2. Wireless Fading Channels

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1 Fundamentals of Wireless Comm. Andreas Biri, D-ITET Introduction Modulation (frequency shift to carrier frequency): - enables multiple (slightly shifted) simultaneous channels - better channel characteristics (less absorbtion) Capacity: maximum rate with error-free communication (asymptotically in the block length; spread redundancy far enough to not affect all) Small-scale fading: displacement in magnitude of waelength results in significant field changes Large-scale fading: due to shadowing & distance. Wireless Fading Channels Transmit signal with complex enelope x(t) x c (t) = Re { x(t) e jπf c t } Receied signal due to multipath propagation α n : path gain ; r c (t) = α n (t) x c (t n (t)) n=0 n : path delay Equialent baseband signal: Doppler shift: r(t) = α n (t) e jπf c n (t) x(t n (t)) n=0 n = f c n Use approximations which hold if B f c / n t With bandlimited signals, we can describe it as r(t) = a n x(t n ) e jπ nt, a n = α n e jπf c n n=0 Linear time-inariant (LTI): time shifts, no frequency shifts h(t, ) = g(), S H (, ) = g() δ() Linear time-ariant (LTV): both time & frequency shifts (do not commute in general) - time shifts: multipath propagation - frequency shifts: moement of Tx, Rx or scatterers Time-arying transfer function (Weyl symbol) L H (t, f) = h(t, ) e jπf d = F f { h(t, )}. Tapped Delay-line Interpretation max r(t) = 0 h(t, ) x(t ) d Systems oeriew Time shifts: due to multipath propagation Frequency shift: due to Doppler shifts as objects include moement and change location oer time Linear time-ariant (LTV): time & frequency shifts Linear time-inariant (LTI): only time shifts, no freq. shifts r(t) = h() x(t), h(t, ) = g() Linear frequency-inariant (LFI): only freq. shifts, no time - modulation of input signal = S H (, ) x(t ) e jπt dd (Delay-Doppler) Spreading function: influence of scatterers S H (, ) = h(t, ) e jπt dt = F t {h(, t)} Time-arying impulse response t h(t, ) = S H (, ) e jπt d = F t { S H (, )} For digital tapped delay: r[n] = h[k] x[n k] k r(t) = m(t)x(t), S H (, ) = M()δ() h(t, ) = m(t) δ() r(t) = h(t, ) x(t ) d

2 .3 WSSUS Channels Wide-sense stationary (WSS): statistics does not change - all tap weights zero-main stationary with respect to time Uncorrelated scattering (US): scattered paths uncorrelated R h (t, t ;, ) = R h (t t, ) δ( ) = E[h (t, ) h (t, ) ] R H (t, t ; f, f ) = R H (t t, f f ) = E[L H (t, f) L H (t, f )] L H is both wide-sense stationary in both time & frequency (US in delay WSS in freq.; US in Doppler shifts WSS in time) Scattering function: aerage output power of the channel (depending on Doppler freq. and delay ) E[ S H (, ) S H (, )] = C H (, ) δ( )δ( ).4 Parameter Characterization for WSSUS Path loss: fraction of input energy arriing at receier Time dispersieness P = C H (, ) dd Power-delay profile (PDP): ag. reflected power at delay Multipath delay spread: q() = C H (, ) d 0 σ = P ( ) q() d Coherence bandwidth B c : width of R H (0, f) = F {q()} Frequency dispersieness Power-Doppler profile: aerage reflected power at p() = C H (, ) d 0 Doppler spread: spectral broadening through moement σ = P ( ) p() d Coherence time T c : width of R H ( t, 0) Slow fading: T c = const σ T signal T c (Time inariant: entire signal sees same channel) Fast fading: channel changes substantially oer signal Frequency Flat fading: B B c (Freq. inariant: all freq. scaled with same factor) Frequency-selectie fading: factor depends on frequency B c const σ B c : spread in frequency; σ : spread in time. Flat (B B c, T T c ): r(t) = c x(t) Because of the uncertainty principle, both cannot be small ( small freq. domain spread large time domain spread) With flat fading, the frequency does not matter and we hae a (time-selectie) modulation of the channel: r(t) c(t) x(t), L H (t, f) c(t). Frequency-selectie (LTI): r(t) = (h x)(t) 3. Time-selectie (LFI): r(t) = m(t)x(t) 4. B > B c, T > T c : r(t) = S H (, ) x(t ) e jπt

3 .5 Probabilistic Characterization of Fading Rayleigh fading (non-los): h(t, ) ~ C(0, σ ) Magnitude: Rayleigh distributed f h(t,) (z) = z z σ e σ Squared magnitude: exponentially distributed f h(t,) (x) = σ e x σ, x 0 Ricean fading (LOS case): h(t, ) ~ μ + C(0, σ ) Ricean K-factor: K = μ /σ.7 Discretized Channel Models Use sampling theorem to get countably infinite number of parameters for discretized channel description Input frequency limitation: band-limited to B Output time limitation: maximal signal duration T Receied signal consists of time-frequency shifted ersions of a band-limited ersion of the input signal The corresponding receied signal can be constructed with Most of the olume of S H is supported oer rectangle [ D B, D + ] x [ V B T, V + T ] V: max. Doppler shift ; D: max. time shift / delay Complete channel characterization with finite parameters Discrete-time channel model Input signal: bandlimited to [ B, +B] Doppler shift limited to [ V, + V] Receied signal: bandlimited to [ B V, B + V] h(t, ) bandlimited to [ B, B] with respect to, as L H (t, f) bandlimited with respect to f Therefore, we can sample the receied signal with f s = (B + V), where f 0 = B Additie White Gaussian oise (AWG) Assume complex zero-mean additie white Gaussian noise - sampled as well with f s - white independent oer time - independent of the paths, influence usually at receier y[n] = m= h[n, m] x[n m] + w[n].7 Identification of LTV Systems Want to extract h(t, ) from response r(t) to a known probing signal x(t) send pilot first LTI systems: just use Dirac pulse x(t) = δ(t) (need to obsere output long enough to identify system) For LTV systems, this deliers h(t, ) only along a 45 line: r(t) = h(t, ) x(t ) d = h(t, t) Assume S H (, ) supported on [ 0, 0 ] x [ 0, 0 ] h(t, ) supported on [ 0, 0 ] in, bandlimited to [ 0, 0 ] with respect to t Solution: Dirac train to track eolution of impulse response x(t) = δ(t l t 0 ) l= r(t) = h(t, t l t 0 ) l=, t 0 0 To reconstruct h(t, ) for all alues from the known samples in t direction, we require a sampling of 0 t 0 0, bandlimited to [ 0, 0 ] For such a solution to exist, we therefore require = H That is, support area of S H (, ) must be smaller than Probing fraction: A = of signal space dim. for probing Probing signal: design as orthogonal as possible (else noise) 3

4 3. Diersity Send signals that carry same information oer multiple independently fading paths more reliable reception Small coherence BW: If I send info oer multiple frequencies, one might fail but others will not diersity: decreased chance of failure 3. Detection in Rayleigh Fading Channel on-coherent detection For a flat-fading channel (LFI), we get y[m] = h[m]x[m] + w[m] w[m] ~ C(0, 0 ), h[m] ~ C(0,) eed either different magnitudes or orthogonal symbols Log-likelihood ratio Λ(y) = ln ( f(y H H = H 0) f(y H ) ) < H = H Optimum noncoherent detection projects the receied signal ectors onto each of the two possible transmitted messages and compares the magnitudes squared P(e) = Coherent detection ( + SR) P(e h ) = Q ( h[0] SR ) Aeraging oer random channel P(e) = ( + SR ) 4 SR = P(e) non coh. AWG channel P(e) = Q( SR ) ~ e SR In comparison to non-coherent & coherent detection with inerse decay with SR, the error probability in the AWG channel decays exponentially with the SR In a fading channel, error performance is poor not because the channel is unknown at the receier, but because the probability that the channel fades is high P(deep fade) = P ( h[0] < SR ) SR If the channel gain is much larger than SR (no deep fade), conditional error probability decays exponentially in SR At high SR, typical error is due to small channel gain and not because of large additie noise P e = P e "deep fade" P df + e SR P n df Diersity: send information oer multiple channels, so that at least one is not in deep fade and can be used - time, frequency & space (antenna) diersity - macro (cellular networks) & multi-user (scheduling) 3. Time Diersity Aeraging oer the fading of the channel oer time Coherence time usually around 0-00 symbols Interleaing: ensure symbols are transmitted oer independently fading branches Coherent detection: project onto known channel ector matched filter : Maximum Ratio Combiner (max SR) - weight receied signal proportional to signal strength - align the phases in the summation (reerse phase shift) L independently fading branches are coherently combined and result in an array & diersity gain P(e h) = Q ( h SR ) L h = h[l] Sum of the squares of L independently real Gaussian RV chi-square distribution with L degrees of freedom (X L ) f XL (x) = l=0 (L )! xl e x, x 0 With those diersity branches, we less error probability, as we are less in deep fade (all channels would hae to be): log P(e) L log SR + C P ( h < SR ) L! SR L Repetition coding: already achiees diersity gain - does not efficiently use degrees of freedom in the system Rotation code: send rotated QAM signal - rotated QAM so faded channel still allows detection - calculate Pairwise Error Probability (PEP) P(e) = 48 min (δ ij) SR = c SR d i,j d diersity order, minimum SR exponent c coding gain, minimum c i in front of PEP Maximize the minimum product distance for the best error probability (minimal oer all is important) Rotation-based constellation has an increased minimum product distance than its comparable 4-PAM repetitionbased code, as it spreads the codewords in -dimensional space rather than -dimensional (e.g. line) 4

5 3.3 Transmission Rate Diersity Tradeoff Define transmission rate as a fraction r of the capacity R = r log SR Using PAM with R = SR r constellation points, we can then find the diersity order as d(r) = r, r [0,/] Using QAM, we can use the real and imaginary dimension, each with R/ constellation points, resulting in d(r) = r, r [0,] 3.4 Frequency diersity Assume LTI system where we only hae time shifts L y[n] = h[m]x[n m] + w[n] m=0 One-shot: send once, then wait L slots for multipath - gies L diersity order (L samples), but ery bad rate Frequency diersity: use multiple paths which can be resoled at the receier, as they arrie separately This creates large Intersymbol interference (ISI) as replicas of earlier symbols oerlap which we need to deal with Direct sequence spread spectrum (DSSS) Symbols are modulated onto pseudonoise (P) and spread across the frequency Delayed replica are nearly orthogonal (shift-orthogonality), simplifying the receier structure Single-carrier Modulation Use shift operator (Dx)[n] = x[n ] Knowing the channel at the receier, we can then do standard ector detection & see that we can get L order diersity if M has full rank (all singular alues strictly positie) Code difference matrix M: M = X i X j X j = [x j Dx j D L x j ] As the columns of M are linearly independent, we hae full rank & therefore full L order diersity een for uncoded transmission for long enough block lengths L Same diersity gain as one-shot communication - much higher data rate (less waste of degrees of freedom) - higher receier complexity Orthogonal Freq. Diision Multiplexing (OFDM) Conert frequency-selectie channel into a set of frequency-flat fading channels through precoding Diersity through coding across symbols in diff. subcarriers LTI: sinusoidal functions are eigenfunctions of the system x(t) = e jπf 0t r(t) = G(f 0 ) e jπf 0t, LTV: not eigenfunctions anymore x(t) = e jπf 0t r(t) = L H (t, f 0 ) e jπf 0t Eliminate all multipath without haing to know the instantaneous realization of H. Transform H into a circulant matrix. Diagonalize it using the DFT matrix F h(t, ) = g() 3. Receie modulation of input without interference f q = ω (q )0 ω (q ) [ ω (q )( ) ], ω = e jπ/ Spectral Decomposition for circulant matrices: Any circulant matrix C C x has eigenectors which are the column of the DFT matrix F and eigenalues λ = [λ 0 λ ] T = F H c, c = [c 0 c ] T Cyclic prefix: transmit last L symbols of the input ector before the input ector x to make H circulant ifft at transmitter, FFT at receier y = F H r = F H FΛF H Fs = Λ s 3.5 Diersity Order Estimates Input band-limited: f < B, Output time-limited: t < T Maximally achieable diersity order is B T = B c T c input signal BW coherence BW 3.6 Infinite Diersity Order output signal duration coherence time For an increasing L, the SR is more concentrated around its mean and eentually becomes a deterministic quantity SR = E L S h[l] 0 l=0 L Error probability conerges to that one of AWG channel ( aerage out channel realisation gies fixed SR) E S 0 5

6 4. Information Theory of Wireless 4. Information theoretic basics Capacity: maximal rate for communication oer channel Capacity of AWG channel C = log ( + P 0 / ), w ~ (0, 0/) Memoryless channel: channel noise corrupts inputs independently, no interferences (freq.-inariant) Each of M messages is mapped onto codeword of length P(e) = P( m m ) R = log M bits symbol Reliable communication at rate R exists, if δ > 0, we can find a codelength so that P(e) < δ ( for small δ) Entropy: uncertainty associated with X ( How much info? ) H(X) 0, H(X) = p X (x) log p X (x) H(X) log K with K = X, X X Entropy is maximal for uniformly distributed alues oer K Joint entropy H(X, Y) = p X,Y (x, y) log P X,Y (x, y) Conditional entropy H(X Y) = p X,Y (x) log p X Y (x y) Chain rule for entropy H(X, Y) = H(X) + H(Y X) = H(Y) + H(X Y) Conditioning reduces entropy H(X Y) H(X) Mutual information ( reduction of uncertainty if I know one ) I(X; Y) = H(Y) H(Y X) = H(X) H(X Y) 0 oisy channel theorem For a reliable channel, we should hae low uncertainty in decoding the input signal based on the output signal: R bit I(X; Y) H( x y) 0 symbol, I(X; Y) log M For a large enough blocklength of the code, we can aerage out the effect of the random noise and get C = max I(x; y) p x (.) Continuous random ariables Differential entropy h(x) = f X (x) log f X (x) dx x h(x Y) = f X,Y (x, y) log f X Y (x y) dxdy x y Usually, a power constraint exists: E[x ] P AWG Channel For a Gaussian RV X ~ (μ, σ ) h(x) = log(πe σ ) As noise & input are Gaussian, the output is also Gaussian: h( y x) = log (πe 0 ), w ~ (0, 0 ) E[y ] P + 0 Gaussian random ariables maximize differential entropy C = log ( + P 0 ) For a continuous-time AWG channel with complex noise w ~ C(0, 0 ) (real- & imaginary part σ = 0 each) C = log ( + P W 0 ) C = W log ( + P 0 W ) Small SR: High SR: Small W: Large W: bit complex dimension bit s linear increase with receied power capacity doubles with eery 3dB increase 3dB increase only yields additional one bit increasing W yields rapid capacity increase bandwidth-limited regime little effect, spread P oer more dimensions power-limited regime (achiee capacity for W ) 6

7 4. Capacity of Fading Channels Slow Fading Channel: y[n] = h x[n] + w[n] Short codeword length compared to coherence time Outage probability P out (R) = P( log( + h SR) < R ) = P ( h < R SR ) Coding can only aerage out noise, but cannot do anything against channel fading (in slow fading, channel is constant) Capacity of this fading channel is zero, as coding cannot guarantee a diminishing error probability Outage capacity Capacity, so that rate lower in ( ε) 00% : ε = P(log ( + h SR) C out,ε ) For small ε (and Rayleigh fading), we get Diersity C out,ε log ( + ε SR), P out = R SR For an effectie channel with L diersity order, we get log P out (R) = L (log SR) + c Optimal diersity-multiplexing tradeoff (DMT) Define rate as constant fraction of capacity Fast Fading Channel y[n] = h[n] x[n] + w[n] Whereas in the slow fading channel, the Shannon capacity was zero, we now hae a positie ergodic capacity C = E[ log ( + h SR ) ] using a random Gaussian codebook with i.i.d. symbols Deriation I(y; x) I(y[n]; x[n]) n= log( + h[n] SR) n= As Gaussian input symbols achiee max. mutual information. This conerges to aboe formula by the law of large numbers. In fast fading case, we can code oer many independent fades of the channel by coding oer many symbols and can therefore aerage out fading channels (Requires ergodic channel where each channel realisation can be seen when listening long enough) Capacity of fading channel is always smaller than AWG channel and only equal for deterministic channel. - low SR: difference is negligible - high SR: Jensen penalty C fading = C AWG 0.83 bit s Hz need.5db more power to achiee the same capacity 5. Multiple Input Multiple Output 5. Maximum Ratio Combining ( beam forming ) Receier MRC (CSIR) With a resulting noise w ~ C(0, ( h 0 + h ) 0 ) By knowing the channel, we get second-order diersity as well as a x array gain Transmit MRC (CSIT) r = x 0 ( h 0 + h ) R = r log SR, P(e) = SR d(r) In the optimal case where P(e) = P out, we get d opt (r) = r QAM is DMT-optimal for scalar fading channels (see 3.3) SR = E x 0 ( h 0 + h ) Also, x array gain as well as a diersity gain MRC uses beam forming to get spatial filtering (only receie from a certain direction) 7

8 5. Alamouti Scheme (CSIR) 5.4 Intentional frequency offset diersity 5.6 MIMO wireless systems Adding new antennas opens up new degrees of freedom; just like with adding bandwidth, this is especially effectie for a small number of antennas Power constraint Capacity E[x H x] = trace[ E[ x x H ] ] P As we hae an ergodic channel (and therefore H i.i.d.), all directions are equally good and we just transmit equally Knowing the channel at the receier, we project y: This results in second-order diersity, but without power gain, as we hae to send each symbol twice for detection We excite the channel in two orthogonal directions; therefore, een if one shoot be perpendicular and anishes after projection onto h, the other one is receied perfectly 5.3 Delay Diersity Conert spatial diersity into frequency diersity looks like frequency-selectie (LTI) channel Diersity order of two, as two channel taps Conert spatial diersity into time diersity looks like time-selectie (LFI) channel 5.5 Space-time coding (CSIR) Generalization of Alamouti for multiple transmit antennas send oer linearly independent channels We find that the diersity is gien by the rank of the difference of two space-time codeword matrices C, E E h [ P(C E h] SR rank(c E) r λ i / Rank criterion: Full diersity is achieed, if rank(c E) = M T {C, E} i= Determinant criterion: C E as orthogonal as possible λ i large coding gain C = E H [log det ( I MR + P M T H H H ) ] For a fixed M R, increasing the transmit antennas ensures the different channels are orthogonal and results in C = M R ( + SR), M T That is, we get an M R -fold increase in capacity due to M R spatial degrees of freedom thanks to rich scattering In general, an M R x M T i.i.d. C(0,) channel gies us C = min(m T, M R ) log ( SR M T ) + const. SIMO and MISO systems do not lead to an increase in the number of degrees of freedom. 8

9 5.7 Capacity of MIMO with CSIT For parallel Gaussian channels, we use waterfilling to correctly distribute power oer the different channels so that capacity is achieed The allocated power with z n ~ (0, σ n ) is P n = max { 0, μ σ n }, P n P We use singular alue decomposition (SVD) to decompose the ector channel into a set of parallel independent scalar Gaussian subchannels: H = U Λ V H, Λ = diag[ λ,, λ ] Sender sends x = V x, at receier use y = U H y Using this and defining = min{ M T, M R }, we get C = max P n P log ( + P n λ n σ ) n= with waterfilling for the optimal power allocation Low SR: allocate all power to the best subchannel power gain of max n λ n High SR: allocate equal power to subchannel with λ n > 0 capacity increases linearly with rank of H: K min{ M T, M R } : number of spatial degrees of freedom 6. Various Uncertainty principle Cannot hae strong limitation in time and frequency domain T 0 = t x(t) dt, WT Theorem T 0 B 0 x(t) /4π B 0 = f X(f) df Signals which are time-limited to [ T, T] and band-limited [ B, B] lie in a 4BT dimensional signal space Pocket Gambler trick H(X) = log M M = H(x) : # of constellation points Good channel: H(x y) = ( H(x y) = 0 ) Bad channel: multiple X s can cause a certain Y can only distinguish which cluster of X may cause Y Resolution How many clusters can I separate? # of X cluster size = H(x) = H(x y) H(x) H(x y) = I(x;y) I(x; y) number of equialence classes / effectie number of bits I can detect Mathematics x(t) = x(t) dt Circularly symmetric complex Gaussian RV U = U R + j U I ~ C(0, σ ) U R, U I i. i. d ~ (0, σ ) Toeplitz matrix Constant along its diagonals - a cyclic matrix is always Toeplitz Eigenfunctions of LTI system Sinusoids are eigenfunctions of an LTI system H = F Λ F H, Λ diagonal By periodically repeating the signals, we can get a circular matrix for the channel matrix Complementary error function Q(x) = π e x Q(x) e x u du Exponentially distributed random ariable f h(t,) (x) = σ e x σ, x 0 Useful approximations E[ e sx ] = + s + x = x + o(x), x 0 e x = + x + o(x) log ( + x) x log e, x 0 log ( + x ) log (x), x 9

Lecture 5: Antenna Diversity and MIMO Capacity Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH

Lecture 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