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Overview of Signals and Systems – Types and differences

You’ll come across a variety of signals and systems in this digital signal processing course. It might get a bit confusing to remember them all. But eventually, you’ll get a hang of them. You can always keep returning to this post for a quick reference. Here, we have enlisted all the types of signals and systems and the differences between them. Let’s start with some elementary signals that you’ll use quite frequently.

Elementary Signals

Here is a table of all the different elementary signals, their graphs and several important properties to take note.

Unit Step Signal

Equation

Equations for Unit Step Signal

Graphs
Graphical representation of the Unit Step Function in Continuous Time
Unit Step Signal in Continuous Time
Graphical representation of the Unit Step Function in Discrete Time
Unit Step Signal in Discrete Time
Properties

The unit step function is denoted by u(t) or u(n). Since it is UNIT step, the amplitude must be 1.

1) { (u(t)) }^{ n }={u(t)} since 1 to the power of nay integer will give you 1.

2) { (u(t-{ t }_{ o })) }^{ n }={ u(t-{ t }_{ o })}

3) u(at)=u(t)

u(at-to )=u(a(t-(to /a)))=u(t-(to /a))

Time scaling is not applicable to a unit step signal.

4)For { u(t-{ t }_{ o })}, we have to perform time shifting. Let us take t-to=0, hence we will get t=to which means t will be shifted from 0 to the value of to .

For example, let to=0 be 2. Therefore u(t-2) will be the figure shown below.

Graph of u(t-2)

Impulse signal

Equation

Impulse Signal Equation

Graphs
Graphical Representation of an Impulse Signal in Continuous Time
Impulse Signal in Continuous Time
Graphical Representation of Impulse Signal in Discrete Time
Impulse Signal in Discrete Time
Properties

The Unit Impulse Function is given by 𝛿(t) or 𝛿[n].

  • \int _{ -\infty }^{ \infty }{ \delta (t)dt\quad =1 }
  • 𝛿(n-k) = 1 (for n=k); 𝛿(n-k) = 0 (for n!=k)
  • 𝛿(n)=u(n)-u(n-1)
  • f(t) 𝛿(t)=f(0)
  • 𝛿(t-to)f(t)=f(to)
  • \delta (kt)=\frac { 1 }{ \left| k \right| } \delta (t)
  • 𝛿(-t)=𝛿(t)

Ramp

Equation

Equation for Ramp Signal

Graphs
Graphical Representation of Ramp Signal in Continuous Time
Ramp Signal in Continuous Time
Graphical Representation of Ramp Signal in Discrete Time
Ramp Signal in Discrete Time
Properties

The Ramp Signal is denoted by r(t)or r(n)

  • r(t)=t.u(t)

Therefore,

    • \int _{ -\infty }^{ \infty }{ u(t)dt=r(t) }
    • \frac { dr(t) }{ dt } =u(t)

also, since \frac { du(t) }{ dt } =\delta (t)

    • \iint { \delta (t)=r(t) }
    • \delta (t)=\frac { { d }^{ 2 }r(t) }{ { dt }^{ 2 } }
  • It is neither an Odd nor an Even Signal
  • It is also neither an Energy nor a Power Signal

Parabolic

Equation

Equation for Parabolic signal

Graphs
Graphical Representation Parabolic Signal in Continuous Time
Parabolic Signal in Continuous Time
Graphical Representation of Parabolic Signal in Discrete Time
Parabolic Signal in Discrete Time
Properties
  • x(t)=\int r(t)\quad =\iint { u(t)\quad =\iiint { \delta (t) } }
  • \frac { dx(t) }{ d(t) } =r(t)
  • It is neither an Odd nor an Even Signal
  • It is also neither an Energy nor a Power Signal

Signum

Equation
Equation for Signum Signal in continuous time Equation for Signum Signal in discrete time
Graphs
Graphical Representation of Signum Signal in Continuous Time
Signum Signal in Continuous Time
Graphical Representation of Signum Signal in Discrete Time
Signum Signal in Discrete Time
Properties
  • At t=0, the Signum Signal is discontinuous, Hence, we cannot differentiate this function at 0.
  • x(t)=2.u(t)-1
  • \frac { dx(t) }{ d(t) } =2\delta (t)
  • Can also be represented as x(t)=\begin{cases} \frac { \left| t \right| }{ t } \quad t\neq 0 \\ 0\quad \quad \quad \quad t=0 \end{cases}
  • x(t)=\frac { d\left| t \right| }{ dt } ,\quad t\neq 0

Exponential

Equation
x(t)=A{ e }^{ \alpha t }
Graphs
Graphical Representation of Exponential Signal when α=0
Exponential Signal when α=0
Graphical Representation of Exponential Signal when α is positive
Exponential Signal when α is Positive (α>0)
Exponential Signal when α is negative
Exponential Signal when α is Negative (α<0)
Properties
  • α>0 -> Exponentially Growing; α<0 -> Exponentially Decaying
  • When multiplied with unit step signal, it gives only the portion on the graph on the right of they-axis on the exponential graph. If the sign is changed, one has to multiply accordingly.

For example u(t)e^{ 3t }

Graphical representation of the Unit Step Function in Continuous Time

multiplied by

 

Graph of e to the power of 3 t

Rectangular

Equation
x(t)=A.rect(\frac { t }{ T } )

A->Amplitude of Rectangle

T->Period of Rectangle

Graphs
Graphical Representation of Rectangular Signal in Continuous Time
Rectangular Signal in Continuous Time
Properties

If given A=5 and T=4

x(t)=5rect(t/4)

Hence, the graph will look like the figure below:

Example of Rectangular Signal with Amplitude 5 and time period 4

Triangular

Equation
x(t)=A(1-\frac { \left| t \right| }{ T } )

A->Amplitude of the Triangular Signal

T->Time period of the Triangular Signal

Graphs
Graphical Representation of a Triangular Signal
Triangular Signal in Continuous Time
Properties

If given A=3 and T=5

x(t)=3(1-|t|/5)

Hence, the graph will look like the figure below:

Example of Triangular Signal with Amplitude 3 and time period 5

Sinusoidal

Equation
x(t)=Asin({ \omega }_{ o }t+\phi )

or

x(t)=Acos({ \omega }_{ o }t+\phi )

A->Amplitude

ωo->Fundamental Frequency

φ->Phase Shift

Graphs

 

Graphical Representation of Sinusoidal Signal in Continuous Time
Sinusoidal Signal in Continuous Time(Sine)
Graph of Cosine Signal
Sinusoidal Signal in Continuous Time(Cosine)
Properties
  • ωo=2πf
  • f=ωo/2π
  • T=1/f=2π/ωo

If given A=1 and ωo=1,

f=1/2π

T=2π

The graph of the signal will be as shown below:

Example of cosine signal with amplitude 1 and time period 2pi

Sinc

Equation
Sinc(\lambda )=\frac { sin(\pi \lambda ) }{ \pi \lambda } .\quad where\quad \lambda =\pm 1,\pm 2,\pm 3...
Graphs
Graphical Representation of a Sinc Signal in Continuous Time
Sinc Signal in Continuous Time
Properties
  • Denoted as sinc(λ)
  • sinc(0)=1
  • It is a normalized function

Types Of Signals

Signals can be classified in many ways to help one understand the characteristics of the signal they are analyzing

What is the difference between continuous-time and discrete-time signals? 

Continuous-Time Signals Discrete-Time Signals
A Continuous-Time Signal is defined for all values of time. X is the dependent variable and t is the independent variable. When there is an X(t) for every single value of t, it is continuous. Discrete-Time Signals are defined only at certain discrete values referred to as n and denoted in square brackets.
For example, sinusoidal graphs which have the time limit of infinity to negative infinity are clearly continuous-time signals.

 

Example of Continuous Signal
Example of Continuous Signal
For example, the graph below is of that of a discrete signal

Example of Discrete Signal
Example of Discrete Signal

 

Each time period following 0 can be taken as t1,t2,t3… The difference between these time periods is represented by 𐤃t. If 𐤃t=t1-t0=t2-t1=…, it is a uniformly sampled. Since 𐤃t is equal for each point on the time period, we assume 𐤃t=T. That is why the T in x(nT) is cancelled out as all the time period values have a T in them, and the graph is denoted as X[n].

Discrete values can also be represented within curly braces, where an arrow points at the value when n=0.

Alternative way to represent discrete signal
Discrete signal
A common misconception is that discrete and digital signals are congruous but they are in fact very different. For discrete-time signals, time is discrete while the amplitude is continuous. However, for digital signals both the amplitude and time are discrete.

What is the difference between deterministic and non-deterministic signals? 

Deterministic signals
Non-Deterministic signals
A signal is said to be Deterministic when there is no doubt with respect to value at any instant of time. It can be easily identified and represented as a function, X(t). A signal is said to be Non-Deterministic when it is basically the opposite of that of a Deterministic signal, which means there is uncertainty with respect to value at any instant of time. For example, random signals such as voice signals, there is no fixed pattern as to how it is going to fluctuate and hence it cannot be represented as a function.
Deterministic signal
Deterministic signal
Non-Deterministic Signal
Non-Deterministic Signal

What is the difference between even and odd signals? 

Even Signals Odd Signals
Even signals remain identical when they are reflected or mirrored across the y-axis. Odd signals are anti-symmetrical when they are reflected or mirrored across the y-axis.
For an even signal, x(-t) = x(t) for every value of t. For an odd signal, x(-t) = -x(t) for every value of t.
For example, let us take a sinusoidal signal x(t) = cos(ωt).

x(-t)=cos(-ωt)=cos(ωt)=x(t)

Therefore, cos(ωt) is an even signal.

For example, let us take a sinusoidal signal x(t) = sin(ωt).

x(-t)=sin(-ωt)=-sin(ωt)=-x(t)

Therefore, sin(ωt) is an even signal.

Similarly, triangular and rectangular signals that have their midpoint at t/n=0 are also examples of even signals x(t)=t, x(t)=t3 are other examples of odd signals.
Sum of two or more even functions, the product of two or more even functions or product of even and odd function results in even function. For a signal to be odd:

  • When t=0, the value of x(t)=0, and the signal always passes through the origin.
  • And also the mean or average of the signal should be 0.
  • Sum of two or more odd functions results in odd. Also, the product of odd numbers of odd functions(such as the product of 3 odd functions or any other odd number) results in odd.

What is the difference between periodic and non-periodic signals? 

Periodic Signals Non-Periodic Signals
A signal is said to be periodic when it repeats itself for a regular interval of time. When it is periodic, it can be represented as below

x(t)=x(t+{ \eta t }_{ 0 })

η-> Integer value

to->fundamental period which is the smallest positive value of time for which signal is periodic, it may or may not be an integer. (this value cannot be 0 or 2π)

A signal is said to be non-periodic when it has no pattern, when it does not repeat itself for a regular interval, no matter how large the time interval. The frequency components change over time.
With the fundamental period, we can find the fundamental frequency and the fundamental angular frequency

fo=1/to   Cycles per second /Hz

ωo=2πfo=2π/to   Radians per second 

For example, the sine signal

Example of cosine signal with amplitude 1 and time period 2pi

It has a period of 2π, hence that is the fundamental time period. And it can be represented as

sinθ=sin(θ+n2π) where n=0,1,2,…

Almost all the signals we see in life are non-periodic such as speech signals.

What is the difference between energy and power signals?

Energy Signals Power Signals
The energy of a signal can be calculated using the formula

E=\int _{ -\infty }^{ \infty }{ { \left| x(t) \right| }^{ 2 }dt } for both periodic and non-periodic signals
The power of a signal can be calculated using the formula

P=\frac { 1 }{ { T }_{ o } } \int { { \left| x(t) \right| }^{ 2 } } dt for a periodic signal

P=\begin{matrix} lim \\ t\rightarrow \infty \end{matrix}\frac { 1 }{ T } \int _{ -T/2 }^{ T/2 }{ { \left| x(t) \right| }^{ 2 } } dt for a non-periodic signal
Energy of power signal is infinity Power of energy signal is 0
Majority of the non-periodic signals are energy signals Majority of the periodic signals are power signals
Energy signals are time-limited, they exist only for a short period of time Power signals are however not time-limited, they exist over an infinite period of time

What is the difference between real and imaginary signals?

Real Signals Imaginary Signals
A signal is said to be real when it satisfies condition x(t)=x*(t) A signal is said to be imaginary when it satisfies condition x(t)=-x*(t)
For a real signal imaginary part must be 0 For an imaginary signal real part must be 0

x*(t) means to find the complex conjugate.

You simply have to change the sign of the imaginary part of the complex conjugate.

Types of Systems

A system is a collection of hardware resources that form a device that processes or modifies an input to give an output. A filter can be considered as a system; a signal is sent through the filter, the filter modifies it based on the required frequency and we have the processed output much like the block diagram below.

Simple Block Diagram of a system
Simple Block Diagram of a system

 

What is the difference between linear and non-linear systems?

Linear Systems Non-Linear Systems
A system is said to be linear if it satisfies the superposition principle which is when it satisfies both the Law of Additivity and the Law of Homogeneity. A system is said to be non-linear if it does not satisfy the Law of Additivity and the Law of Homogeneity and hence the Superposition principle.
Law of Additivity:

y1(t)=x1(t)

y2(t)=x2(t)

y3(t)=x3(t)

y’3(t)=y1(t)+y2(t)

y3(t)=y’3(t), then it is linear

For Additivity:

y1(t)=x1(t)

y2(t)=x2(t)

y3(t)=x3(t)

y’3(t)=y1(t)+y2(t)

y3(t)≠y’3(t), then it is non-linear

For example, system is given by

y(t)=x2(t)

y1(t)=x12(t)

y2(t)=x22(t)

y3(t)=x32(t)

y’3(t)=y1(t)+y2(t)= x12(t)+ x22(t)

Since y3(t)=y’3(t), the system is linear

For example, system is given by

y(t)=2t+x(t)

y1(t)=2t+x1(t)

y2(t)=2t+x2(t)

y3(t)=2t+x3(t)

y’3(t)=y1(t)+y2(t)=4t+x1(t)+x2(t)

Since, y3(t)≠y’3(t), the system is non-linear.

Law of Homogeneity:

y(t)=x(t)

y(t)–>ky(t)

x(t)–>kx(t)

if ky(t)=kx(t), then it is linear

For Homogeneity:

y(t)=x(t)

y(t)–>ky(t)

x(t)–>kx(t)

if ky(t)≠kx(t), then it is non-linear

For example, if a system is given by y(t)=x(sin(t))

ky(t)=kx(sin(t))

kx(t)=kx(sin(t))

Since kx(t)=ky(t), the system is linear

For example, if a system is given by y(t)=2+x(t)

ky(t)=2k+kx(t)

Since kx(t)≠ky(t), the system is non-linear

What is the difference between time-variant and time-invariant systems?

Time-Variant systems Time-Invariant systems
A system is considered time-variant if its input and output characteristics change with time. You can figure this out by adding a -to and substituting t with t-to. If y(t,to)≠y(t-to), then the system is time-variant. A system is considered time-invariant if its input and output characteristics do not change with time. You can figure this out by substituting t with to and to-k. If y(t,to)=y(to-k), then the system is time-invariant.
For example, given a system

y(t)=2t+x(t)

y(t,t0)=2t+x(t-to)

y(t-t0)=2(t-to)+x(t-to)

Since y(t,to)≠y(t-to), system is time-variant.

For example, given a system

y(t)=2+x(t)

y(t,to)=2+x(t-to)

y(t-t0)=2+x(t-to)

Since y(t,to)=y(to-k), the system is time-invariant

What is the difference between linear time-variant and linear time-invariant systems?

Linear Time-Variant Linear Time-Invariant
A system is said to be Linear Time-Variant when it satisfies both Linearity and Time variance. You can check if it satisfies by conducting the test mentioned above. A system is said to be Linear Time-Invariant when it satisfies both Linearity and Time variance. You can check if it satisfies by conducting the test mentioned above.

What is the difference between static and dynamic systems?

Static (Memoryless) systems Dynamic (Memory) systems
A system is said to be Static when the output of the system depends only on the present values of the input. A system is said to be Dynamic when the output of the system depends on past or future values of input at any instant of time.
when y(t)=x(t), it is considered as present input when y(t)=x(t-k) where k is any positive real number, it is considered as past input

when y(t)=x(t+k) where k is any positive real number, it is considered as future input

Therefore, y(t)=x(t)+2, y(t)=x2(t) are examples of Static systems. Therefore, y(t)=sin(t), y(t)=x(2t), y(t)=x(t)x(t-1) are examples of Dynamic systems.

What is the difference between causal and non-causal systems?

Causal systems Non-causal systems
A system is considered causal if the output response is dependent upon the present and past inputs only. A system is considered causal if the output response is dependent upon the present and past inputs only.
Take note all static systems are causal but not all causal systems are static. Take note all non-causal systems are dynamic but not all dynamic systems are non-causal
For example, given a system y(n)=x(n)+1/x(n-1)

This system depends on the present input, x(n) and the past input, x(n-1). Hence, it is a causal system.

For example, given a system y(n)=x(n)+1/2x(n+1)

This system depends on the present input, x(n) and the future input, x(n+1). Hence, it is a non-causal system.

What is the difference between invertible and non-invertible systems?

Invertible systems Non-Invertible systems
A system is considered invertible if there is a one-to-one mapping between input and output at every instant of time. A system is considered Non-invertible if a many-to-one mapping exists between the input and the output
For example,

y(t)=x(t)+2

where when x is 1, y is 3. When x is 2, y is 4 and so on. Hence, one input can only be mapped to one output. The system is invertible.

For example,

y(t)=x2(t)

where when y is 4, x can be either 2 or -2. The output 4 can be mapped to more than one input. Therefore, the given system is non-invertible.

What is the difference between stable and unstable systems?

Stable systems Unstable systems
A system is said to be stable when the output of the system is bounded for a *bounded input at all instants of time, in other words, from infinity to negative infinity. A system is said to be Unstable when the output of the system is not bounded for a bounded input at all instants of time, in other words from infinity to negative infinity.
Examples of bounded signals include unit step signals and sinusoidal signals. Examples of unbounded signals include exponential signals and y(t)=1/x(t).
Given a system y(t)=2x2(t)

let x(t)=u(t),

y(t)=2u2(t)=2u(t)

u(t) is bounded,and a constant value is also a bounded input. Therefore, y(t) will be a bounded output, and also, a stable system.

Given a system y(t)=t.x(t)

Let x(t)=u(t).

y(t)=t.u(t)=r(t) (refer to the table on elementary signals)

r(t) is not a bounded input. Hence, the system is unstable.

Bounded input: If a value is bounded, as the name suggests, it means that there are limits to the value and these limits are essentially finite. If none of the extremes are a finite value(not infinity), that value is said to be unbounded.

Now that we have had a quick revision of all the types of signals and systems, we can get started with the core concepts of this DSP course.

Sources: The signals and Systems Playlist in this link.

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