The problem.
Around the 2 meter amateur band very strong signals can be received from professional networks. In The Netherlands for instance on 154,9875MHz, 159,9875MHz and 164,350 MHz are pager networks active with transmitters up to 100W. 

Modern small amateur VHF transceivers mostly suffer from to little selectivity ahead of the mixer stage. Those signals therefore can cause overloading of the receiver, causing I(nter)M(odulation). If for instance, pager transmitters on 154.9875 and 164.350 MHz are active, one IM-product will be 2x154.9875-164.35 = 145.625 MHz. This interference has a very wide deviation, because it is the mixing product of two FM-modulated signals, the deviation of one of them was already multiplied by a factor of 2. Another IM-product falling within the two meter band is 3x154,9875-2x159,990=144,9825 MHz. The bandwidth of this product can be 80 kHz. Because of their loudly modulated, sharp distorted nature, these interferences can be very annoying. Local FM-broadcast transmitters can overload the receiver too, producing 2nd harmonics, for instance: 2x89,5 MHz = 179 MHz. If the IF-frequency of the receiver is 16.9 MHz, the mirror frequency is 145.2 MHz. The broadcast transmitter then can be heard very distorted, over the whole frequency range of 145.05 - 145.35 MHz. The only remedy is to enhance the HF-selectivity, by placing an extra filter between the receiver and the antenna cable.

    A solution.
The cable notch filters described here are easy to make, need no tuning, have low insertion-loss, good SWR inside the amateur band, can endure sufficient HF-power, need no cabinet, and can be connected outside the receiver using standard plugs.

All described cable lengths cause notches symmetrical above and below 145MHz. So not only signals above, but also below 145MHz are attenuated. In this way we can speak about band pass filtering.

fig. 2, 1/4 wave notch-cable.

    How does a piece of coax cable works like a notch filter?
A 1/4 wave length of a transmission line, open on one end, represents a short circuit on the other end. This is true for every odd number times of this frequency. All energy will be reflected to the source. The bandwidth of this low impedance range is small. So, the one-end-open-cable piece works like a rejection circuit (notch filter), see fig 2.

The cable length can be calculated as follows: L = (75 x v) : f_res.
L in meters, v = velocity factor of the type of cable, f_res in MHz.

    Matching problem:
If this cable is designed to work at 155 MHz, it will introduce some mismatch on 146 MHz, and therefore insertion loss. On every even number of times the 1/4 wavelength resonance-frequency, the cable piece will represent a very high impedance. On those frequencies the cable piece will have no influence on the transmission line attached to it.

    Solution:
If such a cable piece is designed to be an odd number of times longer than for the notch frequency, but at the same time an even number of times longer for 145 MHz (the pass-frequency), it will not have any influence on two meters.

    Calculation example.
F_notch = 155MHz.
De cable needs to be an odd times longer than 1/4 wavelength for 155 MHz.
F_pass = 145MHz.
De cable needs to be an (odd-1) times longer than 1/4 wavelength for 155 MHz.
For 155 : 15 = 10.333 MHz, 14 x 10.333 = 144.666 MHz. This will be close enough to 145 MHz.

    Notch depth.
Fig.2 shows, that notches become less deep at higher frequencies. This is caused by cable losses. So best use low-loss cable, avoid the use of RG58, especially when using long pieces of coax.

    2 notches.
By combining two cable pieces of different length, a filter with two notches will be made. A better attenuation in between the two notch frequencies can be achieved, by separating them with a 1/4 wave cable piece for 160 MHz.

If two pieces of the same length are used, the notch on that frequency will be deeper.

    Results.
For all calculated cable length here below, notches occur at frequencies symmetrical above and below 145 MHz. Table 1 shows test results for RG58CU cable. Using low-loss types of cable, like Aircom+ or Aircell7, better results are obtainable. 

    Construction.
Table 2 shows cable lengths for different types of cable for a 2-notch-filter for 155 and 164.35 MHz. Fig 3 and 4 give further details. Cable pieces can be winded to coils, if their minimum bending-radius is respected.

fig. 3, test setup.	fig.4

Beware: plugs and T-pieces lengthen the cable pieces. Therefore, when assembling,
L1 must be measured between the middles of the T-pieces.
L2 and 3 must be measured between the open end and the middle of the T-piece.

fig. 5, soldering cable pieces.

    A 3-notch filter.
With three different cable pieces in parallel, a 3-notch filter can be realized. Cable lengths in table 3 are optimized for notch frequencies. As the "160 MHz" cable also resonates at 140 and 150 MHz, this filter shows even lesser bandwidth, and helps against strong voice-communication signals too. All three cables were directly connected in parallel using plugs. It also can be done by soldering them together on a piece of PC board, connected to on-soldered (PTFE) BNC-receptacles, screening it afterwards with a piece if thin copper plate. 

fig. 6, 1/2 wave loop.

    FM-broadcast.
If a strong local FM-broadcast transmitter on for instance 89,4 MHz makes trouble, a cable-piece-filter can help.
Example:
f_notch = 89,4 MHz.
f_pass=145MHz (including 1 percent tolerance).

89,4 : 11 = 8,127 MHz; 8,127 MHz x 18 = 146,29 MHz,
close enough to 145 MHz.

1/4 lambda on 8,127 MHz = 9,228 m.
Length Aircom+ : 9,228 x 0,85 = 7,844 m.

Remark: In such a case, a band pass filter for two meters is preferable.
See: "Helix band pass filters" on this site.

    1/2 lambda cable piece.
In order to get deeper notches, 2 or 3 cable pieces of the same length for each frequency could be connected in parallel.

Another method is, to use a 1/2 wave cable piece (see fig. 6). This works like two 1/4 wave pieces in parallel.

Connections could easiest be made like in fig. 5. In this way i made cheap and perfect cable harnesses for repeater duplex filters.

Best use BNC or  N-connectors of good quality at the ends (PFTE insulation).

PTFE cable is easiest soldered, but suffers of greater loss, and therefore less deeper notches.

Many thanks to: Henk pa0hpv, for his contribution about IM-calculations.

73's, Nico, pa0nhc.