Signal and system (Classifications of systems)

Classification of Signals
• Continuous and discrete-time signals
• Deterministic signals, random signals
• Finite and infinite length
• Periodic signals, non-periodic signals
• Even and odd signals

Deterministic signals, random
signals
 Deterministic signals
-There is no uncertainty with respect to its value
at any time. 1) sin(3t) 2) Square wave
 Random signals
- There is uncertainty before its actual
occurrence.
1) EEG 2) Speech 3) Noise

Finite and infinite length
• Finite-length signal : nonzero over a
finite interval tmin< t< tmax
• Infinite-length singal : nonzero over all
real numbers

Even and odd signals
• Even signals : x(-t)=x(t)
• Odd signals : x(-t)=-x(t)
• Even signal is symmetric about vertical axis
• Odd signal is antisymmertic about the time origin

Basic Operations on Signals
A variety of operations can be carried out on signals to
obtain new signals.
These operations can be classified into two categories:
•Operations that are performed on the dependent variable
•Operations that can be performed on the independent
variable.
Recall that for a signal x(t), t is the independent variable
and x(t) is the dependent variable.

Operations performed on dependent
variable
• Amplitude scaling
• Addition
• Multiplication
• Differentiation
• Integration
( ) ( )
y t cx t

1 2
( ) ( ) ( )
y t x t x t
 
1 2
( ) ( ) ( )
y t x t x t
 
( ) ( )
d
y t x t
dx

( ) ( )
t
y t x d
 
 
 

Operations performed on the
independent variable
• Reflection ( ) ( )
y t x t
 
This operation can be visualized as flipping the signal about the origin.

Operations performed on the independent
signals

independent signals
• Time scaling
a>1 : compressed
0<a<1 : expanded
( ) ( )
y t x at


independent signals
- Precedence Rule for time shifting & time
scaling
( ) ( ) ( ( ))
b
y t x at b x a t
a
   

• Notes on the precedence of Time-Shift,
Reflection and Scaling for C-T signals.
There is no order of precedence between Reflection and Scaling. i.e. Reflection can be done first and
then Scaling or Scaling can be done first and then Reflection.
Without loss of generality, any function can be written as
• x(at+b)
where ‘a’ can be positive or negative. If ‘a’ is positive then only scaling is involved; if ‘a’ is negative
then both scaling and reflection are involved. ‘b’ can be any real no. (positive or negative).
• Method I (Direct method)
•
• Step 1: Start with x(at+b) – (note: remove any inner parenthesis)
• Step 2: Apply the Time-Shift by the factor ‘b’
• Step 3: Apply Scaling (and Reflection, if ‘a’ negative) by the factor ‘a’
Method II (Parenthesis method)
• Step 1: Express x(at+b) in the form x(a(t+b/a)). (note that the coefficient of ‘t’ within the
parenthesis become +1)
• Step 2: Apply Scaling (and Reflection, if ‘a’ negative) by the factor ‘a’
• Step 3: Apply the Time-Shift by the factor ‘b/a’

Quiz # 1
Compute
• x(-4t-1)
• 3x(-t+10)
• 2x(t-4)

Energy of signal
• In signal processing, a signal is viewed as a function of time. The
term “size of a signal” is used to represent the “strength of the
signal”. It is crucial to know the “size” of a signal used in a
certain application. Energy is used to compute the strength of a
signal. The Energy of a continuous-time complex signal is
defined as:

Voice activity detection (Energy Example)

Systems Viewed as Interconnection
of Operations
system output
signal
input
signal

Properties of Systems
• Stability
• Memory
• Invertibility
• Causality
• Linearity
• Time Invariance

Stability
• BIBO stable : A system is said to be
bounded-input bounded-output stable
iff every bounded input results in a
bounded output.
| ( ) | | ( ) |
x y
t x t M t y t M
        

Causality
• Causal system : A system is said to be
causal if the present value of the output
signal depends only on the present and/or
past values of the input signal.
• Non-causal system
• (example)
y[n]=x[n]+1/2x[n-1]
y[n]=x[n+1]+1/2x[n-1]

Time Invariance
• Time invariant system : A system is
said to be time invariant if a time delay
or time advance of the input signal
leads to a identical time shift in the
output signal.

Linearity
a1
a2
aN
 H
x1(t)
x2(t)
xN(t)
y(t)
H
H
H
a1
a2
aN

x1(t)
x2(t)
xN(t)
y(t)

Linear Time Invariant (LTI)
Systems
• Linearity and time invariance are two
system properties that greatly simplify the
study of systems that exhibit them. In our
study of signals and systems, we will be
especially interested in systems that
demonstrate both of these properties,
which together allow the use of some of
the most powerful tools of signal
processing.

Signal and system (Classifications of systems)

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