1. Handling Inter-Symbol Interference (ISI): Wave Shaping, Equalization and OFDM
Y. Richard Yang
09/25/2012

2. Outline Admin. and recap Handling ISI Symbol wave shaping equalization
OFDM
PHY

3. Admin. Feedback on coverage approach
Continue to MAC layer
Switch to App layer (basic Android) and then back
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 instances of time
space diversity: send/receive same info at different locations
frequency diversity: send info at different frequencies

5. Outline
Admin. and recap
Inter-Symbol Interference (ISI)

6. ISI
ISI happens when the signaling for one symbol leaks into that of another symbol
Why does ISI happen?
Band limit produced ISI
Multipath produced ISI 1 2

7. Outline Admin. and recap Inter-Symbol Interference (ISI)
Bandlimit produced ISI

8. Spectrum of BPSK Rectangular Pulse
BPSK: g(t) = x(t) cos(2πfct), for t in [0, T]
Since multiplying cos() is just to shift frequency, consider the baseband, which is a rectangular pulse
Fourier transform of a rectangular pulse spans the whole frequency range
This may not be acceptable in real deployment

9. Band Limit Signaling Use band limited signaling functions
Design 1: sinc
W = 1/2T
sinc
G(f)
g(t)

10. ‘sinc’-like Signaling
a g(t)
Effect: signals spread across symbols.
Question: does it “ruin” our demodulation?

11. Received Signal Representation

12. Matched Filter Decoding
Suppose matched filter h(t)

13. Impact on Matched Filter Decoding
Define
Recall that we make decisions at iT instances of time:
Define pk = p(kT)
Question: what is a condition for no ISI?

14. Folded Spectrum Design
Define no ISI as only p0 is non-zero.
Objective: how to design p(t) = g(t) * h(t) to satisfy no ISI?
A design of p(t) has no ISI iff folded spectrum is flat

15. Folded Spectrum Design

16. Folded Spectrum Design: Sufficiency

17. Folded Spectrum Design: Necessary
=>

18. Flat Folded Spectrum

19. Flat Folded Spectrum: Raised Cosine
beta called rolloff factor or excessive bw
smaller beta, lower excessive bw, but slower time-domain dying down

20. From P() to G()
Recall
Assume g(t) symmetric, h(t) = g(-t) = g(t)

21. Root Raised Cosine

22. Apply to Real Life
See dbpsk.py of gnuradio implementation

23. ?

24. Outline Admin. and recap Inter-Symbol Interference (ISI) Bandlimit ISI
Multipath ISI

25. Multipath ISI y3 1 2 3 4 1 2 3 4 1 2 3 4

26. Multipath ISI y3
h(0) x(3) 1 2 3 4 1 2 3 4 1 2 3 4

27. Multipath ISI y3
h(0) x(3)
h(1) x(2) 1 2 3 4 1 2 3 4 1 2 3 4

28. Multipath ISI y3
h(0) x(3)
h(1) x(2) 1 2 3 4 1 2 3 4 1 2 3 4
h(2) x(0)

29. Recall: Representation of Wireless Channels
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:

30. Visualizing ISI

31. 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]

32. 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

33. 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]

34. Likelihood For given sequence x[1], x[2], …, x[L]
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] …
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

35. Likelihood The likelihood of observing y[2]
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] …
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

36. One Technique: Enumeration
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] …
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?

37. 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

38. 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

39. 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
If we take -log of the probability of each edge, the problem becomes identifying the shortest path problem!

40. 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

41. Problems of Viterbi for Multipath ISI
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

42. CPU Cycles In (bit) Out (bit) Compute Mcyles/sec

43. Outline Admin. and recap Inter-Symbol Interference (ISI) Bandlimit ISI
Multipath ISI
Viterbi
OFDM

44. 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

45. Multiple Carrier Modulation

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

47. Orthogonal Frequency Division Multiplexing: Key Idea
Avoid subcarrier interference
Interference of subcarrier i on subcarrier j
Effect of interference after matched filter j i

48. Orthogonal Frequency Division Multiplexing: Key Idea
Frequencies chosen so that an integral number of cycles in a symbol period, i.e.,
subcarrier freq = k 1/T
They do not need to have the same phase, so long integral number of cycles in symbol time T !

49. OFDM: Orthogonal Subcarriers
Frequencies chosen so that an integral number of cycles in a symbol period, i.e.,
subcarrier freq = k 1/T
They do not need to have the same phase, so long integral number of cycles in symbol time T !

50. OFDM Modulation

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

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

53. OFDM: Key Idea 2
freq0
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
freqN-1

54. OFDM Implementation: FFT
channel

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

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

57. Summary of PHY

58. Wireless PHY
PHY

59. Outline Admin. and recap Inter-Symbol Interference (ISI)
Bandlimit ISI
Multipath ISI
Viterbi
OFDM
Delay spread as diversity

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

61. 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

62. 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

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

64. 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

65. 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:

66. Comparison [PAH95]
MCM is OFDM