Directive gain and radiated power of an antenna while attending a lecture on antennas, we were explained an example where it was stated that a lossless antenna with a directive gain of +6 db is going to radiate 4 mW of power if it is fed 1mW of power. However, working on this example myself, I am of the view that the output power should be 1 mW instead, since the antenna is lossless, and so no ohmic power should be dissipated and the radiated power should equal the power fed. I tried browsing the net for finding out an explanation for this, but couldn't find such an example. Can anyone please clarify this doubt of mine, for I need to develop a project based on antennas, and I must have all my concepts crystal clear. Thanks! 
 A: A maximum directive gain (directivity) of +6 db tells you that the power density radiated by the antenna in the direction of maximum power density radiation is four times (6 db) the power density for an antenna which radiated equally in all directions.
A: Your view that the output power of any lossless antenna is the same as the input power fed to it is correct – 1mW in, 1mW out.  Passive antennas cannot amplify the power fed to them, but they can concentrate that power into a preferred direction, usually referred to as the main lobe, main beam, bore-sight, or some such terminology. 
For an isotropic antenna (numeric gain = 1, logarithmic gain = 0 dB) this 1mW of power expands outward equally in all directions.  At a given distance D from an isotropic, you will only be able to capture a fraction of the radiated power, but that fraction will be the same no matter where you stand around the isotropic.
For an antenna of 6dB directive gain this is not the case.  At the same distance D, if you are standing in the main-lobe preferred direction you will capture four times the energy that you would for an isotropic, but only in the preferred direction.  At any other angle away from the 6dB main lobe you will capture much less energy than for an isotropic and at some angles the energy capture will be close to zero.
To put this into the context of your original question; a 6dB directive gain antenna fed 1mW of power radiates the same amount of energy in the main-lobe preferred direction that an isotropic antenna would if it were radiating 4mW of energy equally in all directions.
This means that even for 6dB of directive gain, the main lobe must be fairly accurately aimed toward the receiving antenna in order to realize the full 4 to 1 advantage over an isotropic.
