2. 1 What is a “signal”?
Matlab implementation of
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SignalAnalysisAndProcessing 2019 Chap2-3
- Bu sahifa navigatsiya:
- Fig. 2.8: 3D plot of a complex exponential of frequency
- 2.4.6.1 Matlab implementation of the complex exp.
- 2.4.7 The “Sinc” signal
- We will not use this definition
- Fig. 2.9: Plot of the
- signal and of the functions T t for
- 2.4.7.1 Matlab implementation of
- 2.4.8 The single-sided decreasing exponential signal
- 2.4.9 The Gaussian signal
- Fig. 2.11: a Gaussian signal
- 2.4.10 The constant unit signal and the constant signal
- 2.4.11 The constant “zero” signal
- Fig. 2.12: formal graphical representation of
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2.4.5.2 Matlab implementation of sin
and cos
Matlab has of course built-in sin and cos functions, which accept arrays as input.
As significant as sin and cos, the “complex exponential” is a very important function in this and many other courses. As is well-known, it can be defined in terms of a sin and a cos function, according to Euler’s formula:
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019
( ) ( ) exp(
) cos
sin ,
e jx x j x x = = +
Notice that throughout this class, the complex unity is always going to be written “j” and not “i”. It is well-known that the complex number jx e spans a full circle in the complex plane, of radius one, for x spanning the interval 0, 2 . In addition, jx e is periodic in x , that is, it repeats itself every 2 . This property is obviously inherited from the cos and sin functions that make it up. As a result, we can introduce a complex “rotating” signal which incorporates the concept of “frequency”, as for the cos and sin signals, as follows:
(
( ) 0 2 0 0 0 exp( 2
) cos 2
sin 2 j f t e j f t f t j f t = = +
In the context of the complex exponential signal, the meaning of “frequency” can be viewed in yet another way:
• 0 f tells us how many “turns” the complex point 0 2
f t z e = executes in one second over a circle of radius 1 on the complex plane (on your own prove it as an exercise); note that 0
need not be an integer number because, for instance, we can have “9.352” turns in one second.
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 • 0 0 1/
f = is how long it takes (in time) to accumulate 2 in the argument, that is how long it takes to run one full circle
on the
complex plane
(on your own prove it as an exercise). The quantity 0
is called the “period” of the exponential function, which could be equally well written with its period (rather than its frequency) made explicit in the argument: 0 2 j t T z e = .
0 1
. The black line is the time axis.
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 The complex exponential is no less important than its components sin and cos. In fact, it is even more important in this course, as it allows to handle many calculations more effectively than using sin and cos separately.
You can write a complex exponential in Matlab using the built- in function ‘exp’:
exp(2*pi*j*f0*t) Note that the constants pi and j are pre-defined (you do not have to initialize them). Note also that lately Matlab has started recommending the use of ‘1i’ instead of ‘j’ as the imaginary unit. However, both forms are currently OK. If you use ‘j’, then make sure you are not using it for other purposes, such as the index of a ‘for’ loop, since this would obviously create problems. You can produce the plot of Fig. 2.8 by entering the following Matlab statements:
t=0:0.01:10;z=exp(j*2*pi*t);figure; plot3(t,real(z),imag(z), 'LineWidth' ,2);grid on ;
axis equal
; axis([0,10,-1.2,1.2,-1.2,1.2,]);
xlabel( 'time (s)' ); ylabel( 'real part' );
zlabel( 'imaginary part' ); hold on ; plot3(t,zeros(size(real(z))),zeros(size(imag(z))), 'k' ,
ineWidth' ,1); hold off ;
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 on your own: use Matlab’s “grab and turn” capability to rotate the plot till you make just a sin function or a cosine function appear on screen. 2.4.7 The “Sinc” signal The “Sinc” signal is defined as follows 1 :
sin( ) Sinc . t t t =
Both the numerator and the denominator of the “Sinc” are continuous functions. However, for 0
an undetermined form develops of the 0 0
type. This problem can be solved by assigning to Sinc(0) the limit for 0
of the Sinc itself. It is easy to see (prove it on your own; hint, use for instance de l’Hôpital rule, or expand sin( ) t in a Taylor series) that: 0 0 sin( ) lim Sinc( ) lim 1
→ → = =
t t t t
therefore the complete definition of the Sinc function is:
1 Warning: an alternative definition of the Sinc function is also in wide use: ( ) sin( )
Sinc .
t t =
We will not use this definition. We will use instead Eq. 2-1.
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019
( ) sin( ) , 0 Sinc
. 1, 0 t t t t t = =
With this definition, the Sinc function becomes both continuous and differentiable. In particular, it can be differentiated any number of times. Note that the denominator is never zero (except for 0 = t , which we have already discussed). Instead, the numerator goes to zero every time sin(
) 0 = t , that is for: ,
= → = t k t k k
In other words, ( ) Sinc
0 =
for all integer values of
(except 0 =
). In many Signal Analysis applications, the Sinc function argument is often normalized as follows: sin
Sinc t t T t T T
=
. The resulting Sinc has nulls at all integer multiples of T , excluding zero. In other words: Sinc 0,
t t T T T T =
=
as shown in the figure below.
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019
Sinc
signal
An important property of the Sinc signal is the following (prove it as an exercise):
1 Sinc t T T t
This means that the Sinc signal goes to zero as t → , but it does so only as fast as T t . This can also be seen in Fig. 2.10 below: -11T -10T -9T -8T -7T
-6T -5T -4T
-3T -2T
-T 0 T 2T 3T 4T 5T 6T 7T 8T 9T 10 11T [s]
t Sinc
t T
1
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019
Sinc
T t for 1
Finally, the integral of the Sinc over the whole of is 1: Sinc( )
1 t dt − =
We will prove this result later on. 2.4.7.1 Matlab implementation of ( )
Sinc t Matlab has a built-in implementation of ( ) Sinc t . Note that in Matlab the initial is lowercase: sinc(t) . Using an uppercase letter causes an error. The function accepts an array input. -15 -14 -13 -12 -11 -10 -9 -8 -7
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -0.5
0 0.5
1 1.5
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 2.4.8 The single-sided decreasing exponential signal Another important signal is the so called single-sided or “unilateral” decreasing exponential signal. Its analytical expression is: u( ) ,
at t e a a −
Note that per se the exponential functions of a real negative argument would be non-zero over all , but the presence of the unilateral step signal makes this signal non-zero only for 0
. Note that this signal is jump-discontinuous at 0 t = .
On your own What is its actual value for 0
? Find out. On your own Write a Matlab implementation of the single- sided decreasing exponential signal and plot it.
Another important signal is the Gaussian signal. One possible definition is: 2 2 2 G( )
, 0
T t e T T − =
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019
( ) G t with 1
The Gaussian signal is mathematically non-zero over the whole of . However, it decreases very fast for increasing t . 0 G( ) 1 G( )
0.6 2 G( ) 0.14 3 G( ) 0.011 10 G( ) 0.000000000000000000002 t t t T t t T t t T t t T t = = = = = = The Gaussian function is very important in statistics as well. In that context it typically has a normalization constant in front of it. (On your own Can you remember what is the normalization factor, and why is it used?) On your own Write a Matlab implementation of the single- sided decreasing exponential signal. -5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.1
0.2 0.3
0.4 0.5
0.6 0.7
0.8 0.9
1 [s]
t ( )
G t
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 On your own write the Gaussian function so that its width- scaling parameter is its FWHM.
( )
2 FWHM
log 2 G( )
e t T t e − = 2.4.10 The constant unit signal and the constant signal A constant unit signal is a signal whose value is 1 for all times. We will write it as 1( )
t . From this elementary signal, arbitrarily valued constant signals can be derived by simply multiplying 1( )
:
const( , ) 1( )
t t =
A constant “zero” signal is a signal whose value is zero at all times. We will write it as 0( )
written in textbooks as just “0”, the signal 0( ) t is quite a different object than the number “0”. 2.4.12 On your own On your own, look at how all of the signals introduced so far behave for t → . What classes of signals can you identify, according to their behavior at t → ? Also, try
and classify them in
terms of
their continuity/discontinuity and differentiability features.
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 All of these signals can be re-scaled in time and translated left and right. How do you do it? To practice, consider the following: ( )
s t , ( ) s t , ( / )
s t , 0 ( ) s t t − , 0 ( ) s t t + , ( ) 0
t t − , ( ) 0 / s t t − Take as
( ) s t any of the signals previously introduced, such as ( )
or ( ) t , or any other, and apply the above transformations. Draw the results. If you have trouble doing it, refresh the material and do more exercises: these basic transformations are essential and are taken for granted as pre-established background. Using Matlab, plot some of the elementary signals that we have seen. Try also to draw them using the translations and scaling proposed in the previous paragraphs. Also, the concept of “adding signals” or “multiplying signals” must absolutely be clear in the mind of the students. If you have trouble understanding what it means to do:
( )
( ) ( )
s t v t w t = + ( )
( ) ( )
s t v t w t =
then you need to immediately revise your calculus background. Try to do the above using as ( )
v t and ( )
w t any of the previously introduced signals. Make sure you can easily draw the result on your own. If you are unsure, revise the theory. Then you can check the result using for instance the plot function of Matlab.
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 2.5 Dirac’s “delta” Dirac’s “delta” writing “ ( ) t ” may lead to mistaking it for a conventional “signal”, that is, a conventional function of time. However, Dirac’s delta is not a “proper” function, because its value for 0
is undefined, even though informally such value is said to be “infinity”. Note that, at all other times except 0, ( ) t
is a conventional signal and in particular:
( ) ( ) 0 0 t t t =
As a result, ( ) t is typically represented graphically as follows:
( )
The arrow at the origin, of formal height equal to 1, is meant to clearly show that something peculiar happens at 0
.
In fact, Dirac’s delta is not a function, but a special object that acquires actual meaning only within an integral operator. The t ( )
t 1
Signals and Systems – Poggiolini, Visintin – Politecnico di Torino, 2018/2019 actual “definition” of Dirac’s delta can be written in terms of its essential integral property:
t
( )
( ) (0)
t s t dt s + − = Eq. 2-2
Note that this definition requires 2 that
( ) s t exists and is continuous at 0
. Note also that as a direct consequence of the above definition, by choosing ( ) 1( )
s t t = we have: ( )
1 t dt + − = Download 0.84 Mb. Do'stlaringiz bilan baham: 
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