Blind equalization

Blind equalization is a digital signal processing technique in which the transmitted signal is inferred (equalized) from the received signal, while making use only of the transmitted signal statistics. Hence, the use of the word blind in the name.

Blind equalization is essentially blind deconvolution applied to digital communications. Nonetheless, the emphasis in blind equalization is on online estimation of the equalization filter, which is the inverse of the channel impulse response, rather than the estimation of the channel impulse response itself. This is due to blind deconvolution common mode of usage in digital communications systems, as a means to extract the continuously transmitted signal from the received signal, with the channel impulse response being of secondary intrinsic importance.

The estimated equalizer is then convolved with the received signal to yield an estimation of the transmitted signal.

Problem statement

Noiseless model

Assuming a linear time invariant channel with impulse response

{

h

[

n

]

}

n

=

−

∞

∞

{\displaystyle \{h[n]\}_{n=-\infty }^{\infty }}

, the noiseless model relates the received signal

r

[

k

]

{\displaystyle r[k]}

to the transmitted signal

s

[

k

]

{\displaystyle s[k]}

via

r

[

k

]

=

∑

n

=

−

∞

∞

h

[

n

]

s

[

k

−

n

]

{\displaystyle r[k]=\sum _{n=-\infty }^{\infty }h[n]s[k-n]}

The blind equalization problem can now be formulated as follows; Given the received signal

r

[

k

]

{\displaystyle r[k]}

, find a filter

w

[

k

]

{\displaystyle w[k]}

, called an equalization filter, such that

s

^

[

k

]

=

∑

n

=

−

∞

∞

w

[

n

]

r

[

k

−

n

]

{\displaystyle {\hat {s}}[k]=\sum _{n=-\infty }^{\infty }w[n]r[k-n]}

where

s

^

{\displaystyle {\hat {s}}}

is an estimation of

s

{\displaystyle s}

.

The solution

s

^

{\displaystyle {\hat {s}}}

to the blind equalization problem is not unique. In fact, it may be determined only up to a signed scale factor and an arbitrary time delay. That is, if

{

s

~

[

n

]

,

h

~

[

n

]

}

{\displaystyle \{{\tilde {s}}[n],{\tilde {h}}[n]\}}

are estimates of the transmitted signal and channel impulse response, respectively, then

{

c

s

~

[

n

+

d

]

,

h

~

[

n

−

d

]

/

c

}

{\displaystyle \{c{\tilde {s}}[n+d],{\tilde {h}}[n-d]/c\}}

give rise to the same received signal

r

{\displaystyle r}

for any real scale factor

c

{\displaystyle c}

and integral time delay

d

{\displaystyle d}

.