1. Does an antenna like this have to be SWR ed in?
SWR is something that's associated with transmitting, not receiving (it's the ratio of signal being transmitted vs. signal being reflected back to the transmitter) so it's not something that's applicable to a receive-only situation.
2. Is the RG6 cable a problem?
Perhaps, but for moderate runs - 50 to 75 feet long or so - it should work fine. Be aware that my statement is based on using quality cable with properly installed connectors. Cheap cable and cheap or badly installed connectors will negate my response.
3. I read somewhere that the whip antenna on top of the Discone should be trimmed bit by bit till it works properly.
The whip on the top is only used for the VHF-Low (30 - 50 MHz) band reception. If you're using it to receive on this band, you may need to tune it, but if not it will have minimal or no affect.
The Discone Antenna has a female PL-259 fitting so I used the applicable male connector..
However these were screw on fittings and I not sure if leaving the foil and surrounding wire shielding has to be removed before screwing the connector on or is the copper wire shielded from this some how when soldered into the plug pin?
Screw on connectors are not very reliable and can easily cause issues. Also, RG-6 is designed for crimp on type connectors (generally "F" connectors are used) and will generally work better. You will need an adapter (Female "F" to Male UHF for your antenna) to handle the connection for you. The center conductor should go to the center pin on your connector and the shield (that foil and wire braid) is connected to the outer part of the connector. The foil should be cut back, just enough so it doesn't contact the center connection and the braid connected securely to the outer part of the connector.
I have recently installed a Diamond Discone antenna . And I am having worse reception than using the stubby antenna on my UBCD396XT.
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If you can give some advice it would be greatly appreciated. As I am becoming quite frustrated with the new toy/setup.
First comment: The Diamond Discone is a pretty good wide band antenna, but be aware it has no real gain. The design is to provide you with a wide frequency range, not a powerful signal. Since the gain is low, it's important that you use quality low-loss coax that's as short as you can make it.
Second comment: You don't say what frequency of frequencies you're trying to pick up. The higher the frequency, the harder it is to get a good signal from the antenna to the radio. You may be able to optimize a solution for a narrow band of frequencies (say a single band) by using an antenna designed specifically for that range.
Third comment: It's possible that you may have too much signal (yes, that's possible). Generally this is due to a simulcast system and getting strong signals from multiple towers. It's also possible that the new antenna may be picking up other strong signals (broadcast radio, TV, paging, cell phone signals, or other signals) that are drowning out the ones you're trying to monitor.
If you would provide your location and what you're trying to monitor (not "my local PD" but more specifically like "The Podunk ID PD on 155.325 MHz and their trunked system, the Podunk ID Public Safety System" and if possible, provide links to them from the RR Database). This will help us identify possible problems and perhaps a local user may recognize the issues and help to provide specific suggestions to help you out (e.g. "The Podunk PD's Tac-2 channel is dual-use and each has their own tone. If you program it as CSQ - no tone programmed - you'll pick up any transmission on that channel").