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Lecture-2: Limits of Communication

Explore the limits of communication in terms of achievable transmission bit-rate and bit-error-rate in different channel scenarios. Learn about Shannon theory and its application to MIMO transmission.

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Lecture-2: Limits of Communication

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  1. Lecture-2: Limits of Communication • Problem Statement: Given a communication channel (bandwidth B), and an amount of transmit power, what is the maximum achievable transmission bit-rate (bits/sec), for which the bit-error-rate is (can be) sufficiently (infinitely) small ? - Shannon theory (1948) - Recent topic: MIMO-transmission (e.g. V-BLAST 1998, see also Lecture-1) Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  2. Overview • `Just enough information about entropy’ (Lee & Messerschmitt 1994) self-information, entropy, mutual information,… • Channel Capacity (frequency-flat channel) • Channel Capacity (frequency-selective channel) example: multicarrier transmission • MIMO Channel Capacity example: wireless MIMO Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  3. `Just enough information about entropy’(I) • Consider a random variable X with sample space (`alphabet’) • Self-information in an outcome is defined as where is probability for(Hartley 1928) • `rare events (low probability) carry more information than common events’ `self-information is the amount of uncertainty removed after observing .’ Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  4. `Just enough information about entropy’(II) • Consider a random variable X with sample space (`alphabet’) • Average information or entropy in X is defined as because of the log, information is measured in bits Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  5. `Just enough information about entropy’ (III) • Example: sample space (`alphabet’) is {0,1} with entropy=1 bit if q=1/2 (`equiprobable symbols’) entropy=0 bit if q=0 or q=1 (`no info in certain events’) H(X) 1 q 0 1 Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  6. `Just enough information about entropy’ (IV) • `Bits’ being a measure for entropy is slightly confusing (e.g. H(X)=0.456 bits??), but the theory leads to results, agreeing with our intuition (and with a `bit’ again being something that is either a `0’ or a `1’), and a spectacular theorem • Example: alphabet with M=2^n equiprobable symbols : -> entropy = n bits i.e. every symbol carries n bits Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  7. `Just enough information about entropy’ (V) • Consider a second random variable Y with sample space (`alphabet’) • Y is viewed as a `channel output’, when X is the `channel input’. • Observing Y, tells something about X: is the probability for after observing Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  8. `Just enough information about entropy’ (VI) • Example-1 : • Example-2 : (infinitely large alphabet size for Y) decision device noise 00 01 10 11 00 01 10 11 X Y + noise 00 01 10 11 X Y + Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  9. `Just enough information about entropy’(VII) • Average-information or entropy in X is defined as • Conditional entropy in X is defined as Conditional entropy is a measure of the average uncertainty about the channel input X after observing the output Y Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  10. `Just enough information about entropy’(VIII) • Average information or entropy in X is defined as • Conditional entropy in X is defined as • Average mutual information is defined as I(X|Y) is uncertainty about X that is removed by observing Y Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  11. Channel Capacity (I) • Average mutual information is defined by -the channel, i.e. transition probabilities -but also by the input probabilities • Channel capacity(`per symbol’ or `per channel use’) is defined as the maximum I(X|Y) for all possible choices of • A remarkably simple result: For a real-valued additive Gaussian noise channel, and infinitely large alphabet for X (and Y), channel capacity is signal (noise) variances Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  12. Channel Capacity (II) • A remarkable theorem (Shannon 1948): With R channel uses per second, and channel capacity C, a bit stream with bit-rate C*R (=capacity in bits/sec) can be transmitted with arbitrarily low probability of error = Upper bound for system performance ! Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  13. Channel Capacity (II) • For a real-valued additive Gaussian noise channel, and infinitely large alphabet for X (and Y), the channel capacity is • For a complex-valued additive Gaussian noise channel, and infinitely large alphabet for X (and Y), the channel capacity is Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  14. Channel Capacity (III) Information I(X|Y) conveyed by a real-valued channel with additive white Gaussian noise, for different input alphabets, with all symbols in the alphabet equally likely (Ungerboeck 1982) Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  15. Channel Capacity (IV) Information I(X|Y) conveyed by a complex-valued channel with additive white Gaussian noise, for different input alphabets, with all symbols in the alphabet equally likely(Ungerboeck 1982) Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  16. Channel Capacity (V) This shows that, as long as the alphabet is sufficiently large, there is no significant loss in capacity by choosing a discrete input alphabet, hence justifies the usage of such alphabets ! The higher the SNR, the larger the required alphabet to approximate channel capacity Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  17. Channel Capacity (frequency-flat channels) • Up till now we considered capacity `per symbol’ or `per channel use’ • A continuous-time channel with bandwidth B (Hz) allows 2B (per second) channel uses (*), i.e. 2B symbols being transmitted per second, hence capacity is (*) This is Nyquist criterion `upside-down’ (see also Lecture-3) received signal (noise) power Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  18. s(t) r(t)=Ho.s(t)+n(t) Ho H(f) channel Ho f B -B Channel Capacity (frequency-flat channels) • Example: AWGN baseband channel (additive white Gaussian noise channel) + here n(t) Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  19. s(t) r(t)=Ho.s(t)+n(t) Ho channel Channel Capacity (frequency-flat channels) • Example: AWGN passband channel passband channel with bandwidth B accommodates complex baseband signal with bandwidth B/2 (see Lecture-3) + n(t) H(f) Ho f x x+B Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  20. Channel Capacity (frequency-selective channels) • Example: frequency-selective AWGN-channel received SNR is frequency-dependent! s(t) R(f)=H(f).S(f)+N(f) + H(f) channel H(f) n(t) f -B B Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  21. Channel Capacity (frequency-selective channels) • Divide bandwidth into small bins of width df, such that H(f) is approx. constant over df • Capacity is optimal transmit power spectrum? B 0 H(f) f -B B Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  22. Channel Capacity (frequency-selective channels) Maximize subject to solution is `Water-pouring spectrum’ Available Power L B area Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  23. Channel Capacity (frequency-selective channels) Example : multicarrier modulation available bandwidth is split up into different `tones’, every tone has a QAM-modulated carrier (modulation/demodulation by means of IFFT/FFT). In ADSL, e.g., every tone is given (+/-) the same power, such that an upper bound for capacity is (white noise case) (see Lecture-7/8) Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  24. MIMO Channel Capacity (I) • SISO =`single-input/single output’ • MIMO=`multiple-inputs/multiple-outputs’ • Question: we usually think of channels with one transmitter and one receiver. Could there be any advantage in using multiple transmitters and/or receivers (e.g. multiple transmit/receive antennas in a wireless setting) ??? • Answer: You bet.. Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  25. N1 X1 Y1 + + X2 Y2 N2 MIMO Channel Capacity (II) • 2-input/2-output example A B C D Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  26. MIMO Channel Capacity (III) Rules of the game: • P transmitters means that the same total power is distributed over the available transmitters (no cheating) • Q receivers means every receive signal is corrupted by the same amount of noise (no cheating) Noises on different receivers are often assumed to be uncorrelated (`spatially white’), for simplicity Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  27. N1 X1 Y1 + + first example/attempt X2 Y2 N2 MIMO Channel Capacity (IV) 2-in/2-out example, frequency-flat channels Ho 0 0 Ho Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  28. MIMO Channel Capacity (V) 2-in/2-out example, frequency-flat channels • corresponds to two separate channels, each with input power and additive noise • total capacity is • room for improvement... Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  29. N1 X1 Y1 + + X2 Y2 N2 MIMO Channel Capacity (VI) 2-in/2-out example, frequency-flat channels Ho Ho second example/attempt -Ho Ho Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  30. MIMO Channel Capacity (VII) A little linear algebra….. Matrix V’ Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  31. MIMO Channel Capacity (VIII) A little linear algebra…. (continued) • Matrix V is `orthogonal’ (V’.V=I) which means that it represents a transformation that conserves energy/power • Use as a transmitter pre-transformation • then (use V’.V=I) ... Dig up your linear algebra course notes... Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  32. + + + + MIMO Channel Capacity (IX) • Then… X1 N1 X^1 A Y1 V11 V12 B V21 C X^2 Y2 V22 D transmitter X2 N2 channel receiver Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  33. MIMO Channel Capacity (X) • corresponds to two separate channels, each with input power , output power and additive noise • total capacity is 2x SISO-capacity! Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  34. MIMO Channel Capacity (XI) • Conclusion: in general, with P transmitters and P receivers, capacity can be increased with a factor up to P (!) • But: have to be `lucky’ with the channel (cfr. the two `attempts/examples’) • Example : V-BLAST (Lucent 1998) up to 40 bits/sec/Hz in a `rich scattering environment’ (reflectors, …) Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  35. MIMO Channel Capacity (XII) • General I/O-model is : • every H may be decomposed into this is called a `singular value decompostion’, and works for every matrix (check your MatLab manuals) orthogonal matrix V’.V=I orthogonal matrix U’.U=I diagonal matrix Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  36. MIMO Channel Capacity (XIII) With H=U.S.V’, • V is used as transmitter pre-tranformation (preserves transmit energy) and • U’ is used as a receiver transformation (preserves noise energy on every channel) • S=diagonal matrix, represents resulting, effectively `decoupled’ (SISO) channels • Overall capacity is sum of SISO-capacities • Power allocation over SISO-channels (and as a function of frequency) : water pouring Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  37. MIMO Channel Capacity (XIV) Reference: G.G. Rayleigh & J.M. Cioffi `Spatio-temporal coding for wireless communications’ IEEE Trans. On Communications, March 1998 Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  38. Assignment 1 (I) • 1. Self-study material Dig up your favorite (?) signal processing textbook & refresh your knowledge on -discrete-time & continuous time signals & systems -signal transforms (s- and z-transforms, Fourier) -convolution, correlation -digital filters ...will need this in next lectures Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

  39. Assignment 1 (II) • 2. Exercise (MIMO channel capacity) Investigate channel capacity for… -SIMO-system with 1 transmitter, Q receivers -MISO-system with P transmitters, 1 receiver -MIMO-system with P transmitters, Q receivers P=Q (see Lecture 2) P>Q P<Q Module-3 Transmission Marc Moonen Lecture-2 Limits of Communication K.U.Leuven-ESAT/SISTA

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