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West Bengal University of technology
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1.1 What is a Signal
We are all immersed in a sea of signals. All of us from the smallest living
unit, a cell, to the most complex living organism(humans) are all time time
receiving signals and are processing them. Survival of any living organism
depends upon processing the signals appropriately. What is signal? To deﬁne
this precisely is a diﬃcult task. Anything which carries information is a signal. In this course we will learn some of the mathematical representations of
the signals, which has been found very useful in making information processing systems. Examples of signals are human voice, chirping of birds, smoke
signals, gestures (sign language), fragrances of the ﬂowers. Many of our body
functions are regulated by chemical signals, blind people use sense of touch.
Bees communicate by their dancing pattern.Some examples of modern high
speed signals are the voltage charger in a telephone wire, the electromagnetic
ﬁeld emanating from a transmitting antenna,variation of light intensity in an
optical ﬁber. Thus we see that there is an almost endless variety of signals
and a large number of ways in which signals are carried from on place to
another place.
In this course we will adopt the following deﬁnition for the signal: A signal
is a real (or complex) valued function of one or more real variable(s).
When the function depends on a single variable, the signal is said to be onedimensional. A speech signal, daily maximum temperature, annual rainfall
at a place, are all examples of a one dimensional signal.When the function
depends on two or more variables, the signal is said to be multidimensional.
An image is representing the two dimensional signal,vertical and horizontal coordinates representing the two dimensions. Our physical world is four
dimensional(three spatial and one temporal).
1.2 What is signal processing
By processing we mean operating in some fashion on a signal to extract some
useful information. For example when we hear same thing we use our ears
and auditory path ways in the brain to extract the information. The signal
is processed by a system. In the example mentioned above the system is
biological in nature. We can use an electronic system to try to mimic this
behavior. The signal processor may be an electronic system, a mechanical
system or even it might be a computer program.
The word digital in digital signal processing means that the processing is
done either by a digital hardware or by a digital computer.
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1.3 Analog versus digital signal processing
The signal processing operations involved in many applications like communication systems, control systems, instrumentation, biomedical signal processing etc can be implemented in two diﬀerent ways
(1) Analog or continuous time method and
(2) Digital or discrete time method.
The analog approach to signal processing was dominant for many years. The
analog signal processing uses analog circuit elements such as resistors, capacitors, transistors, diodes etc. With the advent of digital computer and
later microprocessor, the digital signal processing has become dominant now
a days.
The analog signal processing is based on natural ability of the analog system
to solve diﬀerential equations the describe a physical system. The solution
are obtained in real time. In contrast digital signal processing relies on numerical calculations. The method may or may not give results in real time.
The digital approach has two main advantages over analog approach
(1) Flexibility: Same hardware can be used to do various kind of signal processing operation,while in the core of analog signal processing one has to
design a system for each kind of operation.
(2) Repeatability: The same signal processing operation can be repeated
again and again giving same results, while in analog systems there may be
parameter variation due to change in temperature or supply voltage.
The choice between analog or digital signal processing depends on application. One has to compare design time,size and cost of the implementation.
1.4 Classiﬁcation of signals
As mentioned earlier, we will use the term signal to mean a real or complex
valued function of real variable(s). Let us denote the signal by x(t). The
variable t is called independent variable and the value x of t as dependent
variable. We say a signal is continuous time signal if the independent variable
t takes values in an interval.
For example
t (−∞, ∞), or t [0, ∞] or t [T0 , T1 ]
The independent variable t is referred to as time,even though it may not be
actually time. For example in variation if pressure with height t refers above
mean sea level.
When t takes a vales in a countable set the signal is called a discrete time
signal. For example
t {0, T, 2T, 3T, 4T, ...} or t {... − 1, 0, 1, ...} or t {1/2, 3/2, 5/2, 7/2, ...} etc.
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For convenience of presentation we use the notation x[n] to denote discrete
time signal.
Let us pause here and clarify the notation a bit. When we write x(t) it has
two meanings. One is value of x at time t and the other is the pairs(x(t), t)
allowable value of t. By signal we mean the second interpretation. To keep
this distinction we will use the following notation: {x(t)} to denote the continuous time signal. Here {x(t)} is short notation for {x(t), tI} where I is
the set in which t takes the value. Similarly for discrete time signal we will
use the notation {x[n]}, where {x[n]} is short for {x[n], nI}. Note that in
{x(t)} and {x[n]} are dummy variables ie. {x[n]} and {x[t]} refer to the
same signal. Some books use the notation x[·] to denote {x[n]} and x[n] to
denote value of x at time n · x[n] refers to the whole waveform,while x[n]
refers to a particular value. Most of the books do not make this distinction
clean and use x[n] to denote signal and x[n0 ] to denote a particular value.
As with independent variable t, the dependent variable x can take values in
a continues set or in a countable set. When both the dependent and independent variable take value in intervals, the signal is called an analog signal.
When both the dependent and independent variables take values in countable
sets(two sets can be quite diﬀerent) the signal is called Digital signal.When
we use digital computers to do processing we are doing digital signal processing. But most of the theory is for discrete time signal processing where
default variable is continuous. This is because of the mathematical simplicity of discrete time signal processing. Also digital signal processing tries to
implement this as closely as possible. Thus what we study is mostly discrete
time signal processing and what is really implemented is digital signal processing.
Exercise:
1.GIve examples of continues time signals.
2.Give examples of discrete time signals.
3.Give examples of signal where the independent variable is not time(onedimensional).
4.Given examples of signal where we have one independent variable but dependent variable has more than one dimension.(This is sometimes called
vector valued signal or multichannel signal).
5.Give examples of signals where dependent variable is discrete but independent variable are continues.
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1.5 Elementary signals
There are several elementary signals that feature prominently in the study
of digital signals and digital signal processing.
(a)Unit sample sequence δ[n]: Unit sample sequence is deﬁned by
1, n = 0
δ[n] =
0, n = 0
Graphically this is as shown below.
δ[n]
1
-3
-2
0
-1
1
2
3
n
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Unit sample sequence is also known as impulse sequence. This plays role akin
to the impulse function δ(t) of continues time. The continues time impulse
δ(t) is purely a mathematical construct while in discrete time we can actually
generate the impulse sequence.
(b)Unit step sequence u[n]: Unit step sequence is deﬁned by
1, n ≥ 0
u[n] =
0, n < 0
Graphically this is as shown below
u[n]
1
.................
..........
-3
-2
-1
0
1
2
3
4
n
................
(c) Exponential sequence: The complex exponential signal or sequence x[n]
is deﬁned by
x[n] = C αn
where C and α are, in general, complex numbers. Note that by writing
α = eβ , we can write the exponential sequence as x[n] = c eβ n .
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