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Point-to-Point Wireless Communication (II): ISI & Equalization, Diversity (Time/Space/Frequency). Shiv Kalyanaraman Google: “Shiv RPI” email@example.com. Ref: Chapter 3 in Tse/Viswanath texbook.
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Point-to-Point Wireless Communication (II):ISI &Equalization, Diversity (Time/Space/Frequency) Shiv Kalyanaraman Google: “Shiv RPI” firstname.lastname@example.org Ref: Chapter 3 in Tse/Viswanath texbook Based upon slides of P. Viswanath/Tse, Sorour Falahati, Takashi Inoue, J. Andrews, Scott Baxter, & textbooks by Tse/Viswanath, A. Goldsmith, J. Andrews et al, & Bernard Sklar.
Multi-dimensional Fading • Time, Frequency, Space
Plan • First, compare 1-tap (i.e. flat) Rayleigh-fading channel vs AWGN. • i.e. y = hx + wvs y = x + w • Note: all multipaths with random attenuation/phases are aggregated into 1-tap • Next consider frequency selectivity, i.e. multi-tap, broadband channel, with multi-paths • 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
Single-tap, Flat Fading (Rayleigh) vs AWGN Why do we have this huge degradation in performance/reliability?
Looks like AWGN, but… pe needs to be “unconditioned” To get a much poorer scaling Rayleigh Flat Fading Channel BPSK: Coherent detection. Conditional on h, Averaged over h, at high SNR.
BER vs. SNR (cont.) Frequency-selective channel (equalization or Rake receiver) BER Frequency-selective channel (no equalization) “BER floor” AWGN channel (no fading) Flat fading channel SNR means a straight line in log/log scale
Typical 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.
Chi-Squared pdf of Preview: 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)
Recall: BPSK, QPSK and 4-PAM • BPSK uses only the I-phase.The Q-phase is wasted. • QPSK delivers 2 bits per complex symbol. • To deliver the same 2 bits, 4-PAM requires 4 dB more transmit power. • QPSK exploits the available degrees of freedom in the channel better. • A good communication scheme exploits all the available d.o.f. in the channel.
MQAM doesn’t change the asymptotics… • QPSK does use degrees of freedom better than equivalent 4-PAM • (Read textbook, chap 3, section 3.1)
Frequency Selectivity: Multipath fading & ISI Mitigation: Equalization & Challenges
ISI Mitigation: Outline • Inter-symbol 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 multi-tap channel into narrowband single tap channels
Recall: Attenuation, Dispersion Effects: ISI! Inter-symbol interference (ISI) Source: Prof. Raj Jain, WUSTL
path-1 Power path-2 path-3 path-2 Path Delay path-1 path-3 Channel Impulse Response: Channel amplitude |h| correlated at delays . Each “tap” value @ kTs Rayleigh distributed (actually the sum of several sub-paths) Recall: Multipaths: Power-Delay Profile multi-path propagation Mobile Station (MS) Base Station (BS)
Transmitted signal: Received Signals: Line-of-sight: Reflected: The symbols add up on the channel Distortion! Inter-Symbol-Interference (ISI) due to Multi-Path Fading Delays
Multipath: Time-Dispersion => Frequency Selectivity • The impulse response of the channel is correlated in the time-domain (sum of “echoes”) • Manifests as a power-delay profile, dispersion in channel autocorrelation function A() • Equivalent to “selectivity” or “deep fades” in the frequency domain • Delay spread: ~ 50ns (indoor) – 1s (outdoor/cellular). • Coherence Bandwidth: Bc = 500kHz (outdoor/cellular) – 20MHz (indoor) • Implications: High data rate: symbol smears onto the adjacent ones (ISI). Multipath effects ~ O(1s)
BER 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 Frequency-selective channel (no equalization) Gaussian channel (no fading) Flat fading channel S/N
BER vs. S/N performance: Flat Fading Flat fading: z= signal power level Typical BER vs. S/N curves BER Frequency-selective channel (no equalization) Gaussian channel (no fading) Flat fading channel S/N
BER vs. S/N performance: ISI/Freq. Selective Channel Frequency selective fading <=> irreducible BER floor!!! Typical BER vs. S/N curves BER Frequency-selective channel (no equalization) Gaussian channel (no fading) Flat fading channel S/N
BER vs. S/N performance: w/ Equalization Diversity (e.g. multipath diversity) <=> improved performance Typical BER vs. S/N curves BER Gaussian channel (no fading) Frequency-selective channel (with equalization) Flat fading channel S/N
Equalization Step 1 – waveform to sample transformation Step 2 – decision making Demodulate & Sample Detect Threshold comparison Frequency down-conversion Receiving filter Equalizing filter Compensation for channel induced ISI For bandpass signals Baseband pulse (possibly distored) Received waveform Sample (test statistic) Baseband pulse
What 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 (f-domain) filter to cancel out the channel mpath filtering effects
Equalization: Channel is a LTI Filter • ISI due to filtering effect of the communications channel (e.g. wireless channels) • Channels behave like band-limited filters Non-constant amplitude Amplitude distortion Non-linear phase Phase distortion
Taking care of ISI caused by tr. filter Pulse Shaping and Equalization Principles No ISI at the sampling time • Square-Root Raised Cosine (SRRC) filter and Equalizer Taking care of ISI caused by channel * Equalizer: enhance weak freq., dampen strong freq. to flatten the spectrum * Since the channel Hc(f) changes with time, we need adaptive equalization, i.e. re-estimate channel & equalize
Equalization: Channel examples • Example of a (somewhat) frequency selective, slowly changing (slow fading) channel for a user at 35 km/h
Equalization: Channel examples … • Example of a highly frequency-selective, fast changing (fast fading) channel for a user at 35 km/h
Recall: 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 amplitude scale Sensitivity to timing error Timing jitter time scale
Example of eye pattern with ISI:Binary-PAM, SRRC pulse • Non-ideal channel and no noise
Example of eye pattern with ISI:Binary-PAM, SRRC pulse … • AWGN (Eb/N0=20 dB) and ISI
Example of eye pattern with ISI:Binary-PAM, SRRC pulse … • AWGN (Eb/N0=10 dB) and ISI
Equivalent model Equivalent system Detector Equalizer filtered (colored) noise Equalizing filters … • Baseband system model Tx filter Channel Rx. filter Detector Equalizer
Equalizer Types Covered later in slideset Source: Rappaport book, chap 7
n(t) Equalizer Channel 1 Heq(f)» Hc(f) Hc(f) Linear Equalizer • A linear equalizer effectively inverts the channel. • 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 frequency-selective fading channels
Decision Feedback Equalizer (DFE) DFE n(t) • => doesn’t work well w/ low SNR. • Optimal non-linear: MLSE… (complexity grows exponentially w/ delay spread) ^ x(t) x(t) + Forward Hc(f) 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. Þ The DFE has better performance than the linear equalizer in a frequency-selective fading channel. • The DFE is subject to error propagation if decisions are made incorrectly.
Equalization by transversal filtering • Transversal filter: • A weighted tap delayed line that reduces the effect of ISI by proper adjustment of the filter taps. Coeff. adjustment
Transversal equalizing filter … • Zero-forcing equalizer: • The filter taps are adjusted such that the equalizer output is forced to be zero at N sample points on each side: • 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) Adjust Adjust
Equalization: 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)
Summary: 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
Tb Introduction 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 Multiple paths unlikely to fade simultaneously
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/bit-rate one… For capacity: need more degrees-of-freedom (i.e. symbols/s) & packing of bits/symbol (MQAM).
Time 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!
Recover K data packets! >= K of N received Lossy Network Forward Error Correction (FEC): Eg: Reed-Solomon RS(N,K) RS(N,K) FEC (N-K) Block Size (N) Block: of sufficient size: (> coherence time), else need to interleave, or use with hybrid ARQ Data = K
Hybrid ARQ/FEC Model Packets • Sequence Numbers • CRC or Checksum • Proactive FEC Timeout • ACKs • NAKs, • SACKs • Bitmaps Status Reports Retransmissions • Packets • Reactive FEC
Example: 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/voice-frame • 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.
Time-Diversity 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 fc = 900Mhz. • Recall: Tc < 5 ms = 1/(4Ds) = c/(8fcv)
GSM 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 sub-channel to another. • Typical delay spread ~ 1μs => the coherence bandwidth (Bc) is 500 kHz. • The total bandwidth of 25 MHz >> Bc => consecutive frames can be expected to fade independently. • This provides the same effect as having time diversity.