Zero Forcing Equalizer
Encyclopedia
Zero Forcing Equalizer refers to a form of linear equalization
algorithm
used in communication systems which inverts the frequency response
of the channel. This form of equalizer was first proposed by Robert Lucky
.
The Zero-Forcing Equalizer applies the inverse of the channel to the received signal, to restore the signal before 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
(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
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:
This second item is often the more limiting condition.
) for a particular channel is H(s) then the input signal is multiplied by the reciprocal
of it. This is intended to remove the effect of channel from the received signal, in particular the intersymbol interference
(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(j2πf) 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
equalizer, which does not usually eliminate ISI completely but instead minimizes the total power of the noise and ISI components in the output.
Equalization
Equalization, is the process of adjusting the balance between frequency components within an electronic signal. The most well known use of equalization is in sound recording and reproduction but there are many other applications in electronics and telecommunications. The circuit or equipment used...
algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
used in communication systems which inverts the frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
of the channel. This form of equalizer was first proposed by Robert Lucky
Robert Lucky
Robert Wendell Lucky is an electrical engineer, inventor, and research manager who worked at Bell Labs and Telcordia Technologies. He is best known for his writings and speeches about technology, society, and engineering culture. Bob is a Fellow of the IEEE and a member of the National Academy of...
.
The Zero-Forcing Equalizer applies the inverse of the channel to the received signal, to restore the signal before the channel. It has many useful applications. For example, it is studied heavily for IEEE 802.11n
IEEE 802.11n
IEEE 802.11n-2009 is an amendment to the IEEE 802.11-2007 wireless networking standard to improve network throughput over the two previous standards—802.11a and 802.11g—with a significant increase in the maximum net data rate from 54 Mbit/s to 600 Mbit/s with the use of four...
(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
Intersymbol interference
In telecommunication, intersymbol interference 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 similar effect as noise, thus making the communication less reliable...
(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
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
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 zeroes in its frequency response that cannot be inverted at all. (Gain * 0 still equals 0).
This second item is often the more limiting condition.
Algorithm
If the channel response (or channel transfer functionTransfer function
A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...
) for a particular channel is H(s) then the input signal is multiplied by the reciprocal
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
of it. This is intended to remove the effect of channel from the received signal, in particular the intersymbol interference
Intersymbol interference
In telecommunication, intersymbol interference 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 similar effect as noise, thus making the communication less reliable...
(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(j2πf) 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
Minimum mean-square error
In statistics and signal processing, a minimum mean square error estimator describes the approach which minimizes the mean square error , which is a common measure of estimator quality....
equalizer, which does not usually eliminate ISI completely but instead minimizes the total power of the noise and ISI components in the output.