Integration of the Powers of the Trigonometric Functions

In this topic we are going to learn how to integrate certain combinations of trigonometric functions. We will be using some techniques to help us make it easy for us to evaluate each of them. First let's start off with the review of the basic formula of Integration by Substitution, I know at this point you could perfectly evaluate integrals using this method.

Formula:

, where n ≠ –1
, where n = –1

Integration of the powers of sine and cosine functions

• Case 1. Has the form ∫sinⁿax cosax dx. I believe this is the basic and the easiest to evaluate because of the presence of the cosine.

Examples 1: ∫sin⁵3x cos3x dx


Solution: Let u = sin3x, du = 3cos3x dx


=



• Case 2. Has the form ∫cosⁿax sinax dx. In this case instead sine, the one with the exponent is the cosine and it is a lot easier to evaluate because of the presence of sine.

Examples 2: ∫cos³4x sin4x dx


Solution: Let u = cos4x, du = -4sin4x dx

=



• Case 3. Has the form ∫sin^max cosⁿax dx where m is odd integer. The first step is to factor sin x, then change the remaining factor into sin²ax = 1 - cos²ax.

Examples 3:


∫sin³3x dx = ∫sin²3x ∙ sin3x dx
= ∫(1 - cos²3x) sin3x dx
= ∫sin3x dx - ∫cos²3x sin3x dx
=


Examples 4:


∫sin⁵2x cos²2x dx = ∫sin⁴2x ∙ sin2x cos²2x dx
= ∫(sin²2x)² cos²2x sin2x dx
= ∫(1 - cos²2x)² cos²2x sin2x dx
= ∫(1 - 2cos²2x + cos⁴2x) cos²2x sin2x dx
= ∫cos²2x sin2x dx - 2∫cos⁴2x sin2x dx + ∫cos⁶2x sin2x dx

Let u = cos2x, du = -2sin2x dx

=



• Case 4. Has the form ∫sin^max cosⁿax dx where n is odd integer. Here we can factor cos x, then change the remaining factor into cos²ax = 1 - sin²ax.

Examples 5:


∫cos³4x dx = ∫cos²4x ∙ cos4x dx
= ∫(1 - sin²4x) cos4x dx
= ∫cos4x dx - ∫sin²4x cos4x dx
=


Examples 6:


∫sin²x cos³x dx = ∫sin²x cos²x ∙ cosx dx
= ∫sin²x (1 - sin²x) cosx dx
= ∫(sin²x - sin⁴x) cosx dx

Let u = sinx, du = cosx dx

=



• Case 5. Has the form ∫sin^max cosⁿax dx where m and n are even integers. In this case we will be using the half-angle identities below.


and


Examples 7:


∫sin²x dx = ½∫(1 - cos2x)dx
= ½∫dx - ½∫cos2x dx
=


Examples 8:


∫sin²x cos²x dx = ∫½(1 - cos2x)½(1 + cos2x) dx
= ¼∫(1 - cos²2x) dx
= ¼∫[1 - ½(1 + cos4x)] dx
= ⅛∫(1 - cos4x) dx
=



• Case 6. Has the form ∫sinaxcosbx dx, ∫sinaxsinbx dx, ∫cosaxcosbx dx. To find the integrals for this kind of combination of trigonometric functions, we will be needing the identities below.







Examples 9:


∫sin6xcos3x dx = ½∫(sin9x + sin3x) dx
= ½∫sin9x dx + ½∫sin3x dx
=


Examples 10:


∫sin5xsin2x dx = ½∫(cos3x - cos7x) dx
= ½∫cos3x dx - ½∫cos7x dx
=

Integration of the powers of tangent and secant functions

• Case 1. Has the form ∫tan^mu secⁿu du. In this case, when m is odd factor out sec u tan u du and change the remaining tangent into secant using the identity, tan²u = sec²u - 1.

Examples 11:


∫tan³x sec³x dx = ∫tan²x sec²x ∙ secx tanx dx
= ∫(sec²x - 1)sec²x ∙ secx tanx dx
= ∫(sec⁴x - sec²x)secx tanx dx

Let u = secx, du = secx tanx dx

= ∫ u⁴ du - u² du
=


• Case 2. Has the form ∫tan^mu secⁿu du. When n is even greater than 2, factor out sec²u du and replace the remaining secant by tangent using the identity, sec²u = 1 + tan²u.

Examples 12:


∫tan⁵x sec⁴x dx = ∫tan⁵x sec²x ∙ sec²x dx
= ∫tan⁵x(1 + tan²x)sec²x dx
= ∫tan⁵x sec²x dx + ∫tan⁷x sec²x dx

Let u = tanx, du = sec²x dx

= ∫ u⁵ du + u⁷ du
=


• Case 3. Has the form ∫tan^mu secⁿu du. When m is even the integrand is tangent only use the identity, tan²u = sec²u - 1.

Examples 13:


∫tan⁴x dx = ∫tan²x ∙(sec²x - 1) dx
= ∫tan²x sec²x dx - ∫tan²x dx
= ∫tan²x sec²x dx - ∫(sec²x - 1) dx

Let u = tanx, du = sec²x dx

= ∫u² du - ∫du + ∫dx
=

Integration of the powers of cotangent and Cosecant functions

• Case 1. Has the form ∫cot^mu cscⁿu du. When m is odd, factor out csc u cot u du and change the remaining cotangent into cosecant using the identity, cot²u = csc²u - 1.

Examples 14:


∫cot³2x csc³2x dx = ∫cot²2x csc²2x ∙ (cot2x csc2x dx)
= ∫(csc²2x - 1)csc²2x ∙ cot2x csc2x dx
= ∫(csc⁴2x - csc²2x)cot2x csc2x dx

Let u = csc2x, du = -2csc2x cot2x dx

= -½∫u⁴ du + ½∫u² du
=


• Case 2. Has the form ∫cot^mu cscⁿu du. When n is even greater than 2, factor out csc²u du and replace the remaining cosecant by cotangent using the identity, csc²u = 1 + cot²u.

Examples 15:


∫cot⁴x csc⁴x dx = ∫cot⁴x csc²x ∙ csc²x dx
= ∫cot⁴x(1 + cot²x) csc²x dx
= ∫(cot⁴x + cot⁶x) csc²x dx

Let u = cotx, du = -csc²x dx

= -∫u⁴ du - ∫u⁶ du
=

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Virtual-Circuit Networks: Frame Relay and ATM - MCQs Answers

Below are the answers key for the Multiple Choice Questions (Quiz) of Set 1 MCQs of Virtual-Circuit Networks: Frame Relay and ATM .

1. FRAD
2. 2 to 4
3. physical
4. mixed architecture
5. UNI
6. a combination of VPI and VCI
7. AAL2
8. PVC
9. AAL
10. 8
11. three
12. ATM
13. DLCIs
14. two
15. NNI
16. VOFR
17. cell
18. ATM
19. AAL3/4
20. either  SVCs or PVCs
21. 1
22. pure
23. 12
24. neither error nor flow
25. ATM
26. an SVC
27. legacy
28. all of the above
29. the physical and data link layers
30. AAL5
31. ATM
32. AAL1
33. Frame Relay
34. 53

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Virtual-Circuit Networks: Frame Relay and ATM - MCQs

This particular topic discuss the two switched WANs: Frame Relay and ATM. Frame Relay is a virtual-circuit wide-area network that was designed in response to demands for a new type of WAN in the late 1980s and early 1990s. Asynchronous Transfer Mode (ATM) is a cell relay protocol that, in combination with SONET, allows high-speed connections. If you skip the summary and important terminology visit Virtual-Circuit Networks: Frame Relay and ATM .

Begin and Good luck!

1. To handle frames arriving from other protocols, Frame Relay uses a device called a _________.
• A)   MUX
• B)   VOFR
• C)   FRAD
• D)   none of the above
2. In Frame Relay, an address can be ________ bytes.
• A)   2 to 3
• B)   2 to 4
• C)   only 2
• D)   none of the above
3. In ATM, the _____ layer defines the transmission medium, bit transmission, encoding, and electrical-to-optical transformation.
• A)   AAL
• B)   physical
• C)   ATM layer
• D)   none of the above
4. A _______ ATM LAN combines features of a pure ATM LAN and a legacy ATM LAN.
• A)   legacy
• B)   pure
• C)   mixed architecture
• D)   none of the above
5. A(n) ______ is the interface between a user and an ATM switch.
• A)   NNN
• B)   UNI
• C)   NNI
• D)   None of the above
6. In ATM, a virtual connection is defined by _________.
• A)   DLCI
• B)   VPI
• C)   VCI
• D)   a combination of (b) and (c)
7. In ATM, _______is for short packets.
• A)   AAL5
• B)   AAL3/4
• C)   AAL2
• D)   AAL1
8. In Frame Relay, when a _____ is selected, the corresponding table entry is recorded for all switches by the administrator.
• A)   SVC
• B)   PVC
• C)   either (a) or (b)
• D)   neither (a) nor (b)
9. In ATM, the _______layer accepts transmissions from upper-layer services and maps them into ATM cells.
• A)   AAL
• B)   physical
• C)   ATM
• D)   none of the above
10. The VPI of a UNI is _______ bits in length.
• A)   32
• B)   16
• C)   12
• D)   8
11. The ATM standard defines ______ layers.
• A)   five
• B)   four
• C)   three
• D)   two
12. ________ technology can be adapted for use in a LAN (ATM LAN).
• A)   ATM
• B)   X.25
• C)   Frame Relay
• D)   none of the above
13. VCIs in Frame Relay are called ______.
• A)   SVC
• B)   DLCIs
• C)   PVC
• D)   none of the above
14. The AAL is divided into _______ sublayers.
• A)   four
• B)   three
• C)   two
• D)   none of the above
15. _________ is the interface between two ATM switches.
• A)   NNI
• B)   NNN
• C)   UNI
• D)   none of the above
16. Frame Relay networks offer an option called ______________ that sends voice through the network.
• A)   FRAD
• B)   VOFR
• C)   MUX
• D)   none of the above
17. A _______ is defined as a small, fixed-size block of information.
• A)   packet
• B)   cell
• C)   frame
• D)   none of the above
18. In ATM, the ______ layer provides routing, traffic management, switching, and multiplexing services.
• A)   ATM
• B)   AAL
• C)   physical
• D)   none of the above
19. In ATM, _____ is for conventional packet switching (virtual-circuit approach or datagram approach).
• A)   AAL5
• B)   AAL3/4
• C)   AAL2
• D)   AAL1
20. Frame Relay provides ________.
• A)   SVCs
• B)   PVCs
• C)   either (a) or (b)
• D)   neither (a) nor (b)
21. In Frame Relay, the EA field defines the number of bytes; it is _____ in the last byte of the address.
• A)   0
• B)   1
• C)   2
• D)   3
22. In a _____ ATM LAN, an ATM switch connects stations.
• A)   legacy
• B)   pure
• C)   mixed architecture
• D)   none of the above
23. The VPI of an NNI is _______ bits in length.
• A)   24
• B)   16
• C)   12
• D)   8
24. At the data link layer, Frame Relay uses a protocol that supports _____control.
• A)   error
• B)   flow
• C)   either (a) or (b)
• D)   neither (a) nor (b)
25. ________ is the cell relay protocol designed by the corresponding Forum and adopted by the ITU-T.
• A)   Frame Relay
• B)   ATM
• C)   X.25
• D)   none of the above
26. In Frame Relay, when ______ is selected, it requires establishing and terminating phases.
• A)   a PVC
• B)   an SVC
• C)   either (a) or (b)
• D)   neither (a) nor (b)
27. In a _______ ATM LAN, the backbone that connects traditional LANs uses ATM technology.
• A)   legacy
• B)   pure
• C)   mixed architecture
• D)   none of the above
28. In ATM, connection between two endpoints is accomplished through _______.
• A)   VCs
• B)   TPs
• C)   VPs
• D)   all of the above
29. Frame Relay has _______.
• A)   the physical, data link, and network layers
• B)   the physical and data link layers
• C)   only the data link
• D)   only the physical layer
30. In ATM, ______ is for packets requiring no sequencing and no error control mechanism.
• A)   AAL5
• B)   AAL3/4
• C)   AAL2
• D)   AAL1
31. ______ eliminates the varying delay times associated with different-size packets.
• A)   Frame Relay
• B)   ATM
• C)   X.25
• D)   all of the above
32. In ATM, ______ is for constant-bit-rate data.
• A)   AAL5
• B)   AAL3/4
• C)   AAL2
• D)   AAL1
33. ________ is a virtual-circuit wide-area network that was designed in response to demands for a new type of WAN in the late 1980s and early 1990s.
• A)   ATM
• B)   Frame Relay
• C)   X.25
• D)   none of the above
34. The ATM data packet is a cell composed of ______ bytes.
• A)   53
• B)   52
• C)   43
• D)   42

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Virtual-Circuit Networks: Frame Relay and ATM

Definition of Terms

• Virtual-circuit switching is a data link layer technology in which links are shared.
• A virtual-circuit identifier (VCI) identifies a frame between two switches.
• The three phases in virtual circuit switching are setup, data transfer, and teardown. 
• The setup phase can use the permanent virtual circuit (PVC) approach or the switched virtual circuit (SVC) approach.
• Frame Relay is a relatively high-speed, cost-effective technology that can handle bursty data.
• Frame Relay is a virtual-circuit wide-area network that was designed in response to demands for a new type of WAN in the late 1980s and early 1990s.
• Frame Relay operates only at the physical and data link layers.
• Frame Relay does not provide flow or error control; they must be provided by the upper-layer protocols.
• To handle frames arriving from other protocols, Frame Relay uses a device called a
  Frame Relay assembler/disassembler (FRAD).  A FRAD assembles and disassembles
  frames coming from other protocols to allow them to be carried by Frame Relay frames.
• Frame Relay networks offer an option called Voice Over Frame Relay (VOFR) that
  sends voice through the network.
• Local Management Information (LMI) is a protocol added recently to the Frame Relay protocol to provide more management features.
• One of the nice features of Frame Relay is that it provides congestion control and quality of service (QoS).
• Both PVC and SVC connections are used in Frame Relay.
• The data link connection identifier (DLCI) identifies a virtual circuit in Frame Relay.
• VCIs in Frame Relay are called DLCIs.
• Asynchronous Transfer Mode (ATM) is a cell relay protocol that, in combination with SONET, allows high-speed connections.
• A cell is a small, fixed-size block of information.
• The ATM data packet is a cell composed of 53 bytes (5 bytes of header and 48 bytes of payload).
• Note that a virtual connection is defined by a pair of numbers: the VPI and the VCI.
• ATM eliminates the varying delay times associated with different-size packets.
• ATM can handle real-time transmission.
• A user-to-network interface (UNI) is the interface between a user and an ATM switch.
• A network-to-network interface (NNI) is the interface between two ATM switches.
• In ATM, connection between two endpoints is accomplished through transmission paths (TPs), virtual paths (VPs), and virtual circuits (VCs).
• In ATM, a combination of a virtual path identifier (VPI) and a virtual-circuit identifier identifies a virtual connection.
• ATM technology can be adopted for use in a LAN (ATM LAN).
• In a pure ATM LAN, an ATM switch connects stations.
• In a legacy ATM LAN, the backbone that connects traditional LANs uses ATM technology.
• A mixed architecture ATM LAN combines features of a pure ATM LAN and a legacy ATM LAN.
• Local-area network emulation (LANE) is a client/server model that allows the use of ATM technology in LANs.
• LANE software includes LAN emulation client (LECS), LAN emulation configuration server (LECS), LAN emulation server (LES), and broadcast/unknown server (BUS) modules.

The ATM standard defines three layers:

• a.   Application adaptation layer (AAL) accepts transmissions from upper-layer services and maps them into ATM cells. The AAL is divided into two sublayers: convergence sublayer (CS) and segmentation and reassembly (SAR).
• b.  ATM layer provides routing, traffic management, switching, and multiplexing services.
• c.  Physical layer defines the transmission medium, bit transmission, encoding, and electrical-to-optical transformation.

There are four different AALs, each for a specific data type:

• a.  AAL1 for constant-bit-rate stream
• b.  AAL2 for short packets.
• c.  AAL3/4 for conventional packet switching (virtual-circuit approach or datagram approach).
• d.  AAL5 for packets requiring no sequencing and no error control mechanism.

Frame Relay network

Three address formats

Virtual connection identifiers in UNIs and NNIs

An ATM cell

ATM layers

ATM LANs

Note: You can proceed to take the multiple choice exam regarding this topic. Virtual-Circuit Networks: Frame Relay and ATM - Set 1 MCQs

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SONET/SDH - MCQs Answers

Below are the answers key for the Multiple Choice Questions (Quiz) of SONET/SDH .

1. 125
2. either  multipoint or point-to-point
3. 9
4. SDH
5. line
6. regenerator
7. section
8. one-to-one
9. 18
10. from left to the right, top to bottom
11. byte
12. 90
13. path
14. line
15. four-layer
16. either bidirectional or unidirectional
17. SONET
18. three-layer
19. from most significant to the least significant
20. section
21. path
22. photonic
23. synchronous
24. 9
25. four
26. one-plus-one
27. ADMs
28. 270
29. STSs
30. OC-n
31. two-layer
32. 9
33. STMs
34. VTs
35. all of the above
36. 8000 
37. all of the above
38. one-to-many
39. 9

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SONET/SDH - MCQs

This particular topic discuss the Synchronous Optical Network (SONET) which is a standard developed by ANSI for fiber-optic networks while the Synchronous Digital Hierarchy (SDH) is a similar standard developed by ITU-T. If you skip the summary and important terminology visit SONET/SDH .

Begin and Good luck!

1. In SONET each frame lasts _______ microseconds.
• A)   20
• B)   64
• C)   125
• D)   none of the above
2. A linear SONET network can be __________.
• A)   multipoint
• B)   point-to-point
• C)   either (a) or (b)
• D)   neither (a) nor (b)
3. The path overhead consists of ______ bytes.
• A)   18
• B)   9
• C)   6
• D)   1
4. _______ is a standard developed by ITU-T.
• A)   SONET
• B)   SDH
• C)   either (a) or (b)
• D)   neither (a) nor (b)
5. A ______ is the portion of the network between two multiplexers.
• A)   line
• B)   path
• C)   section
• D)   none of the above
6. A ________ is a repeater.
• A)   ADM
• B)   regenerator
• C)   STS multiplexer/demultiplexer
• D)   none of the above
7. The ______ layer is responsible for the movement of a signal across a physical section.
• A)   path
• B)   section
• C)   photonic
• D)   line
8. In ______ APS, there is one working line and one protection line. The data are normally sent on the working line until it fails.
• A)   one-to-one
• B)   one-plus-one
• C)   one-to-many
• D)   none of the above
9. Line overhead consists of ________ bytes.
• A)   18
• B)   9
• C)   6
• D)   1
10. In SONET, for each frame, the bytes are transmitted __________.
• A)   from left to the right, bottom to top
• B)   from left to the right, top to bottom
• C)   from right to the left, top to bottom
• D)   from right to the left, bottom to top
11. Each ________in a SONET frame can carry a digitized voice channel.
• A)   frame
• B)   bit
• C)   byte
• D)   none of the above
12. An STS-1 frame is made ______columns.
• A)   90
• B)   9
• C)   1
• D)   none of the above
13. The ______ layer is responsible for the movement of a signal from its optical source to its optical destination.
• A)   line
• B)   path
• C)   photonic
• D)   section
14. The ____ layer is responsible for the movement of a signal across a physical line.
• A)   line
• B)   path
• C)   photonic
• D)   section
15. An STS multiplexer is a ______ device.
• A)   four-layer
• B)   three-layer
• C)   two-layer
• D)   one-layer
16. A ring SONET network can be _________.
• A)   bidirectional
• B)   unidirectional
• C)   either (a) or (b)
• D)   neither (a) nor (b)
17. _______ is a standard developed by ANSI for fiber-optic networks.
• A)   SDH
• B)   SONET
• C)   either (a) or (b)
• D)   neither (a) nor (b)
18. An add/drop multiplexer is a ______ device.
• A)   four-layer
• B)   three-layer
• C)   two-layer
• D)   one-layer
19. In SONET, for each byte, the bits are transmitted ____________.
• A)   from most significant to the least significant
• B)   from least significant to the most significant
• C)   three at a time
• D)   two at a time
20. A _______ is the optical link connecting two neighbor devices.
• A)   path
• B)   line
• C)   section
• D)   none of the above
21. A ______is the end-to-end portion of the network between two STS multiplexers.
• A)   line
• B)   path
• C)   section
• D)   none of the above
22. The _______ layer corresponds to the physical layer of the OSI model.
• A)   photonic
• B)   path
• C)   line
• D)   section
23. SONET is a _______ TDM system.
• A)   statistical
• B)   asynchronous
• C)   synchronous
• D)   none of the above
24. An STS-3 frame is made of ______ rows.
• A)   27
• B)   9
• C)   1
• D)   none of the above
25. SONET defines _______ layers.
• A)   five
• B)   four
• C)   three
• D)   two
26. In ________ APS, there are normally two lines: one working line and one protection line. Both lines are active all the time.
• A)   one-to-one
• B)   one-to-many
• C)   one-plus-one
• D)   none of the above
27. __________ allow insertion and extraction of signals.
• A)   STS multiplexer/demultiplexers
• B)   regenerators
• C)   ADMs
• D)   none of the above
28. An STS-3 frame is made of ________ columns.
• A)   270
• B)   90
• C)   9
• D)   none of the above
29. SONET has defined a hierarchy of signals called ________.
• A)   STMs
• B)   STSs
• C)   either (a) or (b)
• D)   neither (a) nor (b)
30. An ______ signal is the optical modulation of an STS-n (or STM-n) signal.
• A)   FDM-n
• B)   OC-n
• C)   TDM-n
• D)   none of the above
31. A regenerator is a ________ device.
• A)   three-layer
• B)   one-layer
• C)   two-layer
• D)   four-layer
32. An STS-1 frame is made of ________ rows
• A)   90
• B)   9
• C)   1
• D)   none of the above
33. SDH has defined a hierarchy of signals called ________.
• A)   STMs
• B)   STSs
• C)   either (a) or (b)
• D)   neither (a) nor (b)
34. To make SONET backward-compatible with the current hierarchy, its frame design includes a system of ____.
• A)   VTs
• B)   STSs
• C)   STMs
• D)   OCs
35. SONET network topologies can be __________.
• A)   ring
• B)   mesh
• C)   linear
• D)   all of the above
36. SONET sends ________ frames per second.
• A)   8000
• B)   4000
• C)   2000
• D)   1000
37. A SONET system can use _________.
• A)   regenerators
• B)   add/drop multiplexers
• C)   STS multiplexers
• D)   all of the above
38. In ________APS, there is only one protection line for many working lines. When a failure occurs in one of the working lines, the protection line takes control until the failed line is repaired.
• A)   one-to-one
• B)   one-plus-one
• C)   one-to-many
• D)   none of the above
39. The section overhead consists of ______octets.
• A)   18
• B)   9
• C)   6
• D)   1

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SONET/SDH

Definition of Terms

• Synchronous Optical Network (SONET) is a standard developed by ANSI for fiber-optic networks: Synchronous Digital Hierarchy (SDH) is a similar standard developed by ITU-T.
• SONET was developed by ANSI; SDH was developed by ITU-T.
• Each synchronous transfer signal STS-n is composed of 8000 frames. Each frame is a two-dimensional matrix of bytes with 9 rows by 90 × n columns.
• A SONET STS-n signal is transmitted at 8000 frames per second.
• Each byte in a SONET frame can carry a digitized voice channel.
• In SONET, the data rate of an STS-n signal is n times the data rate of an STS-1 signal.
• STS multiplexers/demultiplexers mark the beginning points and endpoints of a SONET link.
• An STS multiplexer multiplexes signals from multiple electrical sources and creates the corresponding optical signal.
• An STS demultiplexer demultiplexes an optical signal into corresponding electric signals.
• Add/drop multiplexers allow insertion and extraction of signals in an STS. An add/drop multiplexer can add an electrical signals into a given path or can remove a desired signal from a path.
• SONET has defined a hierarchy of signals called synchronous transport signals (STSs). SDH has defined a similar hierarchy of signals called synchronous transfer modules (STMs). 
• An OC_-n signal is the optical modulation of an STS_-n (or STM_-n) signal.
• Pointers are used to show the offset of the SPE in the frame or for justification.
• SONET uses two pointers show the position of an SPE with respect to an STS.
• SONET use the third pointer for rate adjustment between SPE and STS. 
• A regenerator takes a received optical signal and regenerates it.
• The SONET regenerator also replaces some of the existing overhead information with new information.
• SONET defines four layers: path, line, section, and photonic.
• The path layer is responsible for the movement of a signal from its source to its destination.
• The line layer is responsible for the movement of a signal across a physical line.
• The section layer is responsible for the movement of a signal across a physical section.
• The photonic layer corresponds to the physical layer of the OSI model. It includes physical specifications for the optical fiber channel.
• SONET uses NRZ encoding with the presence of light representing 1 and the absence of light representing 0.
• SONET is a synchronous TDM system in which all clocks are locked to a master clock.
• SONET sends 8000 frames per second; each frame lasts 125 µs. 
• Section overhead is recalculated for each SONET device (regenerators and multiplexers).
• Path overhead is only calculated for end-to-end (at STS multiplexers).
• An STS-3c signal can carry 44 ATM cells as its SPE.
• An STS-I frame is made of 9 rows and 90 columns; an STS_-n frame is made of 9 rows and n x 90 columns.
• STSs can be multiplexed to get a new STS with a higher data rate.
• SONET network topologies can be linear, ring, or mesh.
• A linear SONET network can be either point-to-point or multipoint.
• A ring SONET network can be unidirectional or bidirectional.
• To make SONET backward-compatible with the current hierarchy, its frame design includes a system of virtual tributaries (VTs).
• SONET is designed to carry broadband payloads. Current digital hierarchy data rates, however, are lower than STS-1. To make SONET backward-compatible with the current hierarchy, its frame design includes a system of virtual tributaries (VTs). A virtual tributary is a partial payload that can be inserted into an STS-1.

A SONET system can use the following equipment:

• 1.  STS multiplexers
• 2.  STS demultiplexers
• 3.  Regenerators
• 4.  Add/drop multiplexers
• 5.  Terminals

SONET/SDH rates

A simple network using SONET equipment

SONET layers compared with OSI or the Internet layers

Taxonomy of SONET networks

Virtual tributary types

Note: You can proceed to take the multiple choice exam regarding this topic. SONET/SDH - Set 1 MCQs

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