there are two variations for MIMO radar: full/thin and thin/full. Full means all the array elements are operational and thin means not all the array elements are operational.
Question: what is the advantage of thin arrays? Why not always full arrays?
It is about the cost of the array antenna, in situations when is needed a highly directive antenna but with reduced gain.
The array gain will be reduced in approximate proportion to the fraction of the antenna elements removed (or not active), because the total gain of the array is related directly to the area of the illuminated aperture.
The thinned array can make it possible to build a highly directive array antenna with reduced gain, for a fraction of the cost of a filled array.
Usually the antenna elements that are turned off are connected to a matched load to not participate in forming the beam of the array.
Thinning an array always result in higher side lobes, and to minimize this effect a periodic distribution is necessary when choosing the off elements.
Statistically speaking, thinnining works well only for large array antennas (with many antenna elements), and is not justified for small array antennas (with few antenna elements).
As far as I know the MIMO radar beamforming is achieved at receive elments by means of matched filters and delay elements. Is it correct? Which part is costly?
Thinned arrays are used in MR (Minimum Redundancy) MIMO radar arrays, where the transmitter send independent beam forms. So no relation with the receive array in this situation.
In this paper, minimum redundancy multiple-input multiple-output (MR-MIMO) arrays - a class of thinned antenna arrays - are discussed. The concepts of MR-MIMO array design and almost difference sets (ADSs) are introduced, and an analytical method based on ADSs is proposed for MR-MIMO array...
Another reason for thinning: commercial 77GHz microwave MMICs have limited TX/RX channels number.
Eliminating few channels may halve number of required MMICs.
Also RX array thinning will provide more power for receive mixers, not sure if it can be used as advantage.
I don't say it not exist, but I never seen a commercial thinned array at 77GHz.
For example the automotive radars on 77GHz use series feeding linear arrays, and there is no reason of thinning.