Signals and Systems 1

1

• Signals and Systems 1
• Lecture 2
• Dr. Ali. A. Jalali
• August 21, 2002

2
Signals and Systems 1

• Lecture 2
• Introduction to Signals

EE 327 fall 2002
3
Signals

• Classification of Signals
• Deterministic and Stochastic signals
• Periodic and Aperiodic signals
• Continuous time (CT) and Discrete time (DT)
• Causal and anti-causal signals
• Right and left sided signals
• Bounded and unbounded signals
• Even and odd signals

EE 327 fall 2002
4
Classification of Signals

• Deterministic - Predictive (An example is sin
  wave, square wave)
• Stochastic Non predictive (An example is noise
  signal or human voice)

EE 327 fall 2002
5
Classification of Signals

• Periodic and Aperiodic Signals
• Periodic signals have the property that
• x(tT)x(t) for all t.
• The smallest value of T satisfies the definition
  is called the period.
• Shown below are aperiodic signal (left) and a
  periodic signal (right).

EE 327 fall 2002
6
Classification of Signals

• Periodic Signals
• Simple
• Complex

EE 327 fall 2002
7
Classification of Signals

• Aperiodic
• Impulse
• Noise

EE 327 fall 2002
8
Classification of Signals

• For both exponential CT ( ) and
  DT( ) signals, x is a complex
  quantity. To plot x, we can choose to plot either
  its magnitude and angle or its real and imaginary
  parts- whichever is more convenient for analysis.
• For example, suppose s j?/8
• and ,
• then the real parts are

EE 327 fall 2002
9
Classification of Signals

• Continuous-time (CT) and discrete time (DT)
  signals
• CT signals take on real or complex values as a
  function of an independent variable that ranges
  over the real numbers and are denoted as x(t).
• DT signals take on real or complex values as a
  function of an independent variable that ranges
  over the integers and are denoted as xn.

EE 327 fall 2002
10
Classification of Signals

• For example, consider the image shown on the left
  and its DT representation shown on the right.

The image on the left consists of 302 ? 435
picture elements (pixels) each of which is
represented by a triplet of numbers R,G,B that
encode the color. Thus, the signal is represented
by cn,m where m and n are the independent
variables that specify pixel location and c is a
color vector specified by a triplet of hues
R,G,B (red, green, and blue).
EE 327 fall 2002
11
Classification of Signals

• Real and complex signals
• Signals can be real, imaginary, or complex.
• An important class of signals are the complex
  exponentials
• ? the CT signal where s is a
  complex number,
• ? the DT signal where z is a
  complex number.
• Q. Why do we deal with complex signals?
• A. They are often analytically simpler to deal
  with than real signals.

EE 327 fall 2002
12
Classification of Signals

• Causal and anti-causal signals
• A causal signal is zero for t lt 0 and
• anti- causal signal is zero for t gt 0.

EE 327 fall 2002
13
Classification of Signals

• Right- and left-sided signal
• A right-sided signal is zero for t lt T and a
  left-sided signal is zero for t gt T where T can
  be positive or negative.

EE 327 fall 2002
14
Classification of Signals

• Bounded and unbounded signals

EE 327 fall 2002
15
Classification of Signals

• Even and odd signals
• Even signals xe(t) and odd signals xo(t) are
  defined as
• xe(t) xe(-t) and xo(t) -
  xo(-t)

EE 327 fall 2002
16
Classification of Signals

• Any signal is a sum of unique odd and even
  signal. Using
• x(t) xe(t) xo(t) and x(-t)
  xe(t) - xo(t)
• Yields
  and

EE 327 fall 2002
17
Measuring Signals

• Measure Amplitude
• peak to peak amplitude
• mean amplitude (r.m.s.)
• Measure Period and Repetition Frequency
• duration of 1 cycle (s)
• number of cycles per second (Hz)
• How to measure shape?

EE 327 fall 2002
18
Signal Analysis

• Simple Periodic Signals
• called sinusoidal signals
• only need to know 3 things
• period (or frequency), amplitude phase

EE 327 fall 2002
19
Representation of signals Building-block
signals

• We will represent signal as sums of
  building-block signals.
• Important families of building-block signals are
  the eternal (everlasting ), complex exponentials
  and the unit impulse functions.

EE 327 fall 2002
20
Building-block signals

• Eternal, complex exponentials
• These signal have the form
• for all t and
  for all n, where X, s, and z are complex
  numbers.
• In general s is complex and can be written as
  s ? j?,
• where ? and ? are the real and imaginary
  parts of s.

EE 327 fall 2002
21
Building-block signals

• Eternal, complex exponentials- real s
• If s ? is real and X is real then
• 
  ,
• And we get the family of real exponential
  functions.
• Eternal, complex exponentials- imaginary s
• If s j? is imaginary and X is real then
• and we get the family of sinusoidal
  functions.

EE 327 fall 2002
22
Building-block signals

• Eternal, complex exponentials- complex s
• If s ? j? is complex and X is real then
• And we get the family of damped sinusoidal
  functions.

EE 327 fall 2002
23
Building-block signals

• For
  is plotted for different values of s
  superimposed on the complex s-plane.

EE 327 fall 2002
24
Building-block signals

• For
  is plotted for different values of s
  superimposed on the complex s-plane.

EE 327 fall 2002
25
Eternal complex exponentials- why are they
important?

• Almost any signal of practical interest can be
  represented as a superposition (sum) of eternal
  complex exponentials.
• The output of a linear, time-invariant (LTI)
  system (to be defined next time) is simple to
  compute if the input is a sum of eternal complex
  exponentials.
• Eternal complex exponentials are the
  eigenfuntions of characteristic (unforced,
  homogeneous) responses of LTI systems.

EE 327 fall 2002
26
Building block signals Unit impulse definition

• The unit impulse ?(t), is an important signal of
  CT systems. The Dirac delta function, is not a
  function in the ordinary sense. It is defined by
  the integral relation
• And is called a generalized function.
• The unit impulse is not defined in terms of its
  values, but is defined by how it acts inside an
  integral when multiplied by a smooth function
  f(t). To see that the area of the unit impulse is
  1, choose f(t) 1 in the definition. We
  represent the unit impulse schematically as shown
  below the number next to the impulse is its area.

EE 327 fall 2002
27
Unit impulse- narrow pulse approximation

• To obtain an intuitive feeling for the unit
  impulse, it is often helpful to imagine a set of
  rectangular pulses where each pulse has width ?
  and height 1/? so that its area is 1.

The unit impulse is the quintessential tall and
narrow pulse!
EE 327 fall 2002
28
Unit impulse- intuiting the definition

• To obtain some intuition about the meaning of the
  integral definition of the impulse, we will use a
  tall rectangular pulse of unit area as an
  approximation to the unit impulse.
• As the rectangular pulse gets taller and narrower,

EE 327 fall 2002
29
Unit impulse- the shape does not matter

• There is nothing special about the rectangular
  pulse approximation to the unit impulse. A
  triangular pulse approximation is just as good.
  As far as out definition is concerned both the
  rectangular and triangular pulse are equally good
  approximations. Both act as impulses.

EE 327 fall 2002
30
Unit impulse- the values do not matter

• The values of the approximation functions do not
  matter either. The function on the left has unit
  area and takes on the arbitrary value A for t0.
  The function on the right, which we shall
  encounter frequently in later lectures, has the
  property that it has non-zero values at most of
  its values, all but a countably infinite number
  of points, nut still acts as a unit impulse

What all these approximations have in common is
that as ? gets small the area of each function
occupies an increasingly narrow time interval
centered on t0.
EE 327 fall 2002