For example,
x3[0] = 1, since cos(0) = 1,
however this value 1 never happens again for any other
x3[n], ∀n ≠ 0.
Therefore, this signal x3[n], it is not periodic since its
frequency is not multiple of π by a rational number.
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
We can write the condition of eq. (2.16) and eq. (2.17), i.e., (ωo/2π) ∈ Q, in another equivalent way:
If (ωo/2π) ∈ Q, then any N that satisfies
...,2,1,0m,2
mNo
±±=
ωπ⋅= eq. (2.21)
Is a period of x[n].
In fact, if ωo ≠ 0, and if N and m are coprime integers (i.e., mutually prime, they
have no common factors), where N > 0, then the fundamental period is
No = N ,
that is,
ωπ⋅=o
o
2mN
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x(t) ≠ for ≠s values of ωo
x(t) is periodic ∀ ωo
π=ωN
m2o
fundamental frequency of x(t)ωo
fundamental period of x(t)
x[n] repeats to
ωo, (ωo + 2π), (ωo + 4π), etc.
x[n] only periodic if
π=ωN
m2o
for some integers m and N > 0
(m and N are coprime integers)
fundamental frequency of x[n]
(m and N are coprime)
if ωo = 0 does not exist!
if ωo ≠ 0 o
o
2T
ωπ=
fundamental period of x[n]
if ωo = 0 does not exist!
if ωo ≠ 0
ωπ⋅=o
o
2mN
Summarizing, Case 2 for continuous and discrete signals :
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
x(t) = e jω to x[n] = e jω no
C = set of complex numbers
If
C ∈ C and α ∈ C :
C = |C| e jθ (C written in the polar form)
α = |α| e jω (α written in the polar form)
Then, the continuous exponential signal
[ ])nsin(Cj)ncos(C
C nx
oo
nn
n
θ+ω⋅α⋅⋅+θ+ω⋅α⋅=
α=
Hence,
Re{ x[n] } and Im{ x[n] }
| α | = 1
| α | > 1
| α | < 1
discrete sinusoidal signals
discrete sinusoidal signals multiplied by growing exponentials
discrete sinusoidal signals multiplied by decaying exponentials
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
o
The discrete exponential signal:
x[n] = C αn = C eβnCase 3:
[ ] [ ]{ }1
)ncos(nxRen o
n
>α
θ+ω⋅α==σ
[ ] [ ]{ }1
)ncos(nxRen o
n
<α
θ+ω⋅α==σ
Signal Analysis______________________________________________________________________________________________________________________________________________________________________________________
