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Some radars have thousands of antennas, there are also some routers that have multiple antenna systems.
But I want to know how to calculate total gain of multiple antennas in a system? Is it simply equal to the number of antennas times gain of one antenna?
What about beamwidth of the resulting system? If for example I use 64 antennas in a system, all are omni directional, is the total system still omni?
I've never heard of radars using thousands of antennas before but the reasons for antenna diversity is well known. Several antennas can increase the probability that, no-matter what direction a transmission is arriving, and no-matter how that transmission is polarized, it will be received correctly.
This usually means, for each antenna, an amplifier and demodulator i.e. it can be quite complex and there is no apparent multiplication of antenna gain. In fact, depending on the mechanical configuration of the antenns, there can be a reduction in gain for each antenna due to the proximity of the other antennas.
The radiation pattern of an antenna array depends not only on their geometrical arrangement, but also on the phase of the signal that each antenna is fed with. An omnidirectional antenna has a radiation pattern of cos(x) on a polar plot (try out the query "polar plot of cos(x)" in WolframAlpha to see this -- I don't have enough reputation to post more than 2 links in my answer).
Electromagnetic signals "add up". If we have two isotropic (omni-directional) antennas at exactly the same place, fed with exactly the same signal, the the radiation pattern will be cos(x) + cos(x). It is intuitive to see that this the signal is now twice as strong as before. If we take that same antenna system, but we feed the antennas with signals that are 180 degrees (i.e. pi radians) out of phase, we get no radiation. Try "polar plot of (cos(x)) + (cos(x+3.14159))" in WolframAlpha to see this.
Let's illustrate the effect of geometry with a basic example: two isotropic (omni-directional) antennas, fed with an identical signal, that are spaced a half-wavelength apart (λ/2). If we stand on top of one of the antennas, our signals will be cos(x) from the antenna we are standing on top of, but we will be a half-wavelength away (or pi radians) from the second antenna. The signals will cancel out. Anywhere around the antenna system where our distance from each antenna differs by a half-wavelength will have zero signal strength. In the diagram below, the antennas are labelled with an X, and you can see that there is a "null", or point of zero radiation, along the imaginary line that runs through both antennas.
On the other hand, if we stand exactly in between both antennas and move to the right or left, the signals coming from each antenna is exactly the same, and they "add up" to create a signal that is twice as strong.
But I want to know how to calculate total gain of multiple antennas in a system? Is it simply equal to the number of antennas times gain of one antenna?
Gain is a directional parameter. It is different at every angle. When we say gain without specifying a direction, we are referring to the gain in the direction of maximum radiation. The maximum gain of the multiple antenna system will depend on the geometry and phasing of the system. In the example above, the maximum gain of the antenna array is twice what each antenna was on its own.
What about beamwidth of the resulting system? If for example I use 64 antennas in a system, all are omni directional, is the total system still omni?
Maybe, but probably not. You may be able to design an omnidirectional radiation pattern, but again, everything depends on the array geometry and phasing.
My examples above deal with isotropic (omni-directional) antennas. Antennas with more complex radiation patterns will be hard to analyze by hand. A common method used is "pattern multiplication". Check out this page for examples and more information.
This analysis method ignores mutual coupling between antennas. A simulation tool such as FEKO or GNEC is the best approach.
Multiple antennas connected in parallel are just a single antenna with a funny shape, so you'd determine gain and radiation pattern for the entire system by analyzing it as a single entity, including mutual coupling.
Multi-antenna systems in wireless routers are typically separate radios operating on the same frequency. Reception uses both radios in parallel, and the same packet will arrive on both at once, but signal strength is going to be different due to the antennas' different directionality. For weak signals, the packet might be received on one side only, which is still sufficient, while a single-radio single-antenna system would have dropped the packet. Sending uses the antenna where the greatest signal strength was seen for the receiving system, under the assumption that the channel gain is similar in reverse.
Multi-antenna systems in radar and mobile communication are used to create directionality by interference patterns. Here also, the antennas themselves are completely separate, so no "summary gain" can be calculated.
Rotating radar systems (with a highly directional antenna) have a limited reception window, and are limited in speed by the size of the window and the maximum distance of recognized objects, as returns must be received before the window has closed; in contrast, with a radar array the return can be received at any time and the direction it arrived from can be inferred by timing differences.
The same methods are used in mobile networks to increase the capacity by creating directionality via Beamforming and MIMO.
