Zero-forcing equalizer

The zero-forcing equalizer is a form of linear <a href="/facts/Equalization_(communications)/44Mc0WXw">equalization</a> <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> used in <a href="/facts/Telecommunications/vxoj1cxR">communication systems</a> which applies the inverse of the <a href="/facts/Frequency_response/zsHmnCJ4">frequency response</a> of the channel. This form of equalizer was first proposed by <a href="/facts/Robert_Lucky/eSmDSa6q">Robert Lucky</a>.
The zero-forcing equalizer applies the inverse of the channel frequency response to the received signal, to restore the signal after the channel. It has many useful applications. For example, it is studied heavily for <a href="/facts/IEEE_802.11n/3gXmkz43">IEEE 802.11n</a> (MIMO) where knowing the channel allows recovery of the two or more streams which will be received on top of each other on each antenna. The name zero-forcing corresponds to bringing down the <a href="/facts/Intersymbol_interference/6oGRSogt">intersymbol interference</a> (ISI) to zero in a noise-free case. This will be useful when ISI is significant compared to noise.
For a channel with <a href="/facts/Frequency_response/zsHmnCJ4">frequency response</a> 
 
 
 
 F
 (
 f
 )
 
 
 {\displaystyle F(f)}
 
 the zero-forcing equalizer 
 
 
 
 C
 (
 f
 )
 
 
 {\displaystyle C(f)}
 
 is constructed by 
 
 
 
 C
 (
 f
 )
 =
 1
 
 /
 
 F
 (
 f
 )
 
 
 {\displaystyle C(f)=1/F(f)}
 
. Thus the combination of channel and equalizer gives a flat frequency response and linear phase 
 
 
 
 F
 (
 f
 )
 C
 (
 f
 )
 =
 1
 
 
 {\displaystyle F(f)C(f)=1}
 
.
In reality, zero-forcing equalization does not work in most applications, for the following reasons:

<ol><li>Even though the channel impulse response has finite length, the impulse response of the equalizer needs to be infinitely long</li>
<li>At some frequencies the received signal may be weak. To compensate, the magnitude of the zero-forcing filter ("gain") grows very large. As a consequence, any noise added after the channel gets boosted by a large factor and destroys the overall signal-to-noise ratio. Furthermore, the channel may have zeros in its frequency response that cannot be inverted at all. (Gain * 0 still equals 0).</li></ol>
This second item is often the more limiting condition. These problems are addressed in the linear <a href="/facts/Minimum_mean-square_error/hI2lqEMh">MMSE</a> equalizer by making a small modification to the denominator of 
 
 
 
 C
 (
 f
 )
 
 
 {\displaystyle C(f)}
 
: 
 
 
 
 C
 (
 f
 )
 =
 1
 
 /
 
 (
 F
 (
 f
 )
 +
 k
 )
 
 
 {\displaystyle C(f)=1/(F(f)+k)}
 
, where k is related to the channel response and the signal <a href="/facts/Signal-to-noise_ratio/qohClhyG">SNR</a>.

Zero-forcing equalizer open-in-new

Zero-forcing equalizer