All the mechanisms of directional listening are missing when we use headphones. The sound at the right ear will no longer reach the left ear and pinnae-reflections no longer interfere with the original soundwave. Moreover, the headphones are directly attached to our head, and so head movements no longer add information. Reverberation is also not present.

As a result, the sound heard by headphones seems to stick to the inside of our head and to our ears and an unnatural soundfield is created. The brain misses logical clues for direction and this subconsciously results in mental stress. Some people cannot tolerate this stress and are unable to use headphones.

In principle, digital soundprocessors can simulate all the mechanisms for directional listening but the results are, thus far, not very satisfactory. In particular, pinnae-reflections are very complex and listener-specific and impossible to simulate accurately.

Fortunately, the mean directional information is provided by the time delay and level of attenuation of the sounds that reaches the opposite ear. A crossfeed filter can electronically simulate this process and, with appropriate attenuation and delay, add some of the right audiosignal to the left channel and vice -versa. This considerably reduces the adverse symptoms of headphone listening.
In normal daily life people use various mechanisms to locate sources of sound.

Firstly, the sound of a source to the right side of the listener (e.g. the right loudspeaker) not only reaches the right ear but, attenuated and delayed, is also heard by the left ear. The level of attenuation and the delay time of this crossfeed signal provide important directional information.

Secondly, the soundwaves are partly absorbed and partly reflected by the tissues of the head. Reflections at the oracles (pinnae) interfere with the soundwaves that directly enter the ear-channel and amplify or attenuate specific frequency components. Since these reflections depend on the direction of the soundwave the "color" of the sound changes with the direction of the source.

Thirdly, reflections of the soundwaves from the walls, ceiling and floor of our listening room produce reverberation that conveys an extra feeling of space.

The information obtained by these mechanisms is further refined by movements of the head. Changes in sound levels, delay times and sound color refine the sense of direction. For a demonstration, blindfold a friend and ask him to locate a ticking clock that you have hidden in the room. He will start turning his head although he can't see anything. With his head in a fixed position an exact localization is much more difficult.
A straightforward approach to mimic crossfeed is to take the original stereo signal, attenuate its amplitude and have it delayed. Then cross the two channels and add the processed signals to the original stereo signal. In a mathematical formula:

Vleft,out (t) = Vleft,in (t) + a.Vright,in (t-tdelay)

Vright,out (t) = Vright,in (t) + a.Vleft,in (t-tdelay)              a < 1

The factor a normally is made frequency dependant. Crossfeed by its nature is stronger at low frequencies than at high frequencies.

A conventional crossfeed filter mimics the sound of a left or a right sound source most adequately, but the frequency-spectrum of a source in front of the listener is unnecessarily disturbed. For this in-front source, the left and right audio signals are equal: a mono signal. In principle, these signals need no crossfeed. However, with conventional solutions, there still is, and the audio signals at the headphone-transducers become:

Vleft,out (t) = Vright,out (t) = Vleft,in (t) + a.Vleft,in (t-tdelay)       a < 1

Especially in the high frequency range, the delayed crossfeed signal interferes with the original input and attenuates specific frequencies. The frequency-curve is no longer flat but shows a larger number of dips. This is the so-called Comb-filter effect.

A unique feature of the crossfeed circuitry on the CORDA headphone amplifiers is that it "recognizes" the virtual positions of the instruments and singers in a recording. The sound of an instrument in the middle of the soundstage will be equally present in both audio-channels and isn't given any crossfeed. A crossfeed signal is only generated for instruments that are not placed at the center. The more off-center the instrument is placed, the stronger the crossfeed and the longer its delay. The frequency-curve is flat again and the Comb-filter effect is eliminated. This is called "natural crossfeed".
Listening by headphones
CROSSFEED
The natural crossfeed filter
Various versions of the natural crossfeed filter
The original (standard) version is based on the small resistor-capacitor-network shown in the figure to the right. It can be easily recognized that the left channel input signal will also be seen at the right channel output and vice versa. A mono signal will pass unaltered and without any delay to both outputs. This version is found on CORDA amplifiers designed/built till around 2005. Crossfeed only is given for signal components with frequencies upto 1 kHz.

With crossfeed activated lower frequency signals are no longer present in one channel only but are now more evenly distributed over both channels. They are less isolated and become a more integrated part of the soundstage. They no longer stand out and this may feel as if the energy in the frequency range below 1 kHz is slightly reduced. From 2005 till now a second slighly modified filter is used that automatically compensates for this apparent bass losss.

Psychoacoustic studies have shown that our sense of direction is mainly determined by the sonic components with frequencies upto 2 kHz. However, with a simple passive network natural delay times only can be achieved for frequencies upto 1 kHz. In recent years therefore a technically more sophisticated filter was designed that allows crossfeed with appropriate delay times with signal components upto 3 kHz. This extended crossfeed filter can be found in the CLASSIC and in the (now discontinued) STAGEDAC.
The pictures below compare the amplitudes and the delay times of the crossfeed signals for both the standard crossfeed filter and the extended crossfeed filter. The crossfeed signal of the standard filter reaches -6dB at 1,1 kHz at which frequency its delay time has reduced from 320 to 160 microseconds. The crossfeed signal of the extended filter reaches -6dB at 2,9 kHz and at this point the effective delay time has decreased from 300 to 250 microseconds only.

The result of the extended crossfeed filter is a better, more precise positioning of the various sources in the acoustic soundstage. It should be noted though, that with most recordings the effect of crossfeed is very subtle and that the differences between the two crossfeed versions only can be noticed with recordings that have an extreme left-right-panning.
Sample files
Theory is a nice thing, but more important is how crossfeed sounds in real life. Therefore below you can download a few audio samples, recorded with and without crossfeed. Hear and compare for yourself.

Note: All recordings are made with the use of the CORDA DACCORD, which offers three crossfeed settings: stereo, medium, high.
This sample offers you a voice that is first heard in the left and in the right channels consecutively. Through headphones this feels rather awkward. There is an unnatural "pressure" on the ear and the positioning of the narrator is not very accurate. Next the voice comes as a mono-signal from both channels and is clearly placed at the middle of the soundstage.
The second sample shows the effect of the extended natural crossfeed filter at medium level. The "pressure" on the ear is strongly reduced and positioning of the narrator is more accurate. Also note that the "color" of the voice placed at the centre does not change when compared to the recording without crossfeed. The natural crossfeed filter only effects signals placed outside the middle of the soundstage!
The third sample is made with the crossfeed set to a high level. The "pressure" on the ear is eliminated and the voices are well positioned. Again the "color" of the voice at the centre has not changed. Compared to the stereo-recording the off-centre voices may sound a little bit light , but that's only because the low frequency components no longer stick out but are naturally integrated into the soundstage.
Especially early stereo-recordings often have an extreme left-right panning of the musicians which makes it very difficult to listen by headphones. The samples above demonstrate well how crossfeed can alleviate the problems with these recordings.
But also some modern stereo-recordings can have extreme placement of the musicians/instruments and are not easy to listen to by headphones. Please have a listen to the samples below.
This recording presents a number of drums in a wide soundstage. It is easy to give the central instruments an exact position but this is hardly possible for the left-most and right-most instruments. They seem to have a wide "floating" body and produce "pressure" on the ears.
With the natural crossfeed filter set to its high level soundstage has changed considerably. It now is possible to position each instrument more precisely. Pressure on the ears (which easily results in listening fatigue) is eliminated. Listening has become more relaxed.
Note: To prevent the crossfeed to make the overall sound a little bit light "tonal control" on the STAGEDAC was activated.
http://www.freestockmusic.com/2013/international-production-music/free-african-stock-music-african-safari-drums
Most modern stereo-recordings normally have less extreme placement of the musicians and are more easy to listen to by headphones. The effects of a crossfeed filter becomes more subtle and sometimes very hard to hear. However, even then crossfeed does have an advantage. The spatial clues are offered more naturally which is well picked up subconsciously. Listening becomes more relaxed and listening fatigue is reduced. Please have a listen to the samples below. Notice how especially the intro becomes more listenable with crossfeed engaged.
This recording by Gilberto Gil and an ensemble of instruments and vocalists can well be enjoyed by headphones. However, some instruments, especially at the intro, are placed wide outside and do sound somewhat awkward. They're not very easy to the ear.
http://ccmixter.org/files/wired/83
Again, setting the natural crossfeed filter to its high level changes soundstage noticably. The intruments at the intro become more logically integrated into the soundstage. The crossfeed may make overall sound a little bit "lighter" but it was decided to keep the "tonal control" de-activated in order not to alter the color of the prominent central voices and instruments.