2.1 Simple operations on signals
In analyzing discrete-time systems, operations on sequences occur frequently.
Some operations are discussed below.
2.1.1 Sequence addition:
Let {x[n]} and {y[n]} be two sequences. The sequence addition is defined as
term by term addition. Let {z[n]} be the resulting sequence
{z[n]} = {x[n]} + {y[n]} where each term z[n] = x[n] + y[n]
We will use the following notation
{x[n]} + {y[n]} = {x[n] + y[n]}
2.1.2 Scalar multiplication:
Let a be a scalar. We will take a to be real if we consider only the real valued
signals, and take a to be a complex number if we are considering complex
valued sequence. Unless otherwise stated we will consider complex valued
sequences. Let the resulting sequence be denoted by w[n]
{w[n]} = ax[n]
is defined by w[n] = ax[n],
each term is multiplied by a
We will use the notation aw[n] = aw[n]
Note: If we take the set of sequences and define these two operators as
addition and scalar multiplication they satisfy all the properties of a linear
vector space.
2.1.3 Sequence multiplication:
Let {x[n]} and {y[n]} be two sequences, and {z[n]} be resulting sequence
{z[n]} = {x[n]}{y[n]}
1
where z[n] = x[n]y[n]
The notation used for this will be {x[n]}{y[n]} = {x[n]y[n]} Now we consider some operations based on independent variable n.
2.1.4 Shifting
This is also known as translation. Let us shift a sequence {x[n]} by n0 units,
and the resulting sequence by {y[n]}
{y[n]} = z−n0({x[n]})
where z−n0()is the operation of shifting the sequence right by n0 unit. The
terms are defined by y[n] = x[n−n)]. We will use short notation {x[n−n0]} to denote shift by n0.
{x[n]}
1
n
.....
-3 -2 -1 0 1 2 3
{x[n − 2]}
n
.....
{x[n + 1]}
n
.....
-1 0 1 2 3
Figure above show some examples of shifting. A negative value of n0 means
shift towards right.
2
2.1.5 Reflection:
Let {x[n]} be the original sequence, and {y[n]} be reflected sequence, then
y[n] is defined by
y[n] = x[−n]
{x[n]}
n
.....
-3 -2 -1 0 1 2 3
-3 -2 -1 0 1 2 3
.....
.....
.....
1
2
3
4
1
2
3
4
We will denote this by {x[n]}.
When we have complex valued signals, sometimes we reflect and do the complex
conjugation, ie, y[n] is defined by y[n] = x∗[−n], where * denotes complex
conjugation. This sequence will be denoted by {x∗[−n]}.
We will learn about more complex operations later on. Some of these operations
commute, ie. if we apply two operations we can interchange their order
and some do not commute. For example scalar multiplication and reflection
commute.
{x[n]} Scalar multiplication by a{y[n]} reflection {z[n]}
{x[n]} reflection {w[n]} Scalar multiplication {v[n]}
3
Then v[n] = z[n] for all n. Shifting and scaling do not commute.
{x[n]} {y[n]} − {x[n − 1]} {z[n]} = {y[−n]}
{x[n]} shift by n0 {y[n]} reflection {z[n]}
reflection shift by n0
{x[n]} {w[n]} {v[n]}
n
.....
n
.....
n
.....
0 1 2 3 -1 0 1 2
.....
{x[n]} {w[n]} = {x[−n]} {v[n]} = {w[n − 1]}
n
.....
n
.....
n
.....
0 1 2 3 0 1 2
.....
We can combine many of these operations in one step, for example {y[n]} may be defined as y[n] = 2x[3 − n].
2.2 Some properties of signals:
2.2.1 Energy of a Signal:
The total enery of a signal {x[n]} is defined by
Ex =
∞
n=−∞
x[n]2
4
A signal is reffered to as an energy signal, if and only if the total energy of
the signal Ex is finite.
2.2.2 Power of a signal:
If {x[n]} is a signal whose energy is not finite, we define power of the signal
as
Px = lim
N→∞
1
(2N + 1)
N
n=−N
x[n]2
A signal is referred to as a power signal if the power Px satisfies the condition
0 < Px < ∞
An energy signal has a zero power and a power signal has infinite energy.
There are signals which are neither energy signals nor power signals. For
example {x[n]} defined by x[n] = n does not have finite power or energy.
2.2.3 Periodic Signals:
An important class of signals that we encounter frequently is the class of
periodic signals. We say that a signal {x[n]} is periodic period N, where N
is a positive integer, if the signal is unchanged by the time shift of N ie.,
{x[n]} = {x[n + N]}
or x[n] = x[n + N for all n.
Since {x[n]} is same as {x[n + N]}, it is also periodic so we get
{x[n]} = {x[n + N]} = {x[n + N + N]} = {x[n + 2N]}
Generalizing this we get {x[n]} = {x[n+kN]}, where k is a positive integer.
From this we see that {x[n]} is periodic with 2N, 3N, ..... The fundamental
period N0 is the smallest positive value N for which the signal is periodic.
The signal illustrated below is periodic with fundamental period N0 = 4
{x[n]} By change of variable we can write {x[n]} = {x[n +N]} as
{x[m − N]} = {x[m]} and then .........as before, we see that
{x[n]} = {x[n + kN]},
for all integer values of k, positive, negative or zero. By definition, period of
a signal is always a positive integer n.
Except for a all zero signal all periodic signals have infinite energy. They may
5
have finite power. Let {x[n]} be periodic with period N, then the power Px
is given by
P − x = lim
M→∞
1
(2M + 1)
M
n=−M
x[n]2
= lim
M→∞
1
2M+1[
N−1
n=0 x[n]2 +
2N−1
n=N x[n]2 + ...
+
kN−1
n=(k−1)N−1 x[n]2 +
M
n=kN x[n]2 +
−1
n=−N x[n]2 + ...
+
−(k−1)N−1
n=−kN x[n]2 +
−kN−1
n=−M x[n]2]
where k is largest integer such that kN −1 ≤ M. Since the signal is periodic,
sum over one period will be same for all terms. We see that k is approximately
equal to M/N(it is integer part of this) and for large M we get 2M/N terms
and limit 2M/(2M + 1) as M goes to infinite is one we get
Px =
1
N
N−1
n=0
x[n]2
2.2.4 Even and odd signals:
A real valued signal {x[n]} is referred as an even signal if it is identical to its
time reversed counterpart ie, if
{x[n]} = {x[−
Monday, December 1, 2008
DIGITAL SIGNAL PROCESSING
INTRODUCTION
First of all we will try to look into the formal definitions of the terms 'signals'.Then we talk of signal processing in brief, about the classification of signals and some properties of signals.
We would even frame our main objectives in this course.
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), receive signals all the time and continue to process them. Survival of any living organism depends upon its ability to process the signals appropriately.
What is a Signal?
Anything which carries information is a signal. e.g. 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.
Modern high speed signals are: voltage changer 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.
Signals: The Mathematical Way
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 one-dimensional and when the function depends on two or more variables, the signal is said to be multidimensional.
Examples of a one dimensional signal: A speech signal, daily maximum temperature, annual rainfall at a placeAn example of a two dimensional signal: An image is a two dimensional signal, vertical and horizontal coordinates representing the two dimensions. Four Dimensions: Our physical world is four dimensional(three spatial and one temporal).
What is Signal processing?
Processing means operating in some fashion on a signal to extract some useful information e.g. we use our ears as input device and then auditory pathways 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.
The signal processor may be an electronic system, a mechanical system or even it might be a computer program.
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
Analog or continuous time method
Digital or discrete time method..
Analog signal processing
Uses analog circuit elements such as resistors, capacitors, transistors, diodes etc
Based on natural ability of the analog system to solve differential equations that describe a physical system
The solutions are obtained in real time...
Digital signal processing
The word digital in digital signal processing means that the processing is done either by a digital hardware or by a digital computer.
Relies on numerical calculations
The method may or may not give results in real time..
The advantages of digital approach over analog approach
Flexibility: Same hardware can be used to do various kind of signal processing operation,while in the case of analog signal processing one has to design a system for each kind of operation
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 of choosing between analog or digital signal processing depends on the application. One has to compare design time,size and thecost of the implementation.
Classification of signals
We use the term signal to mean a real or complex valued function of real variable(s) and denote the signal by x(t)The variable t is called independent variable and the value x of t as dependent variable.
When t takes a vales in a countable set the signal is called a discrete time signal. For example
t ε {0, T, 2T, 3T, 4T,...}
t ε {....-1, 0 ,1,...}
t ε {1/2, 3/2, 5/2, 7/2,...}
For convenience of presentation we use the notation x[n] to denote discrete time 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 both the dependent and independent variable take value in continous set interval, the signal is called an Analog Signal.
Notation: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.Notation for continous time signal{x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I } where I is the set in which t takes the value. Notation for discrete time signalSimilarly for discrete time signal we will use the notation {x(t)}, where {x(t)} is short for {x(t), n ε I }. 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(t)} 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..Discrete Time Signal Processing and Digital Signal ProcessingWhen 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 dependent variable generally is continuous. This is because of the mathematical simplicity of discrete time signal processing. 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.
Elementary Signals
There are several elementary signals that occur prominently in the study of digital signals and digital signal processing.
(a) UNIT SAMPLE SEQUENCE:
Defined by
Graphically this is as shown below. Unit sample sequence is also known as impulse sequence.
This plays role akin to the impulse function of continous time. The continues time impulse is purely a mathematical construct while in discrete time we can actually generate the impulse sequence.
(b) UNIT STEP SEQUENCE:
Defined by :
Graphically this is as shown below
(c) EXPONENTIALSEQUENCE:
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 eβn
Real exponential signals:
: If C and are real, we can have one of the several type of behavior illustrated below
For α > 1 1$
--> magnitude of the signals grows exponentially, α < 1 1$
--> It is decaying exponential.For α > 1 1$
--> all terms of {x[n]} have same sign, α < 1 sign of terms in {x[n]} alternates.
(d)SINUSOIDAL SIGNAL:
The sinusoidal signal {x[n]} is defined by
Euler's relation allows us to relate complex exponentials and sinusoids as
and
The general discrete time complex exponential can be written in terms of real exponential and sinusiodal signals.
Specifically if we write C and α in polar form and then
Thus for α = 1 , the real and imaginary parts of a complex exponential sequence are sinusoidal. α < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential, 1$
-->α > 1 , they correspond to sinusiodal sequence multiplied by a growing exponential
First of all we will try to look into the formal definitions of the terms 'signals'.Then we talk of signal processing in brief, about the classification of signals and some properties of signals.
We would even frame our main objectives in this course.
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), receive signals all the time and continue to process them. Survival of any living organism depends upon its ability to process the signals appropriately.
What is a Signal?
Anything which carries information is a signal. e.g. 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.
Modern high speed signals are: voltage changer 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.
Signals: The Mathematical Way
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 one-dimensional and when the function depends on two or more variables, the signal is said to be multidimensional.
Examples of a one dimensional signal: A speech signal, daily maximum temperature, annual rainfall at a placeAn example of a two dimensional signal: An image is a two dimensional signal, vertical and horizontal coordinates representing the two dimensions. Four Dimensions: Our physical world is four dimensional(three spatial and one temporal).
What is Signal processing?
Processing means operating in some fashion on a signal to extract some useful information e.g. we use our ears as input device and then auditory pathways 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.
The signal processor may be an electronic system, a mechanical system or even it might be a computer program.
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
Analog or continuous time method
Digital or discrete time method..
Analog signal processing
Uses analog circuit elements such as resistors, capacitors, transistors, diodes etc
Based on natural ability of the analog system to solve differential equations that describe a physical system
The solutions are obtained in real time...
Digital signal processing
The word digital in digital signal processing means that the processing is done either by a digital hardware or by a digital computer.
Relies on numerical calculations
The method may or may not give results in real time..
The advantages of digital approach over analog approach
Flexibility: Same hardware can be used to do various kind of signal processing operation,while in the case of analog signal processing one has to design a system for each kind of operation
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 of choosing between analog or digital signal processing depends on the application. One has to compare design time,size and thecost of the implementation.
Classification of signals
We use the term signal to mean a real or complex valued function of real variable(s) and denote the signal by x(t)The variable t is called independent variable and the value x of t as dependent variable.
When t takes a vales in a countable set the signal is called a discrete time signal. For example
t ε {0, T, 2T, 3T, 4T,...}
t ε {....-1, 0 ,1,...}
t ε {1/2, 3/2, 5/2, 7/2,...}
For convenience of presentation we use the notation x[n] to denote discrete time 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 both the dependent and independent variable take value in continous set interval, the signal is called an Analog Signal.
Notation: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.Notation for continous time signal{x(t)} denotes the continuous time signal. Here {x(t)} is short notation for {x(t), t ε I } where I is the set in which t takes the value. Notation for discrete time signalSimilarly for discrete time signal we will use the notation {x(t)}, where {x(t)} is short for {x(t), n ε I }. 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(t)} 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..Discrete Time Signal Processing and Digital Signal ProcessingWhen 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 dependent variable generally is continuous. This is because of the mathematical simplicity of discrete time signal processing. 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.
Elementary Signals
There are several elementary signals that occur prominently in the study of digital signals and digital signal processing.
(a) UNIT SAMPLE SEQUENCE:
Defined by
Graphically this is as shown below. Unit sample sequence is also known as impulse sequence.
This plays role akin to the impulse function of continous time. The continues time impulse is purely a mathematical construct while in discrete time we can actually generate the impulse sequence.
(b) UNIT STEP SEQUENCE:
Defined by :
Graphically this is as shown below
(c) EXPONENTIALSEQUENCE:
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 eβn
Real exponential signals:
: If C and are real, we can have one of the several type of behavior illustrated below
For α > 1 1$
--> magnitude of the signals grows exponentially, α < 1 1$
--> It is decaying exponential.For α > 1 1$
--> all terms of {x[n]} have same sign, α < 1 sign of terms in {x[n]} alternates.
(d)SINUSOIDAL SIGNAL:
The sinusoidal signal {x[n]} is defined by
Euler's relation allows us to relate complex exponentials and sinusoids as
and
The general discrete time complex exponential can be written in terms of real exponential and sinusiodal signals.
Specifically if we write C and α in polar form and then
Thus for α = 1 , the real and imaginary parts of a complex exponential sequence are sinusoidal. α < 1, they correspond to sinusoidal sequence multiplied by a decaying exponential, 1$
-->α > 1 , they correspond to sinusiodal sequence multiplied by a growing exponential
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