SIGNALS, SYSTEMS, AND SIGNAL

PROCESSING

Signal
 Convey information from one point to another (They may be
generated by electronic means, or by some natural means such
as talking, walking, your heart beating, an earthquake, the sun
heating the sidewalk)
 described by a function of one or more independent variables.
The value of the function (i.e., the dependent variable) can be
real-valued scalar quantity a complex-valued quantity or perhaps
a vector.

Examples: 𝑠1 𝑡 = 𝐴𝑠𝑖𝑛 3𝜋𝑡, 𝑠2 𝑡 = 5𝑡,𝑠3 𝑡 = 5𝑡 2

𝑠4 𝑥, 𝑦 = 3𝑥 + 2𝑥𝑦 + 10𝑦 2
𝑠5 𝑡 = 𝐴𝑒 𝑗3𝜋𝑡 = 𝐴𝑐𝑜𝑠 3𝜋𝑡 + 𝑗𝐴𝑠𝑖𝑛 3𝜋𝑡
𝑠1 𝑡
𝑺6 𝑡 = 𝑠2 𝑡
𝑠3 𝑡
 System

process signals to produce a modified

SIGNALS, or transformed version of the original
signal
SYSTEMS,
AND SIGNAL Example. A filter
used to reduce
defined as a physical the noise and
PROCESSING device that performs interference
corrupting a
an operation on a desired
signal. information-
bearing signal is
called a system.

Signal Processing –the operation on

or analysis of signals, or
measurements of time-varying or
spatially varying physical quantities.
Basic Elements of a Digital Signal
Processing System
Multichannel and
Multidimensional Signals
 Continuous-Time versus
Discrete Time Signals

 Discrete- time signals are defined only at certain specific values

of time. To emphasize the discrete- time nature of a signal, we
shall denote such a signal as 𝑥 𝑛 or x[n] instead of 𝑥 𝑡 . If the
time instants 𝑡𝑛 are equally spaced (i.e., 𝑡𝑛 = 𝑛𝑇), the notation
𝑥(𝑛𝑇) is also used.

Graphical representation of the discrete time signal 𝒙 𝒏 = 𝟎. 𝟖𝒏 for

𝒏 > 𝟎 and 𝒙 𝒏 = 𝟎 for 𝒏 < 𝟎
 Continuous-Time versus
Discrete Time Signals

 Continuous- time signals or analog signals are defined for every

value of time and they take on values in the continuous interval
(a, b), where a can be −∞ and b can be ∞.
 Example. Speech waveform, signals 𝑥1 𝑡 = cos 𝜋𝑡, 𝑥2 𝑡 =
𝑒 − 𝑡 , −∞ < 𝑡 < ∞

Example of a speech
signal
Continuous-Valued versus
Discrete-Valued Signals

 Continuous- Valued Signals- if a signal takes on all possible values

on a finite or an infinite range.
 Discrete –Valued Signal – if the signal takes on values from a finite
set of possible values.
Deterministic vs Random
Signals

Deterministic - Any signal that can be uniquely described by an

explicit mathematical expression, a table of data, or a well-
defined rule.

Random - are signals that either cannot be described to any

reasonable degree of accuracy by explicit mathematical formulas,
or such a description is too com plicated to be of any practical use
Discrete-Time Signal or Sequence

 Graphically, it can be represented as

Discrete Time Signals or Sequence

 Besides the graphical, there are some alternative representation:

1. Functional representation
1, 𝑛 = 1,3
𝑥 𝑛 = ቐ 4, 𝑛=2
0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
2. Tabular representation

n … -2 -1 0 1 2 3 4 …
x[n] … 0 0 0 1 4 1 0 …
Discrete Time Signals or Sequence

 Besides the graphical, there are some alternative representation:

3. Sequence Representation
 An infinite-duration signal or sequence with the time origin (n = 0)
indicated by the symbol ↑ is represented as
x n = {… 0, 0, 1, 4, 1, 0,0, … )

 A sequence x[n], which is zero for 𝑛 < 0, can be represented as

x n = {0, 1, 4, 1, 0,0, … }

 A finite –duration sequence can be represented as

𝑥 𝑛 = {3, −1, −2, 5, 0, 4, 1}
whereas a finite-duration sequence that satisfies the condition x[n] = 0 for n <
0 can be represented as
x[n] = { 0, 1, 4, 1}
 SEATWORK

A discrete time signal is defined as

𝑛
1 + , −3 ≤ n ≤ −1
3,
𝑥 𝑛 =൞ 1 0≤n≤3
, elsewhere
0

A. Give the graphical, tabular and sequential

representation of the given functional
representation.
Some Elementary Discrete Time
Signal

1. Unit Sample Sequence

 a signal that is zero everywhere except at 𝑛 = 0 where its value is
unity. Denoted by 𝛿 𝑛
 It is sometimes referred to as a unit impulse.
 Functional representation
1, 𝑛 = 0
𝛿 𝑛 =ቊ
0, 𝑛 ≠ 0

 Graphical representation

𝛿𝑛
Some Elementary Discrete Time
Signal

2. Unit Step Signal

 Denoted by u[n]
 Functional representation
1, 𝑛 ≥ 0
𝑢 𝑛 =ቊ
0, 𝑛 < 0

 Graphical representation

u𝒏
Some Elementary Discrete Time
Signal

3. Unit Ramp Signal

 denoted by

𝑛, 𝑛 ≥ 0
𝑢𝑟 [𝑛] = ቊ
0, 𝑛 < 0

 Graphical representation

𝑢𝑟 [𝑛]
Some Elementary Discrete Time
Signal

4. Exponential signal
 A sequence of the form 𝑥 𝑛 = 𝑎𝑛 for all 𝑛
 Graphical representation (Parameter a is real)
Some Elementary Discrete Time
Signal

4. Exponential signal (cont…)

 Graphical representation (Parameter a is complex-valued)
𝑎 = 𝑟𝑒 𝑗𝜃 where r and θ are now parameters.
∴ 𝑥 𝑛 = 𝑟 𝑛 𝑒 𝑗𝜃𝑛 = 𝑟 𝑛 (cos 𝜃𝑛 + 𝑗𝑠𝑖𝑛 𝜃𝑛)
Some Elementary Discrete Time
Signal

Graph of amplitude and phase function of a complex-valued

exponential signal
Classification of Discrete-Time Signals

A. Energy signals and Power Signals

1. Energy signals
 The energy E of a signal x[n] is defined as
∞
2
𝐸 = ෍ 𝑥(𝑛)
𝑛=−∞

 It can be finite or infinite.

 If E is finite (i.e , 0 < 𝐸 < ∞), then x[n] is called an energy signal.
 Classification of Discrete-Time
Signals

2. Power Signal
 Many signals that possess infinite energy, have a finite average power.
1
 It is defined as 𝑃 = lim σ∞
𝑛=−∞ 𝑥(𝑛) 2 .
𝑁→∞ 2𝑁+1
 If we define the signal energy of x[n] over the finite interval −𝑁 ≤ 𝑛 ≤ 𝑁 as
𝐸𝑁 =σ𝑁
𝑛=−𝑁 𝑥(𝑛)
2

 Signal energy can be expressed as

𝐸 = lim 𝐸𝑁
𝑁→∞

 The average power of the signal x[n] as

1
𝑃 = lim 𝐸𝑁
𝑁→∞ 2𝑁 + 1

 If E is finite, 𝑃 = 0. On the other hand, if E is infinite, the average power

may be either finite or infinite.
 If P is finite(and nonzero), the signal is called a power signal.
Sample Problems:

1. Determine the power and energy of the unit step sequence.

1 𝑛
2. Find the power and signal energy of 𝑥 𝑛 = 𝑢[𝑛]
4
𝜋𝑛
3. Find the power and signal energy of 𝑥 𝑛 = sin .
3
𝜋𝑛
4. Find the power and signal energy of 𝑥 𝑛 = cos .
2

5. Find the signal energy of 𝑥 𝑛 = 2𝑛 𝑢[−𝑛].

𝑗𝜋𝑛
6. Find the power and signal energy of 𝑥 𝑛 = 𝑒 4
Classification of Discrete-Time Signals

B. Symmetric(Even) and Antisymmetric (Odd)

 A real-valued signal x[n] is called symmetric(even) if 𝒙 −𝒏 = 𝒙 𝒏 .
 A real-valued signal x[n] is called antisymmetric(odd) if 𝒙 −𝒏 = −𝒙 𝒏 .
 Examples of signals with even and odd symmetry.
Classification of Discrete-Time Signals

B. Symmetric(Even) and Antisymmetric (Odd)

 Even Signal Component
𝑥 𝑛 + 𝑥[−𝑛]
𝑥𝑒 𝑛 =
2
 Odd Signal Component
𝑥 𝑛 − 𝑥[−𝑛]
𝑥𝑜 𝑛 =
2
 Any arbitrary signal can be expressed as
𝑥 𝑛 = 𝑥𝑒 𝑛 + 𝑥𝑜 𝑛
 Sample Problem:

1. Find the even and odd part of the function 𝑥[𝑛] = 3𝑛

𝑗𝜋𝑛
2. What are the even and odd parts of the function 𝑥 𝑛 = 3𝑒 5

3. What are the even and odd parts of the function 𝑥 𝑛 =

{2, −2,6, −2}
Classification of Discrete-Time Signals

C. Periodic and Aperiodic Signals

 A signal x[n] is periodic with period 𝑁(𝑁 > 0) if and only if
𝑥 𝑛 + 𝑁 = 𝑥 𝑛 for all n.
 If there is no value of N that satisfies the equation above, the signal is
called non periodic or aperiodic.
 Periodic signals are power signal.
 N = fundamental period in discrete time. N must be a positive integer.
 𝜔 = fundamental angle of frequency or discrete angle of frequency.
It is denoted as 𝜔 = 2𝜋𝑓
𝐾
 Frequency is denoted as 𝑓 =
𝑁
 Simple Manipulation of Discrete-
Time Signals

In this section we consider some simple modifications or

manipulations involving the independent variable and the signal
amplitude (dependent variable).

Transformation of the independent variable(time)

 A signal x[n] may be shifted in time by replacing the
independent variable n by n−k , where k is an integer.
 If k is a positive integer, the time shift results in a delay of the signal
by k units.
 If k is a negative integer, advance the signal by k.
 Simple Manipulation of Discrete-
Time Signals

Transformation of the independent variable(time) (cont…)

Example:
The signal is graphically illustrated below:
x[n]

Show the graphical representation of (a) x n − 3 and (b) x n + 2

 Simple Manipulation of Discrete-
Time Signals

Transformation of the independent variable(time) (cont…)

 Folding or reflection
 Replace the independent variable n by –n.
 The result of this operation is a folding or a reflection of the signal
about the time origin 𝒏 = 𝟎.
Example.
Show the graphical representation of the signal x[−n] and 𝑥[−𝑛 + 2],
where x[n] is the signal illustrated in the previous slide.
 Simple Manipulation of Discrete-
Time Signals

Transformation of the independent variable(time) (cont…)

 Time Scaling or Down Sampling
 A third modification of the independent variable which involves by
replacing by 𝐧 by 𝛍n, where 𝛍 is an integer.
Example. Show the graphical representation of the signal
𝑦 𝑛 = 𝑥[2𝑛] , where x[n] is the signal illustrated below.
 Simple Manipulation of Discrete-
Time Signals

 Addition, multiplication, and scaling of sequences

 Amplitude scaling of a signal by a constant A is accomplished by
multiplying the value of every signal sample by A.
𝐲 𝐧 = 𝐀𝐱 𝐧
 Addition of sequences or signal
 The sum of two signals 𝑥1 𝑛 and 𝑥2 𝑛 is a signal 𝑦 𝑛 , whose value at any
instant is equal to the sum of the values of these two signals at the instant
𝑦 𝑛 = 𝑥1 𝑛 + 𝑥2 𝑛 − ∞ < 𝑛 < ∞

 Product of sequences or signal

 The product of two signals is similarly defined on a sample-to-sample basis as
𝑦 𝑛 = 𝑥1 𝑛 𝑥2 𝑛 − ∞ < 𝑛 < ∞
DISCRETE TIME SYSTEMS

 It is a device or algorithm that operates on a discrete-time signal

(input or excitation) to produce another discrete-time signal
(output or response).

 Block diagram representation

 We say that the input signal x[n] is transformed by the system into
a signal y[n] and express the general relationship between x[n]
and y[n] as
𝒚𝒏 ≡𝓣𝒙𝒏
Input-Output Description of the System

 It consists of a mathematical expressions or rule, which explicitly

defines the relation between the input and output signals (input-
output relationship).
 Aside from the previous figure and equation, alternatively, we use
the notation
𝒯
𝑥 𝑛 → 𝑦[𝑛]
Input-Output Description of the System

Example:
Determine the response of the following the systems to the input signal
𝑛 , −3 ≤ 𝑛 ≤ 3
𝑥 𝑛 =ቊ
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
a. 𝑦 𝑛 =𝑥 𝑛
b. 𝑦 𝑛 =𝑥 𝑛−1
c. 𝑦 𝑛 =𝑥 𝑛+1
1
d. 𝑦 𝑛 =3 𝑥 𝑛+1 +𝑥 𝑛 +𝑥 𝑛−1

e. 𝑦 𝑛 = 𝑚𝑒𝑑𝑖𝑎𝑛 𝑥 𝑛 + 1 , 𝑥 𝑛 , 𝑥[𝑛 − 1]
f. 𝑦 𝑛 = 𝑚𝑎𝑥 𝑥 𝑛 + 1 , 𝑥 𝑛 , 𝑥 𝑛 − 1
g. 𝑦 𝑛 = σ𝑛𝑘=−∞ 𝑥 𝑘 = 𝑥 𝑛 + 𝑥 𝑛 − 1 + 𝑥 𝑛 − 2 + ⋯
 Block Diagram Representation of
Discrete-Time Systems

 Adder
 Block Diagram Representation of
Discrete-Time Systems

 Constant Multiplier
 Block Diagram Representation of
Discrete-Time Systems

 Signal Multiplier
 Block Diagram Representation of
Discrete-Time Systems

 Unit delay element

 Block Diagram Representation of
Discrete-Time Systems

 A unit advance element

EXAMPLE NO. 1

1
x[n] + y[n]

5/6
+ 𝒛−𝟏

-1/6
𝒛−𝟐

Give the input-output relation of the given system above

EXAMPLE NO. 2

 Using basic building blocks introduced, sketch the block

diagram representation of the discrete time system described
by the input-output relation
1 1
𝑦 𝑛 + 𝑦 𝑛 − 1 − 𝑦 𝑛 − 2 = 𝑥 𝑛 + 2𝑥 𝑛 − 1
2 3

 where x[n] is the input and y[n] is the output of the system.
EXAMPLE NO. 3

 Using basic building blocks introduced, sketch the block

diagram representation of the discrete time system described
by the input-output relation
1 1 1
𝑦 𝑛 = 𝑦 𝑛 − 1 + 𝑥 𝑛 + 𝑥[𝑛 − 1]
4 2 2

 where x[n] is the input and y[n] is the output of the system.
Classification of Discrete-Time
Systems

Static versus Dynamic System

 Static system
 A discrete-time system is called static or memoryless if its output at
any instant n depends at most on the input sample at the same time,
but not on past or future samples of the input.
 There is no need to store any of the past inputs or outputs in order to
compute the present output.
 The system described by the following input-output equations
𝑦 𝑛 = 𝑎𝑥 𝑛
𝑦 𝑛 = 𝑛𝑥 𝑛 + 𝑏𝑥 3 𝑛
are both static or memoryless.
Classification of Discrete-Time
Systems
Static versus Dynamic System
 Dynamic system
 System that have a memory.
 If the output of the system at time n is completely determined by the
input samples in the interval from 𝑛 − 𝑁 to 𝑛(𝑁 ≥ 0), the system is said
to have memory of duration N.
 If 0 < 𝑁 < ∞, the system is said to have finite memory.
 If N = ∞, the system is said to have a infinite memory.
 The system described by the following input-output equations
𝑛

𝑦 𝑛 = 𝑥 𝑛 + 3𝑥 𝑛 − 1 𝑦 𝑛 = ෍𝑥 𝑛−𝑘
𝑘=0
∞

𝑦 𝑛 = ෍𝑥 𝑛−𝑘
𝑘=0

are dynamic systems or system with memory.

Classification of Discrete-Time
Systems
Time-Invariant versus Time Variant Systems
 Time Invariant System
 If its input-output characteristic do not change with time.

 Definition. A relaxed system is time-invariant or shift invariant if and

only if
𝒯
𝑥 𝑛 → 𝑦[𝑛]
implies that
𝒯
𝑥 𝑛 − 𝑘 → 𝑦[𝑛 − 𝑘]
for every input signal x[n] and every time shift k.
Classification of Discrete-Time
Systems

Time-Invariant versus Time Variant Systems

How to determine if the system is time variant or time invariant ?
To determine if any given system is time invariant, we need to
perform the test specified by the preceding definition.
1. We excite the system with an arbitrary input sequence x[n], which
produces an output denoted as y[n].
2. We delay the input sequence by some amount k and recomputed
the output. In general, we can write the output as
y[n, k] = 𝒯[𝑥 𝑛 − 𝑘 ].
3. If the output y[n, k] = y[n − k], for all possible values of k, the system is
time- invariant.
4. If the output y n, k ≠ y n − k , even for one value of k, the system is
time variant.
Classification of Discrete-Time
Systems
Time-Invariant versus Time Variant Systems
Example: Determine if the system shown below are time invariant or
time variant.
Classification of Discrete-Time
Systems
Time-Invariant versus Time Variant Systems
Example: Determine if the system shown below are time invariant or
rime variant.
Classification of Discrete-Time
Systems

Linear versus Nonlinear System

Linear System
 It is one that satisfies the superposition principle.
 A relaxed ,linear system with zero input produces a zero output
 Superposition Principle –requires that the response of the system to a weighted
sum of signals be equal to the corresponding weighted sum of the responses
(outputs) of the system to each of the individual input signals.
 Definition. A relaxed 𝒯 system is linear if and only if
𝒯 𝑎1 𝑥1 𝑛 + 𝑎2 𝑥2 𝑛 = 𝑎1 𝒯 𝑥1 𝑛 + 𝑎2 𝒯 𝑥2 𝑛
for any arbitrary input sequence 𝑥1 𝑛 and 𝑥2 𝑛 , and any arbitrary
constant 𝑎1 and 𝑎2 .
Nonlinear System
 If a relaxed system does not satisfy the superposition principle as given by
the definition above.
 If a system produces a nonzero output with a zero input, the system may
be either nonrelaxed or nonlinear
Classification of Discrete-Time
Systems
Linear versus Nonlinear System
 𝒯 is linear if and only if 𝑦 𝑛 = 𝑦 ′ 𝑛

Graphical representation of the superposition principle.

Classification of Discrete-Time
Systems
Linear versus Nonlinear System
Example: Determine if the systems described are linear or nonlinear.
a. 𝑦 𝑛 = 𝑛𝑥 𝑛
b. 𝑦 𝑛 = 𝑥 𝑛2
c. 𝑦 𝑛 = 𝑥2 𝑛
d. 𝑦 𝑛 = 𝐴𝑥 𝑛 + 𝐵
e. 𝑦 𝑛 = 𝑒𝑥 𝑛
Classification of Discrete-Time
Systems
Causal versus Noncausal System
 Definition. A system is said to be causal if the output of the system at
any time n depends only on present and past inputs, but does not
depend on future inputs.
 If a system does not satisfy this definition, is called noncausal.
Example: Determine if the system described are causal or non causal.
a. 𝑦 𝑛 =𝑥 𝑛 −𝑥 𝑛−1
b. 𝑦 𝑛 = σ𝑛𝑘=−∞ 𝑥 𝑘
c. 𝑦 𝑛 = 𝑎𝑥 𝑛
d. 𝑦 𝑛 = 𝑥 𝑛 + 3𝑥 𝑛 + 4
e. 𝑦 𝑛 = 𝑥 𝑛2
f. 𝑦 𝑛 = 𝑥 2𝑛
g. 𝑦 𝑛 = 𝑥 −𝑛
Classification of Discrete-Time
Systems

Stable versus unstable systems

Stability is an important property that must be considered in any practical
application of a system . Unstable systems usually exhibit erratic and extreme
behavior and cause overflow in any practical implementation.
Definition. An arbitrary relaxed system is said to be bounded input-bounded
output (BIBO) stable if and only if every bounded input produces a bounded
output.
Classification of Discrete-Time
Systems
Example: Determine if the system described are stable or unstable.
a. 𝑦 𝑛 = 𝑥 2 [𝑛]
b. 𝑦 𝑛 = 𝑛𝑥 𝑛
c. 𝑦 𝑛 = 𝑐𝑜𝑠𝑛. 𝑥 𝑛
𝑥[𝑛]
d. 𝑦 𝑛 = sin 𝑛