Communications Theory Introduction Manuel A. V azquez January 27, - PowerPoint PPT Presentation

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Communications Theory Introduction Manuel A. V´ azquez January 27, 2020

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Index Course Content 1 Overview 2 Source of information 3 Transmitter 4 Channel 5 The receiver 6 Design of a system and quality metrics 7 Quality Available technologies, Cost and Resources consumption Analog vs Digital communications systems 8

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Contents of the course 1 Noise in communications systems: stochastic processes, white noise, SNR 2 Modulation and detection in Gaussian channels: information modulation, demodulation and detection, error probability 3 Fundamental limits in communications 4 Analog modulation

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital What is the purpose of a communications system? Goal : to transmit information between two points that are somehow connected by some physical structure ...the physical structure might be: a cable, the air, empty space... Applications cellphone - base station base station - TV peer-to-peer radio streaming ...plenty more

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Block diagram When focusing on the functionality , the structure of a typical communications system is: Source of Transmitter Channel Receiver Destination Information message tx signal rx signal rx message message: physical manifestation of the information We study each of the above blocks separately...

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Source of information It aims a communicating/reporting something Information might be in different formats voice text images · · · Sources can be analog digital ...according to the information they produce

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Analog source It produces messages that are modeled as a continuous waveform. This could represent variation in the air pressure, temperature variation, bitcoin price, price of stocks...

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Digital source It produces a sequence of symbols belonging to a finite set (the alphabet ), each one sent during a certain time interval. a symbol “a thing that represents or stands for something else” (Oxford English Dictionary) For us, a symbol translates into a (continuous-time) signal transmitted during a symbol period (usually denoted as T ) the alphabet is a set of symbols

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Digital source: examples I s 1 ( t ) s 2 ( t ) T t Alphabet: A − A t T s 0 ( t ) s 0 ( t ) t T 2 T 3 T 4 T s 1 ( t ) s 1 ( t )

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Digital source: examples II More examples of alphabets { , } { A sin( w 0 t ) , − A sin( w 0 t ) } (the signals are digital in amplitude ) { A sin( w 1 t ) , A sin( w 2 t ) } (the signals are digital in frequency ) � t � t � t � t � t � � � � �� � A Π , − A Π , 3 A Π , − 3 A Π where Π is T T T T T a rectangular pulse of length T centered at 0, i.e., � t � Π T 1 w T 2 (the signals are digital in amplitude )

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Two different kinds of communication systems each kind of source → communications system specifically tailored for it: digital source digital communications system examples: ADSL, HDTV...pretty much everything analog source analog communications system examples: old TV, radio (for how long??) how come we use digital communications system for nearly everything???

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Two different kinds of communication systems each kind of source → communications system specifically tailored for it: digital source digital communications system examples: ADSL, HDTV...pretty much everything analog communications system analog source digital communications system? examples: old TV, radio (for how long??)

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Digitizing signals 3 T s T s sampling to discretize the time axis no information loss if Nyquist condition holds quantization to discretize the amplitude information loss

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Transmitter It shapes up the information coming from the source so that it can traverse the channel It needs to know whether the system is analog or digital ...but also, whether the channel is baseband...e.g., H 1 ( jw ) w W 1 passband...e.g., H 2 ( jw ) - w c w c w w c − W 1 w c + W 1

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Transmission I Here, we model the channel as an LTI system, x ( t ) y ( t ) = x ( t ) ∗ h ( t ) h ( t ) H ( jw ) X ( jw ) Y ( jw ) = X ( jw ) H ( jw ) so, what happens if the spectrum of the signal to be transmitted is X ( jw ) w W Can the signal travel through both channels?

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Transmission II x ( t ) can travel through the baseband channel ( baseband transmission) without distortion if W 1 > W with distortion if W 1 < W (information loss) x ( t ) cannot travel through the passband channel as it is, X ( jw ) X ( jw ) w w - w c w c W × cos( w c t ) but... (time domain) ...and we have passband transmission The above operation is called modulation and cos( w c t ) is the so-called carrier signal

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Channel It is the physical medium though which information propagates In general, it doesn’t let the transmitted signal go through as it is: disturbances noise interference distortions linear time-invariant: ∃ h ( t ) that serves to characterize it → channel is an LTI system non-linear: ∄ h ( t ) characterizing it The channel is usually modeled like this: x ( t ) h ( t ) + r ( t ) n ( t )

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Receiver It must recover the information transmitted as faithfully as possible Among other things, it must 1 Demodulate, i.e., carry the signal back to its original frequency band 2 Reject disturbances 3 Fix channel distortions whenever possible Ideally, we would like to find h − 1 ( t ) such that y ( t ) y n ( t ) h − 1 ( t ) x ( t ) h ( t ) + x ( t ) + v ( t ) receiver n ( t )

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Receiver in an analog system 2 and 3 are challenging in an analog system... Let us assume we receive y ( t ) t Was this the signal actually transmitted?

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Receiver in an digital system We know the alphabet of the system, e.g., s 2 ( t ) s 1 ( t ) A T t t − A T we know disturbances If we receive... and or distortions y ( t ) occurred A we can estimate what t was transmitted (making T − A a decision ) This is the point of digital communication systems!!

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Design of a system When designing a system, we have to take into account (among others): Quality Available technologies Cost Resources consumption ...we briefly review each one of them

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Quality We need a measure of the quality of a system so that we can design it properly and compare it against others Different metrics for the two different kind of systems: analog system → fidelity digital system → error probability

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Quality in an analog system I Fidelity refers to whether the received signal resembles the transmitted one. Example #1 Example #2 transmitted transmitted received received t t On the left, the transmitted signal is recognizable in the received one...no so much on the right

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Quality in an analog system II We need a quantitative measure of fidelity: it is the signal-to-noise ratio (SNR), which is defined as S → power of the signal → power of the noise N Other parameter related to the quality: bandwidth telephone AM radio FM radio HiFi system − quality + quality

Course Content Overview Source Transmitter Channel Receiver Quality Analog vs Digital Quality in a digital system We can count how many symbols were correctly received...and the probability of error is estimated as P e = number of symbols incorrectly received overall number of symbols transmitted Clearly, ↑ quality ⇒ ↓ probability of error ( P e ) Just like in analog systems, the bandwidth also has an impact here ↑ bandwidth ⇒ ↑ quality