The zero-forcing equalizer is a form of linear
equalization algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
used in
communication systems
A communications system or communication system is a collection of individual telecommunications networks, transmission systems, relay stations, tributary stations, and terminal equipment usually capable of interconnection and interoperatio ...
which applies the inverse of the
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
of the channel. This form of equalizer was first proposed by
Robert Lucky
Robert Wendell Lucky (January 9, 1936 – March 10, 2022) was an electrical engineer, inventor, and research manager at Bell Labs and Bell Communications Research (Bellcore). He is best known for his writings and speeches about technology, soci ...
.
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
IEEE 802.11n (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
intersymbol interference
In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have a similar effect as noise, thus making ...
(ISI) to zero in a noise-free case. This will be useful when ISI is significant compared to noise.
For a channel with
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
the zero-forcing equalizer
is constructed by
. Thus the combination of channel and equalizer gives a flat frequency response and linear phase
.
In reality, zero-forcing equalization does not work in most applications, for the following reasons:
# Even though the channel impulse response has finite length, the impulse response of the equalizer needs to be infinitely long
# 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).
This second item is often the more limiting condition. These problems are addressed in the linear
MMSE equalizer
by making a small modification to the denominator of
:
, where k is related to the channel response and the signal
SNR.
Algorithm
If the channel response (or
channel transfer function) for a particular channel is H(s) then the input signal is multiplied by the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of it. This is intended to remove the effect of channel from the received signal, in particular the
intersymbol interference
In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have a similar effect as noise, thus making ...
(ISI).
The zero-forcing equalizer removes all ISI, and is ideal when the channel is noiseless. However, when the channel is noisy, the zero-forcing equalizer will amplify the noise greatly at frequencies ''f'' where the channel response H(j2π''f'') has a small magnitude (i.e. near zeroes of the channel) in the attempt to invert the channel completely. A more balanced linear equalizer in this case is the
minimum mean-square error
In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In ...
equalizer, which does not usually eliminate ISI completely but instead minimizes the total power of the noise and ISI components in the output.
References
{{Reflist
Filter theory