1. DSSS, ISI Equalization and OFDM
Y. Richard Yang
01/19/2011

2. Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization
OFDM

3. Admin. Homework 1 is linked on the schedule page
Please start to think about project

4. 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

5. 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)

6. 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

7. 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

8. DSSS Encoding
chip: -1 1
Data: [ ] -1 1 1 -1

9. 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

10. 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

11. 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

12. 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

13. Outline Admin. and recap Direct sequence spread spectrum
operating view
why does DSSS work?

14. 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

15. 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)

16. Add Noise/Jamming/Channel Loss
Assume noise at carrier frequency f:
Received signal: y(t) + w(t)

17. DSSS/BPSK Decoding

18. 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

19. Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization
OFDM

20. 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:

21. ISI Effects

22. 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]

23. 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

24. 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]

25. 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

26. 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

27. 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?

28. 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

29. 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

30. 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

31. 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!

32. 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

33. Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization
OFDM

34. 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

35. 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

36. Multiple Carrier Modulation

37. Multiple Carrier Modulation (MCM): Problem
Traditional approach of using multiple subcarriers uses guard band to avoid interference among subcarriers
Guardband wastes spectrum

38. Orthogonal Frequency Division Multiplexing: Key Idea
Avoid subcarrier interference by using orthogonal subcarriers

39. 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 !

40. OFDM Modulation

41. 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 !

42. Orthogonal Frequency Division Multiplexing
OFDM allows overlapping subcarriers frequencies
802.11a

43. OFDM Implementation
Take N symbols and place one symbol on each subcarrier (freq.)
Q: any problem with the straightforward implementation strategy?
freq0
freqN-1

44. 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

45. OFDM Implementation: FFT
channel

46. OFDM Implementation
Parallel data streams are used as inputs to an IFFT
IFFT does multiplexing and modulation in one step !

47. OFDM Implementation
OFDM also uses cyclic prefix to avoid intercarrier and intersymbol interference caused by multipath delays
For details see Chap of

48. Outline Admin. and recap Direct sequence spread spectrum
Delay spread and ISI equalization
OFDM
Delay spread as diversity

49. Reducing to Transmit Diversity
Delay spread is really a type of transmit diversity

50. 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

51. 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

52. Rake Receiver Blocks
Correlator
Combiner
Finger 1
Finger 2
Finger 3

53. 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

54. 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:

55. Comparison [PAH95]
MCM is OFDM