Lecture 4 Capacity of Wireless Channels
|
|
- Simon Cobb
- 5 years ago
- Views:
Transcription
1 Lecture 4 Capacity of Wireless Channels I-Hsiang Wang ihwang@ntu.edu.tw 3/0, 014
2 What we have learned So far: looked at specific schemes and techniques Lecture : point-to-point wireless channel - Diversity: combat fading by exploiting inherent diversity - Coding: combat noise, and further exploits degrees of freedom Lecture 3: cellular system - Multiple access: TDMA, CDMA, OFDMA - Interference management: orthogonalization (partial frequency reuse), treat-interference-as-noise (interference averaging)
3 Information Theory Is there a framework to - Compare all schemes and techniques fairly? - Assert what is the fundamental limit on how much rate can be reliably delivered over a wireless channel? Information theory! - Provides a fundamental limit to (coded) performance - Identifies the impact of channel resources on performance - Suggest novel techniques to communicate over wireless channels Information theory provides the basis for the modern development of wireless communication 3
4 Historical Perspective G. Marconi C. Shannon First radio built 100+ years ago Great stride in technology But design was somewhat ad-hoc Information theory: every channel has a capacity Provides a systematic view of all communication problems Engineering meets science New points of view arise 4
5 Modern View on Multipath Fading Channel quality Time Classical view: fading channels are unreliable - Diversity techniques: average out the variation Modern view: exploit fading to gain spectral efficiency - Thanks to the study on fading channel through the lens of information theory! 5
6 Plot Use a heuristic argument (geometric) to introduce the capacity of the AWGN channel Discuss the two key resources in the AWGN channel: - Power - Bandwidth The AWGN channel capacity serves as a building block towards fading channel capacity: - Slow fading channel: outage capacity - Fast fading channel: ergodic capacity 6
7 AWGN Channel Capacity Outline Resources of the AWGN Channel Capacity of some LTI Gaussian Channels Capacity of Fading Channels 7
8 AWGN Channel Capacity 8
9 Channel Capacity Capacity := the highest data rate can be delivered reliably over a channel - Reliably Vanishing error probability Before Shannon, it was widely believed that: - to communicate with error probability 0 - data rate must also 0 Repetition coding (with M-level PAM) over N time slots on AWGN channel: r! 6N M 3 SNR - Error probability Q - Data rate = log M N - As long as M N ⅓, the error probability 0 as N - But, the data rate! =! log N! still 0 as N 3N 9
10 Channel Coding Theorem For every memoryless channel, there is a definite number C that is computable such that: - If the data rate R < C, then there exists a coding scheme that can deliver rate R data over the channel with error probability 0 as the coding block length N - Conversely, if the data rate R > C, then no matter what coding scheme is used, the error probability 1 as N We shall focus on the additive white Gaussian noise (AWGN) channel - Give a heuristic argument to derive the AWGN channel capacity 10
11 AWGN Channel Power constraint: NX x[n] apple NP n=1 x[n] z[n] N (0, ) y[n] =x[n]+z[n] We consider real-valued Gaussian channel As mentioned earlier, repetition coding yield zero rate if the error probability is required to vanish as N Because all codewords are spread on a single dimension in an N-dimensional space How to do better? 11
12 Sphere Packing Interpretation y = x + z R N By the law of large numbers, as N, most y will lie p N(P + ) inside the N-dimensional sphere of radius p N(P + ) p N Also by the LLN, as N, y will lie near the surface of the N-dimensional sphere centered at x with radius p N Vanishing error probability non-overlapping spheres How many non-overlapping spheres can be packed into the large sphere? 1
13 Why Repetition Coding is Bad R N p N(P + ) y = x + z It only uses one dimension out of N! 13
14 Capacity Upper Bound R N p N(P + ) p N y = x + z Maximum # of non-overlapping spheres = Maximum # of codewords that can be reliably delivered p N(P + NR ) N apple p N N =) R apple 1 N log p N(P + ) N p N N! = 1 log 1+ P This is hence an upper bound of the capacity C. How to achieve it? 14
15 Achieving Capacity (1/3) (random) Encoding: randomly generate NR codewords {x1, x,...} lying inside the x-sphere of radius Decoding: Performance analysis: WLOG let x 1 is sent - By the LLN, := P P + y! MMSE! y! Nearest Neighbor! bx y x 1 = w +( 1)x 1 N +( 1) NP = N P - As long as x1 lies inside the uncertainty sphere centered at αy r with radius!!!, decoding will be correct - Pairwise error probability (see next slide) = N P P + P + p NP P + N/ 15
16 Achieving Capacity (/3) x-sphere r p NP N P P + x 1 y When does an error occur? Ans: when another codeword x falls inside the uncertainty sphere of αy What is that probability (pairwise error probability)? Ans: the ratio of the volume of the two spheres Pr {x 1! x } = p NP /(P + p N NP ) N x = P + N/ Union bound: Total error probability apple NR P + N/ 16
17 Achieving Capacity (3/3) Total error probability (by union bound) Pr {E} apple NR P + N/ = N R+ 1 log 1 1+ P!! As long as the following holds, Pr {E}! 0 as N!1 R< 1 log 1+ P Hence, indeed the capacity is C = 1 1+ log P bits per symbol time 17
18 Resources of AWGN Channel 18
19 Continuous- Time AWGN Channel System parameters: - Power constraint: P watts; Bandwidth: W Hz - Spectral density of the white Gaussian noise: N0/ Equivalent discrete-time baseband channel (complex) - 1 complex symbol = real symbols Capacity: Power constraint: NX x[n] apple NP n=1 x[n] z[n] CN(0,N 0 W ) y[n] =x[n]+z[n] C AWGN (P, W) = 1 log 1+ P/ bits per symbol time N 0 W/ SNR := P/N = W log 1+ P 0 W SNR per complex symbol bits/s = log (1 + SNR) bits/s/hz N 0 W 19
20 Complex AWGN Channel Capacity C AWGN (P, W) =W log 1+ P N 0 W bits/s = log (1 + SNR) bits/s/hz Spectral Efficiency The capacity formula provides a high-level way of thinking about how the performance fundamentally depends on the basic resources available in the channel No need to go into details of specific coding and modulation schemes Basic resources: power P and bandwidth W 0
21 Power SNR = P N 0 W log (1 + SNR) Fix W: High SNR: - Logarithmic growth with power Low SNR: - Linear growth with power SNR C = log(1 + SNR) log SNR C = log(1 + SNR) SNR log e 1
22 Bandwidth Fix P C(W )=W log 1+ P N 0 W W P N 0 W log e = P N 0 log e P N 0 log e Power limited region 1 C(W ) (Mbps) Capacity Limit for W 0.4 Bandwidth limited region Bandwidth W (MHz) 5 30
23 Bandwidth- limited vs. Power- limited C AWGN (P, W) =W log SNR = When SNR 1: (Power-limited regime) C AWGN (P, W) W P N 0 W 1+ P N 0 W - Linear in power; Insensitive to bandwidth When SNR 1: (Bandwidth-limited regime) - Logarithmic in power; Approximately linear in bandwidth P N 0 W C AWGN (P, W) W log bits/s log e = P N 0 log e P N 0 W 3
24 Capacity of Some LTI Gaussian Channels 4
25 SIMO Channel x h y y = hx + w C L # h MRC, h # ey = h x + ew, ew CN 0, Power constraint: P w CN 0, I L MRC is a lossless operation: we can generate y from ey : y = ey (h/ h ) Hence the SIMO channel capacity is equal to the capacity of the equivalent AWGN channel, which is C SIMO = log 1+ h P Power gain due to Rx beamforming 5
26 MISO Channel x h h = h 1 y h y = h x + w C, # Tx Beamforming x = xh/ h # y = x h + w x, h C L Power constraint: NX x apple NP n=1 Goal: maximize the received power h* x - The answer is h P! (check. Hint: Cauchy-Schwarz inequality) Achieved by Tx beamforming - Send a scalar symbol x on the direction of h - Power constraint on x : still P Capacity: C MISO = log 1+ h P 6
27 Frequency- Selective Channel y[m] = LX 1 h l x[m l]+w[m] l=0 Key idea 1: use OFDM to convert the channel with ISI into a bunch of parallel AWGN channels - But there is loss/overhead due to cyclic prefix Key idea : CP overhead 0 as N c First focus on finding the capacity of parallel AWGN channels of any finite Nc Then take N c to find the capacity of the frequencyselective channel 7
28 Recap: OFDM x [1] = d[n L + 1] y[1] x [L 1] = d[n 1] y[l 1] d 0 d[0] Cyclic prefix x [L] = d[0] Channel y[l] Remove prefix y[l] ỹ 0 IDFT DFT d N 1 d[n 1] x [N + L 1] = d[n 1] y[n + L 1] y [N + L 1] ỹ N 1 y := y[l : N c + L 1], w := w[l : N c + L 1], h := h 0 h 1 h L T ey n = e h n e dn + ew n, n =0, 1,...,N c 1 Nc parallel AWGN channels ey n := DFT (y) n, e dn := DFT (d) n, ew n := DFT (w) n, e hn := p N c DFT (h) n 8
29 Parallel AWGN Channels e h0 ew 0 [m] Parallel Channels ed 0 [m] ey 0 [m] ey n = e h n e dn + ew n, n [0 : 1 : N c 1] ed 1 [m] e h1 ew 1 [m] ey 1 [m] Equivalent Vector Channel ey = H e d e + ew ew CN 0, eh = diag eh0,..., e h Nc 1 I... ed Nc 1[m] e hnc 1 ew Nc 1[m] ey Nc m =1,,...,M (M channel uses) 1[m] Power Constraint MX d[n] e apple MN c P n=1 Due to Parseval theorem of DFT 9
30 Independent Uses of Parallel Channels One way to code over such parallel channels (a special case of a vector channel): treat each channel separately - It turns out that coding across parallel channels does not help! Power allocation: - Each of the Nc channels get a portion of the total power NX - c 1 Channel n gets power Pn, which must satisfy P n apple N c P For a given power allocation {P n}, the following rate can be achieved: R = NX c 1 n=0 log 1+ e h n P n! n=0 30
31 Optimal Power Allocation Power allocation problem: max P 0,...,P N c 1 subject to NX c 1 n=0 NX c 1 n=0 log 1+ e h n P n It can be solved explicitly by Lagrangian methods!, P n = N c P, P n 0, n =0,...,N c 1 Final solution: let (x)+ := max(x, 0) P n = e h n! +, satisfies NX c 1 n=0 e h n! + = N c P 31
32 Water]illing e h n P* 1 = 0 * P * P 3 Note: e h n = H b nw N c Subcarrier n Baseband frequency response at f = nw/nc 3
33 Frequency- Selective Channel Capacity Final step: making N c - Replace all!!!!! by Hb(f), summation over [0 : Nc 1] becomes integration from 0 to W Power allocation problem becomes Optimal solution becomes P (f) = max P (f) Z W 0 subject to e hn = H b nw N c log H(f) 1+ H(f) P (f) Z W 0 +, df, P (f) =P, P(f) 0, f [0,W] satisfies Z W 0 H(f) + df = P 33
34 Water]illing over the Frequecy Spectrum H(f) P *( f ) W 0.W 0 0.W 0.4W Frequency ( f ) 34
35 Capacity of Fading Channels 35
36 Flat vs. Frequency Selective Fading Frequency selectivity induces ISI converted to parallel flat fading channels Frequency selective underspread channel can be We focus on flat fading channels (single tap): E h[m] =1, 8 m Channel Gain Power constraint: NX x[n] apple NP h[m] w[m] Noise n=1 Channel Input x[m] y[m] Channel Output y[m] =h[m]x[m]+w[m] 36
37 Slow vs. Fast Fading Slow fading (quasi-static) - h[m] = h for all m - h is random, x[m] h[m] w[m] y[m] - h is unknown to Tx (if known to Tx same as AWGN channel!) - h is known to the Rx (if not, it is the non-coherent setting) Fast fading - h[m] = hl for all m within the l-th coherence time (Tc) period - hl : i.i.d. over l, that is, i.i.d. over different coherence time periods - hl is random, could be known or unknown to Tx - hl is known to the Rx (if not, it is the non-coherent setting) 37
38 Slow Fading Channel h[m] = h for all m and h is random - Conditional on h, channel capacity = log(1+ h SNR) - Tx does not know the channel h Suppose Tx send at rate R bits/s/hz: - If R < log(1+ h SNR), Shannon: Pe 0 as N - If R > log(1+ h SNR), Shannon: Pe 1 as N Total Error probability: P (N) e =Pr R<log 1+ h SNR Pr E R<log 1+ h SNR +Pr R>log 1+ h SNR Pr E R>log 1+ h SNR! Pr R<log 1+ h SNR 0 +Pr R>log 1+ h SNR 1 = Pr R>log 1+ h SNR as N!1 38
39 Shannon Capacity = 0 Error probability Pr{R > log(1+ h SNR)} as N Vanishing error probability: impossible due to deep fade! According to Shannon s definition, capacity = 0 Outage probability p out(r) := Pr{R > C(h ; SNR)} - Determines the reliability level of sending at a particular rate R - C(h ; SNR) = log(1+ h SNR) for the point-to-point channel 39
40 Outage Probability: Computation p out (R) :=Pr R>log 1+ h SNR =Pr h < ( R 1)SNR 1 pdf of log(1+ h SNR) =1 e (R 1) SNR Area = p out (R) for h CN(0, 1): Rayleigh fading 0 0 R log(1+ h SNR) 40
41 Outage Capacity Shannon capacity of a slow fading channel is 0 - Because we insist that Pe 0 as N! More realistic capacity measure: - Set a reliability level ϵ - Find the maximum rate R * such that Pout(R) ϵ ϵ-outage capacity: C := max {R p out (R) apple } It s just the inverse function of outage probability! (why?) C = log 1+F 1 (1 ) SNR F (x) :=Pr h >x : complementary CDF of h 41
42 Outage Capacity: Computation p out (C )= () Pr C > log 1+ h SNR = () Pr C 1 SNR 1 > h = () 1 F C 1 SNR 1 = () C 1=F 1 (1 ) SNR () C = log 1+F 1 (1 ) SNR For Rayleigh fading: C log (1 + SNR) when 1 h CN(0, 1) : y = F (x) =e x [0, 1], F 1 (y) = ln y 0 F 1 (1 ) = ln(1 ), for 1 4
43 Fade Margin AWGN vs. ϵ-outage capacity: C AWGN = log (1 + SNR) C = log 1+F 1 (1 ) SNR Typically less than 1 because ϵ 10% Extra Tx power improve deep fade Under a reliability level ϵ, to achieve the same capacity as the AWGN channel, in slow fading channel the Tx has to use an extra 10log10(1/F 1 (1-ϵ)) db of power Fade margin: the extra amount of Tx power to improve the system when channel is in deep fade 43
44 Impact of Fading High SNR: C log F 1 (1 )SNR C AWGN log - Additive loss - Smaller when SNR 1 F 1 (1 ) Low SNR: C F 1 (1 )SNR log e F 1 (1 )C AWGN - Multiplicative loss - For Rayleigh, F 1 (1 ) = ln(1 ), for 1 - Significant loss when SNR is small Impact of fading is also significant at low SNR 44
45 AWGN Capactiy vs Outage Capacity C C AWGN = = SNR (db)
46 Diversity Order Recall: - Error probability Outage probability pout(r) as N Recall from Lecture : - For uncoded transmission and some coding scheme, we see that Error probability ~ SNR 1 for some fixed N - Is this true for the optimal coding scheme and arbitrarily large N? Outage probability at high SNR: p out (R) =1 e (R 1) SNR R 1 SNR when SNR 1 - Even for optimal coding scheme and large N, error prob. ~ SNR 1 Optimal diversity order is 1 46
47 Optimal Diversity Order and pout(r) Hence we are able to define the optimal diversity order for point-to-point slow fading channels: d := lim SNR!1 log p out (R) log SNR where p out (R) :=Pr{C (h; SNR) <R} Note: in taking the limit, we assume that R is a constant This view will be modified in later lectures when we discuss the diversity-multiplexing tradeoff 47
48 Receive Diversity x h y y = hx + w C L P Power constraint: P SNR := w CN 0, I L E h = L C (h; SNR) = log 1+ h SNR Outage probability Outage capacity p out (R) :=Pr log 1+ h SNR <R =Pr h < R 1 SNR Lecture Slide #5 (R C log L! 1) L SNR L = p out (C ) 1+(L!) 1 L ( ) 1 L SNR for Rayleigh faded h s at high SNR 48
49 SIMO Outage Capacity C C SIMO 1 = 1% C SIMO = log 1+ h SNR = log (1 + LSNR) 0.8 L = 5 L = 4 L = L = L = SNR (db) 49
50 Transmit Diversity x h h = h 1 y h y = h x + w C, # Tx Beamforming x = xh/ h # y = x h + w x, h C L Power constraint: NX x apple NP n=1 SNR := P E h = L Tx beamforming is impossible since Tx does not know h - For SIMO, Tx does not need to know h to achieve C(h ; SNR) How to find the optimal outage probability? For i.i.d. Rayleigh fading, it can be shown (cf. Appendix B.8 and Exercise 5.15, 5.16) that the optimal outage probability p out (R) =Pr log 1+ h SNR L <R 50
51 Impact of CSIT: Loss in Power Gain Comparison of outage probability p Tx out(r) =Pr log 1+ h SNR L <R p Rx out(r) =Pr log 1+ h SNR <R - The same diversity order L - SIMO has L-fold power gain over MISO Lack of channel state information at the transmitter (CSIT) Loss in power gain 51
52 Repetition Coding u Time 1 Time h 1 h 1 h u h Equivalent Channel: apple apple y1 h1 = u + y h apple w1 w Projection h! ey = y = h u + ew h Supports rates up to 1 log 1+ h SNR Outage probability: p Repetition out (R) =Pr blocks, but just 1 scalar channel 1 log 1+ h SNR <R 5
53 Alamouti Scheme u 1 Time 1 Time h 1 u h 1 u h u 1 h u 1,u C Equivalent Channel: X = apple u1 u u u 1 space-time codeword eh 1 e h e h1? e h apple y1 y = apple h1 h h h 1 apple u1 u + apple w1 w = u 1 apple h1 h + u apple h h 1 + apple w1 w Projection onto the two column vectors respectively, we can get two clean channels for u1 and u! 53
54 Performance of Alamouti Scheme apple y1 y = u 1 apple h1 h + u apple h h 1 + apple w1 w ey = u h1 e? h e 1h1 e + u h e + ew Projection onto two orthogonal directions ey 1 := e h 1 e h 1 ey = u 1 e h 1 + ew 1 = u 1 h + ew 1 ey := e h e h ey = u e h + ew = u h + ew Power allocation: Lack of CSIT No idea which channel is better Uniform power allocation (P/ each) Each channel supports rates up to Outage probability: achieves optimal p Alamouti out (R) =Pr log log 1+ h SNR Two parallel channels, each for one symbol! 1+ h SNR <R 54
55 Repetition Coding vs. Alamouti Repetition Alamouti C(h;SNR) 1 log 1+ h SNR log 1+ h SNR pout(r) when SNR 1 Cϵ for ϵ 1 ( R 1) SNR 1 log 1+ p SNR ( R 1) SNR log 1+ r SNR Diversity Order = 55
56 Time and Frequency Diversity Recall from Lecture : - Time diversity is obtained by coding + interleaving across multiple (L) coherence time - Frequency diversity is obtained by coding + hopping across multiple (L) coherence bandwidth Hence, time and frequency diversity techniques are equivalent to coding over L parallel channels: y l [m] =h l x l [m]+w l [m], Channel l has power constraint P l, P1+P+ +PL LP, where P is the power constraint of the original channel No CSIT cannot do water-filling l =1,,...,L 56
57 Outage Probability Instead, first use uniform power allocation P l = P, l Given h and SNR, L parallel channels can support up to LX l=1 Subtlety: due to lack of CSIT, coding across parallel channel is necessary log 1+ h l SNR bits/s/hz - Because Tx does not know for each of the L channels, how high the rate should be! ( L ) X Outage probability: p out (R) =Pr log 1+ h l SNR <LR For i.i.d. Rayleigh fading, it can be shown (cf. Exercise 5.17) l=1 that uniform power allocation is optimal! 57
58 Fast Fading Channel without CSIT Block fading model: - h[m] = hl for all m within the l-th coherence time (Tc) period - hl : i.i.d. over l, that is, i.i.d. over different coherence time periods If 1 T c N (comm. time) - L parallel channels; L - For finite L, capacity = 0 because C (h; SNR) := 1 LX log 1+ h l SNR L - is random l=1 But as L, C(h;SNR) x[m] h[m] l = 0 h[m] w[m] l = 1 l = l = 3 E log 1+ h SNR y[m] m 58
59 Ergodic Capacity What if T c 1 does not hold? - In particular, y[m] = h[m]x[m] + w[m], {h[m] m [1:N]}: i.i.d. It turns out that the capacity of such channel is C = lim N!1 1 N NX m=1 log 1+ h[m] SNR By LLN, we get C = E log 1+ h SNR In fact, for any fading process is stationary and ergodic, the capacity of the fading channel is given by the above - Stationary and ergodic the long-term average (over time) converges to the expectation (under the stationary distribution) - Note: for slow fading channel, the fading process is not ergodic 59
60 Ergodic Capacity vs. AWGN Capacity C AWGN = log (1 + SNR), C CSIR = E log 1+ h SNR In the fast fading channel, if channel state information is available only at Rx, then CCSIR CAWGN - Due to Jensen s inequality and the concavity of log(): E log 1+ h SNR apple log 1+E h SNR = log (1 + SNR) Even if Tx can code over multiple coherence time periods and average out the fluctuation of the fading channel, the capacity gets hit by fading With CSIT, the situation is changed 60
61 Transmitter State Information So far we assume that Tx does not know the realization of the fading coefficients (channel state information) Such assumption might be too conservative How to obtain CSI at Tx (CSIT)? - TDD system: channel reciprocity - FDD system: feedback from Rx How to use CSIT? - Slow fading channel: Channel Inversion Control Tx power according to the current channel condition so that a fixed data rate can be supported May not be feasible because many systems are also peak power constrained - Fast fading channel: Water Filling 61
62 Fast Fading Channel with CSIT (1) Idea: with CSIT, Tx should allocate more power when the channel is in good state! L parallel AWGN channels - Power allocation problem 1 max P 1,...,P L L LX log l=1 - Solution: 1+ h l P l P l =, subject to 1 L h l LX P l = P, P l 0, l =1,...,L - The value of ν depends on all L channels {h1,, hl} - Not feasible because it requires the Tx to know the future channel beforehand! l=1 +, satisfies 1 L LX l=1 h l + = P 6
63 Fast Fading Channel with CSIT () Final step: taking L - Due to LLN, optimal solution becomes: P (h) = h +, satisfies E " h + # = P - Now the value of ν only depends on the distribution of h Ergodic capacity of fast fading channel with CSIT: C = E apple log 1+ h P (h) 63
64 Power Allocation at High SNR Optimal Uniform: Near-Optimal h[m] h[m] m m 64
65 Power Allocation at Low SNR Optimal Best Only: Near-Optimal h[m] h[m] m m 65
66 Performance Comparison 7 6 CAWGN > CFull CSI CCSIR C (bits /s / Hz) AWGN Full CSI CSIR CFull CSI > CAWGN CCSIR C AWGN = log P apple, C CSIR = E log SNR (db) 1+ h P, E h =1 0 C Full CSI = E apple log 1+ h P (h) P (h) = h +, satisfies E " h + # = P 66
67 Performance at Low and High SNR High SNR regime: - The optimal power allocation is approximately uniform apple - Hence, C Full CSI C CSIR E log h P = log SNR + E log( h ) C AWGN 0.83 Low SNR regime: - The optimal power allocation is approximately put all the power in the best channel - Hence, C Full CSI Pr h = G max log 1+G max P/Pr h = G max G max SNR log e G max C AWGN! - Beside, C CSIR E h SNR log e = E h SNR log e = SNR log e C AWGN 67
68 Summary A slow fading channel is a source of unreliability - Poor outage capacity - Diversity is needed A fast fading channel with only CSIR: - Ergodic capacity close to AWGN capacity - Decoding delay is long compared to coherence time A fast fading channel with full CSI: - Ergodic capacity can be greater than AWGN capacity - Turn fading into a friend rather than an enemy Opportunistic communication - Send more when the channel is good - Even more powerful in multiuser situations 68
Lecture 4 Capacity of Wireless Channels
Lecture 4 Capacity of Wireless Channels I-Hsiang Wang ihwang@ntu.edu.tw 3/0, 014 What we have learned So far: looked at specific schemes and techniques Lecture : point-to-point wireless channel - Diversity:
More informationLecture 2. Capacity of the Gaussian channel
Spring, 207 5237S, Wireless Communications II 2. Lecture 2 Capacity of the Gaussian channel Review on basic concepts in inf. theory ( Cover&Thomas: Elements of Inf. Theory, Tse&Viswanath: Appendix B) AWGN
More informationLecture 4. Capacity of Fading Channels
1 Lecture 4. Capacity of Fading Channels Capacity of AWGN Channels Capacity of Fading Channels Ergodic Capacity Outage Capacity Shannon and Information Theory Claude Elwood Shannon (April 3, 1916 February
More informationAppendix B Information theory from first principles
Appendix B Information theory from first principles This appendix discusses the information theory behind the capacity expressions used in the book. Section 8.3.4 is the only part of the book that supposes
More informationLecture 5: Antenna Diversity and MIMO Capacity Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH
: Antenna Diversity and Theoretical Foundations of Wireless Communications Wednesday, May 4, 206 9:00-2:00, Conference Room SIP Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication
More informationELEC546 Review of Information Theory
ELEC546 Review of Information Theory Vincent Lau 1/1/004 1 Review of Information Theory Entropy: Measure of uncertainty of a random variable X. The entropy of X, H(X), is given by: If X is a discrete random
More informationLecture 9: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH
: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1 Rayleigh Wednesday, June 1, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication
More informationLecture 6 Channel Coding over Continuous Channels
Lecture 6 Channel Coding over Continuous Channels I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw November 9, 015 1 / 59 I-Hsiang Wang IT Lecture 6 We have
More informationLecture 9: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1
: Diversity-Multiplexing Tradeoff Theoretical Foundations of Wireless Communications 1 Rayleigh Friday, May 25, 2018 09:00-11:30, Kansliet 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless
More informationSingle-User MIMO systems: Introduction, capacity results, and MIMO beamforming
Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming Master Universitario en Ingeniería de Telecomunicación I. Santamaría Universidad de Cantabria Contents Introduction Multiplexing,
More informationLecture 8: MIMO Architectures (II) Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH
MIMO : MIMO Theoretical Foundations of Wireless Communications 1 Wednesday, May 25, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Overview MIMO
More informationLecture 7: Wireless Channels and Diversity Advanced Digital Communications (EQ2410) 1
Wireless : Wireless Advanced Digital Communications (EQ2410) 1 Thursday, Feb. 11, 2016 10:00-12:00, B24 1 Textbook: U. Madhow, Fundamentals of Digital Communications, 2008 1 / 15 Wireless Lecture 1-6 Equalization
More informationChapter 4: Continuous channel and its capacity
meghdadi@ensil.unilim.fr Reference : Elements of Information Theory by Cover and Thomas Continuous random variable Gaussian multivariate random variable AWGN Band limited channel Parallel channels Flat
More informationLecture 7 MIMO Communica2ons
Wireless Communications Lecture 7 MIMO Communica2ons Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2014 1 Outline MIMO Communications (Chapter 10
More information16.36 Communication Systems Engineering
MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.36: Communication
More informationELEC546 MIMO Channel Capacity
ELEC546 MIMO Channel Capacity Vincent Lau Simplified Version.0 //2004 MIMO System Model Transmitter with t antennas & receiver with r antennas. X Transmitted Symbol, received symbol Channel Matrix (Flat
More informationMultiple-Input Multiple-Output Systems
Multiple-Input Multiple-Output Systems What is the best way to use antenna arrays? MIMO! This is a totally new approach ( paradigm ) to wireless communications, which has been discovered in 95-96. Performance
More informationOn the Secrecy Capacity of Fading Channels
On the Secrecy Capacity of Fading Channels arxiv:cs/63v [cs.it] 7 Oct 26 Praveen Kumar Gopala, Lifeng Lai and Hesham El Gamal Department of Electrical and Computer Engineering The Ohio State University
More informationLecture 4 Noisy Channel Coding
Lecture 4 Noisy Channel Coding I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 9, 2015 1 / 56 I-Hsiang Wang IT Lecture 4 The Channel Coding Problem
More informationLecture 5: Channel Capacity. Copyright G. Caire (Sample Lectures) 122
Lecture 5: Channel Capacity Copyright G. Caire (Sample Lectures) 122 M Definitions and Problem Setup 2 X n Y n Encoder p(y x) Decoder ˆM Message Channel Estimate Definition 11. Discrete Memoryless Channel
More informationMultiple Antennas in Wireless Communications
Multiple Antennas in Wireless Communications Luca Sanguinetti Department of Information Engineering Pisa University luca.sanguinetti@iet.unipi.it April, 2009 Luca Sanguinetti (IET) MIMO April, 2009 1 /
More informationDiversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT. ECE 559 Presentation Hoa Pham Dec 3, 2007
Diversity-Multiplexing Tradeoff in MIMO Channels with Partial CSIT ECE 559 Presentation Hoa Pham Dec 3, 2007 Introduction MIMO systems provide two types of gains Diversity Gain: each path from a transmitter
More informationExploiting Partial Channel Knowledge at the Transmitter in MISO and MIMO Wireless
Exploiting Partial Channel Knowledge at the Transmitter in MISO and MIMO Wireless SPAWC 2003 Rome, Italy June 18, 2003 E. Yoon, M. Vu and Arogyaswami Paulraj Stanford University Page 1 Outline Introduction
More informationLimited Feedback in Wireless Communication Systems
Limited Feedback in Wireless Communication Systems - Summary of An Overview of Limited Feedback in Wireless Communication Systems Gwanmo Ku May 14, 17, and 21, 2013 Outline Transmitter Ant. 1 Channel N
More informationLecture 18: Gaussian Channel
Lecture 18: Gaussian Channel Gaussian channel Gaussian channel capacity Dr. Yao Xie, ECE587, Information Theory, Duke University Mona Lisa in AWGN Mona Lisa Noisy Mona Lisa 100 100 200 200 300 300 400
More informationInterleave Division Multiple Access. Li Ping, Department of Electronic Engineering City University of Hong Kong
Interleave Division Multiple Access Li Ping, Department of Electronic Engineering City University of Hong Kong 1 Outline! Introduction! IDMA! Chip-by-chip multiuser detection! Analysis and optimization!
More informationLecture 2. Fading Channel
1 Lecture 2. Fading Channel Characteristics of Fading Channels Modeling of Fading Channels Discrete-time Input/Output Model 2 Radio Propagation in Free Space Speed: c = 299,792,458 m/s Isotropic Received
More informationLecture 4 Channel Coding
Capacity and the Weak Converse Lecture 4 Coding I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw October 15, 2014 1 / 16 I-Hsiang Wang NIT Lecture 4 Capacity
More informationLecture 5 Channel Coding over Continuous Channels
Lecture 5 Channel Coding over Continuous Channels I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw November 14, 2014 1 / 34 I-Hsiang Wang NIT Lecture 5 From
More informationLecture 1: The Multiple Access Channel. Copyright G. Caire 12
Lecture 1: The Multiple Access Channel Copyright G. Caire 12 Outline Two-user MAC. The Gaussian case. The K-user case. Polymatroid structure and resource allocation problems. Copyright G. Caire 13 Two-user
More informationDiversity Combining Techniques
Diversity Combining Techniques When the required signal is a combination of several plane waves (multipath), the total signal amplitude may experience deep fades (Rayleigh fading), over time or space.
More informationEE 4TM4: Digital Communications II. Channel Capacity
EE 4TM4: Digital Communications II 1 Channel Capacity I. CHANNEL CODING THEOREM Definition 1: A rater is said to be achievable if there exists a sequence of(2 nr,n) codes such thatlim n P (n) e (C) = 0.
More informationMulti-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems
Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User
More informationMulti-User Gain Maximum Eigenmode Beamforming, and IDMA. Peng Wang and Li Ping City University of Hong Kong
Multi-User Gain Maximum Eigenmode Beamforming, and IDMA Peng Wang and Li Ping City University of Hong Kong 1 Contents Introduction Multi-user gain (MUG) Maximum eigenmode beamforming (MEB) MEB performance
More informationEE 5407 Part II: Spatial Based Wireless Communications
EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang E-mail: rzhang@i2r.a-star.edu.sg Website: http://www.ece.nus.edu.sg/stfpage/elezhang/ Lecture II: Receive Beamforming
More information2. Wireless Fading Channels
Fundamentals of Wireless Comm. Andreas Biri, D-ITET.07.7. Introduction Modulation (frequency shift to carrier frequency): - enables multiple (slightly shifted) simultaneous channels - better channel characteristics
More informationPrinciples of Communications
Principles of Communications Weiyao Lin Shanghai Jiao Tong University Chapter 10: Information Theory Textbook: Chapter 12 Communication Systems Engineering: Ch 6.1, Ch 9.1~ 9. 92 2009/2010 Meixia Tao @
More informationApproximately achieving the feedback interference channel capacity with point-to-point codes
Approximately achieving the feedback interference channel capacity with point-to-point codes Joyson Sebastian*, Can Karakus*, Suhas Diggavi* Abstract Superposition codes with rate-splitting have been used
More informationHarnessing Interaction in Bursty Interference Networks
215 IEEE Hong Kong-Taiwan Joint Workshop on Information Theory and Communications Harnessing Interaction in Bursty Interference Networks I-Hsiang Wang NIC Lab, NTU GICE 1/19, 215 Modern Wireless: Grand
More informationCapacity of multiple-input multiple-output (MIMO) systems in wireless communications
15/11/02 Capacity of multiple-input multiple-output (MIMO) systems in wireless communications Bengt Holter Department of Telecommunications Norwegian University of Science and Technology 1 Outline 15/11/02
More informationMulti-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems
Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User MIMO Systems Multi-Input Multi-Output Systems (MIMO) Channel Model for MIMO MIMO Decoding MIMO Gains Multi-User
More informationPOWER ALLOCATION AND OPTIMAL TX/RX STRUCTURES FOR MIMO SYSTEMS
POWER ALLOCATION AND OPTIMAL TX/RX STRUCTURES FOR MIMO SYSTEMS R. Cendrillon, O. Rousseaux and M. Moonen SCD/ESAT, Katholiee Universiteit Leuven, Belgium {raphael.cendrillon, olivier.rousseaux, marc.moonen}@esat.uleuven.ac.be
More informationChapter 9 Fundamental Limits in Information Theory
Chapter 9 Fundamental Limits in Information Theory Information Theory is the fundamental theory behind information manipulation, including data compression and data transmission. 9.1 Introduction o For
More informationLecture 8: Orthogonal Frequency Division Multiplexing (OFDM)
Communication Systems Lab Spring 2017 ational Taiwan University Lecture 8: Orthogonal Frequency Division Multiplexing (OFDM) Scribe: 謝秉昂 陳心如 Lecture Date: 4/26, 2017 Lecturer: I-Hsiang Wang 1 Outline 1
More informationWireless Communications Lecture 10
Wireless Communications Lecture 1 [SNR per symbol and SNR per bit] SNR = P R N B = E s N BT s = E b N BT b For BPSK: T b = T s, E b = E s, and T s = 1/B. Raised cosine pulse shaper for other pulses. T
More informationApproximate Capacity of Fast Fading Interference Channels with no CSIT
Approximate Capacity of Fast Fading Interference Channels with no CSIT Joyson Sebastian, Can Karakus, Suhas Diggavi Abstract We develop a characterization of fading models, which assigns a number called
More informationAdvanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung
Advanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung Dr.-Ing. Carsten Bockelmann Institute for Telecommunications and High-Frequency Techniques Department of Communications
More informationEnergy State Amplification in an Energy Harvesting Communication System
Energy State Amplification in an Energy Harvesting Communication System Omur Ozel Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland College Park, MD 20742 omur@umd.edu
More informationMULTICARRIER code-division multiple access (MC-
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 51, NO. 2, FEBRUARY 2005 479 Spectral Efficiency of Multicarrier CDMA Antonia M. Tulino, Member, IEEE, Linbo Li, and Sergio Verdú, Fellow, IEEE Abstract We
More informationMorning Session Capacity-based Power Control. Department of Electrical and Computer Engineering University of Maryland
Morning Session Capacity-based Power Control Şennur Ulukuş Department of Electrical and Computer Engineering University of Maryland So Far, We Learned... Power control with SIR-based QoS guarantees Suitable
More information18.2 Continuous Alphabet (discrete-time, memoryless) Channel
0-704: Information Processing and Learning Spring 0 Lecture 8: Gaussian channel, Parallel channels and Rate-distortion theory Lecturer: Aarti Singh Scribe: Danai Koutra Disclaimer: These notes have not
More informationDiversity-Multiplexing Tradeoff of Asynchronous Cooperative Diversity in Wireless Networks
Diversity-Multiplexing Tradeoff of Asynchronous Cooperative Diversity in Wireless Networks Shuangqing Wei Abstract Synchronization of relay nodes is an important and critical issue in exploiting cooperative
More informationMobile Communications (KECE425) Lecture Note Prof. Young-Chai Ko
Mobile Communications (KECE425) Lecture Note 20 5-19-2014 Prof Young-Chai Ko Summary Complexity issues of diversity systems ADC and Nyquist sampling theorem Transmit diversity Channel is known at the transmitter
More informationL interférence dans les réseaux non filaires
L interférence dans les réseaux non filaires Du contrôle de puissance au codage et alignement Jean-Claude Belfiore Télécom ParisTech 7 mars 2013 Séminaire Comelec Parts Part 1 Part 2 Part 3 Part 4 Part
More informationInteractive Interference Alignment
Interactive Interference Alignment Quan Geng, Sreeram annan, and Pramod Viswanath Coordinated Science Laboratory and Dept. of ECE University of Illinois, Urbana-Champaign, IL 61801 Email: {geng5, kannan1,
More informationON BEAMFORMING WITH FINITE RATE FEEDBACK IN MULTIPLE ANTENNA SYSTEMS
ON BEAMFORMING WITH FINITE RATE FEEDBACK IN MULTIPLE ANTENNA SYSTEMS KRISHNA KIRAN MUKKAVILLI ASHUTOSH SABHARWAL ELZA ERKIP BEHNAAM AAZHANG Abstract In this paper, we study a multiple antenna system where
More informationOne Lesson of Information Theory
Institut für One Lesson of Information Theory Prof. Dr.-Ing. Volker Kühn Institute of Communications Engineering University of Rostock, Germany Email: volker.kuehn@uni-rostock.de http://www.int.uni-rostock.de/
More informationLECTURE 18. Lecture outline Gaussian channels: parallel colored noise inter-symbol interference general case: multiple inputs and outputs
LECTURE 18 Last time: White Gaussian noise Bandlimited WGN Additive White Gaussian Noise (AWGN) channel Capacity of AWGN channel Application: DS-CDMA systems Spreading Coding theorem Lecture outline Gaussian
More informationTitle. Author(s)Tsai, Shang-Ho. Issue Date Doc URL. Type. Note. File Information. Equal Gain Beamforming in Rayleigh Fading Channels
Title Equal Gain Beamforming in Rayleigh Fading Channels Author(s)Tsai, Shang-Ho Proceedings : APSIPA ASC 29 : Asia-Pacific Signal Citationand Conference: 688-691 Issue Date 29-1-4 Doc URL http://hdl.handle.net/2115/39789
More informationHierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks
Hierarchical Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks Ayfer Özgür, Olivier Lévêque Faculté Informatique et Communications Ecole Polytechnique Fédérale de Lausanne 05 Lausanne, Switzerland
More informationSecrecy in the 2-User Symmetric Interference Channel with Transmitter Cooperation: Deterministic View
Secrecy in the 2-User Symmetric Interference Channel with Transmitter Cooperation: Deterministic View P. Mohapatra 9 th March 2013 Outline Motivation Problem statement Achievable scheme 1 Weak interference
More informationHow Much Training and Feedback are Needed in MIMO Broadcast Channels?
How uch raining and Feedback are Needed in IO Broadcast Channels? ari Kobayashi, SUPELEC Gif-sur-Yvette, France Giuseppe Caire, University of Southern California Los Angeles CA, 989 USA Nihar Jindal University
More informationSpace-Time Coding for Multi-Antenna Systems
Space-Time Coding for Multi-Antenna Systems ECE 559VV Class Project Sreekanth Annapureddy vannapu2@uiuc.edu Dec 3rd 2007 MIMO: Diversity vs Multiplexing Multiplexing Diversity Pictures taken from lectures
More informationDiversity Multiplexing Tradeoff in ISI Channels Leonard H. Grokop, Member, IEEE, and David N. C. Tse, Senior Member, IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 55, NO 1, JANUARY 2009 109 Diversity Multiplexing Tradeoff in ISI Channels Leonard H Grokop, Member, IEEE, and David N C Tse, Senior Member, IEEE Abstract The
More informationMaximum Achievable Diversity for MIMO-OFDM Systems with Arbitrary. Spatial Correlation
Maximum Achievable Diversity for MIMO-OFDM Systems with Arbitrary Spatial Correlation Ahmed K Sadek, Weifeng Su, and K J Ray Liu Department of Electrical and Computer Engineering, and Institute for Systems
More informationLecture 6: Gaussian Channels. Copyright G. Caire (Sample Lectures) 157
Lecture 6: Gaussian Channels Copyright G. Caire (Sample Lectures) 157 Differential entropy (1) Definition 18. The (joint) differential entropy of a continuous random vector X n p X n(x) over R is: Z h(x
More informationNearest Neighbor Decoding in MIMO Block-Fading Channels With Imperfect CSIR
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 3, MARCH 2012 1483 Nearest Neighbor Decoding in MIMO Block-Fading Channels With Imperfect CSIR A. Taufiq Asyhari, Student Member, IEEE, Albert Guillén
More informationThe Optimality of Beamforming: A Unified View
The Optimality of Beamforming: A Unified View Sudhir Srinivasa and Syed Ali Jafar Electrical Engineering and Computer Science University of California Irvine, Irvine, CA 92697-2625 Email: sudhirs@uciedu,
More informationEE 4TM4: Digital Communications II Scalar Gaussian Channel
EE 4TM4: Digital Communications II Scalar Gaussian Channel I. DIFFERENTIAL ENTROPY Let X be a continuous random variable with probability density function (pdf) f(x) (in short X f(x)). The differential
More informationEE 5407 Part II: Spatial Based Wireless Communications
EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang E-mail: rzhang@i2r.a-star.edu.sg Website: http://www.ece.nus.edu.sg/stfpage/elezhang/ Lecture IV: MIMO Systems March 21,
More informationChannel State Information in Multiple Antenna Systems. Jingnong Yang
Channel State Information in Multiple Antenna Systems A Thesis Presented to The Academic Faculty by Jingnong Yang In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy School of
More information12.4 Known Channel (Water-Filling Solution)
ECEn 665: Antennas and Propagation for Wireless Communications 54 2.4 Known Channel (Water-Filling Solution) The channel scenarios we have looed at above represent special cases for which the capacity
More informationAnalysis of coding on non-ergodic block-fading channels
Analysis of coding on non-ergodic block-fading channels Joseph J. Boutros ENST 46 Rue Barrault, Paris boutros@enst.fr Albert Guillén i Fàbregas Univ. of South Australia Mawson Lakes SA 5095 albert.guillen@unisa.edu.au
More informationLecture 15: Thu Feb 28, 2019
Lecture 15: Thu Feb 28, 2019 Announce: HW5 posted Lecture: The AWGN waveform channel Projecting temporally AWGN leads to spatially AWGN sufficiency of projection: irrelevancy theorem in waveform AWGN:
More informationON ADAPTIVE TRANSMISSION, SIGNAL DETECTION AND CHANNEL ESTIMATION FOR MULTIPLE ANTENNA SYSTEMS. A Dissertation YONGZHE XIE
ON ADAPTIVE TRANSMISSION, SIGNAL DETECTION AND CHANNEL ESTIMATION FOR MULTIPLE ANTENNA SYSTEMS A Dissertation by YONGZHE XIE Submitted to the Office of Graduate Studies of Texas A&M University in partial
More informationRandom Matrices and Wireless Communications
Random Matrices and Wireless Communications Jamie Evans Centre for Ultra-Broadband Information Networks (CUBIN) Department of Electrical and Electronic Engineering University of Melbourne 3.5 1 3 0.8 2.5
More informationPerformance Analysis of Physical Layer Network Coding
Performance Analysis of Physical Layer Network Coding by Jinho Kim A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Electrical Engineering: Systems)
More informationSolution to Homework 1
Solution to Homework 1 1. Exercise 2.4 in Tse and Viswanath. 1. a) With the given information we can comopute the Doppler shift of the first and second path f 1 fv c cos θ 1, f 2 fv c cos θ 2 as well as
More informationLecture 6 I. CHANNEL CODING. X n (m) P Y X
6- Introduction to Information Theory Lecture 6 Lecturer: Haim Permuter Scribe: Yoav Eisenberg and Yakov Miron I. CHANNEL CODING We consider the following channel coding problem: m = {,2,..,2 nr} Encoder
More informationPhysical Layer and Coding
Physical Layer and Coding Muriel Médard Professor EECS Overview A variety of physical media: copper, free space, optical fiber Unified way of addressing signals at the input and the output of these media:
More informationWhen does vectored Multiple Access Channels (MAC) optimal power allocation converge to an FDMA solution?
When does vectored Multiple Access Channels MAC optimal power allocation converge to an FDMA solution? Vincent Le Nir, Marc Moonen, Jan Verlinden, Mamoun Guenach Abstract Vectored Multiple Access Channels
More informationEstimation of Performance Loss Due to Delay in Channel Feedback in MIMO Systems
MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Estimation of Performance Loss Due to Delay in Channel Feedback in MIMO Systems Jianxuan Du Ye Li Daqing Gu Andreas F. Molisch Jinyun Zhang
More informationMulticarrier transmission DMT/OFDM
W. Henkel, International University Bremen 1 Multicarrier transmission DMT/OFDM DMT: Discrete Multitone (wireline, baseband) OFDM: Orthogonal Frequency Division Multiplex (wireless, with carrier, passband)
More informationMinimum BER Linear Transceivers for Block. Communication Systems. Lecturer: Tom Luo
Minimum BER Linear Transceivers for Block Communication Systems Lecturer: Tom Luo Outline Block-by-block communication Abstract model Applications Current design techniques Minimum BER precoders for zero-forcing
More informationCapacity of Memoryless Channels and Block-Fading Channels With Designable Cardinality-Constrained Channel State Feedback
2038 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 9, SEPTEMBER 2004 Capacity of Memoryless Channels and Block-Fading Channels With Designable Cardinality-Constrained Channel State Feedback Vincent
More informationMMSE estimation and lattice encoding/decoding for linear Gaussian channels. Todd P. Coleman /22/02
MMSE estimation and lattice encoding/decoding for linear Gaussian channels Todd P. Coleman 6.454 9/22/02 Background: the AWGN Channel Y = X + N where N N ( 0, σ 2 N ), 1 n ni=1 X 2 i P X. Shannon: capacity
More informationCyclic Division Algebras: A Tool for Space Time Coding
Foundations and Trends R in Communications and Information Theory Vol. 4, No. 1 (2007) 1 95 c 2007 F. Oggier, J.-C. Belfiore and E. Viterbo DOI: 10.1561/0100000016 Cyclic Division Algebras: A Tool for
More informationSecure Degrees of Freedom of the MIMO Multiple Access Wiretap Channel
Secure Degrees of Freedom of the MIMO Multiple Access Wiretap Channel Pritam Mukherjee Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park, MD 074 pritamm@umd.edu
More informationOn the Relation between Outage Probability and Effective Frequency Diversity Order
Appl. Math. Inf. Sci. 8, No. 6, 2667-267 (204) 2667 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/0.2785/amis/080602 On the Relation between and Yongjune Kim, Minjoong
More informationHybrid Pilot/Quantization based Feedback in Multi-Antenna TDD Systems
Hybrid Pilot/Quantization based Feedback in Multi-Antenna TDD Systems Umer Salim, David Gesbert, Dirk Slock, Zafer Beyaztas Mobile Communications Department Eurecom, France Abstract The communication between
More informationNOMA: Principles and Recent Results
NOMA: Principles and Recent Results Jinho Choi School of EECS GIST September 2017 (VTC-Fall 2017) 1 / 46 Abstract: Non-orthogonal multiple access (NOMA) becomes a key technology in 5G as it can improve
More informationECE Information theory Final (Fall 2008)
ECE 776 - Information theory Final (Fall 2008) Q.1. (1 point) Consider the following bursty transmission scheme for a Gaussian channel with noise power N and average power constraint P (i.e., 1/n X n i=1
More informationMultiple Antennas. Mats Bengtsson, Björn Ottersten. Channel characterization and modeling 1 September 8, Signal KTH Research Focus
Multiple Antennas Channel Characterization and Modeling Mats Bengtsson, Björn Ottersten Channel characterization and modeling 1 September 8, 2005 Signal Processing @ KTH Research Focus Channel modeling
More informationAdaptive Bit-Interleaved Coded OFDM over Time-Varying Channels
Adaptive Bit-Interleaved Coded OFDM over Time-Varying Channels Jin Soo Choi, Chang Kyung Sung, Sung Hyun Moon, and Inkyu Lee School of Electrical Engineering Korea University Seoul, Korea Email:jinsoo@wireless.korea.ac.kr,
More informationOn the Performance of. Golden Space-Time Trellis Coded Modulation over MIMO Block Fading Channels
On the Performance of 1 Golden Space-Time Trellis Coded Modulation over MIMO Block Fading Channels arxiv:0711.1295v1 [cs.it] 8 Nov 2007 Emanuele Viterbo and Yi Hong Abstract The Golden space-time trellis
More informationError Exponent Region for Gaussian Broadcast Channels
Error Exponent Region for Gaussian Broadcast Channels Lihua Weng, S. Sandeep Pradhan, and Achilleas Anastasopoulos Electrical Engineering and Computer Science Dept. University of Michigan, Ann Arbor, MI
More informationPerformance Analysis of Multiple Antenna Systems with VQ-Based Feedback
Performance Analysis of Multiple Antenna Systems with VQ-Based Feedback June Chul Roh and Bhaskar D. Rao Department of Electrical and Computer Engineering University of California, San Diego La Jolla,
More informationChapter 2 Underwater Acoustic Channel Models
Chapter 2 Underwater Acoustic Channel Models In this chapter, we introduce two prevailing UWA channel models, namely, the empirical UWA channel model and the statistical time-varying UWA channel model,
More informationUpper Bounds on the Capacity of Binary Intermittent Communication
Upper Bounds on the Capacity of Binary Intermittent Communication Mostafa Khoshnevisan and J. Nicholas Laneman Department of Electrical Engineering University of Notre Dame Notre Dame, Indiana 46556 Email:{mhoshne,
More informationEE5139R: Problem Set 7 Assigned: 30/09/15, Due: 07/10/15
EE5139R: Problem Set 7 Assigned: 30/09/15, Due: 07/10/15 1. Cascade of Binary Symmetric Channels The conditional probability distribution py x for each of the BSCs may be expressed by the transition probability
More information