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Point-to-Point Wireless Communication ISI & Equalization, Diversity

Point-to-Point Wireless Communication ISI & Equalization, Diversity 43
PointtoPoint Wireless Communication (II): ISI Equalization, Diversity (Time/Space/Frequency) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 1Multidimensional Fading  Time, Frequency, Space Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 2Plan  First, compare 1tap (i.e. flat) Rayleighfading channel vs AWGN.  i.e. y = hx + w vs y = x + w  Note: all multipaths with random attenuation/phases are aggregated into 1tap  Next consider frequency selectivity, i.e. multitap, broadband channel, with multipaths  Effect: ISI  Equalization techniques for ISI complexities  Generalize Consider diversity in time, space, frequency, and develop efficient schemes to achieve diversity gains and coding gains Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 3Singletap, Flat Fading (Rayleigh) vs AWGN Why do we have this huge degradation in performance/reliability Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 4Rayleigh Flat Fading Channel BPSK: Coherent detection. Looks like Conditional on h, AWGN, but… p needs to be e “unconditioned” To get a much Averaged over h, poorer scaling at high SNR. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 5BER vs. SNR (cont.) Frequencyselective channel BER (equalization or Rake receiver) () P e Frequencyselective channel (no equalization) “BER floor” AWGN channel Flat fading channel (no fading) () SNR 0 P14 means a straight line in log/log scale e 0 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 6Typical Error Event Conditional on h, When the error probability is very small. When the error probability is large: Typical error event is due to: channel (h) being in deep fade … rather than (additive) noise being large. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 7Preview: Diversity Gain: Intuition  Typical error (deep fade) event probability:  In other words, h w/x  i.e. hx w  (i.e. signal x is attenuated to be of the order of noise w) ChiSquared pdf of Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 8Recall: BPSK, QPSK and 4PAM  BPSK uses only the Iphase.The Qphase is wasted.  QPSK delivers 2 bits per complex symbol.  To deliver the same 2 bits, 4PAM requires 4 dB more transmit power.  QPSK exploits the available degrees of freedom in the channel better. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 9MQAM doesn’t change the asymptotics…  QPSK does use degrees of freedom better than equivalent 4PAM  (Read textbook, chap 3, section 3.1) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 10Frequency Selectivity: Multipath fading ISI Mitigation: Equalization Challenges Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 11ISI Mitigation: Outline  Intersymbol interference (ISI): review  Nyquist theorem  Pulse shaping (last slide set)  1. Equalization receivers  2. Introduction to the diversity approach  Rake Receiver in CDMA  OFDM: decompose a wideband multitap channel into narrowband single tap channels Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 12Recall: Attenuation, Dispersion Effects: ISI Intersymbol interference (ISI) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Source: Prof. Raj Jain, WUSTL : “shiv rpi” 13Recall: Multipaths: PowerDelay Profile path1 path2 path3 multipath propagation path2 Path Delay path1 path3 Mobile Station (MS) Base Station (BS) Channel Impulse Response: Channel amplitude h correlated at delays . Each “tap” value kTs Rayleigh distributed (actually the sum of several subpaths) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 14 PowerInterSymbolInterference (ISI) due to Multi Path Fading Transmitted signal: Received Signals: Lineofsight: Reflected: The symbols add up on the Delays channel  Distortion Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 15Multipath: TimeDispersion = Frequency Selectivity  The impulse response of the channel is correlated in the timedomain (sum of “echoes”)  Manifests as a powerdelay profile, dispersion in channel autocorrelation function A()  Equivalent to “selectivity” or “deep fades” in the frequency domain  Delay spread: 50ns (indoor) – 1s (outdoor/cellular).  Coherence Bandwidth: Bc = 500kHz (outdoor/cellular) – 20MHz (indoor)  Implications: High data rate: symbol smears onto the adjacent ones (ISI). Multipath effects O(1s) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 16BER vs. S/N performance: AWGN In a Gaussian channel (no fading) BER = Q(S/N) erfc(S/N) Typical BER vs. S/N curves BER Frequencyselective channel (no equalization) Gaussian channel Flat fading channel (no fading) S/N Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 17BER vs. S/N performance: Flat Fading Flat fading: BER BER S N z p z dz   z = signal power level Typical BER vs. S/N curves BER Frequencyselective channel (no equalization) Gaussian channel Flat fading channel (no fading) S/N Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 18BER vs. S/N performance: ISI/Freq. Selective Channel Frequency selective fading = irreducible BER floor Typical BER vs. S/N curves BER Frequencyselective channel (no equalization) Gaussian channel Flat fading channel (no fading) S/N Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 19BER vs. S/N performance: w/ Equalization Diversity (e.g. multipath diversity) = improved performance Typical BER vs. S/N curves BER Gaussian Frequencyselective channel channel (with equalization) Flat fading channel (no fading) S/N Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 20Equalization Step 1 – waveform to sample transformation Step 2 – decision making Demodulate Sample Detect z(T) ˆ m Threshold i r(t) Frequency Receiving Equalizing comparison downconversion filter filter Compensation for For bandpass signals channel induced ISI Baseband pulse Received waveform Sample Baseband pulse (possibly distored) (test statistic) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 21What is an equalizer  We’ve used it for music in everyday life  Eg: default settings for various types of music to emphasize bass, treble etc…  Essentially we are setting up a (fdomain) filter to cancel out the channel mpath filtering effects Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 22Equalization: Channel is a LTI Filter  ISI due to filtering effect of the communications channel (e.g. wireless channels)  Channels behave like bandlimited filters j ( f ) c H ( f ) H ( f ) e c c Nonconstant amplitude Nonlinear phase Amplitude distortion Phase distortion Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 23Pulse Shaping and Equalization Principles No ISI at the sampling time H ( f ) H ( f )H ( f )H ( f )H ( f ) RC t c r e  SquareRoot Raised Cosine (SRRC) filter and Equalizer H ( f ) H ( f )H ( f ) RC t r Taking care of ISI caused by tr. filter H ( f ) H ( f ) H ( f ) H ( f ) r t RC SRRC 1 Taking care of ISI H ( f ) e H ( f ) caused by channel c Equalizer: enhance weak freq., dampen strong freq. to flatten the spectrum Since the channel H (f) changes with time, we need adaptive equalization, c i.e. reestimate channel equalize Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 24Equalization: Channel examples  Example of a (somewhat) frequency selective, slowly changing (slow fading) channel for a user at 35 km/h Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 25Equalization: Channel examples …  Example of a highly frequencyselective, fast changing (fast fading) channel for a user at 35 km/h Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 26Recall: Eye pattern  Eye pattern:Display on an oscilloscope which sweeps the system response to a baseband signal at the rate 1/T (T symbol duration) Distortion due to ISI Noise margin Sensitivity to timing error Timing jitter time scale Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 27 amplitude scaleExample of eye pattern with ISI: BinaryPAM, SRRC pulse  Nonideal channel and no noise h (t) (t) 0.7 (tT) c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 28Example of eye pattern with ISI: BinaryPAM, SRRC pulse …  AWGN (Eb/N0=20 dB) and ISI h (t) (t) 0.7 (tT) c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 29Example of eye pattern with ISI: BinaryPAM, SRRC pulse …  AWGN (Eb/N0=10 dB) and ISI h (t) (t) 0.7 (tT) c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 30Equalizing filters …  Baseband system model a 1 a (t kT) z z(t)  k ˆ Channel Rx. filtera Equalizer k Tx filter r(t) k k h (t) h (t) h (t) h (t) t c e r Detector t kT H ( f ) H ( f ) H ( f ) H ( f ) t c r T e a a 2 3 n(t)  Equivalent model H( f ) H ( f )H ( f )H ( f ) t c r a 1 Equivalent system z k a (t kT) z(t) x(t) z(t) ˆ a  k Equalizer k k h(t) h (t) e Detector t kT H ( f ) e H ( f ) T a a 2 3 ˆ n(t) filtered (colored) noise ˆ n(t) n(t)h (t) r Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 31Equalizer Types Covered later in slideset Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: Rappaport book, chap 7 32Linear Equalizer • A linear equalizer effectively inverts the channel. n(t) Equalizer Channel 1 H (f) H (f) eq c H (f) c • The linear equalizer is usually implemented as a tapped delay line. • On a channel with deep spectral nulls, this equalizer enhances the noise. (note: both signal and noise pass thru eq.) poor performance on frequencyselective fading channels Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 33Noise Enhancement w/ Spectral Nulls Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 34Decision Feedback Equalizer (DFE) DFE n(t) x(t) x(t) Forward + H (f) c Filter Feedback Filter • The DFE determines the ISI from the previously detected symbols and subtracts it from the incoming symbols. • This equalizer does not suffer from noise enhancement because it estimates the channel rather than inverting it.  The DFE has better performance than the linear equalizer in a frequencyselective fading channel. • The DFE is subject to error propagation if decisions are made incorrectly.  = doesn’t work well w/ low SNR.  Optimal nonlinear: MLSE… (complexity grows exponentially w/ delay spread) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 35Equalization by transversal filtering  Transversal filter:  A weighted tap delayed line that reduces the effect of ISI by proper adjustment of the filter taps. N z(t) c x(t n ) nN,..., N k2N,...,2N  n nN x(t)  c c c c N N1 N1 N z(t)  Coeff. adjustment Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 36Training the Filter Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 37Transversal equalizing filter …  Zeroforcing equalizer:  The filter taps are adjusted such that the equalizer output is forced to be zero at N sample points on each side: 1 k 0  Adjust z(k)  N 0 k1,...,N  c n nN  Mean Square Error (MSE) equalizer:  The filter taps are adjusted such that the MSE of ISI and noise power at the equalizer output is minimized. (note: noise is whitened before filter) 2 Adjust min E(z(kT) a ) N k c n nN Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 38Equalization: Summary  Equalizer “equalizes” the channel response in frequency domain to remove ISI  Can be difficult to design/implement, get noise enhancement (linear EQs) or error propagation (decision feedback EQs) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 39Summary: Complexity and Adaptation  Nonlinear equalizers (DFE, MLSE) have better performance but higher complexity  Equalizer filters must be FIR  Can approximate IIR Filters as FIR filters  Truncate or use MMSE criterion  Channel response needed for equalization  Training sequence used to learn channel Tradeoffs in overhead, complexity, and delay  Channel tracked during data transmission Based on bit decisions Can’t track large channel fluctuations Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 40Diversity Techniques: Time, Frequency, Code, Space Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 41Introduction to Diversity  Basic Idea  Send same bits over independent fading paths Independent fading paths obtained by time, space, frequency, or polarization diversity  Combine paths to mitigate fading effects T b t Multiple paths unlikely to fade simultaneously Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 42Diversity Gain: Short Story…  AWGN case: BER vs SNR: (any modulation scheme, only the constants differ) Note: γ is received SNR  Rayleigh Fading w/o diversity:  Rayleigh Fading w/ diversity:  (MIMO): Note: “diversity” is a reliability theme, not a capacity/bitrate one… For capacity: need more degreesoffreedom (i.e. symbols/s) packing of bits/symbol (MQAM). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 43Time Diversity Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 44Time Diversity  Time diversity can be obtained by interleaving and coding over symbols across different coherent time periods. Channel: time diversity/selectivity, but correlated across successive symbols (Repetition) Coding… w/o interleaving: a full codeword lost during fade Interleaving: of sufficient depth: ( coherence time) At most 1 symbol of codeword lost Coding alone is not sufficient Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 45Forward Error Correction (FEC): Eg: ReedSolomon RS(N,K) Recover K = K of N RS(N,K) data packets received FEC (NK) Block Size Lossy Network (N) Block: of sufficient size: ( coherence time), else need to interleave, or use with hybrid ARQ Data = K Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 46Hybrid ARQ/FEC Model Packets • Sequence Numbers • CRC or Checksum • Proactive FEC Timeout • ACKs Status Reports • NAKs, • SACKs • Bitmaps Retransmissions • Packets • Reactive FEC Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 47Example: GSM  The data of each user are sent over time slots of length 577 μs  Time slots of the 8 users together form a frame of length 4.615 ms  Voice: 20 ms frames, rate ½ convolution coded = 456 bits/voiceframe  Interleaved across 8 consecutive time slots assigned to that specific user:  0th, 8th, . . ., 448th bits are put into the first time slot,  1st, 9th, . . ., 449th bits are put into the second time slot, etc.  One time slot every 4.615 ms per user, or a delay of 40 ms (ok for voice).  The 8 time slots are shared between two 20 ms speech frames. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 48TimeDiversity Example: GSM  Amount of time diversity limited by delay constraint and how fast channel varies.  In GSM, delay constraint is 40ms (voice).  To get full diversity of 8, needs v 30 km/hr at f = 900Mhz. c  Recall: T 5 ms = 1/(4D ) = c/(8f v) c s c Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 49GSM contd  Walking speed of say 3 km/h = too little time diversity.  GSM can go into a frequency hopping mode,  Consecutive frames (each w/ time slots of 8 users) can hop from one 200 kHz subchannel to another.  Typical delay spread 1μs = the coherence bandwidth (Bc) is 500 kHz.  The total bandwidth of 25 MHz B c = consecutive frames can be expected to fade independently.  This provides the same effect as having time diversity. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 50Repetition Code: Diversity Analysis After interleaving over L coherence time periods, Repetition coding: for all where and This is classic vector detection in white Gaussian noise. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 51Repetition Coding: Matched Filtering hx only spans a 1 1dimensional space (similar to MPAM, w/ random channel gains instead) h Shivkumar Kalyanaraman Rensselaer Polytechnic Institute Multiply by conjugate = cancel phase : “shiv rpi” 52Repetition Coding: Fading Analysis (contd)  BPSK Error probability: 2  Average over h i.e. over Chisquared distribution, Ldegrees of freedom Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 53Diversity Gain: Intuition  Typical error (deep fade) event probability:  In other words, h w/x  i.e. hx w  (i.e. signal x is attenuated to be of the order of noise w) ChiSquared pdf of Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 54Key: Deep Fades Become Rarer Deep fade ≡ Error event… Note: this graph plots reliability (i.e. BER vs SNR) Repetition code trades off information rate (i.e. poor use of degoffreedom) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 55Beyond Repetition Coding: Coding gains  Repetition coding gets full diversity, but sends only one symbol every L symbol times.  i.e. trades off bitrate for reliability (better BER)  Does not exploit fully the degrees of freedom in the channel. (analogy: PAM vs QAM)  How to do better Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 56Example: Rotation code (L=2) x , x are two BPSK symbols before rotation (each, either a or –a). 1 2 where d and d are the normalized distances between the codewords in the two 1 2 basis directions (axes). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 57ProductDistance Criterion If d = 0 or d = 0, the the diversity gain of the code is only 1. 1 2 product distance Choose the rotation angle to maximize the worstcase product distance to all the other codewords: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 58Rotation vs Repetition Coding Recall repetition coding was like PAM (see matched filter slide before) Rotation code uses the degrees of freedom better Coding gain over the repetition code in terms of a saving in transmit power by a factor of sqrt(5) or 3.5 dB for the same product distance Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 59Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 60Time Diversity + Coding + Fading: The gory details  If we plot this p vs SNR curve vs the one for repetition code, then we can get the e coding gain (for any target p ) e  Note: the squaredproductdistance idea will reappear as a determinant criteria in spacetime codes Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 61Antenna Diversity Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 62Antenna Diversity Transmit Both Receive (MISO) (MIMO) (SIMO) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 63Antenna Diversity: Rx Transmit Both Receive (MISO) (MIMO) (SIMO) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 64Receive Diversity Same mathematical structure as repetition coding in time diversity (), except that there is a further power gain (aka “array gain”). Optimal reception is via matched filtering/MRC (a.k.a. receive beamforming). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 65Array Gain vs Diversity Gain  Diversity Gain: multiple independent channels between the transmitter and receiver, and is a product of the statistical richness of those channels  Array gain does not rely on statistical diversity between the different channels and instead achieves its performance enhancement by coherently combining the actual energy received by each of the antennas.  Even if the channels are completely correlated, as might happen in a line ofsight (LOS) system, the received SNR increases linearly with the number of receive antennas,  Eg: Correlated flatfading:  Single Antenna SNR:  Adding all receive paths: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 66Recall: Diversity Gain: Short Story…  AWGN case: BER vs SNR: (any modulation scheme, only the constants differ) Note: γ is received SNR  Rayleigh Fading w/o diversity:  Rayleigh Fading w/ diversity:  (MIMO): Note: “diversity” is a reliability theme, not a capacity/bitrate one… For capacity: need more degreesoffreedom (i.e. symbols/s) packing of bits/symbol (MQAM). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 67Receive Diversity: Selection Combining  Recall: Bandpass vs matched filter analogy.  Pick max signal, but don’t fully combine signal power from all taps. Diminishing returns from more taps. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: J. Andrews et al, Fundamentals of WIMAX 68Receive Beamforming: Maximal Ratio Combining (MRC) Weight each branch SNR: MRC Idea: Branches with better signal energy should be enhanced, whereas branches with lower SNR’s given lower weights Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: J. Andrews et al, Fundamentals of WIMAX 69Recall: Maximal Ratio Combining (MRC) or “Beamforming” … is just Matched Filtering in the Spatial Domain  Generalization of this fdomain picture, for combining multitap signal Weight each branch SNR: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: J. Andrews et al, Fundamentals of WIMAX 70Selection Diversity vs MRC Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: J. Andrews et al, Fundamentals of WIMAX 71Antenna Diversity: Tx Transmit Both Receive (MISO) (MIMO) (SIMO) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 72Transmit Diversity If transmitter knows the channel, send: maximizes the received SNR by inphase addition of signals at the receiver (transmit beamforming), i.e. closedloop Tx diversity. Reduce to scalar channel: same as receive beamforming. What happens if transmitter does not know the channel Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 73OpenLoop Tx Diversity: SpaceTime Coding Alamouti Code:  Alamouti : Orthogonal spacetime block code (OSTBC).  2 × 1 Alamouti STBC  Rate 1 code:  Data rate is neither increased nor decreased;  Two symbols are sent over two time intervals.  Goal: harness spatial diversity. Don’t care about ↑ rate Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 74Alamouti Scheme Over two symbol times: Projecting onto the two columns of the H matrix yields: •double the symbol rate of repetition coding. •3dB loss of received SNR compared to transmit beamforming (i.e. MRC or matched filtering). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 75What was that, again Alamouti STBC  Flat fading channel.  h (t), h (t) are the complex channel gains from antenna 1 1 2 antenna 2  Channel is constant over 2 symbol times, i.e. h (t = 0) = h (t = T) = h . 1 1 1 Received Signal: Receiver: Project on columns of H: Like MRC, but 3dB (i.e. ½) lower power Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 76Spacetime Codes  Note: Transmitter does NOT know the channel instantaneously (openloop)  Using the antennas one at a time and sending the same symbol over the different antennas is like repetition coding.  Repetition scheme: inefficient utilization of degrees of freedom  Over the two symbol times, bits are packed into only one dimension of t the received signal space, namely along the direction h , h . 1 2  More generally, can use any timediversity code by turning on one antenna at a time.  Spacetime codes are designed specifically for the transmit diversity scenario.  Alamouti scheme spreads the information onto two dimensions along t t the orthogonal directions h , h and h,−h . 1 2 2 1 Alamouti: Repetition: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 77Spacetime Code Design: In Brief A spacetime code is a set of matrices Full diversity is achieved if all pairwise differences have full rank. Coding gain determined by the (min) determinants of Timediversity codes have diagonal matrices and the determinant reduces to squared product distances. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 78STCoding Design: Details  Spacetime code as a set of complex codewords X , where i each X is an L by N matrix. i  L: number of transmit antennas  N: block length of the code. Alamouti: Repetition:  Normalize the codewords so that the average energy per symbol time is 1, hence SNR = 1/N . 0  Assume channel constant for N symbol times Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 79STCoding Design: Details Note: λ here instead of d l l in rota Sti hiv on kuc mode ar Ka a lyna ana ly rasis man Rensselaer Polytechnic Institute : “shiv rpi” 80ST Coding Design: Details 2  If all the λ are strictly positive for all the codeword l differences, then the maximal diversity gain of L is achieved. 2  Number of positive eigenvalues λ equals the rank of the l codeword difference matrix, this is possible only if N ≥ L. (Recall: determinant = product of evalues) Mindeterminant over codeword pairs controls the coding gain (detcriterion) If X etc are diagonal, then the determinant = squaredproddistance A For Alamouti, mindet is 4; Repetition STcode: mindet = 16/25 = Alamouti coding gain: factorof6 (or 7.8 dB) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 81Spacetime Code Design: Summary A spacetime code is a set of matrices Full diversity is achieved if all pairwise differences (eg: X – X have full rank (i.e. all evalues positive). A B Coding gain determined by the (min) determinants of Timediversity codes have diagonal matrices and the determinant reduces to squared product distances. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 82Code Design Degrees of Freedom Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 83Antenna Diversity: Tx+Rx = MIMO Transmit Both Receive (MISO) (MIMO) (SIMO) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 84MIMO: w/ Repetition or Alamouti Coding  Transmit the same symbol over the two antennas in two consecutive symbol times (at each time, nothing is sent over the other antenna).  ½ symbol per degree of freedom (d.f.)  MRC combining w/ repetition:  Alamouti scheme used over the 2 × 2 channel:  Sends 2 symbols/2 symbol times (i.e. 1symbol/d.f),  Same 4fold diversity gain as in repetition.  But, the 2x2 MIMO channel has MORE degrees of freedom Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 85MIMO: degrees of freedom  Degrees of freedom = dimension of received signal space  1xL: Onedimensional  2x2: Has 2 dimensions  h : vector of channel gains j from Tx antennas.  Space gives new degrees of freedom.  A “spatial multiplexing” scheme like VBLAST can leverage the additional d.f. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 86Spatial Multiplexing: VBLAST  Transmit independent uncoded symbols over antennas and over time  VBLAST: poorer diversity gain than Alamouti. But exploits spatial degrees of freedom better  Spaceonly coding: no Tx diversity. Diversity order only 2.  Coding gain possible by coding across space time (increased degrees of freedom) with spatial multiplexing Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 87MIMO Receiver Issues  VBLAST uses joint ML reception (complex)  Zeroforcing linear receiver loses one order of diversity.  Interference nuller, decorrelator  Noise samples correlated (colored). Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 88Summary: 2x2 MIMO Schemes  Need closedloop MIMO to be able to reap both diversity and d.f. gains Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 89Frequency Diversity: MLSD, CDMA Rake, OFDM Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 90Frequency Diversity  Resolution of multipaths provides diversity.  Full diversity is achieved by sending one symbol every L symbol times.  But this is inefficient (like repetition coding).  Sending symbols more frequently may result in intersymbol interference.  Note: ISI is not intrinsic, but frequencydiversity is  Challenge is how to mitigate the ISI while extracting the inherent diversity in the frequencyselective channel. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 91Approaches  Timedomain equalization (eg. GSM)  Directsequence spread spectrum (eg. IS95 CDMA)  Orthogonal frequencydivision multiplexing OFDM (eg. 802.11a, FlashOFDM) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 92ISI Equalization  Suppose a sequence of uncoded symbols are transmitted.  Maximum likelihood sequence detection is performed using the Viterbi algorithm.  Can full diversity be achieved Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 93Reduction to Transmit Diversity (FlatFading) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 94MLSD Achieves Full Diversity Spacetime code matrix for input sequence Difference matrix for two sequences first differing at is full rank. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 95Uncoded Max Likelihood Seq. Detection (MLSD) MLSD:  Tradeoff: MLSD too complex Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 96MLSD: Viterbi Algorithm  A bruteforce exhaustive search would require a complexity that grows exponentially with the block length n.  Key: exploit the structure of the problem and should be recursive in n so that the problem does not have to be solved from scratch for every symbol time.  Solution: Viterbi algorithm.  Key Observation: memory in the frequencyselective channel can be captured by a finite state machine.  At time m, define the state (an L dimensional vector) L  states is M , where M is the constellation size L: of taps (diversity order) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 97MLSD: Viterbi Algo (Contd)  Rewrite MLSD, conditioned on states si, instead of input sequence x  Conditional independence =  MLSD ≡ finding the shortest path through an nstage trellis  the cost associated with the mth transition (or “hop”) is Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 98MLSD/Viterbi: Trellis  Note: a trellis is a state diagram that evolves with time as well. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 99Viterbi: Dynamic Programming  We only consider the states that the finite state machine can be in at stage m− 1  Subset of shortest path, also a shortest path  The complexity of the Viterbi algorithm is linear in the number of stages n L  Complexity is also proportional to the size of the state space, which is M ,  … where M is the constellation size of each symbol Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 100Rake Receiver for Frequency Diversity Detour: Spread Spectrum, CDMA, Ref: Chapter 3 4, Tse/Viswanath book, Chap 13, 15: A. Goldsmith book Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 101What is CDMA spread spectrum Baseband Spectrum Radio Spectrum Code B Code A B B A Code A A C C B C B B A B A A A C B Time Sender Receiver Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 102Types of CDMA Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 103Spread Spectrum  Spreadspectrum modulation is considered “secondary” modulation after the usual primary modulation. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 104Direct Sequence Spread Spectrum  Bit sequence modulated by chip sequence S(f) s(t) s (t) S (f) c S(f) S (f) c c 1/T b 1/T c T T =KT c b c  Spreads bandwidth by large factor (K)  Despread by multiplying by s (t) again (s (t)=1) c c  Mitigates ISI and narrowband interference Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 105Chips Spreading Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 106Processing Gain / Spreading Factor Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 107Processing Gain Shannon  With 8K vocoders, above 32 users, SNR becomes too low.  Practical CDMA systems restrict the number of users per sector to ensure processing gain remains at usable levels Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 108ISI and Interference Rejection  Narrowband Interference Rejection S(f) I(f) S(f) S(f) S (f) c I(f)S (f) c Despread Signal Receiver Input Info. Signal  Multipath Rejection (Two Path Model) aS(f) S(f)S (f) a(t)+b(t) S(f) c bS’(f) Despread Signal Receiver Input Info. Signal Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 109How to Spread Spectrum Direct Sequence (DS) user data TIME data rate Modulation (primary modulation) spreading sequence Baseband (spreading code) Frequency 10110100 Tx Radio Frequency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 110 Power Power Density Density Spreading (secondary modulation)Spreading: TimeDomain View Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 111Spreading: FreqDomain View Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 112Demodulation 1/2 If you know the correct spreading sequence (code) , received signal TIME 10110100 10110100 01001011 spreading sequence 10110100 (spreading code) Radio Frequency you can find the spreading timing which gives the 10110100 10110100 10110100 maximum detected gathering energy power, and Accumulate for one bit duration 00000000 11111111 00000000 Demodulated data 0 0 1 Baseband Frequency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 113 Power DensityDemodulation 2/2 If you don’t know the correct spreading sequence (code) ••• received signal TIME 10110100 10110100 01001011 spreading sequence (spreading code) 01010101 01010101 01010101 Radio Frequency you cannot find the spreading 10101010 10101010 10101010 timing without correct spreading code, and 10110100 10110100 10110100 Accumulate for No data can be detected one bit duration Demodulated data Baseband Frequency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 114 Power DensitySecurity Aspects of Spread Spectrum Privacy, Security Power density of SSsignals could be lower than the noise density. Noise Radio Radio Baseband Frequency Frequency Frequency With correct code (and carrier frequency), de transmitted SSsignal received signal data can be detected. modulator With incorrect code (or carrier frequency), Noise SSsignal itself cannot be detected. They cannot perceive the existence of communication, Baseband because of signal behind the noise. Frequency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 115 Power Density •••••• Power Density •••••• Power Power Density DensitySpreading: Details Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 116Spreading: Mutually Orthogonal, Walsh Codes Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 117Spreading: Walsh Codes Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 118Walsh Codes (Contd) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 119Numerical Example: Walsh Codes 1 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 120Properties of Walsh Codes Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 121Multiplexing using Walsh Code Modulator Code for 00 Code for 01 Data Code for 10 Demodulator Code for 11 Code for 00 T dt  0 Select maximum Code for 01 value T dt  0 Code for 10 T dt  0 Code for 11 T dt  0 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 122DSCDMA System Overview (Forward link) CDMA is a multiple spread spectrum. Freq. Freq. Freq. Freq. BPF BPF Data A Despreader Data A MSA Code A Code A Freq. Freq. Freq. Freq. BPF BPF Data B Despreader Data B Code B MSB Code B BS Difference between each communication path is only the spreading code Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 123 ••• •••The IS95 CDMA (2G) Forward Link Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 124Synchronous DSCDMA Synchronous CDMA Systems realized in Point to Multipoint System. e.g., Forward Link (Base Station to Mobile Station) in Mobile Phone. Forward Link (Down Link) Synchronous Chip Timing A A A Less Interference for A station B Signal for B Station (after respreading) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 125The IS95 Reverse Link Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 126Asynchronous DSCDMA Reverse Link Asynchronous Chip (Up Link) Timing A Big Interference from A station B A B Signal for B Station (after respreading) Signals from A and B are interfering each other. In asynchronous CDMA system, orthogonal codes have bad crosscorrelation. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 127CrossCorrelation: PN Sequences Spreading Code A Spreading Code A 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 1 one data bit duration one data bit duration Spreading Code A Spreading Code B 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 SelfCorrelation CrossCorrelation for each code is 16/16. between Code A and Code B = 5/16 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 128Preferable Codes In order to minimize mutual interference in DSCDMA , the spreading codes with less crosscorrelation should be chosen. Synchronous DSCDMA : Orthogonal Codes are appropriate. (Walsh code etc.) Asynchronous DSCDMA : • Pseudorandom Noise (PN) codes / Maximum sequence • Gold codes Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 129Generating PN Sequences Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 130MSequences Autocorrelation: like impulse Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 131NearFar Problem: Power Control Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 132Power Control (continued) Open Loop Power Control Closed Loop Power Control ① ② ② measuring decide transmit power control received power transmission command power estimating path about 1000 times loss per second measuring transmit calculating received power transmission ① power transmit receive Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 133Effect of Power Control Effect of Power Control • Power control is capable of compensating the fading fluctuation. • Received power from all MS are controlled to be equal. ... NearFar problem is mitigated by the power control. from MS B from MS A Time A B Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 134 Detected PowerCDMA: Issues Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 135Key: Interference Averaging Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 136Voice Activity: Low Duty Cycle Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 137Variable Rate Vocoders Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 138Sector Antennas in CDMA Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 139Capacity Comparison Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 140Soft Handoff Handoff : • Cellular system tracks mobile stations in order to maintain their communication links. • When mobile station goes to neighbor cell, communication link switches from current cell to the neighbor cell. Hard Handoff : • In FDMA or TDMA cellular system, new communication establishes after breaking current communication at the moment doing handoff. Communication between MS and BS breaks at the moment switching frequency or time slot. switching Cell A Cell B Hard handoff : connect (new cell B) after break (old cell A) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 141Soft Handoff Soft Handoff : • In CDMA cellular system, communication does not break even at the moment doing handoff, because switching frequency or time slot is not required. transmitting same signal from both BS A and BS B simultaneously to the MS Σ Cell Cell A B Soft handoff : break (old cell A) after connect (new cell B) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 142Soft vs Hard Handover  Hard handover: the connection to the current cell is broken, and then the connection to the new cell is made.  "breakbeforemake" handover.  Universal freq. reuse in CDMA  "makebeforebreak" or "soft" handover.  Soft handovers require less power, which reduces interference and increases capacity.  Mobile can be connected to more that two BTS the handover.  "Softer" handover is a special case of soft handover where the radio links that are added and removed belong to the same node. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 143CDMA: Rake Receiver for Frequency Diversity Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 144FrequencySelective Fading in nonCDMA Broadband System path1 With low timeresolution, different signal paths cannot be discriminated. path2 ••• path3 These signals sometimes strengthen, and sometimes cancel out each other, Path Delay depending on their phase relation. ••• This is “fading”. ••• In this case, signal quality is damaged Detected Power when signals cancel out each other. In other words, signal quality is dominated by the probability for detected power Time to be weaker than minimum required level. This probability exists with less than two paths. In nonCDMA system, “fading” damages signal quality. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 145 Power PowerSynchronization Adder Fading in CDMA System: Rake Principle Because CDMA has high timeresolution, path1 different path delay of CDMA signals path2 can be discriminated. ••• path3 Therefore, energy from all paths can be summed by adjusting their phases and path delays. Path Delay ••• This is a principle of RAKE receiver. path1 interference from path2 and path3 CDMA Receiver path3 CODE A Path Delay path2 with timing of path1 path1 CDMA Receiver path2 CODE A with timing of path2 Path Delay Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 146 Power ••• Power Power ••• PowerFading in CDMA System (continued) In CDMA system, multipath propagation improves the signal quality by use of RAKE receiver. path3 path2 path1 Detected Power Time RAKE receiver Less fluctuation of detected power, because of adding all energy . Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 147 Power PowerFrequency Diversity via Rake Receiver (details)  Consider a simplified situation (uncoded).  Each information bit is spread into two pseudorandom sequences x and x (x = x ). A B B A  Each tap of the match filter is a finger of the Rake. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 148Frequency Diversity via Rake Receiver  Project y … (assuming h is known)  What the Rake actually does is take inner products of the received signal  … with shifted versions of the candidate transmitted sequences.  Each output is then weighted by the channel tap gain of the appropriate delay and summed.  The signal path associated with a particular delay is sometimes called a finger of the Rake receiver. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 149Recall: Maximal Ratio Combining (MRC), “Beamforming” , Rake Receiving: are just Matched Filtering operations  Generalization of this fdomain picture, for combining multitap signal Weight each branch SNR: Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” Source: J. Andrews et al, Fundamentals of WIMAX 150Rake Receiver: MaxRatioCombiner  Due to hardware limitations, the actual number of fingers used in a Rake receiver may be less than the total number of taps L in the range of the delay spread.  = a tracking mechanism in which the Rake receiver continuously searches for the strong paths (taps) to assign the limited number of fingers to. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 151Rake Receiver: Summary  CounterIntuitive: Increase rate and bandwidth  PN Code Autocorrelation attenuates ISI  Not particularly effective for wideband signals (no spreading gain) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 152ISI vs Frequency Diversity  In narrowband systems, ISI is mitigated using a complex receiver.  In asynchronous CDMA uplink, ISI is there but small compared to interference from other users.  But ISI is not intrinsic to achieve frequency diversity.  The transmitter needs to do some work too Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 153MultiCarrier Modulation and OFDM Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 154Frequency Diversity Multicarrier Modulation, i.e. OFDM  Key Idea: Since we avoid ISI if Ts Tm, just send a large number of narrowband carriers  M subcarriers each with rate R/M, also have Ts’ = TsM. Total data rate is unchanged. channel carrier subchannel frequency Figure courtesy B. Evans Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 155 magnitudeMulticarrier Modulation R/N bps QAM x R bps Modulator Serial To cos(2pf t) 0 ParallelS Converter R/N bps QAM x Modulator cos(2pf t) N  Breaks data into N substreams  Substream modulated onto separate carriers  Substream bandwidth is B/N for B total bandwidth  B/NB implies flat fading on each subcarrier (no ISI) c  Can overlap substreams (OFDM) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 156Multicarrier vs Equalizers  Equalizers use signal processing in receiver to eliminate ISI.  Linear equalizers can completely eliminate ISI (ZF), but this may enhance noise. MMSE better tradeoff.  Equalizer design involves tradeoffs in complexity, overhead, and performance (ISI vs. noise).  Number of filter taps, linear versus nonlinear, complexity and overhead of training and tracking  Multicarrier is an alternative to equalization  Divides signal bandwidth to create flatfading subchannels. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 157Multicarrier: Time vs Freq. Domain  Multicarrier: interesting interpretation in both time and frequency domains.  In the time domain, the symbol duration on each subcarrier has increased to T = LTs, …  … so by letting L grow larger, it can be assured that the symbol duration exceeds the channel delay spread,  … which is a requirement for ISIfree communication.  In the frequency domain,  …the subcarriers have bandwidth B/L Bc,  … which assures “flat fading”, …  the frequency domain equivalent to ISIfree communication. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 158OFDM: Parallel Tx on Narrow Bands Channel Channel impulse transfer function Frequency response Time (Freq selective fading) Signal is 1 Channel (serial) Frequency “broadband” 2 Channels Frequency 8 Channels Frequency Channels are “narrowband” (flat fading, ↓ ISI) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 159Multicarrier ISI Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 160Issues w/ Multicarrier Modulation Ch.1 Ch.2 Ch.3 Ch.4 Ch.5 Ch.6 Ch.7 Ch.8 Ch.9 Ch.10 Conventional multicarrier techniques frequency  1. Large bandwidth penalty since the subcarriers can’t have perfectly rectangular pulse shapes and still be timelimited.  2. Very high quality (expensive) low pass filters will be required to maintain the orthogonality of the subcarriers at the receiver.  3. This scheme requires L independent RF units and demodulation paths.  OFDM overcomes these shortcomings Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 161OFDM  OFDM uses a computational technique known as the Discrete Fourier Transform (DFT)  … which lends itself to a highly efficient implementation commonly known as the Fast Fourier Transform (FFT).  The FFT (and its inverse, the IFFT) are able to create a multitude of orthogonal subcarriers using just a single radio. Ch.2 Ch.4 Ch.6 Ch.8 Ch.10 Ch.1 Ch.3 Ch.5 Ch.7 Ch.9 Saving of bandwidth 50 bandwidth saving Orthogonal multicarrier techniques frequency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 162Concept of an OFDM signal Ch.1 Ch.2 Ch.3 Ch.4 Ch.5 Ch.6 Ch.7 Ch.8 Ch.9 Ch.10 Conventional multicarrier techniques frequency Ch.2 Ch.4 Ch.6 Ch.8 Ch.10 Ch.1 Ch.3 Ch.5 Ch.7 Ch.9 Saving of bandwidth 50 bandwidth saving Orthogonal multicarrier techniques frequency Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 163Spectrum of the modulated data symbols Magnitude  Rectangular Window of duration T T 0 0  Has a sincspectrum with zeros at 1/ T 0  Other carriers are put in these zeros  subcarriers are Frequency orthogonal Subcarrier orthogonality must be preserved Compromised by timing jitter, frequency offset, and fading. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 164OFDM Symbols  Group L data symbols into a block known as an OFDM symbol.  An OFDM symbol lasts for a duration of T seconds, where T = LTs.  Guard period delay spread  OFDM transmissions allow ISI within an OFDM symbol, but by including a sufficiently large guard band, it is possible to guarantee that there is no interference between subsequent OFDM symbols.  The next task is to attempt to remove the ISI within each OFDM symbol Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 165Circular Convolution DFT/IDFT  Circular convolution:  Circular convolution allows DFT  Detection of X (knowing H): (note: ISI free Just a scaling by H) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 166Cyclic Prefix: Eliminate intrasymbol interference  In order for the IFFT/FFT to create an ISIfree channel, the channel must appear to provide a circular convolution  If a cyclic prefix is added to the transmitted signal, then this creates a signal that appears to be xn , and so yn = xn hn. L  The first v samples of y interference from preceding OFDM symbol = discarded. cp  The last v samples disperse into the subsequent OFDM symbol = discarded.  This leaves exactly L samples for the desired output y, which is precisely what is required to recover the L data symbols embedded in x. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 167Cyclic Prefix (Contd)  These L residual samples of y will be equivalent to  By mimicking a circular convolution, a cyclic prefix that is at least as long as the channel duration (v+1)… … allows the channel output y to be decomposed into a simple multiplication of the channel frequency response H = DFTh and the channel frequency domain input, X = DFTx. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 168Cyclic Prefix Circular Convolution Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 169Circulant Matrix DFT Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 170Recall: DFT/Fourier Methods ≡ Eigen Decomposition  Applying transform techniques is just eigen decomposition  Discrete/Finite case (DFT/FFT):  Circulant matrix C is like convolution. Rows are circularly shifted versions of the first row  C = FΛF where F is the (complex) fourier matrix, which happens to be both unitary and symmetric, and multiplication w/ F is rapid using the FFT.  Applying F = DFT, i.e. transform to frequency domain, i.e. “rotate” the basis to view C in the frequency basis.  Applying Λ is like applying the complex gains/phase changes to each frequency component (basis vector)  Applying F inverts back to the timedomain. (IDFT or IFFT) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 171Cyclic Prefix overhead Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 172Cyclic Prefix Overhead: final thoughts  OFDM overhead = length of cyclic prefix / OFDM symbol time  Cyclic prefix dictated by delay spread.  OFDM symbol time limited by channel coherence time.  Equivalently, the intercarrier spacing should be much larger than the Doppler spread.  Since most channels are underspread, the overhead can be made small. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 173OFDM Implementation  1. Break a wideband signal of bandwidth B into L narrowband signals (subcarriers) each of bandwidth B/L. The L subcarriers for a given OFDM symbol are represented by a vector X, which contains the L current symbols.  2. In order to use a single wideband radio instead of L independent narrow band radios, the subcarriers are modulated using an IFFT operation.  3. In order for the IFFT/FFT to decompose the ISI channel into orthogonal subcarriers, a cyclic prefix of length v must be appended after the IFFT operation. The resulting L + v symbols are then sent in serial through the wideband channel. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 174OFDM Block Diagram Transmitter I/Q I/Q Channel Symbol OFDM Guard coding / mapping modulation interval interleaving (modulation) (IFFT) OFDM spectrum for N = 128, N = 12, N = 24, oversampling = 1 FFT w in guard 0110 010101001 10 N symbols 0 10 1 OFDM symbol 20 Receiver 30 40 FFTpart 50 Decoding / symbol de OFDM Guard 60 40 20 0 20 40 60 f MHz deinter mapping demod. interval time domain signal (baseband) 0.2 leaving (detection) (FFT) removal I/Q I/Q Channel 0.1 time impulse response: Channel est. Time sync. 0 0.1 imaginary real 0.2 Shivkumar Kalyanaraman Rensselaer Polytechnic Institute 0 20 40 60 80 100 120 140 160 180 200 sample nr. : “shiv rpi” 175 power spectrum magnitude dBOFDM in WiMAX Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 176OFDM in Wimax (Contd)  Pilot, Guard, DC subcarriers: overhead  Data subcarriers are used to create “subchannels”  Permutations clustering in the timefrequency domain used to leverage frequency diversity before allocating them to users. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 177Example: Flash OFDM (Flarion)  Bandwidth = 1.25 Mz  OFDM symbol = 128 samples = 100  s  Cyclic prefix = 16 samples = 11  s delay spread  11 overhead. • Permutations for frequency diversity for each user (gaps filled by other users) • Recall: like repetition coding • Efficiency gained across users •(multiuser frequency diversity) Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 178Summary: OFDM vs Equalization Shivkumar Kalyanaraman Rensselaer Polytechnic Institute CMAC: complex multiply and accumulate operations per received symbol : “shiv rpi” 179Summary: An OFDM Modem N subchannels 2N real samples quadrature amplitude add D/A + Bits S/P modulation NIFFT cyclic P/S transmit 00110 (QAM) prefix filter encoder TRANSMITTER multipath channel RECEIVER N subchannels 2N real samples invert Receive channel QAM remove filter = P/S demod NFFT S/P cyclic + frequency decoder prefix A/D domain equalizer Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 180Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 181OFDM: summary Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 182Channel Uncertainty  In fast varying channels, tap gain measurement errors may have an impact on diversity combining performance.  The impact is particularly significant in channel with many taps each containing a small fraction of the total received energy. (eg. Ultrawideband channels)  The impact depends on the modulation scheme. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 183Summary: Diversity  Fading makes wireless channels unreliable.  Diversity increases reliability and makes the channel more consistent.  Smart codes yields a coding gain in addition to the diversity gain.  This viewpoint of the adversity of fading will be challenged and enriched in later parts of the course. Shivkumar Kalyanaraman Rensselaer Polytechnic Institute : “shiv rpi” 184
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