Improving your Channel , Bro! use SVD / Channel Performance Factorz

The capacity can be composed by decomposing the vector channel into a set of parallel independent scalar Gaussian sub-channel.

Using SVD ::

H = U Λ V*
U and V are uniary matrix
Λ is a rectangular matrix whose diagonal elements are non-negative and the others are zero.

|λ1                    |
|    λ2                |
|        λ3            |
|             λmin  |

λi*λi is the eigen values of H.H*

u is eigen vector of H.H*
v is the eigen vector of H*.H

y = ((U Λ V*). x)+n

Let y’ = U* . y
y’ = U*. ((U Λ V*). x)+n
=  Λ V* . x + U*.n        (lets put V*. x =  x’  and  U*. n = n’)
= y’ = Λ . x’ + n’  (where Λ => (λ1 >= λ2 >= λ3 >= … >= λmin)

We can represent this operation using the parallel sub-channels ::

(yi)’ = λi . (Xi)’ + (ni)’ , (i = 1,2,3…. nmin)

X’—>(multiplier: λ1)—>(adder: n1′)—->y1′
.                     .                                 .                    .
.                     .                                 .                    .
.                     .                                 .                    .
X<nmin>’—>(multiplier: λ<nmin>)—>(adder: n<nmin>’)—->  y<nmin>’

If full CSI ,

Introduce waterfilling scheme to MIMO ::
Pi* = (μ – [N0/(λi*λi)] ) ^ t   with μ  chosen to satisfy the total power constraint

Summation(i=0 to <nmin>) Pi* = P
C = Summation(i=1 to <nmin>) log ( 1+ ( [Pi*.(λi*λi)]/N0)    //like , seriously improved channel.

—————————————————————————
So , after doing all of this – improving the capcity and all that – we also need to consider –
The parameters that determine the performance.

Ofcourse – the best place to start is with SNR (high or low)

1.High SNR – equal power allocation has almost the same performance as waterfilling. Therefore in each subchannel , the

transmit capacity (considering k = nmin) :: P / k
total C = summation(i=1:k) log(1+ P/k *   (λi*λi)/N0)

This k is basically minimum of (nt,nr)<— number of spatial degrees of freedom.
in case of high SNR , we ignore the 1 in the C equation , so we get ::
C = summation(i=1:k) log(P/k *   (λi*λi)/N0)  further simplifying…
C=  summation(i=1:k) log(P/N0) + summation(i=1:k)  log(λi*λi)/k)  …
but we know ..  log(P/N0) = SNR

SO…  C=  summation(i=1:k) log SNR(Cawgn for high SNR) + summation(i=1:k)  log(λi*λi)/k) :: HIGH SNR

So..  A MIMO channel provides Nmin spatial degrees of freedom.

If we go ahead and divide C by k , we get this whole thing where the summation pretty much turns into an expectation E of log(1 + [P.(λi*λi) / k.N0])
Then if we go ahead and use the Jensen’s Inequality (cuz log is a Concave function)

E(f(x)) < f(E{x})

C/k = log {1 +   P/k.No *  (1/k * summation(i=1 to k) (λi*λi) ) }

where … [summation(i=1 to k) (λi*λi)] = Trace [ H.H*] = summation(i,j) { |h<i,j>|*|h<i,j>| }

So , the total power gain of the matrix channel among the channels with the same total power gain , the one which has the higher capacity is the one with all the singular values equal.

The condition number of H is the ratio of => max λi : min λi => 1 cuz when all the singular values are equal , min λi = max λi = 1 . Sooo… —> H is said to be well-conditioned and also in this case we have the highest capacity.
Well-Conditioned channel matrix facilitate communication in the high SNR regime.

Phew… so much for high SNR.  What about LOW SNR ?

Using the Waterfilling scheme to allocate power to the highest (λi * λi) (that is the best channel)
Best Channel = max of (λi*λi).
Best Channel Capacity = C = log ( 1 +  [  P . max<i>(λi*λi) ] / N0)
C ~ P/N0 . max<i>(λi*λi) . log(base 2) e …..  cuz [ log (1+X) ~ X log(base 2) e     at small X )
C~ { max<i>(λi*λi) . log(base 2) e  }. SNR

Fast Fading : L. Tc (time diversity)
MIMO : Spatial Diversity
C > Cawgn
Want to read more?  : Read here

More insights to come….

This Channel

Capacity of Wireless Channel – Part One.Five

Channel has fading. That’s just the way the cookie crumbles.

 

Slow Fading – Tc(coherence time) > Delay
C = log (1 + |h|*|h|* SNR)  , R < C  or R > C

Fast Fading – Tc(coherence time) < Delay

Outage Probability – Pout(R) = Pr[ log (1 + |h|*|h|* SNR) < R] = Pr [ (|h|*|h|) < ((math.pow(2,R)-1)/SNR)

Cε (capacity for epsilon outage) = log (1+ (math.pow(F,-1)*(1-ε)*SNR) 

For High SNR – Cε ~ Cawgn – log( 1/(math.pow(F,-1)*(1-ε)))

For Low SNR  – Cε ~ Cawgn and also – Cε < Cawgn

——————————————————–
Now , if you want to improve the capacity you need to introduce diversity techniques.

1. Receive Diversity  (using h = [h1,h2,h3….]^t)
C =  log (1+ ||h||*||h|| * SNR)              where ||h||*||h|| = |h1|*|h1| + |h2|*|h2|

2.Transmit Diversity
If we don’t know the transmitter info ,use equal partition. If you know all the transmitter characteristics , use Completely correlated signals.

These are for slow fading channels. For fast fading channels …

Model the fast fading channel as L parallel sub-channels which fade independently.

 

Transmission Side –
TX { CSIR (equal power allocation)              CSI (Tx & Rx)= Full CSI}

For full CSI , power allocation scheme is :
1)Channel Inversion Scheme (Slow fading)
2)Waterfilling  (fast fading)

Channel Inversion – Very straightforward – P = (1/ |h|*|h|)     where C  : a constant
Advantage: No outage , disadvantage : channel is bad => huge power p (in-feasible)

Waterfilling scheme –
Gmax = max (over all L) {|hL|*|hL|)
C ~ Gmax * Cawgn
E{ |hL|*|hL| } = 1 & Gmax > 1
=> C > Cawgn (cuz it takes the advantage of peaks in fast fading because of the channel gain Gmax)

Waterfilling Scheme

 

More coming through..