Numerical relays provide several advantages over electromechanical and solid state relays including improved reliability, flexibility, and performance. They allow for advanced protection schemes through programming and use of microprocessors. Numerical relays also simplify interfacing with current and potential transformers, enable advanced functions like phasor measurement, and provide time-stamped fault data for analysis. Their main advantages are reliability, security, dependability, and ability to implement new protection schemes through programming.

1. Numerical Relaying

2. Why Numerical Relaying
.
• The first and foremost driving force for advances in relaying systems is the need to
improve reliability.
• In turn, this implies increase in dependability as well as security.
• This need to improve reliability propelled the development of solid state relays.
• Solid state relays have inherent self checking facility which was not available with
electromechanical relays.
• This feature is also available with numerical relays
• For example, when we boot a computer, it goes through a self checking phase
where in it checks RAM, hard disk, etc.
• Also, with the reduced cost of computer hardware, and an exponential growth in
processing capability, numerical relays can provide high performance at moderate
costs.
• Since, numerical relays are based on digital technology, they are more or less
immune to variation or drift in parameters of individual components like OP-AMPS
etc. due to changes in temperature, ageing etc.

3. Why Numerical Relaying
• Numerical relays also help in reducing burden (volt-amperes) of Current
Transformer (CT) and Voltage Transformer (VT).
• This is desirable because ideally sensors should not consume any power.
• If a sensor consumes energy from the measure end, it will automatically
distort the signal.
• This problem is further aggravated in CTs due to non-linearity of iron core.
• Numerical relays offer very low impedance to the secondary of CT and
hence reduce burden on CT.

6. Why Numerical Relaying
• Numerical relays permit much more flexibility than their electromechanical and
solid state counterparts.
• In electromechanical relays, the constructional details like magnetic path, air gap
etc., are used to design various operating characteristics.
• Since, solid state relays mainly use analog circuit, they permit more innovation than
corresponding electromechanical relays which are no doubt robust.
• However, solid state relays can not have the kind of flexibility that computer aided
relaying can have.
• For example, providing magnitude scaling and phase shift to a voltage signal to
generate line to line voltage from phase to neutral voltage is much simpler with
computer aided relaying because it can be handled by the program.
• A computer relay can be programmed. Further, due to the programming feature, it
is possible to have generic hardware for multiple relays, which reduces the cost of
inventory.
•

7. Why Numerical Relaying
• The first protection devices based on microprocessors were employed in 1985.
• The widespread acceptance of numerical technology by the customer and the
experiences of the user helped in developing the second generation numerical relays
in 1990.
• Modern power system protection devices are built with integrated functions.
• Multi-functions like protection, control, monitoring ,measuring ,event recording,
and communication are available today in numeric power system protection
devices.
• Also, the communication capability of these devices facilitates remote control,
monitoring and data transfer.
• The relays’ volumes as well as the panel wiring was significantly reduced due to the
integration of functions and the use of serial communication.
• Traditionally, electromechanical and static protection relays offered single-function,
single characteristics, whereas modern numeric protection offers multi-function and
multiple characteristics.
• Some protections also offer adaptable characteristics, which dynamically change
the protection characteristic under different system conditions by monitoring the
input parameters.

8. Why Numerical Relaying
• The measuring principles and techniques of conventional relays (electromechanical
and static) are fewer than those of the numerical technique, which can differ in
many aspects like the type of protection algorithm used, sampling, signal
processing, hardware selection, software discipline, etc.
• First generation numerical relays were mainly designed to meet the static relay
protection characteristic, whereas modern numeric protection devices are capable of
providing complete protection with added functions like control and monitoring.
• The second generation of numerical protection relay is equipped with powerful
digital signal processor, more reliable surface-mount construction, improved
algorithms and protection elements.
• more reliable, more secure, and more dependable operation.
• These relays have built-in test routines, a “watchdog”, that signals an alarm output,
& the operators know when the relay fails.
• The use of global positioning system (GPS) allows providing time synchronization
to ±1μs accuracy, a thing which proves the high precision character of this
technique.
• Numerical protection devices offer several advantages in terms of protection,
reliability, and trouble shooting and fault information.
• Numerical protection devices are available for generation, transmission and
distribution systems.

9. Why Numerical Relaying
• Perhaps the most attracting feature of these numerical relays was the ability of
communication that offered new horizons for protection and protection related
applications.
• At present, there are many advanced communication techniques & protocols which
can be used to improve protection, control, operation analysis, maintenance
functions and planning, which allows the engineer to operate, test, maintain &
access realtime relay information.
• Information is the basic constituent in any protection scheme.
• It is critical to communicate & process this information in an efficient and
economical way.
• The key element in a communication system is the physical medium used to convey
information through the system.
• There are many different types of communication media such as twisted pair cable,
coaxial cable, fibre optic cable and wireless communication.
• The wireless networks are by far the most popular choice for new network
algorithm.

10. Why Numerical Relaying
• Numerical relaying along with developments in fiber optic communication
have pioneered development of automated substations.
• Once, the analog signals from CTs and VTs are digitized, they can be
converted to optical signals and transmitted on substation LAN using fiber
optic network.
• With high level of EMI immunity offered by fiber optic cable, it has
become the transmission medium by choice in substation environment.
• Numerical relays can be nicely interfaced with a substation LAN.
• This in turn should be contrasted with legacy substations where in lead
wires have to run from each CT and VT to the control panel.
• This not only reduces wiring complexity in the substation but also reduces
burden on the CT as resistances of long lead wires are eliminated.
• Further, a single fiber optic LAN permits multiplexing of multiple analog
signals which is not possible with legacy arrangement.

12. Why Numerical Relaying
• Numerical relays also permit development of new functions as well as development of
adaptive relaying schemes.
• Traditionally, relaying systems are designed and set in a conservative manner.
• They represent compromise between:
• economy and performance
• dependability and security
• complexity and simplicity
• speed and accuracy
• credible and conceivable
• Adaptive relaying is meant to minimize such compromises and also allow relays to fine-tune
to existing system conditions.
• Numerical relays also permit storage of pre and post fault data (of the order of few cycles).
• This data can also be time stamped, now-a-days by Geographical Positioning System (GPS).
• GPS systems (a cluster of 24 satellites of pentagon, USA) not only provides positional
information but also a time pulse every second for synchronization of sampling.

13. Why Numerical Relaying
• Efficient power transmission and distribution would benefit from synchronized near-real-time
measurements of voltage and current phasors at widely dispersed locations in an electric
power grid.
• Such measurements also have the promise to enable effective real-time system monitoring
and control, which have been considered to be the key to preventing wide-scale cascading
outages
• A phasor measurement unit (PMU), also known as synchrophasor, is a device capable of
measuring power system voltage and current phasors at a rate of thousands of samples per
second
• The samples are time-stamped with 1 microsecond or better accuracy to a common absolute
time reference provided by the Global Positioning System (GPS) receivers attached to PMUs
• IRIG-B is the most commonly used time standard in US substations.
• IRIG-B is a standard for representing time
• It does not define the physical port.
• Can be also be transmitted over RS232, RS485 and serial fiber opt
• IRIG-B is used to synchronize the internal clocks of RTUs and IEDs(Intelligent Electronic
Device)
• IRIG-B can ensure accuracy of time-stamped events from different substation to be accurate
to 1ms.

14. Why Numerical Relaying
• Thus, in principle, every sample and every event like closing or opening of breakers
can be time stamped.
• This helps in postmortem analysis which is used to determine whether (1) a relay
operated correctly (or incorrectly) and (2) any other relaying system or device (like
circuit breaker) has failed to operate.
• Time stamping of relay operation allows us to capture the sequence of relay
operations.
• Thus, in a complex situation like catastrophic failure of the power system (brown
out or black out), it is now possible to precisely determine the sequence of relay
operations.
• This helps engineers to capture and simulate the disturbance using transient
stability, (EMTP) programs.
• Such simulation studies help in understanding shortcomings of the existing systems
and thereby improving them.
• In this role, a numerical relay is analogous to a fault data recorder (FDR).

15. Why Numerical Relaying
•Numerical relays also simplify
interfacing with CTs and VTs.
•Consider a protective function which
requires zero sequence voltage.
•Traditionally, it would be generated by
open delta VT connection shown in
fig.
•If zero sequence current is also
required, it is obtained by using an
additional CT in the ground wire.
• With numerical relays, zero sequence
voltages and currents can be derived
inside the processor from the phase
voltage (Va, Vb, Vc) and line currents
(Ia, Ib and Ic).

16. Why Numerical Relaying
• In differential protection e.g., three phase transformer protection, traditional
protection schemes also require additional care to handle polarity, scaling and
phase shifting problems.
• This may even necessitate use of an auxiliary CT.
• Such complications can be resolved with ease when numerical relays are used.

17. Numerical relay models
• Numerical relay models can be divided into two categories.
• The models of the first category consider only the fundamental frequency
components of voltages and currents.
• Phasor based models were the first to be widely used to design and apply relays.
• The models of the second category take into consideration the high frequency and
decaying DC components of voltages and currents in addition to the fundamental
frequency components.
• These models are called transient models of relays.
• Due to processing capacity of numerical relays many protection criteria can be
implemented.
• Protection relays, such as other monitoring and control equipments have taken the
advantage from the increasing improvement of the semiconductor industry and the
enormous number of digital signal processing and control algorithms.
• The latest generations of protective relays be provided with a large capacity of
processing capabilities become more efficient and can perform a numerous number
of functions such as fault locators, integrated monitoring and control functions.

18. Drawback of numerical relay
Short life cycle

19. Susceptibility to transients

21. Relay Hardware
• Block Diagram
• Fig below shows the functional block diagram of a digital relay.
• It can be seen that a digital relay consists of:
• Analog input subsystem
• Digital input subsystem
• Digital output subsystem
• A processor along with RAM (data scratch pad), main memory (historical data file) and power
supply.

22. Analog Input Subsystem
• The 3-Ø voltage and current signals are analog in nature.
• Isolation and analog signal scaling
• Current and voltage waveforms from instrument transformers are acquired and scaled down
to convenient voltage levels for use in the digital and numerical relays.
• Finally, an antialiasing filter is used after signal conditioning hardware.
• Analog anti-aliasing filtering
• Low-pass filters are used to avoid the phenomena of aliasing in which the high frequency
components of the inputs appear to be parts of the fundamental frequency components.
• Anti aliasing filter is a low pass filter (LPF) used to cut off the high frequency content
(including noise) in the input signal.
• The analog inputs must be applied to low-pass filters and their outputs should be sampled and
quantized.
• The use of low-pass filter is necessary to limit the effects of noise and unwanted components
of frequencies.
• The cutoff frequency of LPF and the sampling rate have to be properly matched.

24. Analog-to-digital conversion (ADC)
• Because digital processors can process numerical or logical data
only, the waveforms of inputs must be sampled at discrete times. To
achieve this, each analog signal is passed through a sample- and-
hold module, and conveyed, one at a time, to an Analog-to-Digital
Converter (ADC) by a multiplexer (Mux), as shown in figure.
• This functionality is provided by the analog input subsystem.
• Typically, it consists of sample and hold circuit, Analog to Digital
Converter (ADC) and multiplexer interfaced to the processor.
• The digital input data consists of Circuit Breaker (CB) status (open
or close).
• The digital output is relay's operate / do not operate decision.
• Once, the data is acquired within RAM, it is filtered by a digital
filter and processed by the relay logic

25. Sample and Hold Circuit
• The analog information is held by a Sample and Hold circuit.
• Any A/D converter requires a finite conversion time.
• A S & H circuit which conceptually is a shunt capacitor with a switch holds the
information (in terms of voltage).
• While the conversion takes place, switch is in open position.
• This is known as the `hold' state. When the switch is closed, the Vout of S and H
follows the Vin.

26. Sampling Scheme
• There are two commonly used schemes for configuring the analog input
subsystem.
• One is known as the 'simultaneous' & other 'non-simultaneous' scheme.
• Non-Simultaneous Sampling Scheme
• Fig below illustrates non-simultaneous sampling scheme.
• In this scheme, a multiplexer selects the analog channel sequentially.
• Typically, power system applications involve more than one analog input.
• To reduce the cost of the hardware, multiple channels are multiplexed through
analog multiplexer to a single ADC.
• An analog multiplexer permits a single output line to mirror the signal at the
selected input, say one of the 3 voltages/ 3 currents.

27. Non-Simultaneous Sampling Scheme
In the scheme illustrated in figure, it can be
observed that In the Non-simultaneous
sampling scheme, relative pharos information
between two signals is not preserved.
•This is because the samples from different inputs
are not obtained at same instant of time.
•One way to overcome, this hardware limitation is
to interpolate the value of the sample from
previous values.
•Fig illustrates the concept.
•Let Va(t) be sampled first and then Vb(t) be
sampled.
•The first two samples of 'a' & 'b' phases are given
by points 'A' and 'C'. After one sampling interval,
samples 'B' and 'D' are obtained, for phases 'a' and
'b' respectively.
•The problem is to estimate value of Vb(t) at the
sampling instant for 'a' i.e. say at sample 'B'.
•This can be obtained by linear interpolation for
samples 'C' and 'D' and corresponding to point 'E'.

28. Simultaneous Sampling Scheme
•Fig shows a simultaneous sampling
scheme.
• In this scheme, all S&H amplifiers are set
to hold state simultaneously.
•This preserves the relative phase
information between multiple analog
signals. Then, the multiplexer selects the
channel sequentially.
• Typically, digital relays use successive
ADC which have a conversion time of 15-
30 µs.
• The sampling rate must satisfy Nyquist
criteria.
•
simultaneous sampling scheme

29. Source Impedance
• Most signal sources have impedances of less than 1.5 k Ω, so such a maximum source
impedance is usually not a problem.
• However, faster multiplexer rates require lower source impedances.
• For example, a 1 MHz multiplexer in a 12-bit system requires a source impedance less than
1.0 k Ω.
• When the source impedance exceeds this value, buffering is necessary to improve accuracy.
A buffer is an amplifier with a high input impedance and extremely low output impedance.
(See Figure below)
• A buffer on each channel located between the transducer and the multiplexer ensures higher
accuracies by preventing the multiplexer’s stray capacitance from discharging through the
impedance of the transducer.
Buffering signals ahead of the multiplexer increases accuracy,
especially with high-impedance sources of fast multiplexers

30. Multiplexer
• Thus, multiplexer is a collection of analog switches.
• Each channel can be selected by supplying appropriate binary code to the multiplexer e.g. for 8-
channel multiplexer, 3 bit address space is required.
• A chip disable line permits parallel expansion if external logic is used to select desired multiplexer.
• A multiplexer has two inputs (terminals) for a single channel.
• It provides better noise immunity.
• Accuracy of the analog multiplexer depends on load impedance at the output terminal.
• Typical recommended value is 107 to 108.
• As Sample (S) and Hold (H) circuit has impedance in the range 108 - 1012, no problem is encountered.
• Channel-to-channel cross talk is another non-ideal characteristic of analog switching networks,
especially integrated circuit multiplexers.
• Cross talk develops when the voltage applied to any one channel affects the accuracy of the reading in
another channel.
• Conditions are optimum for cross talk when signals of relatively high frequency and high magnitude
such as 4 to 5 V signals are connected to one channel while 100 mV signals are connected to an
adjacent channel.
• High frequency multiplexing also aggravate cross talk because the signals couple through a small
capacitance between switch channels.
• Low source impedance minimizes the cross talk and eliminates the charge injection.

31. Digital filter
• Digital filters have, for many years, been the most common application of digital
signal processors.
• There are two basic forms of digital filters, the Finite Impulse Response (FIR) filter
and Infinite Impulse Response (IIR) filter.
• The main draw back to the use of IIR filters in digital protection relays is that the
group delay cannot be specified in the design process.
• This makes their use in protection somewhat tedious in general, FIR filters are
usually the preferred type.

32. Relaying hardware for Metering
• In principle, the hardware setup shown in previous fig. can be used for both
measurement and protection function.
• However, considering the order of difference between current magnitudes in case of
fault and load, there can be loss of accuracy during metering applications.
• Consider a hypothetical case where in maximum load current is 100A and maximum
fault current is 20 times this load current (2000A).
• Let a 12 bit unipolar ADC be used for sampling current signal.
• This implies that resolution of ADC is 2000/(2**12-1)=0.488 A.
• This resolution may be inadequate for metering purposes.
• One solution is to increase resolution i.e. the number of bits in ADC.
• For example, one may use 16 bit ADC in place of 12 bit ADC.
• However, increasing the number of bits of ADC also affects the selection of processor.
• A good design guideline is to choose a processor with double the number of bits of
ADC.
• This ensures that truncation and numerical precision problems associated with finite
precision arithmetic do not cause significant loss of accuracy.

33. Relaying hardware for Metering
• For example, with 16 bit ADC, 32 bit processor is the natural choice.
• Alternatively, a variable gain amplifier can be used along with the ADC.
• At low currents, high gain setting is used and at high currents low gain setting
is preferred.
• However, during the change from one setting to another, loss of information
can take place.
• Therefore, a simple solution would be to keep metering and protection
functionality separately.

34. Open System Relaying
• Open system relaying motivated by experiences from energy management field where
in a plethora of manufacturers specific equipment has led to difficulty in expanding
the system without changing the entire existing SCADA (system control and data
acquisition) system.
• Open system movement encourages standard based development, thereby permitting
incremental or evolutionary growth.
• This has to be contrasted with proprietary solutions that required either a complete
changeover or force the utility to a vendor.
• Consider a case of two vendors (A and B) supplying a Remote Terminal Units (RTUs)
to a utility C.
• Let us consider that initially, the utility had procured the SCADA system from the
manufacturer A.
• At a later date, the utility wants to add RTUs from the vendor B because it has cost
and performance benefits.
• If the initial solution provided by vendor A was proprietary, it will not be possible for
RTUs of vendor B to be interfaced with SCADA system supplied by vendor A.
• This restricts cross migration and hence it is unfair.
• On the other hand, if the initial SCADA system was based on open standards, then the
device of another vendor using the same standard could be interfaced with ease.

35. Open System Relaying
• This is the basic idea behind any open systems movement.
• An open relay conceptually consists of two separate 'boxes'.
• The first box is the well known standard computer.
• The second box encloses the scalable analog input subsystem.
• The processing board which may have multiple DSPs is plugged onto PC
motherboard and once programmed can run independently of the PC.
• Such a relay may be interfaced with substaion LAN using standard protocol.

36. Why Digital Signal Processing?
• Digital relaying involves digital processing of one or more analog signals. It
involves following three steps:
• 1. Conversion of analog signal to digital form.
• 2. Processing of digital form.
• 3. Boolean decision to trip or not to trip.
• Usually in DSP, after processing information in discrete domain, it has to be
converted back to analog domain.
• However, for us the step- 3 does not involve conversion of processed signals back
to analog form.

37. Why Digital Signal Processing?
• At this point, a worthwhile observation is that direct analog signal processing is
conceptually much simpler.
• However, advantages of digital processing far outweigh analog processing.
• Some of the advantages of digital processing are as follows:
• Operation of digital circuits do not depend on precise values of digital signals.
• As a result, a digital circuit is less sensitive to tolerances of component values.
• A digital circuit has little sensitivity to temperature, aging and other external
parameters.
• In terms of economics of volume, a digital circuit can be reproduced easily in
volume quantities.
• With VLSI circuits, it is possible to integrate highly sophisticated and complex
digital signal processing systems on a single chip.
• In DSP, accuracy of computation can be increased by increasing word length.
• With the availability of floating point arithmetic in digital signal processors,
dynamic ranges of signal and coefficients can be increased.
• .

38. Why Digital Signal Processing?
• A signal processor can process many signals, reducing processing cost per signal.
• Digital implementation allows the realization of certain characteristics not possible
with analog implementation; such as polygon in R-X plane for distance relaying.
• Digital signals can be stored indefinitely without loss of accuracy.
• There are also some disadvantages with DSP.
• One of them is that DSP contains active devices.
• Active devices are less reliable than passive components.
• Passive components consume less power than active devices.
• However, advantages of digital relays (i.e. relaying using digital signal processing)
are far more significant than the disadvantages.

39. ADC

40. Nyquist Theorem
• Transforming a signal from the time domain to the frequency domain requires the application of
the Nyquist theorem.
• The Nyquist sampling theorem states that if a signal only contains frequencies less than cutoff
frequency, fc, all the information in the signal can be captured by sampling it at a minimum
frequency of 2fc.
• This means that capturing a signal with a maximum frequency component of fmax requires that
it must be sampled at 2fmax or higher.
• However, common practice dictates that while working in the frequency domain, the sampling
rate must be set more than twice and preferably between five and ten times the signal’s highest
frequency component.
• Waveforms viewed in the time domain are usually sampled 10 times the frequency being
measured to faithfully reproduce the original signal and retain accuracy of the signal’s highest
frequency components.

41. Nyquist Theorem

42. Nyquist Theorem

43. Nyquist Theorem

44. Nyquist Theorem

45. Aliasing
• Any analog signal consists of components at various frequencies. The simplest case
is the sinewave, in which all the signal energy is concentrated at one frequency.
• In practice, analog signals usually have complex waveforms, with components at
many frequencies.
• The highest frequency component in an analog signal determines the bandwidth of
that signal.
• The higher the frequency, the greater the bandwidth, if all other factors are held
constant.
• Suppose the highest frequency component, in hertz, for a given analog signal is
fmax.
• According to the Nyquist Theorem, the sampling rate must be at least 2fmax, or
twice the highest analog frequency component.
• The sampling in an analog-to-digital converter is actuated by a pulse generator
(clock).
• If the sampling rate is less than 2fmax, some of the highest frequency components in
the analog input signal will not be correctly represented in the digitized output.

46. Aliasing
• When such a digital signal is converted back to analog form by a digital-to-
analog converter, false frequency components appear that were not in the
original analog signal.
• This undesirable condition is a form of distortion called aliasing.
• Aliasing occurs when a signal is under sampled. If the signal sampling rate
is too low, we get frequency-domain aliasing.
• Nyquist Theorem: Sampling rate (f s) > 2 * highest frequency component
(of interest) in the measured signal The Nyquist theorem states that a
signal must be sampled at a rate greater than twice the highest frequency
component of interest in the signal to capture the highest frequency
component of interest; otherwise, the high-frequency content will alias at
a frequency inside the spectrum of interest (pass-band).
• If a signal is sampled at a sampling rate smaller than twice the Nyquist
frequency, false lower frequency component(s) appears in the sampled
data. This phenomenon is called Aliasing

47. Aliasing
• When sampling an ac signal at less than two times the Nyquist frequency, the original waveform
cannot be reproduced faithfully.
• When ac inputs are sampled more than twice the Nyquist frequency of the sine wave, the
frequency content of the signal is preserved, and all the Fourier components of the periodic
waveform are recovered.
• When input signals are sampled at less than the Nyquist rate, ambiguous signals that are much
lower in frequency than the signal being sampled can appear in the time domain.
• This phenomenon is called aliasing.
• Conversely, input frequencies of half or more of the sampling rate will also generate aliases.
• To prevent these aliases, a lowpass, anti-aliasing filter is used to remove all components of
these input signals.
• The filter is usually an analog circuit placed between the signal input terminals and the ADC.
• Although the filter eliminates the aliases, it also prevents any other signals from passing
through that are above the stop band of the filter, whether they were wanted or not.
• In other words, when selecting a data acquisition system, make certain that the per channel
sampling frequency is more than twice the highest frequency intended to be measured.

48. Aliasing
• These types of systems primarily utilize low-pass filters, digital filters or a
combination of both.
• With the analog low-pass filter, high frequency noise and interference can be
removed from the signal path prior to the analog-to-digital (A/D) conversion.
• In this manner, the digital output code of the conversion does not contain
undesirable aliased harmonic information.
• An Anti-Aliasing filter is a low pass filter (LPF) used to cut off the high frequency
content in the input signal, connected after signal conditioning hardware

52. • Example
Assume fs, the sampling frequency, is 100 Hz and that the input signal contains the following frequencies:
25 Hz, 70 Hz, 160 Hz, and 510 Hz. These frequencies are shown in the following figure.
As shown in the following figure, frequencies below the Nyquist frequency (fs/2 = 50 Hz) are sampled correctly.
Frequencies above the Nyquist frequency appear as aliases. For example, F1 (25 Hz) appears at the correct frequency,
but F2 (70 Hz), F3 (160 Hz), and F4 (510 Hz) have aliases at 30 Hz, 40 Hz, and 10 Hz, respectively.
Original frequencies present in the input signal
Original and aliased frequencies that appear in the measured signal after passing through a ADC
Alias F2 = |100 – 70| = 30 Hz
Alias F3 = | (2)100 – 160| = 40 Hz
Alias F4 = | (5)100 – 510| = 10 Hz

54. An interesting analog can be drawn by considering a room having many mirrors each
reflecting image from one to another. It is seen that if a person is standing in such a
room, another observer cannot distinguish him from his image. The difficulty can be
resolved if the observer has an idea of location or coordinates of the real person. In the
same manner, we can identify the original lobe from replicated lobes if we have an idea
of the frequency content of original signal. In fig 28.5, notice that lobes are distinctly
separated because ƒs > 2B Hz . On the other hand, if ƒs = 2B Hz , then as seen in fig 28.6,
lobes will just touch each other. If however, ƒs < 2B Hz, then lobes will overlap (fig 28.7)
and this will lead to distortion of replicated frequency spectrum. Thus, it is necessary
that ƒs the sampling frequency should atleast equal to 2B Hz.

55. Thus, we can classify sampling frequency into three categories.
1. Sampling at a rate
2. Sampling at a rate
3. Sampling at a rate