Wireless Indeed: Channel Equalizers

What are channel equalizers ?

If you search the word equalizers on google, it says "a thing that has an equalizing effect" . The same applies in relation to communication theory. When the transmitted signal pass through a dispersive channel , the symbols interfere between them resulting in Information loss. We call this as ISI or inter symbol interference. Equalizers are filters that help us reduce the ISI and equalize the symbols.

Linear Equalizer:

Zero Forcing(ZF) Equalizer
Minimum Mean Square Error(MMSE) Equalizer

Decision Feedback Equalizer :

Improves the Linear Equalizers by adding a new feedback filter which uses the feedback from prior decisions to cancel the interference due to past symbols. This in turn reduce the noise enhancement as compared to ZF and MMSE Linear Equalizers.

Lets try to understand it mathematically.

we can model a linear modulated signal over a dispersive channel as below:

where,
${B_n}$ - Transmitted bits
${G_{transmitter}}$ - Transmitter filter
${G_{Channel}}$ - Channel filter
AWGN- Additive white gausian noise n(t)
Rx- Received Symbols $r[n]$

We can write the receiver model as, $r[n] = Ub[n] + W[n]$

where ,
   $r[n]$   - received symbol vector
$U$ - Vector , whose each column gives the response on each
symbol from all the filters through which it passes. mathematically
$U = {G_{transmitter}} \otimes {G_{Channel}} \otimes {G_{{{\rm Re}\nolimits} ceiver}}$ .
G-Corresponding filters Models of Transmitter, Channel and Receiver.
All the elements of U Matrix are shifted in acyclic so that it gives contribution to the corresponding transmitted vector.
B[n]- transmitted symbols Vector. $(b[n - {k_1}],.........b[n],b[n + 1],..........b[n + {k_2}])$
W[n]- AWGN noise with Power Spectral density $\sigma _w^2$

we can write received symbols r[n] as,
   $r[n] = b[n]{u_0} + \sum\limits_{j \ne 0}^n {b[n + j]{u_j}} + w[n]$
Now the idea of Linear Equalizer is to design a filter such that the second term $\sum\limits_{j \ne 0}^n {b[n + j]{u_j}}$ is significantly smaller as compared to $b[n]{u_0}$ . To understand lets expand the Model stated above,

$r[n] = \left( {\begin{array}{ccccccccccccccc} {{a_{32}}}&{{a_{11}}}&0\\ 0&{{a_{22}}}&0\\ 0&{{a_{32}}}&{{a_{11}}} \end{array}} \right)b[n] + {w_n}$
$\Rightarrow r[n] = \left( {\begin{array}{ccccccccccccccc} {{a_{32}}}&{{a_{11}}}&0\\ 0&{{a_{22}}}&0\\ 0&{{a_{32}}}&{{a_{11}}} \end{array}} \right){[b[n - 1],b[n],b[n + 1]]^T} + w[n]$
$\Rightarrow r[n] = {[{a_{32}}b[n - 1] + {a_{11}}b[n],{a_{22}}b[n],{a_{32}}b[n] + {a_{11}}b[n + 1]]^T} + w[n]$
$\Rightarrow {r_1}[n] = {a_{32}}b[n - 1] + {a_{11}}b[n] + {w_1}[n]$ , similarly ${r_2}[n]$ and ${r_3}[n]$ .

The work of equalizer filter is to give us $b[n]$   from ${r_1}[n]$   and remove the influence of $b[n - 1]$ .

ZF : In case of Zero forcing equalizer , we design the ZF filter such that the term

$\sum\limits_{j \ne 0}^n {b[n + j]{u_j}}$ = 0 and $b[n]{u_0}$ has only contribution to $r[n]$ plus the additive noise for sure.

MMSE: The above explanation shows that the ZF Equalizer ignores the effect of noise at the output, thus to have a trade off between ISI and effect of Noise is the MMSE Equalizer. We design the equalizer filter such that Mean Square Error(MSE) , $MSE = E[|{F_{MMSE}}r[n] - b[n]{|^2}]$ is minimized to get the Filter ${F_{MMSE}}$ .

Wireless Indeed

Sunday 22 March 2015

Channel Equalizers

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