1. Advanced Wireless Networks
Lecture 5: : Diversity Techniques in Multi-Carrier System
Here, as in Lecture 4, instead of one carrier, we consider N carriers.
This allows us to have the following advantages. Thus, as was shown in previous
lecture, we define for the channel with flat slow or fast fading the average SNR as:
Each branch has an instantaneous SNR=
and then probability of error .
The probability that a single branch has SNR less than some threshold is the
cumulative distribution function defined as:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

2. Diversity Techniques in Multi-Carrier System
Now, the probability that all M independent diversity branches receive signals which are simultaneously less than some specific SNR threshold is:
If any branch with number I achieve SNR>
then the probability of that is
The PDF is found as derivative of CDF of all braches to achieve threshold, i.e.,
Then, the mean SNR
can be defined as
and
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

3. Diversity Techniques in Multi-Carrier System
Example: Consider four branches, which each branch is independent Rayleigh fading channel. The average SNR
Find: The probability that the SNR will drop below threshold
Solution:
We see that
Then using above formulas, we get for four independent branches:
When diversity is not used, this formula can be evaluated using M=1, i.e.,
It is clear to see that without diversity the SNR drops below the specified threshold of 10 dB with a probability that is three order of magnitude greater than if four branches diversity is used.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

4. Diversity Techniques in Multi-Carrier System
Frequency Diversity
Why OFDM?
Frequency diversity transmits information on more than one carrier frequency. Here frequencies are separated by more than coherent bandwidth of the channel and, therefore, will not experience the same fades.
This means that the channel bandwidth (coherence bandwidth) of each sub-channel denoted as bw, will be much smaller than
, that is,
By use of N carriers, we finally have:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

5. Frequency Diversity – OFDM (Cont.)
In Figure is shown example on how we can split the channel bandwidth
on N sub-channels with bandwidth bw very narrow to exclude effects of deep
fading.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

6. Frequency Diversity – OFDM (Cont.)
Then, each independent sub-signal will have in frequency domain a rectangular shape of power spectral density (PSD) which in frequency domain has sharp
presentation f .
PSD
And conversely, rectangular shaping function in time domain (a pulse),
has in frequency domain a shape of ideal Nyquist filter
time
shaping function
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

7. Frequency Diversity – OFDM (Cont.)
We now assume that the transmitted signal passing the fading channel
consisting N sub-carriers (e.g., N paths) can be generally presented
via the attenuation factor for signal received in nth sub-channel
and its own independent phase, , i.e.,
where
The sub-carriers of OFDM are orthogonal on the interval , from
which is followed that
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

8. Frequency Diversity – OFDM (Cont.)
The corresponding splitting allows us to obtain a signal for the each carrier in
the following manner:
where
and is a set of constellation’s points containing L points .
and
OFDM implementation is based on direct and inverse fast Fourier transform
(denoted FFT and IFFT, respectively), which we will described below.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

9. Frequency Diversity – OFDM (Cont.)
The corresponding discrete-form presentation of IFFT/FFT algorithm is the following:
where
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

10. Frequency Diversity – OFDM (Cont.)
Here we denoted the following parameters:
Finally, we get:
or schematically:
Then, at the receiver just FFT on sampled is used.
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

11. Time Diversity
One modern implementation of time diversity involves the use of RAKE receiver,
working as n-delay line through which the received signal is passed.
Its action is somewhat analogous to an ordinary garden rake, and consequently,
the name "RAKE receiver" has been used for this receiver structure by Price
and Green in 1958.
The RAKE receiver collects the time-shifted versions of the original signal by provide a
separate correlation receiver for each of the multipath signals. Usually for CDMA,
a RAKE receiver utilizes multiple correlators to separately detect the M multipath
components with deep fading (i.e., strongest multipath components). .
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

12. Time Diversity – OFDM (Cont.)
The outputs of the M correlators are denoted as
They are weighted by , respectively.
The weighting coefficients are based on the power or the SNR from each correlator
output. As in the combining diversity scheme, the overall signal is given
The weighting coefficients are normalized to the output signal power of the correlator
in such a way that the coefficients sum limits to unity, shown by the following formula:
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

13. Time Diversity – OFDM (Cont.)
Choosing weighting coefficients based on the actual outputs of the correlators yields better RAKE receiver performance. This performance gives a conditional error probability in the form of
where
for the orthogonal signals, and
for the antipodal signals.
Here
For the Rayleigh flat fading channel, we get probability for instantiations SNR .
where, as above,
is an average SNR for the kth path (kth
sub-channel).
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

14. Spatial Diversity
One modern implementation of time diversity involves the use of RAKE receiver,
working as n-delay line through which the received signal is passed.
Its action is somewhat analogous to an ordinary garden rake, and consequently,
the name "RAKE receiver" has been used for this receiver structure by Price
and Green in 1958.
The RAKE receiver collects the time-shifted versions of the original signal by provide a
separate correlation receiver for each of the multipath signals. Usually for CDMA,
a RAKE receiver utilizes multiple correlators to separately detect the M multipath
components with deep fading (i.e., strongest multipath components). .
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

15. MIMO Systems for Spatial Diversity
Propagation Channel M N
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

16. QOS Performance for MIMO
Systems
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

17. Signal Data Parameters in the MIMO
Systems
Shannon Formulation
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

18. K-Parameter of Fading Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

19. QOS Performance for MIMO
Systems
Bit Error Rate
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

20. Bit Error Rate (Cont.) Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.

21. QOS Performance for MIMO Systems – Numerical Results
Lectures 1 & 2: Overview
Adv. Wireless Comm. Sys.