Signals & Systems – 17EE54
Module-I
Introduction
Definition of Signal:
A signal is a function representing a physical quantity or variable, and
typically it contains information about the behaviour or nature of the
phenomenon.
For instance, in a RC circuit the signal may represent the voltage across the
capacitor or the current flowing in the resistor. Mathematically, a signal is
represented as a function of an independent variable ‘t’. Usually ‘t’represents
time. Thus, a signal is denoted by x(t).
Definition of System:
A system is a mathematical model of a physical process that relates the
input (or excitation) signal to the output (or response) signal. Let x and y be the
input and output signals, respectively, of a system. Then the system is viewed as
a transformation (or mapping) of x into y. This transformation is represented by
the mathematical notation
y= T(x) -----------------------------------------(1.1)
where T is the operator representing some well-defined rule by which x is
transformed into y. Relationship (1.1) is depicted as shown in Fig. 1-1(a). Multiple
input and/or output signals are possible as shown in Fig. 1-1(b). We will restrict
our attention for the most part in this text to the single-input, single-output case.
Fig1.1 :System with single or multiple input and output signals
Classification of signals:
Basically seven different classifications are there:
1. Continuous-Time and Discrete-Time Signals
2. Analog and Digital Signals
3. Real and Complex Signals
4. Deterministic and Random Signals
5. Even and Odd Signals
6. Periodic and Non periodic Signals
7. Energy and Power Signals
Continuous-Time and Discrete-Time Signals
A signal x(t) is a continuous-time signal if t is a continuous variable. If t is a
discrete variable, that is, x(t)is defined at discrete times, then x(t) is a discrete-
time signal. Since a discrete-time signal is defined at discrete times, a discrete-
time signal is often identified as a sequence of numbers, denoted by {x,) or x[n],
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where n = integer. Illustrations of a continuous- time signal x(t) and of a discrete-
time signal x[n] are shown in Fig. 1-2.
Fig 1.2 Graphical representation of (a) continuous-time and (b) discrete-
time signals
2.Analog and Digital Signals
If a continuous-time signal x(t) can take on any value in the continuous
interval (a, b), where a may be - ∞ and may be +∞ then the continuous-time
signal x(t) is called an analog signal. If a discrete-time signal x[n] can take on
only a finite number of distinct values, then we call this signal a digital signal.
3.Real and Complex Signals
A signal x(t) is a real signal if its value is a real number, and a signal x(t) is
a complex signal if its value is a complex number. A general complex signal x(t) is
a function of the form
x (t) = x1(t) + jx2 (t) --------------------------------(1.2)
where x1 (t) and x2 (t) are real signals and j = √-1
Note that in Eq. (1.2) ‘t’represents either a continuous or a discrete variable.
Deterministic and Random Signals:
Deterministic signals are those signals whose values are completely
specified for any given time. Thus, a deterministic signal can be modelled by a
known function of time ‘t’.
Random signals are those signals that take random values at any given time and
must be characterized statistically.
Even and Odd Signals:
A signal x ( t ) or x[n] is referred to as an even signal if x (- t) = x(t)
x [-n] = x [n] -------------(1.3)
A signal x ( t ) or x[n] is referred to as an odd signal if x(-t) = - x(t)
x[- n] = - x[n]
--------------(1.4)
Examples of even and odd signals are shown in Fig. 1.3
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Figuer1.3 Examples of even signals (a and b) and odd signals (c and d).
Any signal x(t) or x[n] can be expressed as a sum of two signals, one of which is
even and one of which is odd. That is,
Similarly for x[n],
X[n] = Xe[n] + Xo[n] -------(1.7)
Where,
Xe[n]=1/2{x[n] +x[-n]}
Xo[n]=1/2{x[n] -x[-n]} -------(1.8)
Note that the product of two even signals or of two odd signals is an even signal
and that the product of an
even signal and an odd signal is an odd signal.
Periodic and Non periodic Signals :
A continuous-time signal x ( t ) is said to be periodic with period T if there is
a positive nonzero value of T for which
………(1.9)
An example of such a signal is given in Fig. 1-4(a). From Eq. (1.9) or Fig. 1-4(a) it
follows that
for all t and any integer m. The fundamental period T, of x(t) is the smallest
positive value of
T for which Eq. (1.9) holds. Note that this definition does not work for a constant
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Figure 1.4
Examples of
periodic
signals.
signal x(t)
(known as a dc
signal). For a
constant signal
x(t) the
fundamental
period is
undefined since
x(t) is periodic for
any choice of T
(and so there is
no smallest
positive value).
Any continuous-
time signal which
is not periodic is called a non periodic (or a periodic) signal.
Periodic discrete-time signals are defined analogously. A sequence (discrete-time
signal) x[n] is periodic with period N if there is a positive integer N for which
An example of such a sequence is given in Fig. 1-4(b). From Eq. (1.11) and Fig. 1-
4(b) it follows that
for all n and any integer m. The fundamental period No of x[n] is the smallest
positive integer N for which Eq.(1.11) holds. Any sequence which is not periodic is
called a non periodic (or a periodic sequence).
Note that a sequence obtained by uniform sampling of a periodic continuous-time
signal may not be periodic.
Note also that the sum of two continuous-time periodic signals may not be
periodic but that the sum of two periodic sequences is always periodic.
Energy and Power Signals :
Consider v(t) to be the voltage across a resistor R producing a current i(t). The
instantaneous power p(t) per ohm is defined a
Total energy E and average power P on a per-ohm basis are
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For an arbitrary continuous-time signal x (t), the normalized energy content E of
x(t) is defined as
The normalized average power P of x(t) is defined as
Similarly, for a discrete-time signal x[n], the normalized energy content E of x[n]
is defined as
…………….(1.17)
The normalized average power P of x[n] is defined as
Basic Operations on signals:
The operations performed on signals can be broadly classified into two
kinds
1 Operations on dependent variables
2 Operations on independent variables
Operations on dependent variables
The operations of the dependent variable can be classified into five types:
amplitude scaling, addition,multiplication, integration and differentiation.
Amplitude scaling
Amplitude scaling of a signal x(t) given by equation 1.19, results in
amplification of
x(t) if a >1, and attenuation if a <1.
y(t) =ax(t) ……..(1.20)
Addition:
The addition of signals is given by equation of 1.21.
y(t) = x1(t) + x2 (t) ……(1.21)
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Multiplication:
The multiplication of signals is given by the simple equation of 1.22.
y(t) = x1(t).x2 (t) ……..(1.22)
Operations on independent variables:
Time scaling
Time scaling operation is given by equation 1.26
y(f) = x(af) . ..............1.26
This operation results in expansion in time for a<l and compression in time for
a>1.as evident from the ex amp1es of figure 1.10.
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Time reflection:
Time reflection is given by equation (1.27), and some examples are contained in
fig1.11.
y(t) = x(−t) ………..1.27
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Figure 1.11 Examples of time reflection of a continuous time signal
Time shifting:
The equation representing time shifting is given by equation (1.28), and
examples of this operation are given in figure 1.12.
y(t) x(t- t0) ..............1.28
Figure 1.12 Examples of time shift of a continuous time signal
Time shifting and scaling:
The combined transformation of shifting and scaling is contained in equation
(1.29), along with examples in figure 1.13. Here, time shift has a higher
precedence than time scale.
y(t) x(at –t0) .................1.29
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Figure 1.13 Examples of simultaneous time shifting and scaling. The
signal has to be shifted first and then time scaled.
Example for operations on signals:
For the signal x(t) shown in Fig. 1.5, sketch x(3t - 5), x(1 - t).
Solution:
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Example on periodic or not:
2. Consider the signal . It can be shown that
Solution:
Therefore, x(t) is a periodic signal.
3. Determine the fundamental period of the following signals:
Solution:
Examples on Even and odd signals:
3. Let us consider the signal x(t) = et
solution:
Similarly, we can define even and odd parts of a discrete-time signal x[n]:
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It is easy to check that
Elementary signals
Exponential signals:
The exponential signal given by equation (1.29), is a monotonically increasing
function if
a>0, and is a decreasing function if a < 0.
It can be seen that, for an exponential signal,
…………………..(1.30)
Hence, equation (1.30), shows that change in time by ±1/ a seconds, results in
change in magnitude by e±1 .The term 1/ a having units of time, is known as the
time-constant. Let us consider a decaying exponential Signal
X(t)=e-at ……………(1.31)
This signal has an initial value x(0) =1, and a final value x(∞) = 0 . The
magnitude of this signal at five times the time constant is,
x(5/a)=6.7x10-3…………….(1.32)
while at ten times the time constant, it is as low as,
The unit impulse:
The unit impulse usually represented as δ (t) , also known as the dirac delta
function, is given by,
From equation (1.38), it can be seen that the impulse exists only at t = 0 , such
that its area is 1. This is a function which cannot be practically generated. Figure
1.16, has the plot of the impulse function
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The unit step:
The unit step function, usually represented as u(t) , is given by,
Fig 1.17 Plot of the unit step function along with a few of its
transformations
The unit ramp:
The unit ramp function, usually represented as r(t) , is given by,
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System viewed as interconnection of operation:
This basically deals with system connected in series or parallel Further
these systems are connected with adders/subtractor, multipliers etc.
FIG:Interconnection of systems. (a) A series or cascade interconnection
of two systems; (b)
A parallel interconnection of two systems
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System Properties:
Memoryless
Definition : A system is memoryless if the output at time t (or n) depends only
on the input at time t (or n).
Examples.
CASUAL:
STABLE:
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TIME INVARIANT:
EXAMPLES:
LINEAR:
A system is linear if it is additive and scalable. That is,
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Module-II
Time-Domain Representations For LTI Systems
Time domain representation of LTI Systems
•Impulse response: characterizes the behavior of any LTI system
•Linear constant coefficient differential or difference equation: input output
behavior
•Block diagram: as an interconnection of three elementary operations
An important subclass of linear time invariant system is one where the
input and output sequences satisfy constant coefficient linear difference equation
--------1
The constants,x(n) is input sequence and y(n) is output sequence
-----------2
Problems:
1.Consider the difference equation
Solution:
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then
Differential and Difference equation
•General form of differential equation is
Block diagram representations
• A block diagram is an interconnection of elementary operations that act on the
input signal
• This method is more detailed representation of the system than impulse
response or
differential/difference equation representations
• The impulse response and differential/difference equation descriptions
represent only the input-output behavior of a system, block diagram
representation describes how the operations are ordered
• Each block diagram representation describes a different set of internal
computations used to determinethe system output
• Block diagram consists of three elementary operations on the signals:
– Scalar multiplication: y(t) = cx(t) or y[n] = x[n], where c is a
scalar
– Addition: y(t) = x(t)+w(t) or y[n] = x[n]+w[n].
•Block diagram consists of three elementary operations on the signals
Integration for continuous time LTI system:
Time shift for discrete time LTI system: y[n] = x[n−1]
•Scalar multiplication: y(t) = cx(t) or y[n] = x[n], where c is a scalar
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Example 1:
•Consider the system described by the block diagram and its difference
equation is y[n]+(1/2)y[n−1]+(1/4)y[n−2]= x[n−1]
Block
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Block diagram representation is not unique, direct form II structure of
Example 1
•We can change the order without changing the input output behavior
Let the output of a new system be f [n] and given input x[n] are related
By f[n] = −a1 f [n−1]−a2 f [n−2]+x[n]
•The signal f [n] acts as an input to the second system and output of
second system is y[n] = b0 f [n]+b1 f [n−1]+b2 f [n−2].
•The block diagram representation of an LTI system is not unique
Example 2: Direct form I
Direct form
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