SIGNAL THEORY
Lecture Notes
This book is intended as supplementary material for the lecture
Signal Theory
given by dr Tomasz Grajek
for students of Electronics and Telecommunications
at Poznan University of Technology.
Note: This book may contain copyrighted material.
Any copying, distribution, and non-authorised usage is strictly
prohibited.
In particular, electronic distribution over Internet is prohibited
1. Fundamental Concepts and Measures
1.1.
Signals and Their Models
In engineering, a signal is any observable and measurable quantity that is changing in a stochastic or deterministic way, and usually describes a parameter(s) of an object(s), or a state of
a process. In Electronics and Telecommunications, a signal is usually considered to be a (possibly time-varying) measure of an electric current; however, it may also refer to a non-electric
quantity that is changing with respect to some independent variable.
A model is a descriptive idealized representation of an existing phenomenon. In engineering, mathematical models are usually employed that describe the essence of a process or
objects by means of analytical expressions. In fact, we usually refer to the model as the signal
itself. In signal theory, as employed within electrical and electronic engineering, the actual
nature of the phenomenon described by the signal model is often irrelevant; e.g., f (t ) may
denote either electrical current or voltage. Please observe that, assuming a reference impedance of R  1 , both quantities are described by the same value (disregarding the units).
u(t)
R=1
i(t)
R=1
f (t)=u(t)=i(t) (equal in value)
A typical model for a deterministic signal is a function, i.e., a mapping from the time
(or space) domain to the domain of deterministic values,
f : t  f  f (t ),
(1.1)
where t is an independent real variable, and f is a dependent variable from the set of real or
complex numbers.
Non-deterministic signals cannot be expressed in an analytical form. Random signals
are often natural manifestations of (at least partially) unpredictable events which are not analytically, though statistically or stochastically describable. A very common model for a nondeterministic signal is a stochastic process of certain distribution X that is characterized by its
(possibly time-varying) parameters,
f : t  f  X (t ), (t ), (t ),... .
(1.2)
In this course, both t and f are scalar quantities, however, in general, they may be multidimensional (or vector). Multidimensional signal processing is a branch of signal processing that
handles these kinds of signals and has many applications, such as image and video processing,
spatial acoustics, seismography, meteorology, ocean science, etc.
1.2.
Signal Classes and Examples
1.2.1. Continuous, Discrete, Analogue, Quantized and Digital Signals
Continuous signals (pictured below, left) are defined over a continuous domain, e.g., f (t )
has a value in every instant of time in a finite or infinite range. The function does not have to
be strictly continuous, i.e., it may have a countable number of discontinuities.
Discrete time signals (also known as discrete signals, pictured below, middle) are
signals defined over a discrete domain (a countable set of arguments). Discrete signals may be
considered as consisting of samples from continuous signals. Sampling is in general an irreversible process of taking the instantaneous value of a continuous signal at specific time instances called sample positions. A sampled signal may be converted to a continuous signal by
using one of the many forms of interpolation. Interpolation determines all intermediate signals
between the known sample values based on one of many interpolating formulas. These interpolated values are in general not the values of the original continuous signal.
Quantized signals (also known as discrete value, pictured below, right) are signals
defined in a countable set of values, so that only values from a countable set are allowed for
any signal argument. Obviously, quantized signals change their value in a discontinuous way.
The quantization of a signal that is continuous in value requires the use of a device called a
quantizer. The process is not reversible, as quantization introduces an error (the difference
between the original and quantized value) that cannot be removed.
f (t)
f (tn)
t
fq (t)
tn
t
As opposed to analog signals that are continuous in time and value, digital signals are
signals that are both discrete time and discrete value. Only digital signals may be represented
by a sequence of numbers with limited numerical representation (a finite number of bits per
sample, a finite number of samples per second), stored in digital memory, and processed by
digital computers. Therefore, the application of any digital signal processing (DSP) system
to process analog signals requires an analog-to-digital converter (ADC) at the front end, and
usually requires a digital-to-analog converter (DAC) at the end of the chain. The purpose of
ADC is sampling and quantization, while the purpose of DAC is reconstruction and interpolation.
1.2.2. Periodic Signals
Periodic signals are an important class of signals that satisfy the condition

0 T  

t R
f (t  T )  f (t ) .
(1.3)
For a periodic signal, the smallest value of T for which the above is true, is called the fundamental period. Fundamental frequency f0, and fundamental angular frequency ω0, are defined
respectively by
f0 
1
2
, 0 
 2  f0 .
T
T
(1.4)
Phase is the measure of progression of a periodic signal w.r.t. the origin (the starting point of
a cycle). Phase is usually expressed in degrees or radians and relates to a circular motion, particularly described by sinusoidal signals (an important subclass of periodic signals). Angular
frequency is thus a measure of the rate of phase increase.
The initial phase 0 of a signal represents the angular shift w.r.t. the origin of the
function. For example, f (t )  sin 0t  0  has an initial phase 0 . The shift of the origin may
be determined in the time domain by solving
0t0  0  0  t0  
0
.
0
(1.5)
f (t)
t0
0
t
The fundamental period (and frequency) of a periodic signal does not change with
linear operations (such as multiplying by a constant, shifting in time, differentiating and integrating). Nonlinear operations may result in a change of the period, especially if the signal
exhibits some symmetries within a cycle.
Example:
The signal |f(t)| is a result of a nonlinear operation y = |x| performed on f(t). The original period of f(t) is significantly modified
f (t)
| f (t) |
t
t
T| f |
Tf
□
A linear combination of two periodic signals may be a periodic or non-periodic signal,
depending on the ratio of their periods, T1:T2, being a rational number, or not. In the former
case, the resulting period is
T
lcm (T1 , T2 )
,
N
(1.6)
where lcm( ) denotes the least common multiple, and N is some positive integer.
Example 1:
The sum of f1 (t )  sin(3t ) and f 2  sin( 4t ) is a periodic signal, because T1 : T2  23 : 24  4 : 3
is a rational number. The resulting period is T  2 , as it can be observed in the plot.
f1 (t)+f2 (t)





T
t
□
Example 2:
The sum of f(t) and g(t) is a periodic signal. Its period is shorter, because certain components
of f(t) and g(t) are cancelled in the sum, and these components yield a longer period of f(t) and
g(t).
f (t )
-A
Tf
t
Tg
t
g(t)
-A
f (t)+g(t)
-A
Tf+g
t
□
1.2.3. Sinusoidal Signals: Real and Complex
Sinusoidal signals (sometimes called harmonic signals) are periodic signals having the form
f (t )  A cos 0t  0  .
(1.7)
Depending on the value of 0 (the initial phase), a sinusoidal signal may be equivalent to the
sine function, or cosine function, or any other intermediate function of the same shape. A very
important property of sinusoidal signals is that any weighted sum of such signals having the
same angular frequency is also a sinusoidal signal of the same angular frequency, i.e.
A1 cos 0t  1   A 2 cos 0t  2   A 0 cos 0t  0  ,
(1.8)
where A0 and 0 may be determined from
A0 
A cos( )  A cos( )  A sin ( )  A sin ( )
2
1
1
2
 0  arctan
2
1
1
A1 sin (1 )  A 2 sin (2 )
A1 cos(1 )  A 2 cos(2 )
2
.
2
2
(1.9)
(1.10)
f1 (t)+f2 (t)
t
Complex exponential signals (also known as complex sinusoidal signals) have the form
f (t )  A e j ( 0 t   0 )  A  cos 0t  0   j sin 0t  0   ,
(1.11)
where j is the imaginary unit, j   1 . This exponential form is based on the Euler’s identity
e j x  cos( x)  j sin( x) which may also be expressed using a pair of equivalent identities,
cos ( x) 
e jx  e  jx
e jx  e  jx
.
, and sin ( x) 
2
2j
(1.12)
Observe that the properties of a complex exponential function (1.11) are very different from
the properties of a real exponential function. First of all, a complex exponential is periodic,
and neither does it approach zero, nor infinity, unlike a real exponential. Secondly, the
absolute value (the modulus) of a complex exponential is constant for all arguments,
e j x  cos( x)  j sin( x)  cos 2 ( x)  sin 2 ( x)  1 .
(1.13)
The complex exponential function is sometimes called a phasor, because for a varying
argument, the value represents a vector (rotating in the complex plane) that draws a perfect
circle. The plot of a complex sinusoidal signal is a spiral curve. Its projection on the real axis
is a cosine function, and the projection on the imaginary axis is a sine.
Re{ Ae jt }
Im
A
A
Ae jt
 t
t
Re
Im{ Ae jt }
1.2.4. Non-Periodic Signals
Non-periodic signals are obviously those which do not satisfy the condition of periodicity
(1.3). A common example of a non-periodic signal is the step function 1(t) (also known as the
Haeviside function) which is a model of a rapid (or instantaneous) change of signal value,
t0
0

1 (t )  1 / 2 t  0
1
t0

(1.14)
1(t)
1
1/2
t
Note that this signal exhibits one discontinuity, however, its value at each t0 is always a mean
of the left and right limits at t t0.
For the sake of picture clarity, all discontinuities in this book will be shown as vertical lines.
An important subclass of non-periodic signals may be distinguished, called impulse
signals. Impulse signals satisfy the condition
lim 

/ 2
f (t ) d t   .
 / 2
(1.15)
In other words, for impulse signals, the total area under the curve of f(t) is limited.
A common example of an impulse signal is the rectangular impulse, (t) (also known
as the rectangle gate, or gate) defined by
| t | 1 / 2
1

 (t )  1 / 2 | t | 1 / 2
0
| t | 1 / 2

(1.16)
(t)
1
-1/2
1/2
t
Other common impulse signal is the triangular impulse, (t) (also known as the triangle gate) defined by
1 | t | | t | 1
(t )  
| t | 1
 0
1
(1.17)
(t)
-1
1
t
Note that the support of (t) is twice as wide as the support of (t).
It is a common source of confusions and mistakes in all kinds of calculations.
Yet another example of an impulse signal is the exponential impulse,
 0 t0

f exp (t )  1 / 2 t  0
 et t  0

(1.18)
fexp(t)
1
t
Note, that the exponential impulse is asymptotically decreasing towards zero, and, despite
having an infinite support, it still satisfies the condition for an impulse signal.
A very important impulse signal in signal theory is Sa(t) defined as
 sin(t )
Sa (t )   t
 1
t0
t0
.
(1.19)
A popular alternative name for Sa(t) is sinc(t), however, the latter name is inconsistently used
in the literature to denote both Sa(t) and Sa(t). Therefore, the unambiguous name of Sa(t)
will be used in this book.
Sa(t)
1

 



t
Note that Sa(t) is a strictly continuous function. The value of Sa(t) at t=0 may be computed by
the use of the limit calculus,
sin (t ) L ' H
cos(t )
 lim
 1.
t 0
t 0
t
1
lim
(1.20)
Another very important impulse signal is the Dirac impulse, (t), which is an example
of a generalized function, or distribution. The Dirac impulse has peculiar properties – it is an
infinitely narrow impulse, while the total integral of (t) is 1. Certainly, this implies that the
instantaneous value of (t) at t=0 is infinite, which is signaled by an arrow in the figure.
(t)
t
The Dirac impulse may be approximated by a limit of a parameterized impulse of an arbitrary
shape, the width and height of which are inversely scaled with a total area of 1 being maintained, e.g.,
1 t
(t )  lim    ,
0 

(1.21)
1 
,
 t 2  2
(1.22)
1
t
Sa   .
  0 

(1.23)
(t )  lim
 0
(t )  lim
A very important property of the Dirac impulse is called the sampling property. An integral
of any signal f (t ) multiplied by (t - t0) (the Dirac impulse shifted in time to the position of
t = t0) yields the value of f (t ) sampled at t = t0,




f (t ) (t  t0 ) d t  f (t0 )  (t  t0 ) d t  f (t0 ) .

f (t)
f (t 0)
(1.24)
f (t 0) (t-t0 )
(t-t0 )
t0
t
Observe that the product of f (t ) with a Dirac impulse is still a Dirac impulse, however the
area under this impulse is modified by the exact value of f (t0 ) .
1.3.
Basic Signal Metrics
1.3.1. Amplitude
In physics, the amplitude of a wave is the maximum departure from the neutral state. In signal
theory, the neutral state is always zero, hence the amplitude of a signal is a non-negative value
unambiguously defined as
A  max f (t ) .
(1.25)
t
Example:
The amplitude of the signal shown is 3.
1
f ( t)
t
-3
□
Many signal processing operations may modify signal amplitude. Also, the amplitude of
a result of simple arithmetic may be difficult to predict. Multiplying a signal by a scalar constant value results in a signal the amplitude of which is scaled accordingly. However, a sum of
two signals with amplitudes A1 and A2, respectively, usually does not have an amplitude equal
to A1+A2.
1.3.2. The Mean Value
The mean value of a current is equal to the value of a constant current that in the entire time
span brings the same electrical charge. The charge may be computed by integrating the current over time. The equivalent value of a constant current is achieved by dividing the charge
by the length of the integrating segment. Following this reasoning, the mean value of an abstract signal f (t ) is the integral over a time span divided by the length of the time span.
Computing the mean value for periodic signals is quite straightforward, taking into
account that they pass on exactly the same charge in each period,
f 
1
f (t ) d t .
T T
(1.26)
The above integral may be evaluated over an arbitrarily selected segment of time, as long as it
spans exactly one period of the periodic signal. The mean value may be either positive or negative.
Computing the mean value of non-periodic signals is more cumbersome, since it requires calculating the integral over the entire time span, which may result in an infinite value,
and the denominator is also infinite. Therefore the computation requires evaluating a limit
ratio,
1 /2
f (t ) d t .
     / 2
f  lim
(1.27)
Note that the above expression approximates the mean value of a non-periodic signal, as the
range of integration approaches both positive and negative infinity. The result is a non-zero
value only for signals that have an infinite area under the curve, therefore, the mean value for
most impulse signals is zero.
Example 1:
Compute the mean value of the periodic sawtooth signal shown in the figure below
A
...
-T
f (t)
...
T
t
Since we integrate over one period only, it’s beneficial to select the range of integration where
the definition of f (t ) is simplest. Note that the signal is defined by a linear function with a
zero constant term, f (t ) 
A
t for t  0T . Hence,
T
T
A
1 TA
A t2
A
(T 2  0)  .

f  
t dt  2
2
0
2
T 2 0 2T
T
T
□
Example 2:
Compute the mean value of an exponential impulse f (t )  A 1(t ) e  a t shown in the figure
below
f (t)
using the formula for the mean of a non-periodic signal,
/2
1 /2
A  / 2 a t
A e a t
A 0 1
e

lim
 lim
 0. □
f  lim 
A 1(t ) e  a t d t  lim
d
t

 / 2
0
 
 
   a
   a
0
1.3.3. Energy of Signal
Total energy of a signal (or in short - signal energy) is defined by
E  lim 

/2
 / 2
f (t )
2
dt .
(1.28)
Note that the above expression approximates the signal energy of f (t ) as the range of integration approaches both positive and negative infinity. The symbol of an absolute value is
2
particularly important for signals that are complex-valued (note that z  z z *  z 2 for complex z).
Example :
Compute the energy of the triangle impulse shown in the figure,
f (t)
A
-t0
E  lim 

t
t0
2
/ 2
 / 2
f (t )
2

 t 
d t    A    d t .
t0
 t0  

t0
Note, that the triangle impulse is symmetrical, hence the computation may be simplified by
integrating over one half of its time span and multiplying the result by 2. Furthermore, the left
slope of the triangle, when squared will result in a segment of a parabola. The integral may be
further simplified by shifting the parabola (which does not affect the area under the curve),
2
t0
 t
2 A2t 0
2 A2 t 3

.
E  2   A  d t  2
0
t0 3 0
3
 t0 
t0
□
For every signal, its energy is always a non-negative value and, depending on the particular signal, may be either finite or infinite. The energy of many finite support signals is
finite, with the exception of distributions, e.g., the Dirac impulse has an infinite energy. The
energy of most non-trivial periodic signals is infinite.
1.3.4. Power of Signal
For signals with an infinite energy, a more convenient and meaningful measure is the average
power, defined by
1 /2
2
f (t ) d t .
     / 2
P  lim
(1.29)
The above expression approximates the average power value of the signal f (t ) as the range
of integration approaches both positive and negative infinity. Note that for signals of a finite
energy, the average power is always zero.
For periodic signals, the computation of the average power may be greatly simplified,
based on the observation that the energy over a single cycle is always the same, thus
P
1
2
f (t ) d t .

T T
(1.30)
The above integral may be evaluated over an arbitrarily selected segment of time, as long as it
spans exactly one period of the periodic signal.
Example :
Compute the average power of a sinusoidal signal, f (t )  A cos (0 t ) . Since it is a periodic
signal, we are using a simplified formula,
Pcos 
A20 2  1 1
0 2

2
A
t
d
t

 cos (2 0 t )  d t 



cos
(
)
0



2 0
2 0  2 2

0
0
2
 A2 A20
A20  20
A2

0
 sin (4  )  sin (0)  .

d t   sin (2 0 t ) d t  


0
4   0
4
2
 2
Observe that the resulting value depends on the amplitude of the cosine, and is completely
independent of the frequency, or period.
□
1.3.5. Effective Value of Signal (RMS)
The effective value of an electrical current is the equivalent value of a constant current that
yields the same energetic effect, i.e., produces the same power. Since the power of a constant
current is related to its squared value, the effective value may be calculated as the square root
of average power, often denoted RMS (root of the mean squared). This quantity usually applies to periodic signals, and may be computed according to
U
1
2
f (t ) d t .

T
T
(1.31)
It is important to stress that in general, the effective value of a sum of two signals is not equal
to the sum of their respective effective values. This property follows from (a  b) 2  a 2  b 2 .
Example:
The effective value of any sinusoidal signal is U cos 
A 2
, where A is the amplitude. □
2
1.4.
Energy Signals vs Power Signals
All signals may be classified based on the combination of their energy and average power into
4 categories, as shown in the table:
Class
I
II
III
IV
Energy
E=0
0<E<
E=
E=
Power
P=0
P=0
0<P<
P=
Signals of class I are trivial, having only zero values, with a possible exception of at most
countable number of points, where their value is non-zero and bounded. Signals of class II are
called energy signals. Signals of class III are called power signals. Signals of class IV do not
exist in real world, and their theoretical significance is limited to distribution calculus.
In most cases, non-trivial periodic signals fall into the category of power signals, while
non-periodic signals may be either energy or power signals.
1.5.
Orthogonal Signals
An inner product of two signals, f (t ) and g (t ) over an interval T is defined as
f , g L2   f (t ) g * (t ) d t ,
T
T
(1.32)
where * denotes a complex conjugate. Two signals are called orthogonal if their inner
product is zero. The concept of orthogonality is very fundamental in signal theory, and many
theorems are built upon it.
For orthogonal signals, the energy (in the case of energy signals) or power (in the case
of power signals) of their sum or difference is respectively equal to the sum of their energies,
or powers. Consider calculating the power of a sum of two real periodic signals having a
common period T:
1
1
2

f (t )  g (t )  d t   f 2 (t )  2 f (t ) g (t )  g 2 (t ) d t 

T T
T T
1
2
1
2
  f 2 (t ) d t   f (t ) g (t ) d t   g 2 (t ) d t  Pf 
f , g L2  Pg .
T
T T
T T
T T
T
Pf  g 
Hence, the power of the sum will be equal to the sum of powers if and only if the signals are
orthogonal.
Example 1:
Energy signals f (t ) and g (t ) are not orthogonal, while f (t ) and h (t ) are orthogonal.
3
h(t)
g(t)
f (t)
2
1 2
t
2
1
-1
-3
2
t
1
2
1
2
0
1
1
0
t
E f   32 d t   32 d t  18, E g   2 2 d t  8, Eh   2 2 d t  8
1
f , g    3 2 d t  6
0
1
2
0
1
f , h    3 2 d t    3 2 d t  0
f (t)+g(t)
f (t)+h(t)
5
5
2
1 2
-1
1 2
-1
t
t
-3
1
1
2
1
E f  g   2 2 d t   52 d t   (3) 2 d t  38  E f  E g ,
1
0
1
2
0
1
E f  h   52 d t   (1) 2 d t  26  E f  Eh .
□
Example 2:
Consider f (t )  sin (n t ) , g (t )  sin (k t ) , assume n and k to be integer nonzero numbers,
n  k . Check, if f (t ) and g (t ) are orthogonal over the range t  0...2  :
1 2
1 2
cos (n  k ) t  d t   cos (n  k ) t  d t 

0
2 0
2 0
1
 sin (n  k ) 2   sin(0)  1  sin (n  k ) 2   sin(0)  0 .

2 (n  k )
2 (n  k )
2
f , g 2    sin(n t ) sin(k t ) d t 
Hence, these signals are orthogonal. A very similar result is obtained while testing a pair of
cosinusoidal signals, as well as a sine against a cosine.
1.6.
Signal Components
1.6.1 DC and AC Signal Components
Every signal may be decomposed into its DC components and an AC component. The DC
component is, by definition, equal to the mean value of the signal, and is constant. The AC
component is the difference between the original signal and its DC component,
~
f (t )  f (t )  f .
(1.33)
DC and AC components are linearly independent. The DC and AC components of a
sum (or difference) of two (or more) signals are equal to the respective sum (or difference) of
the DC and AC components of these signals.
DC and AC components of any signal are always orthogonal. The energy (or power)
of the signal is equal to the sum of the energies (or powers) of its DC and AC components.
1.6.2 Odd and Even Signal Components
Every signal may be decomposed into its odd and even components,
1
 f (t )  f (t )  ,
2
1
f odd (t )   f (t )  f (t )  .
2
f even (t ) 
(1.34)
An even component exhibits an even symmetry, i.e., f even (t )  f even (t ) , while an odd component exhibits an odd symmetry, i.e., f odd (t )   f odd (t ) . Even and odd components of any
signal are always orthogonal over an interval that is symmetric w.r.t. zero.
Example 1:
For a ramp impulse f (t ) shown below, the even and odd components are also shown.
f (t )
fodd(t)
1
1/2
feven (t)
-1
1/2
1
t
-1
1
1
t
-1/2
t
□
Example 2:
For a complex exponential signal f (t )  e j0 t , we get


1 j 0 t  j 0 t
e e
 cos (0 t ) ,
2
1
f odd (t )  e j0 t  e  j0 t  j sin (0 t ) .
2
f even (t ) 


□