Chapter 1 Systems and Signals








Continuous-Time and Discrete-Time Signals
Classification of Signals
Transformations of the Independent Variables
Exponential and Sinusoidal Signals
Unit Impulse and Unit Step Functions
Continuous-Time and Discrete-Time Systems
Basic System Properties
Summary
1. Signals

Signals
 Any
physical quantity that varies with time, space or any other
independent variable.
 Signals are represented mathematically as functions of one
or more independent variables.
 In this course, time is usually the only independent variable.
 Continuous-time signals are defined for every value of time.
 Discrete -time signals are defined at discrete values of time.
2. Classification of Signals

Periodic Signals
2. Classification of Signals

Even and Odd Signals
 Even
signal: x(-t)=x(t) or x[-n]=x[n]
 Odd signal: x(-t)=-x(t) or x[-n]=-x[n]

Any signal can be broken into a sum of an even signal
1
and an odd signal
x [n]  ( x[n]  x[n])
e
x[n] = xe[n] + xo[n]
2
1
xo [n]  ( x[n]  x[n])
2
2. Classification of Signals
2. Classification of Signals

Complex-valued Signals
 Conjugate
symmetric signal: x*(-t)=x(t) or x*[-n]=x[n]
 Conjugate antisymmetric signal :x*(-t)=-x(t) or x*[-n]=-x[n]

Decomposition
 Conjugate
symmetric-antisymmetric decomposition: Any
signal may be expressed as the sum of a conjugatesymmetric component and a conjugate antisymmetric
1
component as
x e [ n ]  ( x [ n ] x *[  n ])
x[n] = xe[n] + xo[n]
2
1
x o [ n ]  ( x [ n ] x *[  n ])
2
2. Classification of Signals

Instantaneous Power across a resistor R

Energy

Average power
2. Classification of Signals

The total energy is defined as
T /2
E  lim 
T  T / 2


x (t )dt   x 2 (t )dt
2

Time Averaged, Power is defined as
1 T /2 2
P  lim  x (t )dt
T  T T / 2
2. Classification of Signals

Signal Energy and Power
A signal for which 0<E< 
 Power signal: A signal for which 0<P< 
 If P= , or  if E= but P=0, then the signal is neither
energy signal nor power signal
 Energy signal:
3. Basic Operation on Signals– Operations
Performed on Dependent Variables

Amplitude Scaling

Addition

Multiplication

Differentiation

Integration
3. Basic Operation on Signals-Transformations of the Independent Variables

Time Shift
3. Basic Operation on Signals–
Transformations of the Independent Variables

Time Reversal
3. Basic Operation on Signals–
Transformations of the Independent Variables

Time Scaling
3. Basic Operation on Signals–
Transformations of the Independent Variables
4. Exponential and Sinusoidal Signals

Real Exponential Signals
4. Exponential and Sinusoidal Signals



DT real exponential signals: x[n] = Br n, where B and r are real
numbers
Decaying in amplitude: 0<|r|<1; growing in amplitude: |r|>1
Non-alternating in sign: r>0; alternating in sign: r<0
4. Exponential and Sinusoidal Signals

Definition
Xa(t) = A cos(  t+ ), - * <t<*
 A is the amplitude of the sinusoid.
  is the frequency in radians per second.
  is the phase in radians.
 f=  /2p is the frequency in cycles per second or hertz.

Time
Remarks


The fundamental period is P=1/F.
For every fixed value of F, f(t) is periodic
–


f(t+P) = f(t), P=1/F
Continuous-time sinusoidal signals with distinct frequencies are themselves
distinct.
Increasing the frequency F results in an increase in the rate of oscillation.
4. Exponential and Sinusoidal Signals

CT Complex Exponential Signals

Euler’s Identity
4. Exponential and Sinusoidal Signals

Rectangular Form vs Polar Form
4. Exponential and Sinusoidal Signals

Discrete-Time Form
4. Exponential and Sinusoidal Signals

Discrete-Time Sinusoidal Signals
= A cos(  n+  ), n =1, 2, ...
 A is the amplitude of the sinusoid
  is the frequency in radians per sample
  is the phase in radians
 f= /2p is the frequency in cycles per sample or hertz
 x(n)
X(n) = A cos(  n+ )
4. Exponential and Sinusoidal Signals
 Discrete-Time Sinusoidal Signals
 A discrete-time sinsoidal is periodic only if its
frequency f is a rational number
–
x(n+N) = x(n), N=m/f, where m, N are
integers.
 Discrete-time sinusoidals where frequencies are
separated by an integer multiple of 2p are identical
–
–
x1(n) = A cos( 0 n)
x2(n) = A cos( (0 2p) n)
 The highest rate of oscillation in a discrete-time
sinusoidal is attained when =p or (=-p), or
equivalently f=1/2.
–
X(n) = A cos(( 0+p)n) = -A cos((0+p)n
X(n) = A cos(  n+ )
4. Exponential and Sinusoidal Signals
5. Unit Impulse and Unit Step Functions

Unit Impulse Signals
5. Unit Impulse and Unit Step Functions


Dirac Delta function, δ(t), often referred to as the unit
impulse or delta function, is the function that defines the
idea of an unit impulse.
This function is one that is infinitely narrow, infinitely tall,
yet integrates to unity,
5. Unit Impulse and Unit Step Functions

Unit Step Signals
5. Unit Impulse and Unit Step Functions



Perhaps the simplest way to visualize this is
x (t )
as a rectangular pulse from a−ε/2 to a+ε/2
with a height of 1/ε.
As we take the limit of this, lim 0, we see that
the width tends to zero and the height tends
to infinity as the total area remains constant
at one.
/2
The impulse function is often written as δ(t) .
 (t )  lim x (t )
 0
/2
5. Unit Impulse and Unit Step Functions

Properties



5. Unit Impulse and Unit Step Functions

Properties
 Shifting property
 Time-Scaling
1
 (at )  lim x (at )   (t )
 0
a
6. Continuous-Time and Discrete-Time
Systems
6. Continuous-Time and Discrete-Time
Systems
6. Continuous-Time and Discrete-Time
Systems
7. Basic System Properties

Systems
 Mathematically
a transformation or an operator that maps
an input signal into an output signal
 Can be either hardware or software.
 Such operations are usually referred as signal processing.

E.x.
y( n ) 
n
n
k 
k 
 x(k )   x(k  1)  x(n)  y(n  1)  x(n)
n
n
Discrete-Time
System H
7. Basic System Properties

Types of Systems
 CT
systems: input and output are CT signals
 DT systems: input and output are DT signals
 Mixed systems: CT-in and DT-out (e.g., A/D converter), DT-in
and CT-out (e.g., D/A converter)
7. Basic System Properties

Time-Invariant versus Time-Variant Systems

A system H is time-invariant or shift invariant if and only if

x(n) ---> y(n)
implies that
x(n-k) --> y(n-k)
 for every input signal x(n) x(n) and every time shift k.


Causal versus Noncausal Systems

The output at any time depends on values of the input at only the
present and past time.
y(n) = F[x(n), x(n-1), x(n-2), ...].
 where F[.] is some arbitrary function.

7. Basic System Properties

Linear versus Nonlinear Systems

u1(n)
a
+
A system H is linear if and only if
u2(n)
 H[a1x1(n)+
a2 x2 (n)] = a1H[x1 (n)] + a2H[x2 (n)]
 for any arbitrary input sequences x 1(n) and
x2(n), and any arbitrary constants a1 and a2.

Multiplicative or Scaling Property
 H[ax(n)] =

a H[x(n)]
Additivity Property
H[x1(n) + x2 (n)] = H[x1 (n)] + H[x2 (n)]
u1(n)
u2(n)
Linear
systems
y(n)
b
Linear
systems
Linear
systems
a
+
b
y(n)
7. Basic System Properties

Stable versus Unstable Systems
An arbitrary relaxed system is said to be bounded-inputbounded-output (BIBO) stable if and only if every bounded
input produces a bounded output.
 Ex.

 y(t)=tx(t)
 y(t)=ex(t)
7. Basic System Properties

Memory versus Memoryless Systems


A system is referred to as memoryless system if the output for
each value of the independent variable depends only on the
input at the same time.
Examples
Memory
 y[n] = x[n-1] (Delay system)
 y[n] = y[n-1]+x[n] (Accumulator)
 Memoryless


y(t)=R⋅x(t)
7. Basic System Properties

Inverse (or Invertible) System
 If
distinct inputs lead to distinct outputs
8. Matlab






Periodic Signals,1.38
Exponential Signals, 1.39
Sinusoidal Signals, 1.40
Exponentially Damped Sinusoidal Signals, 1.41
Step, Impulse and Ramp Functions
User Defined Function
9. Remarks








Continuous-Time and Discrete-Time Signals
Classification of Signals
Transformations of the Independent Variables
Exponential and Sinusoidal Signals
Unit Impulse and Unit Step Functions
Continuous-Time and Discrete-Time Systems
Basic System Properties
Summary
Homeworks