CHAPTER 1

CLASSIFICATION OF SIGNALS AND SYSTEM

Introduction to signals and systems:

A signal is defined as single-valued function of one or more independent variables

which contain some information.

Any physical quantity that varies with time, space, or any other independent variable.

If a signal depends on only one independent variable -one dimensional signal,

If a signal depends on two independent variable -two dimensional signal.

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A system is defined as an entity that acts on an input signal and transforms it into an
output signal.

It may also be defined as a set of elements or functional blocks which are connected
together and produces an in response to an input signal.

Continuous Time (CT) and Discrete Time (DT) signals

#continuous time signal

A signal of continuous amplitude or time is known as continuous signal or analog

signal.

This signal is having some value at every instant of time.

Ex; sinewave, cosinewave, triangular wave etc.

Mathematical expression

Characteristics:

• For every fix value of t, x(t) is periodic in nature

• If the frequency (1/t) is increased then the rate of oscillation also changes.

# Discrete time signal

In this case the value is specified only at a specific time i.e. discrete interval of time.

It is generated from continuous time signal by using sampling operation.

Characteristics

• They are identical when their frequencies are separated by integer multiple of 2
pi’
• If the frequency of discrete time sinusoidal is a rational number then its periodic
in nature.
• The highest oscillation is obtained when angular frequency

Periodic and Non periodic signals

# Periodic signal

A signal which repeats itself after fixed time period or interval is called periodic signal.

This is called as condition of periodicity.

For discrete time signal,

# Non periodic signal

A signal which does not repeat itself after a fixed time period or does not repeat at all.

IMP CONDITION FOR PERIODICITY

A discrete time sinusoidal signal is periodic only if its frequency is rational. That means
frequency should be in the form of two integers.
Even and Odd signals

An even signal is also called as symmetrical signal.

A continuous time signal x(t) is said to be symetrical or even if it satisfies the following
condition.

A discrete time real valued signal is said to be symmetric if ,

Odd (anti symmetric) signal

A continuous time signal x(t) is said to be anti-symmetric if it satisfies the following

condition.

Similarly for discrete time signal,

Energy and Power signals

#Power signals

Average
Instantaneous Normalised
normalised
power power
power

a. Instantaneous power
 b. Normalised power

c. Average normalised power

#Energy signal
Deterministic and Random signals

Deterministic signals

A signal which can be described by a mathematical expression, loop up table or some

well defined rule is called as deterministic signal.

Ex: cosine wave, square wave, etc.

Random signal

A signal which cannot be described by any mathematical expression is called as

random signal.

Due to this it is not possible to predict about the amplitude of such signals at a given
instant of time.

Ex : noise
Standard signals:

Impulse

Step
Ramp
Pulse,

Real and Complex exponential

Real Exponential Signals

An exponential signal or exponential function is a function that literally represents an

exponentially increasing or decreasing series.

Continuous-Time Real Exponential Signal

A real exponential signal which is defined for every instant of time is called continuous
time real exponential signal. A continuous time real exponential signal is defined as
follows −

𝑥(𝑡) = 𝐴𝑒𝛼𝑡

Depending upon the value of 𝛼, we obtain different exponential signals as −

• When 𝛼 = 0, the exponential signal x(t) is a signal of constant magnitude for all
times.

• When 𝛼 > 0, i.e., 𝛼 is positive, then the exponential signal x(t) is a growing
exponential signal.

• When 𝛼 < 0, i.e., 𝛼 is negative, then the signal x(t) is a decaying exponential
signal.
Discrete-Time Real Exponential Signal

A real exponential signal which is define at discrete instants of time is called a discrete-
time real exponential signal or sequence. A discrete-time real exponential sequence is
defined as −

𝑥(𝑛) = 𝑎𝑛 for all 𝑛

Depending upon the value of a the discrete time real exponential signal may be of
following type −

• When a < 1, the exponential sequence x(n) grows exponentially.

• When 0 < a < 1, the exponential signal x(n) decays exponentially.

• When a < 0, the exponential sequence x(n) takes alternating signs.

These three signals are graphically represented in Figure-2.

Complex Exponential Signals

An exponential signal whose samples are complex numbers (i.e., with real and
imaginary parts) is known as a complex exponential signal.

Continuous-Time Complex Exponential Signal

A continuous time complex exponential signal is the one that is defined for every instant
of time. The continuous time complex signal is defined as −

𝑥(𝑡) = 𝐴𝑒𝑠𝑡
Where,

• A is the amplitude of the signal.

• s is a complex variable.

The complex variable s is defined as,

𝑠 = 𝜎 + 𝑗𝜔

Therefore, the continuous time complex function can also be written as

𝑥(𝑡) = 𝐴𝑒(𝜎+𝑗𝜔)𝑡 = 𝐴𝑒𝜎𝑡𝑒𝑗𝜔𝑡

⟹ 𝑥(𝑡) = 𝐴𝑒𝜎𝑡(cos 𝜔𝑡 + 𝑗 sin 𝜔𝑡)

Depending upon the values of 𝜎 and 𝜔, we obtain different waveforms as shown in

Figure-3.

Discrete-Time Complex Exponential Sequence

A complex exponential signal which is defined at discrete instants of time is known as

discrete-time complex exponential sequence. Mathematically, the discrete-time
complex exponential sequence is defined as,

x(n)=anej(ω0n+φ)=ancos(ω0n+φ)+jansin(ω0n+φ)x(n)=anej(ω0n+φ)=ancos⁡(ω0n+φ)+j
ansin⁡(ω0n+φ)

Depending on the magnitude of a, we obtained different types of discrete-time complex

exponential signals as,

• For |𝑎| = 1, both the real and imaginary parts of complex exponential sequence
are sinusoidal.
 • For |𝑎| > 1, the amplitude of the sinusoidal sequence increases exponentially.

• For |𝑎| < 1, the amplitude of the sinusoidal sequence decays exponentially.

The graphical representation of these signals is shown in Figure-4.

Sinusoidal.
Classification of systems:

1. Continuous-Time (CT) and Discrete-Time (DT) Systems:

• CT Systems: Process signals that are continuous functions of time.

o Example: Analog filters, RC circuits.

• DT Systems: Process signals that are defined at discrete points in time.

o Example: Digital filters, microprocessor-based systems.

2. Memory and Memoryless Systems:

• Memory Systems: Output depends not only on the present input but also on
past inputs.

o Example: Integrators, delay systems.

• Memoryless Systems: Output depends only on the present input.

o Example: Resisters, multipliers.

3. Causal and Non-causal Systems:

• Causal Systems: Output at any time depends only on present and past inputs.

o Example: Real-time systems.

• Non-causal Systems: Output at a given time depends on future inputs.

o Example: Systems that process recorded data.

4. Inverse Systems:

• A system is invertible if there exists another system that can undo the operation
of the original system.

o Example: The inverse of a delay system is an advance system.

5. Stable and Unstable Systems:

• Stable Systems: Bounded input produces a bounded output.

o Example: Low-pass filters.

• Unstable Systems: Bounded input can produce an unbounded output.

o Example: Systems with positive feedback.

6. Time-Variant and Time-Invariant Systems:

• Time-Invariant Systems: System characteristics do not change with time.

o Example: Linear time-invariant (LTI) systems.

• Time-Variant Systems: System characteristics change with time.

o Example: Systems with time-varying coefficients.

7. Linear and Non-linear Systems:

• Linear Systems: Obeys the superposition principle (additivity and homogeneity).

o Example: LTI systems.

• Non-linear Systems: Does not obey the superposition principle.

o Example: Amplifiers with saturation.