The radiation patterns can not be combined because they are obtained assuming a time harmonic solutions to Maxwell's equations and these solutions are independent and you should use every pattern with its related frequency. But antennas can be used in different frequencies with different patterns and this is sometimes very helpful. Consider a case in which you have two receivers in different angles relative to the transmitter. You can use an antenna whose pattern main lobe is rotated by changing frequency as the transmitter, then you can set the receivers in two frequencies that correspond to the frequencies that the main lob of transmitter is directed toward them so you can use one transmitter and two receiver in different directions simultaneously.

if you ignore different frequencies, you can just compare the two radiation patterns for performance comparison of antennas.

Whenever two antennas working on different frequencies placed together there will be interference in electric and magnetic field pattern of those antennas that will be either constructive or destructive depending upon antennas frequencies and distances so its always required to consider the radiation pattern of both antennas before drawing the final conclusion of antennas radiation pattern.One can anylze the radiation pattern of two antennas excited at different frequencies by giving the frequency sweep over whole band and computing the antennas radiation pattern at particular desired frequency

Waqas gave a proper response. You can combine them, but the result will depend a lot on what you are looking for. They will add in a form of modulation that will vary with space. If you had 2 beams of the same amplitude over space, but of different frequencies, the result in every direction would be a double-sideband signal.

This case occurs, for example, in multipath propagation, where the direct ray and ground-reflected ray are combined together to form the multipath interference. The combination is a vector addition of the two components of the field which have the same frequency. When there are two field components having two different frequencies, then combination can also occurs. In such case the results will vary rapidly  with time as the difference in the two frequencies increase. Destructive and constructive interference will be noticed.  The case is similar to adding two sinusoidal signals having different frequencies.

Time domain is the way to go - you can do anything. Please see R. Ziolkowski,  UoP, publications.

Basically, if you use 2 separated sources, the 2 signals are orthogonal and will not influence each other. In order to know what is received you can use superposition, that is just to combine the contribution of the sources at the receiver. Each source will radiate according to its pattern that will not be affected from the other pattern. If you have any non-linearity (in the sources side or the receiver side), than this approach does not fit.

Not at all

Thanks

My comments to Sayidmarie, Stan Podgorski, Einat and Vishvakarma;

Is their a chance of any sort of interaction between the fields of the two different frequency sources radiation patterns, i.e., Destructive and constructive? If yes then according to your explanation there is a logic in the radiation beam pattern combination.

I agree with Vishvakarma. NO, It's not possible. It doesn't mean anything.

Two co-located antennas working at different frequency, will interact if :

1. There is a dimensional relation between them i.e. to say if one is half wave length at frequency 'f1' and dimension of other antenna is in close proximity with 'f1' (in terms of wave length).

2. The interaction between them is also possible based on their position, feed technique, etc.

However, If radiation patterns at two frequencies for two different antennas are combined then there will be no information in that combine radiation pattern as it will be difficult to differentiate the effects  of both the antennas on each other. It will be best to see the radiation pattern of both the antennas at same frequency.

Even If one want to compare the radiation pattern of two antennas, then also same frequency should be used.

A simple response is - Yes. However, one has to include an analysis of beam spreading and use two TEM-horn antennas with absorbera.