CHAPTER 2
SIGNALS AND SIGNAL SPACE
EENG 3810/CSCE 3020
Instructor: Oluwayomi Adamo
Signals and Systems
• What is a signal?
– A set of data or information e.g telephone or TV signal
• What is a System?
– An entity that processes a set of signals (inputs) to yield
another set of signals (outputs)
– Example: a system that estimates the position of a target
based on information from a radar
– A system could be hardware (electrical, mechanical or
hydraulic) or software (algorithm)
• How do we quantify a signal that varies with time?
• How do you device a measure V for the size of
human?
H
– Assuming Cylinder with radius r: V   0 r 2 (h)dh
Signal Energy and Power
• Measure of a signal g(t):
– Could be area under a signal g(t)

• Signal Energy Eg : Eg   g 2 (t )dt


Egc   | g (t ) |2 dt

– g(t) is squared to prevent the positive and negative areas
from cancelling out
– Signal energy must be finite
• For a signal to be finite:
– Signal amplitude -> 0 as |t| -> ∞ if not Eg will not
converge
Signal Energy and Power
Figure 2.1 Examples of signals: (a) signal with finite energy;
(b) signal with finite power.
•If Eg is not finite (infinite)
-Average power Pg (mean squared) : Time average of
energy (if it exists)
1 T2 2
Pg  lim
T g (t ) dt

t  T
2
-RMS (root mean square) value of g(t) = P g
-Pg exists if g(t) is periodic or has statistical regularities
Signal Energy and Power
– Average may not exist if the above condition is not
satisfied e.g a ramp g(t) = t, energy or power does not
exist
•
Measure of signal strength and size
– Signal energy and power - inherent characteristic
• Good indicator of signal quality
– The signal to noise ratio (SNR) or ratio of the message
signal and noise signal power
• Standard unit of signal energy and power
– Joule (J) and watt (W) respectively,
– Logarithmic scales used to avoid zeros and decimal
points
– Signal with average power of P watts (10 log10P) dBw or
(30+10.log10P) dbM
Determine the suitable measure for this signal?
• This signal approaches 0 as |t|  , therefore use the
energy equation.
• This signal does not approach  0 as |t|   and it is a
periodic wave, therefore use the power equation where
g2 is replaced with t2.
• What is the RMS value of this signal?
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Determine the power and rms value of
•Periodic signal with T0  2
• Suitable measure of size is power
RMSvalue C / 2
- First term on the right hand side equals C2/2
- Second term is zero –integral appearing in this term is area under a sinusoid.
-Area is at most the area of half cycle – positive and negative portion cancels
-A sinusoid of amplitude C has a power of C2/2 regardless of angular frequency
Determine the power and rms value
b) g (t) = C1 cos (1t + 1) + C2 cos (2t + 2)
1 ≠ 2
•
•
•
This signal is the sum of two sinusoid signals.
Therefore, use the power equation.
Therefore, Pg = (C12 / 2) + (C22 / 2)
This Can be generalized
What is the suitable measure for this signal?
g (t) = Dejt
• The signal is complex and periodic. Therefore, use the
power equation averaged over T0.
• |ejt| = 1 so that |Dejt|2 = |D|2 and
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Classification of Signals – Continuous and
Discrete time Signal
•
•
•
•
Continuous time and discrete time signals
Analog and digital signals
Periodic and aperiodic signals
Deterministic and probabilistic signals
Continuous Time Signal: A signal that is specified for every
value of time t .eg audio and video recordings
Discrete Time Signal: A signal that is specified only at discrete
points of t=nT e,g quarterly gross domestic product (GDP) or
stock market daily averages
Figure 2.3 (a) Continuous time and (b) discrete time signals.
Analog and Digital Signals
• Is analog signal and continuous time signal the same? What of
discrete time signal and digital signal?
• What is Analog Signal?
– A signal whose amplitude can have values in continuous range
(values can take on infinite (uncountable) values
• What is Digital Signal?
– A signal whose amplitude can take only finite number of
values.
– For a signal to qualify for digital, the values don’t have to be
restricted to two values.
– A digital signal whose values can take on M values is an M-ary
signal
• Continuous and discrete time signal qualify a signal along the xaxis while Analog and digital signal qualify the signal in terms of
the amplitude (y-axis)
Identify the signals above
Periodic and Aperiodic signal
• What is a Periodic Signal?
– A signal is said to be periodic if there exists a positive
constant T0
g (t )  g (t  T0 )
– For all t. The smallest values of T0 that satisfy the
equation (periodicity condition) above is the period of the
signal g(t)
– A periodic signal remains unchanged if time shifted by 1
period. Must start at -∞ and continue forever
• What is an Aperiodic Signal?
– A signal that is not periodic
Figure 2.5 Periodic signal of period T0.
Energy and Power Signal
• Energy Signal: A signal with finite energy. Satisfies:



| g (t ) |2 dt  
• Power Signal: A signal with finite and non zero power (mean
square value). Fulfills:
1
0  lim

T
T
T
2
T 
2
| g (t ) |2 dt  
• Power is the time average of the energy
• Since the averaging is over a large interval, a signal with finite
energy has zero power
• A signal with finite power has infinite energy
• Ramp signal has infinite power and are neither energy nor
power signal. Not all power is periodic
Deterministic and Random Signal
• Deterministic Signal: A signal whose physical description
(mathematical or graphical) is known.
• Random Signal: Signal known by only its probabilistic
description such as mean value, mean squared value and
distributions.
• All message signals are random signals for it to convey any
information.
Useful Signal Operators
• Time Shifting, Time Scaling, and Time inversion
• If T is positive, the shift is to the right (delayed)
• If T is negative, the shift is to the left (advanced)
Time Scaling
• Time Scaling: The compression or expansion of a signal in
time
• The signal (t ) in figure 2.7, g(t) is compressed in time by a
factor of 2
• Whatever happens in g(t) at some instant t will be happening at
the instant t/2
t
 ( )  g (t )
2
 (t )  g ( 2t )
• If g(t) is compressed in time by a factor a>1, the resulting
signal is:
 (t )  g (at)
• If expanded
t
a
 (t )  g ( )
Figure 2.7 Time scaling a signal.
Time Inversion
• A special case of time inversion where a=-1
 (t )  g (t )
 (t )  g (t )
• Whatever happens at some instant t also happens at the instant –t
• The mirror image of g(t) about the vertical axis is g(-t).
Time Inversion Example
• For the signal g(t) in (a) below, the sketch of g(-t) is
shown in (b)
Unit Impulse Signal
• Unit impulse function (Dirac delta)
• A unit Impulse
 (t )  0

  (t ) 1
t0
– Visualized as a tall, narrow rectangular pulse of unit area
– Width ε is very small, height is a large value 1/ε
– Unit impulse is represented with a spike.
Unit Impulse Signal
• Multiplication of unit impulse by a function  (t ) that is
continuous at t = 0
 (t ) (t )   (0) (t )
• Multiplication of a function  (t ) with an impulse  (t  T )
(an impulse located at t=T) ( (t ) must is defined at t=T)
 (t ) (t  T )   (T ) (t  T )




 (t ) (t  T )dt   (T )  (t  T )dt   (T )

• Area under the product of a function with an impulse
is equal to the value of that function at the instant where
 (t )
the unit impulse is located (Sampling or sifting
property)
Unit Step Function u(t)
A signal that starts after t=0 is called a causal signal. A
signal g(t) is causal if:
g(t) = 0 t<0
Figure 2.12 (a) Unit step function u(t). (b) Causal exponential e−atu(t).
Signals and Vectors
• Signal Representation
– As series of orthogonal functions (Fourier
series)
– Fourier series allows signal to be represented
as points in a generalized vector space
(signal space)
– Information can therefore be viewed in
geometrical context
Signal and Vectors
• Any vector A in 3 dimensional space can be
expressed as
A = A1a + A2b + A3c
– a, b, c are vectors that do not lie in the same plane and are
not collinear
– A1, A2, and A3 are linearly independent
– No one of the vectors can be expressed as a linear
combination of the other 2
– a, b, c is said to form a basis for a 3 dimensional vector
space
– To represent a time signal or function X(t) on a T interval
(t0 to t0+T) consider a set of time function independent of
x(t) 1 (t ),2 (t ),3 (t )............N (t )
Signal and Vectors
• X(t) can expanded as
N
xa (t )   xnn (t )
n 0
• N coefficients Xn are independent of time
and subscript xa is an approximation
Signals and Vectors
• Signal g can be written as N dimensional vector
g = [g(t1) g(t2) ………… g(tN)]
• Continuous time signals are straightforward
generalization of finite dimension vectors
lim g  g (t )
N 
t  [ a, b]
• In vector (dot or scalar), inner product of two realvalued vector g and x:
– <g,x> = ||g||.||x||cosθ θ – angle between vector g and x
– Length of a vector x:
||x||2 = <x.x>
29
Component of a Vector in terms of another vector.
• Vector g in Figure 1 can be expressed in terms of vector x
g = cx + e
g  cx
e = g - cx (error vector)
Figure 1
• Figure 2 shows infinite possibilities to express vector g in terms of
vector x
Figure 2
g = c 1 x + e 1 = c2 x + e 2
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Scalar or Dot Product of Two Vectors
•  is the angle between vectors g and x.
• The length of the component g along x is:
• Multiplying both sides by |x| yields:
• Where:
• Therefore:
• If g and x are Orthogonal (perpendicular):
• Vectors g and x are defined to be Orthogonal if the dot product of
the two vectors are zero.
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Components and Orthogonality of Signals
• Concepts of vector component and orthogonality can be extended to
CTS
• If signal g(t) is approximated by another signal x(t) as :
• The optimum value of c that minimizes the energy of the error signal
is:
c
• We define real signals g(t) and x(t) to be orthogonal over the interval
[t1, t2], if:
• We define complex signals* x1(t) and x2(t) to be orthogonal over the
interval [t1, t2]:
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Example
• For the square signal g(t) find the component in g(t)
of the form sin t. In order words, approximate g(t) in
terms of sin t so that the energy of the error signal is
minimum
•
Energy of The Sum of Orthogonal Signals
• If vectors x and y are orthogonal and z = x + y, then:
• If signals x(t) and y(t) are orthogonal over the interval [t1, t2]
and if z(t) = x(t) + y(t), the energy is:
=
0
Signal Comparison: Correlation Coefficient
• Two vectors g and x are similar if g has a large
component along x
• Correlation Coefficient for real signals:
• Correlation Coefficient for complex signals:
• The magnitude of the Correlation Coefficient is never
greater than unity (-1  Cn  1). If the two vectors are
equal then Cn = 1. If the two vectors are equal but in
opposite directions then Cn = -1. If the two vectors are
orthogonal then Cn = 0.
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Signal Comparison: Correlation Coefficient
• Use Eq. (2.48) to compute Cn.
Cn =
Example
Application of Correlation
• Signal processing in radar, sonar, digital
communication, electronic warfare etc
• In radar, if transmitted pulse is g(t), received radar
return signal is :
g (t  t0 )  w(t )
z (t )  
w(t )
Target present
Target absent
• Detection is possible if:

t2
t1
w(t ) g * (t  t0 )dt  0
t1  t 0  t 2

α is target reflection and attenuation loss,
t0 – round trip propagation delay,
w(t) –noises and interferences
 t2 | g (t  t ) |2 dt E
 t
p
0
z (t ) g * (t  t0 )dt   1

0
0
Target present
Target absent
Application of Correlation
• Digital Communication
– Detection of the presence of one or two known
waveform in the presence of noise
– Antipodal Scheme: Selecting one pulse to be the
negative of the other pulse where cn will be -1
– If noise is present in the received signal. Threshold
detector is used to detect signal.
– Large margins should be used to prevent detection
error.
– Antipodal has the highest performance in terms of
guarding against channel noise and pulse distortion
Correlation Functions
• Cross-correlation function of two complex signals
g(t) and z(t):




 zg ( )   z(t ) g * (t  )dt   z(t   ) g * (t )dt  0
• Autocorrelation Function
– Correlation of a signal with itself.
– Measures the similarity of the signal g(t) with its own
displaced version
– Autocorrelation function of a real signal is:
Orthogonal Signal Set
• Vector can be represented as a sum of orthogonal
vectors
• Orthogonality of a signal x1(t) x2(t) x3(t)…..xN(t) over
time domain [t1, t2]:
• If all signal energies En = 1, then the set is
normalized and called orthonormal set
• An orthogonal set can be normalized by dividing
xn(t) by En
• If orthogonality is complete:
N
eN  g (t )  [c1 x1 (t )  c2 x2 (t )......cN x N (t )]  g (t )   cn xn (t )
n 1
lim 
N 
t2
t1
| eN (t ) |2 dt  0
Orthogonal Signal Set

• A signal g(t) can be represented by g (t )   cn xn (t )
n 1
• This is called generalized Fourier series of g(t) with
respect to xn(t)
• Energy of the sum of orthogonal signals is equal to the
sum of their energies (sum of individual components)
• This is called Parseval’s theorem
Exponential Fourier Series
• Examples of orthogonal sets are trigonometric
(sinusoid) functions, exponential (sinusoid) functions
• The set of exponentials e jn t (n  0,1,2,.....) is orthogonal
over any interval of duration T0  2 / 
0
• A signal g(t) can be expressed over an interval of T0
seconds as an exponential Fourier series:
Exponential Fourier Series
• The compact trigonometric Fourier series of a
periodic signal g(t) is given by
• In exponential Fourier series where C0 = D0:
• Exponential Fourier series will be used in this course
because it is more compact, expression for
derivation is also compact
Find the exponential Fourier series for the signal
T0   ,2f 0  2 / T0  2
Exponential Fourier Spectra
• Coefficients Dn is plotted as a function of ω
• If Dn is complex, two plots are required: real and
imaginary parts of Dn or
• Plot of amplitude (magnitude) and angle of Dn
• To plot |Dn| versus ω and Dn versus ω and Dn must
jD
|
D
|
e
be expressed in polar form n
n
•