Communication_System_presentation_Slides.pptx

Full review of a Communication system

1. Communication systems 1

2. Outline 2

3. Fundamentals of Signals and Systems Signal: a function of one or more variables that convey information on the nature of a physical phenomenon .(or) Anything that varies in time and space and carries information from source to destination. Examples : v(t ), i (t), x(t ),heartbeat, blood pressure, temperature, vibration. One-dimensional signals: function depends on a single variable, e.g., speech signal Multi-dimensional signals: function depends on two or more variables, e.g., image 3

4. System: an entity or operator that manipulates one or more signals to accomplish a function, thereby yielding new signals . Commonly encountered systems : communications systems Automatic speaker recoginition system Aircraft landing system Input signal Output signal System 4

5. Introduction Communication system is a system which describes the exchange of information or data between two stations , i.e. between transmitter and receiver. Communication: It is the process of conveying or transferring information from one point to another. ( Or) It is the process of establishing connection or link between two points for information exchange. To transmit signals in communication system, it must be first processed by several stages, beginning from signal representation, to signal shaping until encoding and modulation. 5

6. 6 Elements of Communication Systems Communication: - Transfer of information (or message) - Involves electronic transmitting /receiving /processing - Signal is the electrical form of the message Communication System: An Integrated structure of - Hardware devices (e.g., electronic circuits, antennas, computer processors etc.) - Software algorithms (e.g., digital signal processing algorithms , network protocols)

7. Elements of Communication System : Information source : The message or information to be communicated originates in information source . Message can be words, group of words, code, data, symbols, signalsetc . Transmitter : The objective of the transmitter block is to collect the incoming message signal and modify it in a suitable fash Channel : Channel is the physical medium which connects the transmitter with that of the receiver. The physical medium includes copper wire, coaxial cable, fibre optic cable, wave guide and free space or atmosphere. ion ( ifneeded ),suchthat,itcanbetransmittedviathechosenchanneltothereceivingpoint. 7

8. Channel : Channel is the physical medium which connects the transmitter with that of the receiver. The physical medium includes copper wire, coaxial cable, fibre optic cable, wave guide and free space or atmosphere . Receiver : The receiver block receives the incoming modified version of the message signal from the channel and processes it to recreate the original (non-electrical) form of the message signal . 8

9. Modes of Communication: S implex, Half-Duplex and Full-Duplex) Simplex (SX) – one direction only, e.g. TV Half Duplex (HDX) – both directions but not at the same time, e.g. CB radio Full Duplex (FDX) – transmit and receive simultaneously between two stations, e.g. standard telephone system. Full/Full Duplex (F/FDX) - transmit and receive simultaneously but not necessarily just between two stations, e.g. data communications circuits 9

10. Medias for Communication •Telephone Channel •Mobile Radio Channel •Optical Fiber Cable •Satellite Channel 10

11. Modulation In modulation, a message signal, which contains the information is used to control the parameters of a carrier signal, so as to impress the information onto the carrier . It is the process of varying the characteristics of high frequency carrier in accordance with instantaneous values of modulating or message or baseband signal . (Or) It is a frequency translation technique which converts baseband or low frequency signal to bandpass or high frequency signal . The Messages The message or modulating signal may be either: analogue – denoted by m ( t ) digital – denoted by d ( t ) – i.e. sequences of 1's and 0's The message signal could also be a multilevel signal, rather than binary; this is not considered further at this stage . The Carrier The carrier could be a 'sine wave' or a 'pulse train'. Consider a 'sine wave' carrier: vc (t) = Vc cos ( ωct + φc )  •If the message signal m ( t ) controls amplitude – gives AMPLITUDE MODULATION AM •If the message signal m ( t ) controls frequency – gives FREQUENCY MODULATION FM •If the message signal m ( t ) controls phase- gives PHASE MODULATION PM or 11

12. Benefits or Need of Modulation To reduce the length or height of antenna For multiplexing or narrow banding or to use antenna with single or same length To reduce noise effect To avoid equipment limitation or to reduce the size of the equipment. 12

13. Demodulation Demodulation is the reverse process (to modulation) to recover the message signal m ( t ) or d ( t ) at the receiver. 13

14. Frequency spectrum 14

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16. Low frequencies is used in submarines because low frequencies can penetrate in water more effectively. 16

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18. Advantages of a Communication System 1. Speedy transmission: It requires only a few seconds to communicate through electronic media because it supports quick transmission. 2. Wide coverage: World has become a global village and communication around the globe requires a second only. 3 . Low cost: Electronic communication saves time and money. For example, Text SMS is cheaper than the traditional letter. 4. Exchange of feedback: Electronic communication allows the instant exchange of feedback. So communication becomes perfect using electronic media. 5. Managing global operation: Due to the advancement of electronic media, business managers can easily control operation across the globe. Video or teleconferencing e-mail and mobile communication are helping managers in this regard 18

19. Disadvantages of a Communication System 1.The volume of data: The volume of telecommunication information is increasing at such a fast rate that business people are unable to absorb it within the relevant time limit. 2 . The cost of development: Electronic communication requires huge investment for infrastructural development. Frequent change in technology also demands further investment. 3. Legal status: Data or information, if faxed, may be distorted and will cause zero value in the eye of law. 4 . Undelivered data: Data may not be retrieved due to system error or fault with the technology. Hence required service will be delayed 19

20. Analog Communication 20

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23. Types of AM 23

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56. NOISE 56

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58. NOISE 58

59. NOISE IN COMMUNICATION SYSTEMS Noise :It is an unwanted signal which tends to interfere with the modulating signal . Types of noise : Noise is basically divided into , 1.ExternalNoise 2.InternalNoise 59

60. Classification of Noise 1.External Noise : Atmospheric Noise : Radio noise caused by natural atmospheric processes, primarily lightening discharges in thunderstorms . Extraterrestrial Noise : Radio disturbances from sources other than those related to the Earth. Cosmic Noise : Random noise that originates outside the Earth’s atmosphere . Solar Noise : Noise that originates from the Sun is called Solar noise Industrial Noise : Noise generated by auto mobile ignition, aircrafts, electric motors, Switchgears, welding etc. 2. Internal Noise : Shot Noise : Random motion of electrons in the semiconductor devices generates shot noise. Thermal or Johnson’s Noise : Random motion of electrons in the resistor is called Thermal noise . Pn =K(T )B Where, K=Boltzmann constant T =Absolute temperature, B=Bandwidth 60

61. Noise Temperature and Noise Figure 61

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63. Noise equivalent bandwidth 63

64. Figure of Merit 64

65. Receiver Model for Noise calculation •The receiver is combination of Band Pass Filter (BPF) and Demodulator . • The BPF is combination of RF Tuned Amplifier, Mixer and Local Oscillator whose bandwidth is equal to bandwidth of modulated signal at transmitter . • Channel Interconnects transmitter & receiver. Channel adds noise to the modulated signal while transmitting and it is assumed to be white noise whose Power Spectral Density is uniform . • BPF converts white noise into color or Bandpass noise or narrow bandpass noise . 65

66. Receiver model for noise calculation 66

67. Communication system model for noise calculation • The communication system model for noise calculation contains transmitter, channel and receiver . • Transmitter is replaced by modulator which converts low frequency modulating signal x(t) into high frequency bandpass signal with the help of carrier signal . • Channel is replaced or modelled as additive noise which adds white noise with PSD η/2 and it contains all frequencies . 67

68. Radio receiver measurements The important characteristics of super heterodyne radio receiver are , •Sensitivity •Selectivity •Fidelity Sensitivity : •It is defined as the ability of receiver to amplify weak signals •It is defined in terms of voltage which must be applied at the receiver input terminals to provide a standard output power at the receiver output. Sensitivity is expressed in milli volts • For practical receivers sensitivity is expressed in terms of signal power required to produce minimum acceptable output with minimum acceptable noise. 68

69. Selectivity : It is defined as the ability of receiver to reject unwanted signals. Selectivity depends on • Receiving frequency • Response of IF section 69

70. Fidelity : It is the ability of a receiver to reproduce all the modulating frequenciese qually . 70

71. Digital Communication 71

72. WHY DIGITAL? Digital systems are less sensitive to noise than analog. For long transmission lengths, the signal may be regenerated effectively error-free at different points along the path. With digital systems, it is easier to integrate different services, for example, video and soundtrack into the same transmission scheme The transmission scheme is independent of the source. For example, a digital transmission scheme that transmits voice at 10kpbs can also be used to transmit data at 10kpbs Digital circuits are easy to manufacture and less sensitive to physical effects, such temperature Digital signals are simpler to characterize and do not have the same amplitude range and variability as analog signals. Good processing techniques are available for digital signals, such as medium . Data compression (or source coding ) Error Correction(or channel coding )( A/D conversion ) Equalization Security There are techniques for adding controlled redundancy to a digital transmission such that errors that occur during transmission may be corrected at the receiver. These techniques are called channel coding.Channel compensation techniques, such as equalization, are easy to implement . Easy to mix signals and data using digital techniques 72

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74. Information Representation Communication system converts information into electrical electromagnetic/optical signals appropriate for the transmission medium .  Analog systems convert analog message into signals that can propagate through the channel .  Digital systems convert bits(digits, symbols) into signals 74

75. Types of information: Voice, data, video, music, email etc. Types of communication systems: Public Switched Telephone Network (voice , fax, modem ) Satellite systems Radio , TV broadcasting Cellular phones Computer networks (LANs, WANs, WLANs) 75

76. 76 Analog vs. Digital Data Some Terms: Data/Message Signal/ Message signal : Information in electrical form Signaling : Act of propagating the signal Transmission : Involves propagation and processing both Messages or data can be classified : Analog A physical quantity that varies with “time”, usually in a smooth or continuous fashion e,g speech Digital Takes discrete values e.g text

77. 77 Analog vs. Digital Signal Analog Signals Continuously varying electromagnetic wave Values are taken from an infinite set Digital Signals Sequence of voltage pulses Values are taken from a discrete set Binary Signals Digital signals with just two discrete values t t t 1 1 1

78. 78 Signal is a function of time and frequency Time Domain - Continuous or Discrete - Periodic or Aperiodic S(t)= Asin (2 ft+ Ø ) Signal Characterization t t t t t A in volts T=1/f

79. 79 Signal Characterization Frequency Domain t T 1 =1/f 1 t T 2 =1/f 2 t T 1 =1/f 1 f A 1 f1 f A 2 f2 f A 2 f2 A 1 Fourier Analysis : Any signal can be expressed as a combination of various sinusoids Some interesting points: Fundamental Frequency : All the frequency components are integer multiple of one frequency Period of total signal =Period of fundamental freq.

80. 80 Signal Characterization Phase is a relative measure. If two signals overlap with each other then they are in phase. Otherwise they are out of phase α

81. 81 Signal Characterization t f f 1 f 2 f n Spectrum : Range of frequencies contained Bandwidth: - Absolute : f n -f 1 - Effective : f k -f 1 Signals may have infinite bandwidth but transmission media can accommodate only limited BW DC Component: Freq. term at f=0 f k

82. 82 Spectra , Bandwidth and Data Rate T T A pulse in time domain Amplitudes of frequency components in freq domain If T =1 ms 1000 such pulses can be sent in 1 sec i e data rate is 1 kb/s Strong spectral components are below 1/T Hz i e the signal BW is 1KHz Channel BW 1000 Hz If data rate is to be increased T should be decreased This increases the BW ( For 10 times higher rate BW required increase 10 times)

83. 83 Medium Speed Cost Twisted Wire 300 Bps - 10 Mbps Low Microwave 256 Kbps - 100 Mbps Satellite 256 Kbps - 100 Mbps Coaxial Cable 56 Kbps - 200 Mbps Fiber Optics 500 Kbps - 6.4 Tbps High Bps: Bits Per Second Kbps: Kilobits Ps, Mbps: Megabits Ps, Gbps : Gigabits Ps, Tbps : Terabits Ps Speeds & Cost

84. Digital Communication System source source coding channel coding modulation Tx source decoding channel decoding demodulation Rx Channel o/p noise 84

85. Digital symbols are transmitted as modulated carrier A carrier is a sinusoidal time varying signal represented typically by s(t) = A sin(2  ft +  ) Information can be transmitted by varying one or more of the three defining parameters of the carrier namely – Amplitude (A), Frequency (f) or the phase (  ) Hence we have ASK, FSK and PSK as the basic modulation schemes Modulation 85

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87. Bit-rate, Baud and the Bandwidth Bit-rate bits / sec Baud symbols / sec Bandwidth baud rate, pulse shape 87

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89. Types of Digital Modulation 89

90. Types of Digital Modulation Amplitude Shift Keying (ASK) Frequency Shift Keying (FSK) – Binary, M-ary Phase Shift Keying (PSK) – Binary, QPSK Quadrature Amplitude Modulation – QAM Advanced- OQPSK, /4-QPSK, MSK, GMSK, TCM 90

91. Classification of Digital Modulation Techniques Linear Modulation BPSK, M-ary PSK, QAM, OQPSK, /4-QPSK Constant Envelope Modulation FSK, MSK, GMSK 91

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95. Types of Digital Modulation - Summary Unmodulated FSK ASK PSK ASK + PSK 95

96. Polar Form Representation of Digitally Modulated Signals s(t) = A sin(2  ft +  )  s(A, ) s(A, ) Amplitude Change Phase Change Amp & Ph Change Frequency Change Polar Display 96

97. BPSK 97

98. Typical Tx – Rx for I/Q Modulation Tx Rx Composite signal 98

99. Generation Using I/Q Design Example QPSK S/P i/p bits  A sin(2  ft +  )  A cos(2  ft +  )  1  1 (0,1) (1,1) (0,0) (1,0) 99

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105. Variants of QPSK QPSK Offset QPSK /4 QPSK  = 0,  90,  180  = 0,  90  =  45 ,  135 105

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107. 16 - QAM 107

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109. 16-QAM 4 bits per symbol Symbol rate = 1/ 4 Bit Rate 32-QAM 5 bits per symbol Symbol rate = 1/ 5 Bit Rate Vector Diagram Constellation Diagram As M increase, Constellation becomes denser >> BER increases Effect of Increasing M (# Constellation Points) # bits per Symbol = log 2 M 109

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113. GMSK 113

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118. Effect of Change in BT on GMSK Spectrum 118

119. Effect of Distortion on Digital Modulation 119

120. Effect of Distortions on Signal Constellation 120

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122. Eye Diagram 122

123. Eye Diagram - Contd 123

124. Eye Diagram - Contd 124

125. Eye Diagrams for GMSK BT = 0.3 BT = 0.5 BT = 1.0 125

126. Theoretical BPSK Raised Cosine  = 0.5 126

127. Power Efficiency Vs Bandwidth Efficiency 127

128. Shannon’s Channel Capacity Theorem R  C = B * log 2 {1 + (S / N)} C – Capacity (bits/s), B – Bandwidth (Hz) S – Signal Power (W), N – Noise Power (W) R/B  log 2 [ 1 + (E b / N )(R/B) ] ( E b - - Energy Per Bit, N – Noise Power Density ) E b /N  ( 2 R/B -1 ) / (R/B) = ( 2  -1 ) /  Lim (B  ) E b / N = log e 2 = - 1.6 dB Capacity of Communication Systems 128

129. -2 -1 6 12 18 24 30 16 8 4 2 1/2 1/4        M = 8 M = 16 M = 64 M = 4  MPSK  MQAM Bandwidth Limited Region Power Limited Region Region for R > C Capacity Boundary R = C Shannon Limit -1.59 dB E b / N (dB) R/B ( b/s/Hz ) Bandwidth / Power Efficiency Plot R/B  log 2 [ 1 + (Eb/ N )(R/B) ] 129

130. M-QAM Vs M-PSK Modulation E b / N R / B 8 PSK 14.0 3 8 QAM 10.6 3 16 PSK 18.3 4 16 QAM 14.5 4 64 PSK 26.5 6 64 QAM 18.8 6 130

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132. Modulation Technique Selection Criterion Three-way trade-off Power B/W BER 132

133. Summary 133

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138. Source Coding 138

139. Source Coding Theory • What is the minimum number of bits that are required transmit a particular symbol? • How can we encode symbols so that we achieve (or atleast come arbitrarily close to) this limit? • Source encoding: concerned with minimizing the actual number of source bits that are transmitted to the user • Channel encoding: concerned with introducing redundant bits to enable the receiver to detect and possibly correct errors that are introduced by the channel . 139

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145. Channel Coding 145

146. Channel Coding Theorem Shannon, “A mathematical theory of communication ,”1948 • For any channel, there exists an information capacity C (whose calculation is beyond the scope of this course). • If the transmission rate R ≤ C , then there exists a coding scheme such that the output of the source can be transmitted over a noisy channel with an arbitrarily small probability of error. Conversely, it is not possible to transmit messages without error if R > C . • Important implication: – The basic limitation due to noise in a communication channel is not on the reliability of communication, but rather, on the speed of communication . 146

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149. Error Correction: Basic Principle 149

150. Error Control Techniques Channel Coding in Digital Communication Systems Three approaches can be used to cope with data transmission errors : 1 . Automatic Repeat reQuest (ARQ) : Error detection When a receiver circuit detects errors in a block of data, it requests that the block be retransmitted. The receiver sends a feedback to the transmitter: Error is detected ( NACK : Not-Acknowledgement) in the received packet, then retransmit that data block, or if no errors detected ( ACK : Acknowledgement), don’t resend. The transmitter retransmits the previously sent packet if it receives a NACK. Uses extra/redundant bits merely for error detection . Full-duplex (two-way) connection between the Transmitter and the Receiver. Result : Constant reliability, but varying data rate throughput due to retransmit . 2.Forward Error Correction (FEC): Error detection and correction. The transmitter’s encoder adds extra/redundant bits to a block of message data bits to form a codeword , so the receiver can both detect errors and automatically correct errors incurred during transmission, without retransmission of the data. Simplex (one-way) connection between the Transmitter and the Receiver. Result : Varying reliability, but constant data rate throughput . 3.Hybrid ARQ (ARQ+FEC): Error detection and correction. Full-duplex connection required between the Transmitter and the Receiver. Uses error detection and correction codes. 4.In general, wire-line communications (more reliable) adopts ARQ scheme, while wireless communications (relatively less reliable) adopts FEC scheme. 150

151. Types of Forward Error Control Codes Channel Coding Major categories of Error Detection and Error Correction Codes: 1.Error Detection Codes : Parity Check codes. ARC : Arithmetic Redundancy Check codes. CRC : Cyclic Redundancy Check codes. 2.Block Error Correction Codes : Hamming linear block error correcting codes. BCH (Bose- Chaudhuri - Hocquenghem ) cyclic block codes. Reed-Solomon cyclic block codes. Turbo Product Codes (TCP). 3.ConvolutionalError Correction Code : Tradition , ViterbiDecoding . Turbo ConvolutionalCode (TCC). Low Density Parity Check Code. 4.Concatenated Error Correction Codes : Inner/Outer codes. Reed-Solomon Error Correction Codes/ Viterbi algorithm . 151

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157. Thank You Any Questions ?? 157

158. Advanced 158

159. Trellis Coded Modulation (TCM) 159

160. TCM – Basic Concept 160

161. Trellis Coded Modulation Generalized Block Diagram 161

162. Trellis Coded Modulation Using QPSK QPSK Constellation Trellis Diagram 162

163. Spread Spectrum Modulation 163

164. Shannon’s Channel Capacity Theorem C = B * log 2 {1 + (S / N)} C – Capacity (bits/s), B – Bandwidth (Hz) S – Signal Power (W), N – Noise Power (W) A given information transfer rate can be achieved by increasing the bandwidth (B) of the transmitted signal while reducing the required signal-to-noise ratio (S/N) Capacity of Communication Systems 164

165. Why Spread Spectrum ?? Low Probability of Intercept Anti-Jam or Jam-resistant Interference Rejection Multi-path Rejection Multiple Access Wireless LAN Security Immunity to freq selective fading 165

166. Spread Spectrum Techniques Direct Sequence (DS) Carrier is modulated by digital code Frequency Hop (FH) The carrier frequency is shifted in discrete increments in a pattern generated by code Time Hop (TH) The transmission slot is governed by code Hybrid DS + FH 166

167. DSSS- Basic principles c(t) b(t) Input b(t) c(t) Code Interference i(t) r(t)  m(t) = c(t) b(t) Transmit Signal m(t) Code c(t) r(t) = m(t) + i(t) Output b ' (t) C H A N N E L Spreading De-spreading r(t) Received Signal 167

168. DSSS T b = L T c T c t +1 -1 1 1 Information is multiplied with higher rate digital sequence (spreading code). The sequence has many “chips” for every data bit. The resultant signal modulates the RF carrier. 168

169. Processing Gain Processing Gain is a measure of the amount of rejection offered by spread spectrum system to the interference Processing Gain (G p ) = Information bit period (T b ) Chip Period (T c ) Spreaded Bandwidth Input signal Bandwidth = 169

170. Noise Noise Noise Noise Noise Signal Signal Signal Signal Signal A Signal Signal B Interference Interference Signal A Signal B Low Probability of Intercept Interference Rejection / Anti-Jam DSSS Properties DSSS Rx DSSS Rx DSSS Rx Multiple Access 170

171. FHSS time f 1 f 2 f 3 f n-2 f n-1 f n Frequency Hopping pattern of transmission frequency is selected based on a PN code T 2T 3T frequency 171

172. DS Vs FH DSSS Advantages Most difficult to intercept Better AJ performance in presence of wideband jammer Disadvantages Large Bandwidth Long Acquisition Time Suffers from near-far effect FHSS Advantages Better AJ performance in presence of narrow band jammer No Near-far effect (being freq avoidance) Disadvantages Complex frequency synthesizer Requires Error correction By using a Hybrid System (DSFH) benefits of both can be combined. The system can have a low probability of interception and negligible near-far effect at the same time. 172

173. Application of Spread Spectrum Military Communication Multiple Access GPS Wireless LAN Indoor Wireless Communication 173

174. OFDM 174

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176. Case Study Some Voice Grade Modems 176

177. Some Standard Modems V.22 bis 16 QAM, 600 baud, carrier = 1200 / 2400 Hz FDX, Adaptive Equalizer V.32 16 QAM (TCM), 2400/1200 baud FDX, Echo Canceller, Adaptive Equalizer V.26 ter QPSK, 1200 baud, carrier = 1800 Hz FDX, Echo Canceller, Adaptive Equalizer 177

178. Thank You Any Questions ?? 178

179. 1. CT and DT signals: Classification of signals 179

180. Classification of signals (cont.) For many cases, x [ n ] is obtained by sampling x ( t ) as: x [ n ] = x ( nT ) , n =0, + 1, + 2,… Are there any requirements for the sampling? 180

181. Classification of signals (cont.) 2. Even and odd signals: Even: x ( − t ) = x ( t ) x [ − n ] = x [ n ] Odd: x ( − t ) = − x ( t ) x [ − n ] = − x [ n ] Any signal x ( t ) can be expressed as x ( t ) = x e ( t ) + x o ( t ) ) x ( − t ) = x e ( t ) − x o ( t ) where x e ( t ) = 1/2 ( x ( t ) + x ( − t )), x o ( t ) = 1/2 ( x ( t ) − x ( − t )) 181

182. Classification of signals (cont.) 3. Periodic and non-periodic signals: CT signal: if x ( t ) = x ( t + T ) , then x ( t ) is periodic. Smallest T=Fundamental period: T o Fundamental frequency f o = 1 /T o (Hz or cycles/second) Angular frequency: o = 2 /T o ( rad /seconds) DT signal: if x [ n ] = x [ n + N ] , then x [ n ] is periodic. min ( N o ) : fundamental period F o = 1 /N o (cycles/sample) =2 /N ( rads /sample). If the unit of n is designated as dimensionless, then is simply in radians. Note: A sampled CT periodic signal may not be DT periodic. Any Condition addition of two periodic CT signals, resultant must be periodic signal ? 182

183. Classification of signals (cont.) 4. Deterministic and random signals. Deterministic signal: No uncertainty with respect to its value at any time Completely specified at any time Random signal: Uncertain before it occurs. E.g., thermal noise. 183

184. Classification of signals (cont.) Energy and power signals: CT signal x ( t ) : Energy: E = Power: P = 184

185. Classification of signals (cont.) DT signal x [ n ] : Energy: E = Power: Energy signal: if < E < Power signal: if < P < 185

186. Classification of signals (cont.) Analog Signal and Digital Signal 186

187. Basic operations on signals Basic Operations on Signal 187

188. Rule for time shifting and time scaling: See figure below. Find y ( t ) = x (2 t + 3) . Basic Operations on Signal(cont.) 188

189. Elementary signals 1. Exponential 2-Sinusoidal 189

190. Elementary signals(cont.) 3. Step function 190

191. 5.Unit ramp function Elementary signals(cont.) 4.Unit impulse function 191

192. System Properties 192

193. 2.Memory /Memoryless Memory system: present output value depend on future/past input. Memoryless system: present output value depend only on present input. Example System Properties(cont.) 193

194. System Properties(cont.) 194

195. System Properties(cont.) 195

196. Invertibility x(t) x(t) y(t) H H System Properties(cont.) 196

197. Series(cascade) Interconnection Parallel, Interconnection Interconnection of systems System 1 System 2 System 1 System 2 + Input Output Input Output 197

198. Interconnection of systems Feedback Interconnection System 1 System 2 Input Output 198