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Multiuser Communications
DIMITRIOS VLASTARAS
Introduction to Multiple Access Techniques
Many transmitters to one receiver
One transmitter to many
Receivers (broadcast)
Different kinds of multiuser systems
Introduction to Multiple Access Techniques
Store-and-forward network
Two-way communication system
More kinds of multiuser systems
Introduction to Multiple Access Techniques
How do we add multiple users?
• Divide them in frequency (FDMA)
• Divide them in time (TDMA)
Problems with bursty access methods? Yes!
• Periods of no transmission!
• Therefore, FDMA and TDMA tend to be inefficient
But, what does that mean?
Introduction to Multiple Access Techniques
How do we solve FDMA and TDMA problems?
• By use of Direct-sequence spread spectrum signals (DSSS)
• Each user is assigned a unique code sequence (pref. orthogonal)
• Separated at the receiver by cross correlation
• Sequences must have small cross correlations
to minimize crosstalk
• Completely overlap in time and frequency
• This multiple access method is called Code Division Multiple
Access (CDMA)
• Some times called Spread Spectrum Multiple Access (SSMA)
Introduction to Multiple Access Techniques
Alternative nonspread random access methods to CDMA
Transmissions collide
Loss of data
Protocols for retransmission
Capacity of Multiple Access Methods
Let’s refresh out memory about capacity
• Ideal band-limited AWGN channel
• 𝐾 users
• Each user has an average power 𝑃𝑖 = 𝑃 for 1 ≤ 𝑖 ≤ 𝐾
•
𝑁0
2
is the PSD of the additive noise
Capacity of Multiple Access Methods
In FDMA each user has bandwidth
𝑊
𝐾
, so capacity per user is
And the total capacity is
Therefore, the total capacity is the same as for a single
user with average power 𝑃𝑎𝑣 = 𝐾𝑃
What about FDMA capacity?
Capacity of Multiple Access Methods
Interesting fact: For a fixed 𝑊 the total capacity goes to infinity as
𝐾 increases. However, each user is allocated a smaller bandwidth
𝑊
𝐾
and the capacity per user decreases.
Normalized capacity as function of
𝜀𝑏
𝑁0
for FDMA Total capacity per hertz as a function of
𝜀𝑏
𝑁0
for FDMA.
Given 𝐶𝑛 = 𝐾
𝐶𝑘
𝑊
∶
Capacity of Multiple Access Methods
What about TDMA capacity?
In TDMA each user transmits for
1
𝐾
through the channel of
bandwidth 𝑊 with average power 𝐾𝑃. So, capacity per user is
Which is the same capacity as for an FDMA system. However, it
may not be possible for the transmitters to sustain a power of 𝐾𝑃
when 𝐾 is very large.
Capacity of Multiple Access Methods
What about CDMA capacity?
In CDMA each user transmits a pseudorandom signal of
bandwidth 𝑊 and average power 𝑃. The capacity depends the
level of cooperation among the 𝐾 users.
Noncooperative CDMA: Receiver for each user does not know
the codes of the other users. So, capacity per user is
Or normalized
Normalized capacity as a function of
𝜀𝑏
𝑁0
for
Noncooperative CDMA.
Capacity of Multiple Access Methods
What about CDMA capacity?
In CDMA each user transmits a pseudorandom signal of
bandwidth 𝑊 and average power 𝑃. The capacity depends the
level of cooperation among the 𝐾 users.
”Cooperative” CDMA: 𝐾 users cooperate by transmitting their
coded signals synchronously in time. This is called multiuser
detection and decoding. Each user is assigned a rate 𝑅𝑖, 1 ≤ 𝑖 ≤
𝐾, and a code book containing a set of 2
𝑛𝑅𝑖
codewords of power
𝑃. In each signal interval, each user selects an arbitrary
codeword, say 𝑿𝒊, from its own code book, and all users transmit
their codewords simultaneously.
Conclusion: Capacity similar to that of FDMA and TDMA.
Look in the book (p. 1034) for derivations!
Capacity of Multiple Access Methods
Capacity region of two-user CDMA multiple
access Gaussian channel.
Multiuser Detection in CDMA Systems
• CDMA Signal and Channel Models
• The Optimum Multiuser Receiver (Exponentially growing
complexity)
• Suboptimum Detectors (Lower computational complexity)
• Successive Interference Cancellation
• Other Types of Multiuser Detectors
• Performance Characteristics of Detectors
• User signature waveform:
• All 𝐾 signature waveforms have energy:
• Cross correlations important metrics for
detector performance:
• Lowpass transmitted waveform where
{𝑏𝑘(𝑚)} is the information sequence
of the 𝑘th user:
• Composite transmitted signal for 𝐾 users
where {𝜏𝑘} are transmission delays:
• The received signal assuming that the
transmitted signal is corrupted by AWGN:
CDMA Signal and Channel Models
CDMA channel shared by 𝑲 simultaneous users
The Optimum Multiuser Receiver
The optimum (maximum-likelihood) receiver is the receiver
that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤
𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed
over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻.
Synchronous transmission Each user/interferer produces
exactly one symbol which interferes with the desired symbol.
The received signal is
Compute the log-likelihood function
And select the sequence {𝑏𝑘 1 , 1 ≤ 𝑘 ≤ 𝐾} that minimizes Λ b
Correlations
The Optimum Multiuser Receiver
The optimum (maximum-likelihood) receiver is the receiver
that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤
𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed
over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻.
Synchronous transmission
The Optimum Multiuser Receiver
The optimum (maximum-likelihood) receiver is the receiver
that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤
𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed
over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻.
Asynchronous transmission In this case, there are exactly two
consecutive symbols from each interferer that overlap a desired
symbol.
The log-likelihood function may be expressed in terms of a
correlation metric where
Y U SO COMPLEX?
The Optimum Multiuser Receiver
The optimum (maximum-likelihood) receiver is the receiver
that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤
𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed
over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻.
Asynchronous transmission In this case, there are exactly two
consecutive symbols from each interferer that overlap a desired
symbol.
The log-likelihood function may be expressed in terms of a
correlation metric where
The Optimum Multiuser Receiver
The optimum (maximum-likelihood) receiver is the receiver
that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤
𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed
over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻.
Block processing approach (Brute force)
Compute 2𝑁𝐾 correlation metrics
Sequence estimation employing
the Viterbi algorithm
Feasible only when 𝐾 < 10
Use suboptimum detectors whose
complexity grows linearly with 𝐾!
Suboptimum Detectors
Suboptimum detectors have computational complexity that
grown grows linearly with the number of users 𝐾, in
comparison to the optimal multiuser receiver, whose
complexity grows exponentially with 𝐾.
In conventional single-user detection, the receiver consists of a
demodulator that correlates (match-filters) the received signal
with the signature sequence of the user and passes the output to
the detector, which makes a decision.
It ignores other users and assumes the noise plus interference to
be white and Gaussian. In synchronous transmissions, if the
signature squences are orthogonal
The interference from other users given by the middle term
vanishes.
Suboptimum Detectors
Conventional single-user detector
If one or more signature sequences are not orthogonal, the
interference from other users can become excesive depending
on the received signal energies.
This is called the near-far problem and necessitates some power
of power control.
In asynchronous transmission, the conventional detector is more
vulnerable to interference. This is because it is not possible to
design signature sequences that are orthogonal.
Suboptimum Detectors
Conventional single-user detector
It has linear computational complexity, like the conventional
detector, but does not suffer from other user interference.
The results of the minimization of the likelihood function yields
Then the detected symbols are the sign of each element 𝒃𝐾
0
Suboptimum Detectors
Decorrelating detector (ML)
The transformation 𝑹𝑠
−1 eliminates the interference. Therefore the
near-far problem is eliminated.
It is called the decorrelating detector since the received signal is
correlated with modified signature waveforms. Thus, we have
tuned out or decorrelated the multiuser interference.
For asynchronous transmission the minimization yields
Which is also a decorrelating detector, since 𝒃0
in an unbiased
estimate of 𝒃 and the multiuser interference has been eliminated.
Suboptimum Detectors
Decorrelating detector (ML)
Another solution is obtained if we seek the linear transformation
𝒃0 = 𝑨𝒓, where the matrix 𝑨 is to be determined so as to
minimize the mean square error (MSE).
For synchronous communication
And asynchronous communication
where the output is again 𝒃 = sgn(𝐛0
)
Suboptimum Detectors
Minimum mean-square-error (MMSE) detector
Successive Interference Cancellation
Successive Interference Cancellation (SIC) is another
multiuser detection technique. The interfering signal
waveforms are removed one at a time as they are detected.
The user with the strongest received signal is demodulated and
detected first. Then this information is used to substract the
signal of the particular user from the received signal.
For synchronous transmission
Which is a suboptimum detector. The jointly optimum
interference canceller may be defined as
This is a technique that employs multiple iterations in detecting
the user bits and cancelling the interference.
Example with two users and synchronous transmission:
First stage
(decorrelating detector):
Second stage:
Third stage:
Iterations stop when there is no change in the decisions. This is a
suboptimum detector that does not converge to the jointly
optimum multiuser detector.
Successive Interference Cancellation
Multistage interference cancellation (MIC)
Performance Characteristics of Detectors
Bit error probability is generally the desirable performance
measure in multiuser communications.
The probability of a bit error for a single-user receiver for the
optimum detector is extremely difficult to evaluate. We use the
average probability between a lower and an upper bound.
For synchronous communications (p. 1050)
Also it is shown that
Performance Characteristics of Detectors
Bit error probability is generally the desirable performance
measure in multiuser communications.
The probability of a bit error for a single-user receiver for the
optimum detector is extremely difficult to evaluate. We use the
average probability between a lower and an upper bound.
For asynchronous communications
Where 𝜎𝑘
2
is the variance of the noise
Performance Characteristics of Detectors
Asymptotic efficiencies of optimum (Viterbi) detector,
conventional detector, MMSE detector and linear ML detector in
a two-user synchronous DS/SSMA system.
Asymptotic efficiency represents the performance loss due
to multiuser interference.
Multiuser MIMO Systems for Broadcast Channels
• Linear Precoding of the Transmitted Signals
• Nonlinear Vector Precoding
Precoding is a generalization of beamforming to support
multi-stream (or multi-layer) transmission in multi-antenna
wireless communications.
The communication system configuration is
Hence, the received signal vector is
where 𝑯 is the channel and 𝑨𝑇 is the precoding matrix.
Linear Precoding of the Transmitted Signals
There are two linear precoders; one based on the zero-
forcing and one on the MMSE criteria.
Linear Precoding of the Transmitted Signals
In zero-forcing the precoding matrix 𝑨𝑇 is set to the inverse of the
channel 𝑯−1.
However, the major drawback with zero-forcing is when the
channel matrix 𝑯 is ill-conditioned. Then the power 𝑯−1 2 ≫ 1.
How do we solve this?
Here
Linear Precoding of the Transmitted Signals
By using the linear MSE criterion!
Here the precoding matrix is selected as 𝑨𝑇 ≈ 𝑯−1 so that
𝑨𝑇
2 ≪ 𝑯−1 2.
Such a precoding matrix can be found by minimizing the cost
function
The solution to the MMSE criterion is the precoding matrix
𝑨𝑻
𝟐
≪ 𝑯−𝟏 𝟐
Linear Precoding of the Transmitted Signals
Performance of zero-forcing (left) vs MMSE (right) linear
precodings with 𝑁𝑇 = 𝐾 = 4, 6, 10. Probability of symbol error
increases as the number of users 𝐾 increases (left) / decreases
(right).
Linear Precoding of the Transmitted Signals
Comparison of the sum capacity for the linear precoder as a
function of the number of users 𝐾 for an 𝑆𝑁𝑅 = 10dB.
Nonlinear Vector Precoding
We consider a modification of the zero-forcing precoder. Each
element of the data vector 𝒔 is offset by some wisely chosen
integer such as 𝒔’ = 𝒔 + 𝜏𝒑, as seen in the figure below.
𝜏 is an integer selected so that it results in a symmetric decoding
region around the real and imaginary components of every signal
constellation symbol. Such a choice is
𝜏 = 2 𝑠𝑘 𝑚𝑎𝑥 + Δ
where 2 𝑠𝑘 𝑚𝑎𝑥 is the signal contellation symbol with the largest
magnitude and Δ is the distance between adjacent symbols.
Nonlinear Vector Precoding
Now we select a vector 𝒑 so that it minimizes the power in the
transmitted signal
𝒑 = arg min
p
𝛼𝑯−1 𝒔 + 𝜏𝒑 2
4PAM Example
Transmit 𝒙 = 𝑯−1(𝒔 + 8𝒑)
Receive 𝒚 = 𝒔 + 8𝒑
𝒔 can now be obtained by 𝐬 = 𝒚 𝑚𝑜𝑑 8 since 𝒑
is a vector of integers.

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chpater16.pptxMMMMMMMMMMMMMMMMMMMMMMMMMMM

  • 2. Introduction to Multiple Access Techniques Many transmitters to one receiver One transmitter to many Receivers (broadcast) Different kinds of multiuser systems
  • 3. Introduction to Multiple Access Techniques Store-and-forward network Two-way communication system More kinds of multiuser systems
  • 4. Introduction to Multiple Access Techniques How do we add multiple users? • Divide them in frequency (FDMA) • Divide them in time (TDMA) Problems with bursty access methods? Yes! • Periods of no transmission! • Therefore, FDMA and TDMA tend to be inefficient But, what does that mean?
  • 5. Introduction to Multiple Access Techniques How do we solve FDMA and TDMA problems? • By use of Direct-sequence spread spectrum signals (DSSS) • Each user is assigned a unique code sequence (pref. orthogonal) • Separated at the receiver by cross correlation • Sequences must have small cross correlations to minimize crosstalk • Completely overlap in time and frequency • This multiple access method is called Code Division Multiple Access (CDMA) • Some times called Spread Spectrum Multiple Access (SSMA)
  • 6. Introduction to Multiple Access Techniques Alternative nonspread random access methods to CDMA Transmissions collide Loss of data Protocols for retransmission
  • 7. Capacity of Multiple Access Methods Let’s refresh out memory about capacity • Ideal band-limited AWGN channel • 𝐾 users • Each user has an average power 𝑃𝑖 = 𝑃 for 1 ≤ 𝑖 ≤ 𝐾 • 𝑁0 2 is the PSD of the additive noise
  • 8. Capacity of Multiple Access Methods In FDMA each user has bandwidth 𝑊 𝐾 , so capacity per user is And the total capacity is Therefore, the total capacity is the same as for a single user with average power 𝑃𝑎𝑣 = 𝐾𝑃 What about FDMA capacity?
  • 9. Capacity of Multiple Access Methods Interesting fact: For a fixed 𝑊 the total capacity goes to infinity as 𝐾 increases. However, each user is allocated a smaller bandwidth 𝑊 𝐾 and the capacity per user decreases. Normalized capacity as function of 𝜀𝑏 𝑁0 for FDMA Total capacity per hertz as a function of 𝜀𝑏 𝑁0 for FDMA. Given 𝐶𝑛 = 𝐾 𝐶𝑘 𝑊 ∶
  • 10. Capacity of Multiple Access Methods What about TDMA capacity? In TDMA each user transmits for 1 𝐾 through the channel of bandwidth 𝑊 with average power 𝐾𝑃. So, capacity per user is Which is the same capacity as for an FDMA system. However, it may not be possible for the transmitters to sustain a power of 𝐾𝑃 when 𝐾 is very large.
  • 11. Capacity of Multiple Access Methods What about CDMA capacity? In CDMA each user transmits a pseudorandom signal of bandwidth 𝑊 and average power 𝑃. The capacity depends the level of cooperation among the 𝐾 users. Noncooperative CDMA: Receiver for each user does not know the codes of the other users. So, capacity per user is Or normalized Normalized capacity as a function of 𝜀𝑏 𝑁0 for Noncooperative CDMA.
  • 12. Capacity of Multiple Access Methods What about CDMA capacity? In CDMA each user transmits a pseudorandom signal of bandwidth 𝑊 and average power 𝑃. The capacity depends the level of cooperation among the 𝐾 users. ”Cooperative” CDMA: 𝐾 users cooperate by transmitting their coded signals synchronously in time. This is called multiuser detection and decoding. Each user is assigned a rate 𝑅𝑖, 1 ≤ 𝑖 ≤ 𝐾, and a code book containing a set of 2 𝑛𝑅𝑖 codewords of power 𝑃. In each signal interval, each user selects an arbitrary codeword, say 𝑿𝒊, from its own code book, and all users transmit their codewords simultaneously. Conclusion: Capacity similar to that of FDMA and TDMA. Look in the book (p. 1034) for derivations!
  • 13. Capacity of Multiple Access Methods Capacity region of two-user CDMA multiple access Gaussian channel.
  • 14. Multiuser Detection in CDMA Systems • CDMA Signal and Channel Models • The Optimum Multiuser Receiver (Exponentially growing complexity) • Suboptimum Detectors (Lower computational complexity) • Successive Interference Cancellation • Other Types of Multiuser Detectors • Performance Characteristics of Detectors
  • 15. • User signature waveform: • All 𝐾 signature waveforms have energy: • Cross correlations important metrics for detector performance: • Lowpass transmitted waveform where {𝑏𝑘(𝑚)} is the information sequence of the 𝑘th user: • Composite transmitted signal for 𝐾 users where {𝜏𝑘} are transmission delays: • The received signal assuming that the transmitted signal is corrupted by AWGN: CDMA Signal and Channel Models CDMA channel shared by 𝑲 simultaneous users
  • 16. The Optimum Multiuser Receiver The optimum (maximum-likelihood) receiver is the receiver that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤ 𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻. Synchronous transmission Each user/interferer produces exactly one symbol which interferes with the desired symbol. The received signal is Compute the log-likelihood function And select the sequence {𝑏𝑘 1 , 1 ≤ 𝑘 ≤ 𝐾} that minimizes Λ b Correlations
  • 17. The Optimum Multiuser Receiver The optimum (maximum-likelihood) receiver is the receiver that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤ 𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻. Synchronous transmission
  • 18. The Optimum Multiuser Receiver The optimum (maximum-likelihood) receiver is the receiver that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤ 𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻. Asynchronous transmission In this case, there are exactly two consecutive symbols from each interferer that overlap a desired symbol. The log-likelihood function may be expressed in terms of a correlation metric where
  • 19. Y U SO COMPLEX? The Optimum Multiuser Receiver The optimum (maximum-likelihood) receiver is the receiver that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤ 𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻. Asynchronous transmission In this case, there are exactly two consecutive symbols from each interferer that overlap a desired symbol. The log-likelihood function may be expressed in terms of a correlation metric where
  • 20. The Optimum Multiuser Receiver The optimum (maximum-likelihood) receiver is the receiver that selects the most probable sequence of bits {𝒃𝒌 𝒏 , 𝟏 ≤ 𝒏 ≤ 𝑵, 𝟏 ≤ 𝒌 ≤ 𝑲} given the received signal 𝒓(𝒕) is observed over the time interval 𝟎 ≤ 𝒕 ≤ 𝑵𝑻 + 𝟐𝑻. Block processing approach (Brute force) Compute 2𝑁𝐾 correlation metrics Sequence estimation employing the Viterbi algorithm Feasible only when 𝐾 < 10 Use suboptimum detectors whose complexity grows linearly with 𝐾!
  • 21. Suboptimum Detectors Suboptimum detectors have computational complexity that grown grows linearly with the number of users 𝐾, in comparison to the optimal multiuser receiver, whose complexity grows exponentially with 𝐾.
  • 22. In conventional single-user detection, the receiver consists of a demodulator that correlates (match-filters) the received signal with the signature sequence of the user and passes the output to the detector, which makes a decision. It ignores other users and assumes the noise plus interference to be white and Gaussian. In synchronous transmissions, if the signature squences are orthogonal The interference from other users given by the middle term vanishes. Suboptimum Detectors Conventional single-user detector
  • 23. If one or more signature sequences are not orthogonal, the interference from other users can become excesive depending on the received signal energies. This is called the near-far problem and necessitates some power of power control. In asynchronous transmission, the conventional detector is more vulnerable to interference. This is because it is not possible to design signature sequences that are orthogonal. Suboptimum Detectors Conventional single-user detector
  • 24. It has linear computational complexity, like the conventional detector, but does not suffer from other user interference. The results of the minimization of the likelihood function yields Then the detected symbols are the sign of each element 𝒃𝐾 0 Suboptimum Detectors Decorrelating detector (ML)
  • 25. The transformation 𝑹𝑠 −1 eliminates the interference. Therefore the near-far problem is eliminated. It is called the decorrelating detector since the received signal is correlated with modified signature waveforms. Thus, we have tuned out or decorrelated the multiuser interference. For asynchronous transmission the minimization yields Which is also a decorrelating detector, since 𝒃0 in an unbiased estimate of 𝒃 and the multiuser interference has been eliminated. Suboptimum Detectors Decorrelating detector (ML)
  • 26. Another solution is obtained if we seek the linear transformation 𝒃0 = 𝑨𝒓, where the matrix 𝑨 is to be determined so as to minimize the mean square error (MSE). For synchronous communication And asynchronous communication where the output is again 𝒃 = sgn(𝐛0 ) Suboptimum Detectors Minimum mean-square-error (MMSE) detector
  • 27. Successive Interference Cancellation Successive Interference Cancellation (SIC) is another multiuser detection technique. The interfering signal waveforms are removed one at a time as they are detected. The user with the strongest received signal is demodulated and detected first. Then this information is used to substract the signal of the particular user from the received signal. For synchronous transmission Which is a suboptimum detector. The jointly optimum interference canceller may be defined as
  • 28. This is a technique that employs multiple iterations in detecting the user bits and cancelling the interference. Example with two users and synchronous transmission: First stage (decorrelating detector): Second stage: Third stage: Iterations stop when there is no change in the decisions. This is a suboptimum detector that does not converge to the jointly optimum multiuser detector. Successive Interference Cancellation Multistage interference cancellation (MIC)
  • 29. Performance Characteristics of Detectors Bit error probability is generally the desirable performance measure in multiuser communications. The probability of a bit error for a single-user receiver for the optimum detector is extremely difficult to evaluate. We use the average probability between a lower and an upper bound. For synchronous communications (p. 1050) Also it is shown that
  • 30. Performance Characteristics of Detectors Bit error probability is generally the desirable performance measure in multiuser communications. The probability of a bit error for a single-user receiver for the optimum detector is extremely difficult to evaluate. We use the average probability between a lower and an upper bound. For asynchronous communications Where 𝜎𝑘 2 is the variance of the noise
  • 31. Performance Characteristics of Detectors Asymptotic efficiencies of optimum (Viterbi) detector, conventional detector, MMSE detector and linear ML detector in a two-user synchronous DS/SSMA system. Asymptotic efficiency represents the performance loss due to multiuser interference.
  • 32. Multiuser MIMO Systems for Broadcast Channels • Linear Precoding of the Transmitted Signals • Nonlinear Vector Precoding Precoding is a generalization of beamforming to support multi-stream (or multi-layer) transmission in multi-antenna wireless communications.
  • 33. The communication system configuration is Hence, the received signal vector is where 𝑯 is the channel and 𝑨𝑇 is the precoding matrix. Linear Precoding of the Transmitted Signals There are two linear precoders; one based on the zero- forcing and one on the MMSE criteria.
  • 34. Linear Precoding of the Transmitted Signals In zero-forcing the precoding matrix 𝑨𝑇 is set to the inverse of the channel 𝑯−1. However, the major drawback with zero-forcing is when the channel matrix 𝑯 is ill-conditioned. Then the power 𝑯−1 2 ≫ 1. How do we solve this? Here
  • 35. Linear Precoding of the Transmitted Signals By using the linear MSE criterion! Here the precoding matrix is selected as 𝑨𝑇 ≈ 𝑯−1 so that 𝑨𝑇 2 ≪ 𝑯−1 2. Such a precoding matrix can be found by minimizing the cost function The solution to the MMSE criterion is the precoding matrix 𝑨𝑻 𝟐 ≪ 𝑯−𝟏 𝟐
  • 36. Linear Precoding of the Transmitted Signals Performance of zero-forcing (left) vs MMSE (right) linear precodings with 𝑁𝑇 = 𝐾 = 4, 6, 10. Probability of symbol error increases as the number of users 𝐾 increases (left) / decreases (right).
  • 37. Linear Precoding of the Transmitted Signals Comparison of the sum capacity for the linear precoder as a function of the number of users 𝐾 for an 𝑆𝑁𝑅 = 10dB.
  • 38. Nonlinear Vector Precoding We consider a modification of the zero-forcing precoder. Each element of the data vector 𝒔 is offset by some wisely chosen integer such as 𝒔’ = 𝒔 + 𝜏𝒑, as seen in the figure below. 𝜏 is an integer selected so that it results in a symmetric decoding region around the real and imaginary components of every signal constellation symbol. Such a choice is 𝜏 = 2 𝑠𝑘 𝑚𝑎𝑥 + Δ where 2 𝑠𝑘 𝑚𝑎𝑥 is the signal contellation symbol with the largest magnitude and Δ is the distance between adjacent symbols.
  • 39. Nonlinear Vector Precoding Now we select a vector 𝒑 so that it minimizes the power in the transmitted signal 𝒑 = arg min p 𝛼𝑯−1 𝒔 + 𝜏𝒑 2 4PAM Example Transmit 𝒙 = 𝑯−1(𝒔 + 8𝒑) Receive 𝒚 = 𝒔 + 8𝒑 𝒔 can now be obtained by 𝐬 = 𝒚 𝑚𝑜𝑑 8 since 𝒑 is a vector of integers.