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DSSS, ISI Equalization and OFDM
Y. Richard Yang 01/19/2011
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Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization OFDM
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Admin. Homework 1 is linked on the schedule page
Please start to think about project
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Recap: Main Story of Flat Fading
Communication over a wireless channel has poor performance due to significant probability that channel is in a deep fade, or has interference Reliability is increased by using diversity: more resolvable signal paths that fade independently time diversity: send same info (or coded version) at different times space diversity: send/receive same info at different locations frequency diversity: send info at different frequency frequency hopping; direct sequence
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Direct Sequence Spread Spectrum (DSSS)
One symbol is spread to multiple chips the number of chips is called the expansion factor examples IS-95 CDMA: 1.25 Mcps; 4,800 sps how many chips per symbol? 802.11: 11 Mcps; 1 Msps how may chips per symbol? The increased rate provides frequency diversity (explores frequency in parallel)
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Effects of Spreading and Interference
dP/df f sender dP/df f un-spread signal spread signal Bb Bs : num. of bits in the chip * Bb
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DSSS Encoding/Decoding: An Operating View
spread spectrum signal transmit signal user data X modulator chipping sequence radio carrier transmitter correlator sampled sums products received signal data demodulator X low pass decision radio carrier chipping sequence receiver
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DSSS Encoding chip: -1 1 Data: [ ] -1 1 1 -1
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DSSS Encoding tb: bit period tc: chip period tb user data d(t) 1 -1 X
chipping sequence c(t) -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 = resulting signal -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 tb: bit period tc: chip period
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DSSS Decoding chip: Data: [1 -1] inner product: 6 -6 decision: 1 -1 -1
Trans chips -1 1 1 -1 decoded chips -1 1 1 -1 Chip seq: -1 1 -1 1 inner product: 6 -6 decision: 1 -1
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DSSS Decoding with noise
chip: -1 1 Data: [ ] Trans chips -1 1 1 -1 decoded chips -1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 Chip seq: -1 1 -1 1 inner product: 4 -2 decision: 1 -1
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DSSS Decoding (BPSK): Another View
bit time take N samples of a bit time sum = 0; for i =0; { sum += y[i] * c[i] * s[i] } if sum >= 0 return 1; else return -1; y: received signal c: chipping seq. s: modulating sinoid compute correlation for each bit time
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Outline Admin. and recap Direct sequence spread spectrum
operating view why does DSSS work?
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Assume no DSSS Consider narrowband interference
Consider BPSK with carrier frequency fc A worst-case scenario data to be sent x(t) = 1 channel fades completely at fc (or a jam signal at fc) then no data can be recovered
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Why Does DSSS Work: A Decoding Perspective
Assume BPSK modulation using carrier frequency f : A: amplitude of signal f: carrier frequency x(t): data [+1, -1] c(t): chipping [+1, -1] y(t) = A x(t)c(t) sin(2 ft)
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Add Noise/Jamming/Channel Loss
Assume noise at carrier frequency f: Received signal: y(t) + w(t)
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DSSS/BPSK Decoding
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Why Does DSSS Work: A Spectrum Perspective
sender dP/df dP/df f ii) user signal broadband interference narrowband interference i) f receiver dP/df dP/df dP/df iii) iv) v) f f f i) → ii): multiply data x(t) by chipping sequence c(t) spreads the spectrum ii) → iii): received signal: x(t) c(t) + w(t), where w(t) is noise iii) → iv): (x(t) c(t) + w(t)) c(t) = x(t) + w(t) c(t) iv) → v) : low pass filtering
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Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization OFDM
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Recall: Representation of Wireless Channels
So far we considered inter-symbol interference small: (also called flat fading channel) In the general case, received signal at time m is y[m], hl[m] is the strength of the l-th tap, w[m] is the background noise:
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ISI Effects
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ISI Problem Formulation
The problem: given received y[m], m = 1, …, L+2, where L is frame size and assume 3 delay taps (it is easy to generalize to D taps): y[1] = x[1] h0 + w[1] y[2] = x[2]h0 + x[1] h1 + w[2] y[3] = x[3]h0 + x[2]h1 + x[3] h2 + w[3] y[4] = x[4]h0 + x[3]h1 + x[2] h2 + w[4] y[5] = x[5]h0 + x[4]h1 + x[3] h2 + w[5] … y[L] = x[L]h0 + x[L-1]h1 + x[L-2]h2 + w[L] y[L+1] = x[L]h1 + x[L-1]h2 + w[L+1] y[L+2] = x[L]h2 + w[L+2] determine x[1], x[2], … x[L]
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ISI Equalization: Given y, what is x?
y[1] = x[1] h0 + w[1] y[2] = x[2]h0 + x[1] h1 + w[2] y[3] = x[3]h0 + x[2]h1 + x[3] h2 + w[3] y[4] = x[4]h0 + x[3]h1 + x[2] h2 + w[4] y[5] = x[5]h0 + x[4]h1 + x[3] h2 + w[5] … y[L] = x[L]h0 + x[L-1]h1 + x[L-2]h2 + w[L] y[L+1] = x[L]h1 + x[L-1]h2 + w[L+1] y[L+2] = x[L]h2 + w[L+2] x y
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Solution Technique Maximum likelihood detection:
if the transmitted sequence is x[1], …, x[L], then there is a likelihood we observe y[1], y[2], …, y[L+2] we choose the x sequence such that the likelihood of observing y is the largest y[1] = x[1] h0 + w[1] y[2] = x[2]h0 + x[1] h1 + w[2] y[3] = x[3]h0 + x[2]h1 + x[3] h2 + w[3] y[4] = x[4]h0 + x[3]h1 + x[2] h2 + w[4] y[5] = x[5]h0 + x[4]h1 + x[3] h2 + w[5] … y[L] = x[L]h0 + x[L-1]h1 + x[L-2]h2 + w[L] y[L+1] = x[L]h1 + x[L-1]h2 + w[L+1] y[L+2] = x[L]h2 + w[L+2]
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Likelihood For given sequence x[1], x[2], …, x[L]
Assume white noise, i.e, prob. w = z is What is the likelihood (prob.) of observing y[1]? it is the prob. of noise being w[1] = y[1] – x[1] h0
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Likelihood The likelihood of observing y[2]
it is the prob. of noise being w[2] = y[2] – x[2]h0 – x[1]h1, which is The overall likelihood of observing the whole y sequence (y[1], …, y[L+2]) is the product of the preceding probabilities
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One Technique: Enumeration
foreach sequence (x[1], …, x[L]) compute the likelihood of observing the y sequence pick the x sequence with the highest likelihood Question: what is the computational complexity?
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Viterbi Algorithm Objective: avoid the enumeration of the x sequences
Key observation: the memory (state) of the wireless channel is only 3 (or generally D for D taps) Let s[0], s[1], … be the states of the channel as symbols are transmitted s[0]: initial state---empty s[1]: x[1] is transmitted, two possibilities: 0, or 1 s[2]: x[2] is transmitted, four possibilities: 00, 01, 10, 11 s[3]: x[3] is transmitted, eight possibilities: 000, 001, …, 111 s[4]: x[4] is transmitted, eight possibilities: 000, 001, …, 111 We can construct a state transition diagram If we know the x sequence we can construct s, and vice versa
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observe y[1] observe y[2] observe y[3] observe y[4] s[0] s[1] s[2] s[3] s[4] x[1]=0 x[2]=0 x[3]=0 00 000 000 x[3]=1 001 001 x[1]=1 x[2]=1 x[3]=0 01 010 010 x[3]=1 011 011 x[2]=0 x[3]=0 1 10 100 100 x[3]=1 x[2]=1 101 101 x[3]=0 11 110 110 x[3]=1 111 111 prob. of observing y[1]: w[1] = y[1]-x[1]h0 prob. of observing y[2]: w[2] = y[2]-x[1]h0-x[2]h1 prob. of observing y[4]: w[4] = y[4]-x[4]h0-x[3]h1-x[2]h2
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Viterbi Algorithm Each path on the state-transition diagram corresponds to a x sequence each edge has a probability the product of the probabilities on the edges of a path corresponds to the likelihood that we observe y if x is the sequence sent Then the problem becomes identifying the path with the largest product of probabilities
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Viterbi Algorithm: Largest Product to Shortest Path
If we take -log of the probability of each edge, the problem becomes identifying the shortest path problem!
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Viterbi Algorithm: Summary
Invented in 1967 Utilized in CDMA, GSM, , Dial-up modem, and deep space communications Also commonly used in speech recognition, computational linguistics, and bioinformatics Original paper: Andrew J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, April
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Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization OFDM
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Orthogonal Frequency Division Multiplexing: Motivation
Viterbi algorithm handles ISI Problem? Its complexity grows exponentially with D, where D is the number of multipaths taps relative to the symbol time If we have a high symbol rate, then D can be large, and we need complex receivers
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Multiple Carrier Modulation
Uses multiple carriers modulation (MCM) each subcarrier uses a low symbol rate reduce symbol rate and reduce ISI for N parallel subcarriers, the symbol time can be N times longer spread symbols across multiple subcarriers also gains frequency diversity
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Multiple Carrier Modulation
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Multiple Carrier Modulation (MCM): Problem
Traditional approach of using multiple subcarriers uses guard band to avoid interference among subcarriers Guardband wastes spectrum
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Orthogonal Frequency Division Multiplexing: Key Idea
Avoid subcarrier interference by using orthogonal subcarriers
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OFDM: Orthogonal Subcarriers
Frequencies chosen so that an integral number of cycles in a symbol period They do not need to have the same phase, so long integral number of cycles in symbol time T !
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OFDM Modulation
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OFDM: Orthogonal Subcarriers
Frequencies chosen so that an integral number of cycles in a symbol period They do not need to have the same phase, so long integral number of cycles in symbol time T !
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Orthogonal Frequency Division Multiplexing
OFDM allows overlapping subcarriers frequencies 802.11a
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OFDM Implementation Take N symbols and place one symbol on each subcarrier (freq.) Q: any problem with the straightforward implementation strategy? freq0 freqN-1
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OFDM: Key Idea 2 Straightforward implementation can be expensive if we use one oscillator for each subcarrier Consider data as coefficients in the frequency domain, use inverse Fourier transform to generate time-domain sequence
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OFDM Implementation: FFT
channel
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OFDM Implementation Parallel data streams are used as inputs to an IFFT IFFT does multiplexing and modulation in one step !
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OFDM Implementation OFDM also uses cyclic prefix to avoid intercarrier and intersymbol interference caused by multipath delays For details see Chap of
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Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization OFDM Delay spread as diversity
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Reducing to Transmit Diversity
Delay spread is really a type of transmit diversity
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Multipath Diversity: Rake Receiver
Instead of considering delay spread as an issue, use multipath signals to recover the original signal Used in IS-95 CDMA, 3G CDMA, and Invented by Price and Green in 1958 R. Price and P. E. Green, "A communication technique for multipath channels," Proc. of the IRE, pp , 1958
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Multipath Diversity: Rake Receiver
LOS pulse multipath pulses Use several "sub-receivers" each delayed slightly to tune in to the individual multipath components Each component is decoded independently, but at a later stage combined this could very well result in higher SNR in a multipath environment than in a "clean" environment
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Rake Receiver Blocks Correlator Combiner Finger 1 Finger 2 Finger 3
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Rake Receiver: Matched Filter
Impulse response measurement Tracks and monitors peaks with a measurement rate depending on speeds of mobile station and on propagation environment Allocate fingers: largest peaks to RAKE fingers
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Rake Receiver: Combiner
The weighting coefficients are based on the power or the SNR from each correlator output If the power or SNR is small out of a particular finger, it will be assigned a smaller weight:
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Comparison [PAH95] MCM is OFDM
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