1
Digital Signal Processing
Applications of DSP
?
Speech processing
?
Enhancement
–
noise filtering
?
Coding
?
Text

to

speech (synthesis)
?
Recognition
?
Image processing
?
Enhancement, coding, pattern recognition (e.g. OCR)
?
Multimedia processing
?
Media transmission, digital TV, video conferencing
?
Communications
?
Biomedical engineering
?
Navigation, radar, GPS
?
Control, robotics, machine vision
What is a
signal?
A
signal
is defined as any physical quantity that varies with time, space
,
or any
other independent variable or variables. Mathematically, we
describe a signal as
a function of one or more independent variables. For
example
,
the functions
Describe
two signals
,
one that varies linearly with the indepen
dent
variable
t
(time)
and a second that varies quadratically with
t .
As another
example, consider the
This function describes a signal of two independent variables x and
y
that could
represent the two spatial coordin
ates in a plane.
The signals
described
above
belong to a class of signals that
are precisely defined by
specifying the functional dependence on the independent
variable.
However, there are cases where such a functional relationship is unknown
or too highly
complicated to be of any practical use.
2
For example, a speech signal (see Fig. 1.1) cannot be described
functionally
by expressions such as (1.1.1). In general, a segment of
speech may be represented
to a high degree of accuracy as a sum of
several sin
usoids of different amplitudes
and frequencies, that is, as
Where
(A
i
(t), F
i
(t), and (O
i
(t)) are the sets of (possibly time

varying)
amplitudes,
frequencies, and phases, respectively, of the sinusoids. In
fact, one way to interpret
the information co
ntent or message conveyed
by any short time segment of the speech signal is to measure the
amplitudes
,
frequencies, and phases contained in
the short time segment
of the signal.
Speech, electrocardiogram. and electroencephalogram signals are
examples
of in
formation

bearing signals that evolve as functions of a
single independent
variable.
Namely
. time. An example of a signal that is
a function of two independent
variables is an ima
g
e signal. The
independent variables in this case are
the spatial coordinates
. These are
but a few examples of the countless number of
natural signals
encountered in practice.
Associated with natural signals are the means by which such signals are
generated.
For
example
,
speech signals are generated by forcing air
through the vocal
cords. Images are obtained by exposing a photographic
film to a scene or an object.
Thus signal generation is usually associated
with a
system
that responds to a
stimulus or force. In speech signal
,
the
system consists of the vocal cords and
the vocal tra
ct, also called the
vocal cavity. The stimulus in combination with the
system is called a
signal source.
Thus,
we have speech sources, images sources
,
a
nd
various
other types of signal sources.
Figure 1.1 Example of a speech signal.
Mu
ltichannel and Multidimensional Signals
3
A
signal is described by a function of one or more
independent
variables. The value of the function (
i
.e.. the dependent va
ri
able) can
be a
real

valued scalar quantity, a complex

valued quantity. or perhaps a
vector
.
For example
,
the signal
is a real

valued signal. However, the signal
is complex valued.
In some applications, signals are generated by multiple sources or
multiple
sensors. Such signals. in turn. can be represented in vector form.
We refer to s
uch a vector of signals as a
miilrichannel signal.
In
electrocardiography
,
for example
,
3

lead and 12

lead electrocardiograms
(ECG) are often used in
practice
,
which result in 3

channel and 12

channel signals.
If the signal is
a function of a single inde
pendent variable. the
signal is called a
one

dimensional
signal. On the other hand. a signal is
called
M

dimensional
if its value is a function
of M independent
variables.
The picture is an example of a two

dimensional signal. Since
the
intensity or brigh
tness
I ( x
,
y
) at each point is a function of two
independent
variables. On the other hand. a black

and

white television
picture may be represented
as
I ( x ,
y
,
t
) since the brightness is a
function of time. Hence the TV
picture may be treated as a thr
ee

dimensional signal. In contrast. a color TV picture
may be described by
three intensity functions of the form
I
r
( x
,
y
,
t
).
I
g
(x
,
y
,
t
),
and
I
b ( x
,
y
,
t
) , corresponding to the brightness of the three principal colors
(red
,
green, blue) as functions
of time. Hence the color TV picture is a
three

channel.
three

dimensional signal, which can be represented by the vector
What is a System?
A
system
may also be defined as a physical device that performs an
operation
on a signal. For example, a filte
r used to reduce the noise and
interference
corrupting a desired information

bearing signal is called a
system. In this case the
filter performs some operation(s) on the signal,
4
which has the effect of reducing
(filtering) the noise and interference
from t
he desired information

bearing signal.
When we pass a signal through a system, as in filtering. we say that
we have
processed the signal. In this case the processing of the signal
involves filtering the
noise and interference from the desired signal. In
ge
neral, the system is characterized
by the type of operation that it
performs on the signal. For example
,
if
the operation is linear, the system
is called linear. If the operation on the signal
is nonlinear, the system is
said to be nonlinear, and so forth.
Such operations are
usually referred to
as
signal processing.
It is convenient to broaden the definition of a system to
include not
only
physical
devices
,
but also software realizations of operations on
a
signal. In digital processing of signals on a dig
ital computer
,
the
operations performed
on a signal consist of a number of mathematical
operations as specified by
a software
program
. In this case, the program
represents an implementation of the
system in
so
ftw
are
,
t
hus we have a
system that is realized
on a digital computer
by means of a sequence of
mathematical operations: that is, we have a
digital
signal processing
system realized in software. For example. a
digital
computer can
be
programmed to perform digital filtering. Alternatively, the
digital
pr
ocessing
on the signal map be performed by digital
hardware
(logic
circuits) configured to
perform the desired specified operations. In such a
realization, we have a physical
device that performs the specified
operations. In a broader sense, a digital syst
em
can be implemented as a
combination of digital hardware and software
,
each of
which performs its
own set of specified operations.
Types of signals
The independent variable may be either continuous
or
discrete
.
1

Continuous

time signals
Continuous

time
signals
or
analog signals
are defined for every
value of time and
they take on values in the continuous interval (a. b)
,
where
a
can be

∞
and
b
can be
∞
.
Mathematically
,
these
signals can be
described by functions of a continuous
variable. The speech waveform in
Fig. 1.1 and the signal
x
1
( t )
= cos
t are examples of analog signals.
(i) Periodic functions
(ii) Nonperiodic functions
(iii)
Random functions
PERIODIC FUNCTIONS
A signal v(t) is periodic with period T if
5
Four types of periodic functions which are specified for one period T and
corresponding graphs are
as follows:
(a)
Sine wave:
(b)
Periodic pulse:
(c)
Periodic tone burst:
where T = KA and K is an integer.
6
Periodic signals may be very complex. They may be
represented by a
sum of sinusoids. This type of function will be developed in the following
sections.
SINUSOIDAL FUNCTIONS
A sinusoidal
signal
v(t)
is given by
where V
0
is the amplitude,
is the angular velocity, or angular
frequency, and
is the phase angle.
The angular
velocity
may be expressed in terms of the period T
or the frequency f, where f
=
1
/
T.
The frequency is given in hertz, Hz, or cycles/s. S
ince cos
t
=
cos
(
t
+
2
)
,
and T are related by
T
=
2
. And since it takes T seconds for
v
(t)
to return to its original value, it goes through 1
/
T cycles
in one
second.
In summary, for sinusoidal functions we have
EXAMPLE: Graph each of the following
functions and specify period
and frequency.
7
8
TIME SHIFT AND PHASE SHIFT
If the function
v(t) =
cos
t is delayed by
seconds, we get
v(t

)
=
cos
(t

)
=
cos
(
t

)
where
=
.
The delay shifts the graph of
v(t)
to the right by an
amount of
seconds, which
corresponds to a phase lag
of
=
=2
.
A time
shift of
seconds to the left on the graph
produces
v(t
+
)
, resulting in a leading phase angle called an advance.
Conversely, a phase shift of
corresponds to a time shift of
. Therefore,
for a given phase shift the
higher is the frequency, the smaller is the
required time shift.
EXAMPLE: Plot
This is a cosine function with period of 12s, which is advanced in time by
1s. In other words, the graph is shifted to
the left
by 1 s or 30
as shown
below.
9
COMBINATIONS OF PERIODIC FUNCTIONS
The sum of two periodic functions with respective periods
T
1
and
T
2
is a periodic function if a
common period
T
=
n
1
T
1
=
n
2
T
2
, where
n
1
and
n
2
are integers, can be found. This requires
T
1
/
T
2
=
n
2
/
n
1
to be a
rational number. Otherwise, the sum is not a periodic function.
EXAMPLE: Find the period of
v(t) =
cos5t
+
3 sin
(3
t
+
45
).
The period of cos5t is T1
=
2
/5
and
t
he period of
3sin
(3
t
+
45
)
,
is T2
=
2
/
3. Take T
=
2
=
5T1
=
3T2
which i
s the smallest common integral
multiple of T1 and T2. Observe that
v(t+
T
) =
v
(t)
since
Therefore, the period of v
(t)
is 2
.
QUESTION1
:
Is v
(t) =
cost
+
cos2
t periodic? Discuss.
QUESTION2:
Given p
=
3
.
14, find the period of
V(t) =
cos t
+
cos2pt.
NONPERIODIC FUNCTIONS
A nonperiodic function cannot be specified for al
l times by simply
knowing a finite segment. Example
s
of nonperiodic functions are
10
Several of these functions are used as mathematical models and building
blocks for actual signals in
analysis and design of
sytem
s. Examples are
discussed in the follow
ing sections.
THE UNIT STEP FUNCTION
The dimensionless
unit step function
, is defined by
THE UNIT IMPULSE FUNCTION
Consider the function s
T
(t)
of Fig.
below
(a), which is zero for t < 0 and
increases uniformly from 0 to
1 in T sec
onds. Its derivative d
T
(t)
is a
pulse of duration T and height 1
/
T, as seen in Fig. (b).
11
If the transition time T is reduced, the pulse in Fig. (b) becomes narrower
and taller, but the
area under the pulse remains equal to 1. If we let T
approach zero
, in the limit function s
T
(t) be
come
a unit step u
(t)
and its
derivative d
T
(t)
becomes a unit pulse
(t)
with zero width and infinite
height. The unit impulse
(t)
is shown in Fig. (c). The unit impulse or
unit delta function is defined
by
12
THE E
XPONENTIAL FUNCTION
The function f(t) = e
st
with
s
a complex constant is called exponential. It
decays with time if the
real part of s is negative and grows if the real part
of
s
is positive. We will discuss exponential
s
a
t
in
which the constant
a
is
a rea
l number.
The inverse of the constant
a
has the dimension of time and is
called the time constant
=
1/
a. A
decaying exponential
e

t/
is plotted
versus t as shown
below
. The function decays from one at
t
=
0 to zero at
t
=
. After
seconds the function
e

t/
is reduced to e

1
=
0
.
368. For
=
1,
the
function e

t
is called a normalized exponential
.
DAMPED SINUSOIDS
A damped sinusoid, with its amplitude decaying exponentially has the
form
13
RANDOM SIGNALS
So far we have dealt with signals which are
completely specified. For
example, the values of a
sinusoidal waveform, such as the line voltage,
can be determined for all times if its amplitude, frequency,
and phase are
known. Such signals are called deterministic.
There exists another class of signals which can be specified only partly
through their time averages,
such as their mean, rms value, and frequency
range. These are called random signals. Random signals
can carry
information and should not be mistaken with
noise, which normally
corrupts the information
contents of the signal.
EXAMPLE
:
Samples from a random signal
x(t)
are recorded every 1 ms
and designated by x
(n)
. Approximate
the mean and rms values of x
(t)
from samples given in Table
below
.
14
EXAM
PLE : A binary signal v
(t)
is either at 0.5 or

0
.
5 V. It can change
its sign at 1

ms intervals. The sign
change is not known a priori, but it has
an equal chance for positive or negative values. Therefore, if measured
for a
long time, it spends an equal a
mount of time at the 0.5

V and

0
.
5

V
levels. Determine its average and effective
values over a period of 10 s.
During the 10

s period, there are 10,000 intervals, each of 1

ms duration,
which on average are equally divided
between the 0.5

V and

0
.
5

V
lev
els. Therefore, the average of v
(t)
can be approximated as
The effective value of v
(t)
is
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