This is not important at all. This is a total misconception created by older radio engineers. It came from a time when impedances were not easily matched and it was extremely important to have the feed cable resonate on the frequency to create a proper transfer of the antenna impedance to the transmitter thus reducing power losses. Now that antennas (good ones that is) are so well constructed and low impedances are repeatedly achieved in designs the SWR (standing wave ratio) present on the feed cable is very close to 1:1. When such a low SWR is on the cable its length becomes insignificant as the impedance is transferred properly, no matter at what point along the cable. I strongly suggest getting hold of a radio amateur handbook for further study, this is explained almost ad nauseam.
It must be said, however, there are situations where coaxes are cut to specific lengths but this is when there is a desire to force the coax to become a tuned circuit or act as an situations conversion. The above statement is when a 50 ohm antenna is being fed from a 50 ohm transmitter with a 50 ohm cable.
Let me start with an overheard conversation "RG58 is half the thickness of RG213 so it is 25 ohms so I cannot use it to extend a connection to an aerial" - Er, ugh! to his English and, as to his technical ability, what a shame to the radio engineering world.
It is perfectly acceptable to use different types or thicknesses of coax between the transceiver and the antenna. Many installations will have RG58 from the device to the lightning protection panel (it is thin and very flexible), half-inch 'heliax' up the antenna mast, and finally RG214 from the heliax to the antenna (heliax is a solid form of coax and does not bend easily).
There are only two rules
With these two rules firmly adhered to there is absolutely no reason why mixtures of coax cannot be employed in a professional installation. Ah, since you asked, the impedance of a coax is determined by the ratio of the thickness of the inner conductor to the inside diameter of the outer conductor. The formula is
138 * log(D ÷ d)
This confuses most folk but is easily explained. Without the antenna you were reading all noise generated, for the most part, in the RF amplifier and diode switching circuit up at the front of the receiver. Electrons are free to run around and upon doing so create RF noise. This thermal/kinetic noise starts at about 50MHz and increases with frequency. Below 50MHz the noise is there but other noises start existing that override the thermal noise of electrons.
When the antenna is connected the noise is "shorted out" with the 50 ohm load (being the antenna). Electrons are now not that free to move around as they have to "work against" the 50 ohms now presented to the front of the receiver. If you want to measure background noise take your first reading with a 50 ohm LAN termination connected, then change this for the antenna. Please note this may not exactly give you background noise as there may be a noise off frequency dampening the RF amp when you connect the antenna.
The proper method is to use a directional coupler and signal generator. Assuming you have the equipment to hand then you connect the coupler to your receiver and terminate the other end with a 50 ohm load (such as a LAN termination resistor). Connect your signal generator to the coupler input and increase the output till you obtain a 12dB SINAD reading. Remove the termination and replace this with the antenna and read the SINAD. Now increase the output till 12dB SINAD is again read. Your drop in SINAD should be the amount you needed to increase the output of the generator to re-obtain the 12dB SINAD point. If not, you are likely experiencing some strong signal that is affecting the front end of the receiver. You are more than likely going to have to use a spectrum analyzer to find the source.
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