Power and energy theorems , parseval's power theorem proof example , Rayleigh's energy

By   November 14, 2019

In this article, let us discuss two important theorems regarding the energy and power of signals.
these two theorems are as under:
(i)      Parseval’s power theorem and
(ii)     Rayleigh’s energy theorem.
1.19.1 Parseval’s Power Theorem: Definition
This theorem relates the average power P of a periodic signal to its Fourier series coefficients. The Paraseval’s power theorem states that the total average power of a periodic signal x(t) is equal to the sum of the average powers of the individual Fourier coefficients i.e., Cn.
Thus, average power of x(t) = (Power of C1) + (Power of C2) + ………
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NOTE: Note that the equations for the auto-correlation function and cross-correlation function are identical except for the fact that it has been defined for two different power signals x1(t) and x2(t) whereas the auto-correlation is defined for the same signal x(t).
■       Communication is the process of establishing connection or link between two points for information exchange.
■       The electronic equipments which are used for communication purpose, are called communication equipments. Different communication equipments when assembled together form a communication system.
■       Typical examples of communication system are line telephony and line telegraphy, radio telephony and line telegraphy, radio broadcasting, point-to-point communication and mobile communication, computer communication, radar communication, television broadcasting, radio telemetry, radio aids to navigation, radio aids to aircraft landing etc.
■       In the most fundamental sense, communication involves the transmission of information from one point to another through a succession of process.
■       That the function of information source is to produce required message which has to be transmitted.
■       A transducer is a device which converts one form of energy into another form. The message from the information source may or may not be electrical in nature. In a case when the message produced by the information source is not electrical in nature, an input transducer is used to convert it into a time-varying electrical signal.
■       The function of the transmitter is to process the electrical signal from different aspects. For example in radio broadcasting the electrical signal obtained from sound signal, is processed to restrict its range of audio frequencies (upto 5 kHz in amplitude modulation radio broadcast) and is often amplified.
■       That the function of the channel is to provide a physical connection between the transmitter and the receiver.
■       There are two types of channels, namely point-to-point channels and broadcast channels. Examples of point-to-point channels are wirelines,
microwave links and optical fibres. Wirelines operate by guided electromagnetic waves and they are used for local telephone transmission. In case of microwave links, the transmitted signal is radiated as an electromagnetic wave in free space.
■       Noise is an unwanted signal which tend to interfere with the required signal. Noise signal is always random in character. Noise may interfere with signal at any point in a communication system. However, the noise has its greatest effect on the signal in the channel.
■       The main function of the receiver is to reproduce the message signal in electrical form from the distorted received signal. This reproduction of the original signal is accomplished by a process known as the demodulation or detection. Demodulation is the reverse process of modulation carried out in transmitter.
■       Destination is the final stage which is used to convert an electrical message signal into its original form. For example in radio broadcasting, the destination is a loudspeaker which works as a transducer i.e. it converts the electrical signal in the form of original sound signal.
■       The telecommunication environment is dominated by the following four important sources of information: (i) speech (ii) television (iii) facsimile, and (iv) personal computers
■       Several sources of information which are responsible for the generation of message signals. These message signals need to be transmitted to the destination point. Now, let us discuss various communication channel that carryout the transmission.
■       Basically, we shall consider following channels:
(i) telephone channels                       (ii)     optical fibers
(iii) mobile radio channels, and                   (iv)    satellite channels.
■      A telephone network makes use of a switching mechanism. This switching mechanism is known as circuit switching and it is used to establish an end-to-end communication link on a temporary basis.
■       The second communication channel, an optical fiber, is a dielectric waveguide which transports light signals from one place to another just as a metallic wire pair or a co-coxial cable, transports electrical signals
■       An optical fiber consists of a central core within which the propagating electromagnetic field is confined and which is surrounded by a cladding layer, which is itself surrounded by a thin protective jacket.
■       The third communication channel, mobile radio channel, extends the capability of the public telecommunications network by introducing mobility into the network by virtue of its ability to broadcast.
■       A satellite channel provides broad-area coverage in a continental as well as intercontinental sense. Moreover, access to remove areas not covered by conventional cable or fiber communications is also a distinct feature of satellites.
■       Communications satellites in geostationary orbit offer the following unique system capabilities:
(a)     broad-area coverage
(b)     reliable transmission links
(c)      wide transmission bandwidth.
■       The message signal generated from the information source is known as baseband signal. This baseband signal may be a combination of two or more message signals. If the baseband signal is transmitted directly, then
it is known as baseband transmission. This means that the baseband transmission does not used modulators and demodulators. The baseband signal may be both analog as well as digital.
■       The analog baseband signal varies continuously with time and has continuous amplitude. The digital baseband signal is discrete in both time and amplitude. The baseband transmission is preferred at low frequencies and for short distances. However, Inter Symbol Interference (ISI) is the major problem associated with the baseband transmission.
■       Regarding the mode of propagation, communication may be divided in the following two forms:
(i)      Line Communication
(ii)     Wireless or Radio Communication
■       In line communication, the medium of transmission is a pair of conductors called transmission line. This is also called as line channel.
■       In wireless or radio communication, a message is transmitted through open space by electromagnetic waves called as radio waves. Radio waves are radiated from the transmitter in open space through a device called antenna.
■       Modulation may be defined as the process by which some characteristic of a signal called carrier is varied in accordance with the instantaneous value of another signal called modulating signal. Signals containing information or intelligence are referred as modulating signals. This information bearing signal is also called baseband signal. The carrier frequency is greater than the modulating frequency. The signal resulting from the process of modulation is called modulated signal.
■       When the carrier wave is continuous in nature, the modulation process is known as continuous wave (CW) modulation or analog modulation. Examples of continuous wave modulation are Amplitude Modulation and Angle Modulation.
■       When the carrier wave is a pulse-type waveform, the modulation process is known as pulse modulation.
■       In a communication system, there are following two primary communication resources to be employed:
(i)      transmitted power, and
(ii)channel bandwidth
■       The transmitted power refers to the average power of the transmitted signal. The channel bandwidth is defined as the band of frequencies allocated for the transmission of the message signal.
■       In communication systems, sometimes, it happens that we are available with analog signal, however, we have to transmit a digital signal for a particular application. In such cases, we have to convert an analog signal into digital signal.
■       Depending upon the message signal, communication may also be classified as under:
(i)      Analog Communication
(ii)     Digital Communication
■       Analog communication is that type of communication in which the message or information signal to be transmitted is analog in nature. This means that in analog communication the modulating signal (i.e., baseband
signal) is an analog signal. This analog message signal may be obtained from sources such as speech, video shooting etc.
■       In digital communication, the message signal to be transmitted is digital in nature. This means that digital communication involves the transmission of information in digital form.
■       Following are the advantages of digital communication:
(i)    The digital communication systems are simpler and cheaper compared to analog communication systems because of the advances made in the IC technologies.
(ii)   In digital communication, the speech, video and other data may be merged and transmitted over a common channel using multiplexing.
(iii)  Using data encryption, only permitted receivers may be allowed to detect the transmitted data. This property is of its most importance in military applications.
(iv)  Since the transmission is digital and the channel encoding is used, therefore the noise A not accumulate from repeater to repeater in long distance communications.
(v)   Since the transmitted signal is digital in nature, therefore a large amount of noise interference may be tolerated.
(vi)  Since in digital communication, channel coding is used. therefore the errors may- be detected and corrected in the receivers.
(vii) Digital communication is adaptive to other advanced branches of data processing such as digital signal processing, image processing and data compression etc.

  1. What are the basic constituents of a communication system?
  2. Draw the block diagram of a communication system and explain the function of each block.
  3. What is meant by the term ‘Channel’ as applied to a communication system.
  4. Why is a high-frequency carrier needed in a communication system?
  5. Explain the need for modulation in a communication system?
  6. Explain the radio frequency spectrum used in communication system.
  7. Explain the fundamental limitations of a communication system.
  8. What are the basic constituents of a communication system?
  9. Draw the block diagram of a communication system and explain the function of each block.
  10. What is meant by the term ‘Channel’ as applied to a communication system?
  11. Why is a high-frequency carrier needed in a communication system?
  12. Explain the need for modulation in a communication system?
  13. Explain the radio frequency spectrum used in communication system.
  14. Explain the difference between Analog and Digital communications.
  15. How will you convert an analog signal into a digital signal.

16.     Write the advantages and disadvantages of a digital communication system.

  1. Explain the difference between baseband transmission and bandpass transmission.
  2. Draw the block diagram of a digital communication system and explain the function of each block.


  1. The higher modulating frequency used in AM broadcast system is

(a) 10 kHz                                         (c) 5 kHz
(b) 15 kHZ                                        (d) 2 MHz

  1. The high frequency (HP’) range extends from

(a) 300-3000 kHz                                       (b) 3-30 MHz
(c) 30-300 MHz                                 (d) 300-3000 MHz

  1. The very high frequency (VHF) range extends from

(a) 3-30 MHz                                    (b) 30-300 MHz
(c) 300-3000 MHz                             (d) 3000-30,000 MHz

  1. The ultra high frequency (UHF) range extends from

(a) 3-30 MHz                                    (b) 30-300 MHz
(c) 300-3000 MHz                             (d) 3000-30,000 MHz

  1. The function of the input transducer in a communication system is

(a)   to transmit the message signal
(b)   to modulate the message signal
(c)      to convert message sound signal into electrical signal
(d)     none of the above

  1. Communication is the process of

(a)   keeping in touch
(b)   broadcasting
(c)   exchanging information
(d)   entertainment by electronics

  1. Two key harriers to human comm-unication are

(a) distance                                        (b) cost
(c) ignorance                            (d) language

  1. Electronics communication was discovered in which century?

(a) sixteenth                             (b) eighteenth
(c) nineteenth                           (d) twentieth

  1. Which of the following is not a major communication medium?

(a) free space                           (b) water
(c) wires                                   (d) fibber-optic cable

  1. The communications medium causes the signal to be

(a) amplified                            (b) modulated
(c) attenuated                           (d) interferred with

  1. The process of transmitting two or more information signals simultaneously over the same channel is called

(a) multiplexing                       (b) telemetry
(c) detection                             (d) modulation

  1. Recovering information from a carrier in known as

(a) demultiplexing                             (b) modulation
(c) detection                             (d) carrier recovery

  1. Radio signals are made up of

(a) voltages and currents
(b) electric and magnetic fields
(c) electrons and protons
(d) noise and data

  1. (c) 2. (b) 3. (b)                    4. (c)
  2. (c) 6. (c) 7. (a & b)              8. (c)
  3. (b) 10. (c) 11. (a)                   12. (c)
  4. (b)

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