Despite their low gain, many of the low-band receiving antennas exhibit strikingly good directivity. Conventionally, we might think that one of the available versions of a front-to-back ratio might suffice to characterize the directivity adequately. However, if we review the various front-to-back ideas, we may soon learn why they may not be suitable to the special needs of low-band receiving antennas.

**Conventional Front-to-Back Ratios**

The language of upper HF directional antennas has grown very conventionalized over the decades. It rests on a 2-dimensional graphic portrayal of the far-field pattern of a directional beam, such as a Yagi-Uda array. **Fig. 1** provides some of the key elements in the usual pattern description that we find in much literature. We find variations in some of the terms and in the style of the graphics used to present the pattern, but the terms shown in the sketch are very usual ones.

One key to our discussion is the clear directivity of the pattern that allows us to distinguish forward and rearward lobes that represent gain maximums in the various directions. The pattern appears in normalized form, that is, with the maximum antenna gain just reaching the outer ring of the background scale. Other presentations either with a different level of gain for the outer ring or with a different scale for inner rings are possible and often useful. The key property of the pattern and its parts is the fact that it is a 2-dimensional portrayal. In free-space, the pattern represents the E-plane of the antenna, in this case a 3-element Yagi array. Over ground, the pattern would use a constant elevation angle. We normally select either the take-off angle, that is, the elevation angle of maximum gain, or some other elevation angle of special interest, such as the elevation angle dictated by a propagation forecast for strongest signals into or out of a target communications area.

The 2-dimensional nature of the pattern has yielded the concept of the front-to-back ratio as a measure of directivity. First, not everyone uses the basic term in the same way. So we shall find some refinements in the terminology. Second, not everyone who uses the refined terminology uses it in the same way. **Table 1** and **Fig. 2** will be our guides, but only for part of the journey. Both the table and the graphic present information on the rearward performance of 3 sample antennas. Numbers and pictures do not always determine how people use words. Our first step will be to present some initial definitions (with modifications to come). These definitions will coincide with the labels in **Table 1**. The 180° front-to-back ratio is the main lobe forward gain (or the maximum antenna gain) minus the gain of the lobe (however big or small) that is 180° away from the heading of the maximum forward gain. This value of front-to-back ratio is most commonly used in general antenna literature and is the one shown in most NEC antenna software. If the main forward lobe is split or does not align with the graph heading, the 180° front-to-back ratio is 180° away from the direction of maximum pattern strength. Hence, the value may not be for a heading directly to the rear of the antenna structure. Since a Yagi is usually symmetrical, the maximum gain will normally be directly forward, and the 180° front-to-back ratio will indicate the relative strength to the direct rear. Note that if we use a normalized scale, we can read the front-to-back ratio directly from the plot--between 25 and 30 dB relative to the maximum gain of the antenna in the leftmost pattern.

In **Fig. 2**, the leftmost pattern comes from **Fig. 1**. The strongest rearward lobe is 180° from the main lobe. However, the center pattern shows a 180-degree gain of very tiny proportions. Hence, the 180° front-to-back ratio is very large (over 40 dB compared to a "mere" 27 dB for the leftmost pattern). Yet, we find rearward lobes that have considerable strength. The line through one of those lobes indicates the direction of maximum strength. It is only about 22 dB weaker than the maximum gain. Some sources call this the worst-case front-to-back ratio, and its value is the maximum forward gain minus the highest value of gain in either rearward quadrant. For this antenna, the 180° front-to-back ratio does not give a true picture of the QRM levels from the rear, so some folks prefer to use this figure as a better indicator. The worst-case front-to-back ratio provides the most conservative value for rearward suppression of QRM. The rightmost graphic in **Fig. 2** shows that the 180° and the worst-case front-to-back values do not require separate lobes, even thought the values differ. (We may debate elsewhere whether the 8-element Yagi main rearward radiation is a single main lobe or a junction of 3 overlapping lobes.) When we find the two ratios related to the same rearward lobe, we usually do not find much difference in their value.

We are not done with front-to-back ratios. Each sketch in **Fig. 2** contains an arc going from 90° on one side of the line of maximum gain around the rear to the other point that is 90° from the maximum gain line. Suppose that we add up all of the gain values at the headings that pass through the arc. Next take their average value. Subtract the average gain value to the rear from the maximum forward gain and you arrive at what some call the front-to-rear ratio. Others call this the averaged front-to-back ratio. **Table 1** performs this task at 5° intervals, which is sufficient for this sampling. If you compare the front-to-rear ratio with the other front-to-back ratios, you can see why an antenna maker might use it. The value is higher than all of the other values (with the exception of the 180° front-to-back ratio for the 3-element short-boom Yagi). The rationale behind using the front-to-rear ratio is that it provides an averaged total picture of the rearward QRM suppression.

The 2-dimensional scheme works reasonably well in characterizing the directivity of antennas used from the middle of the HF region through the UHF portion of the radio spectrum. In most cases, we are concerned with the rearward quadrants at angles equal or close to the elevation angle that we select for the forward lobe. However, even within this region, the scheme has limitations, especially the versions of the front-to-back ratio intended to overcome limitations of the 180° version. **Fig. 3** offers just two samples.

The conventions of front-to-back ratios arose largely with the Yagi array in mind. One feature of these antennas is that in the E-plane, the array exhibits a very deep null 90° away from the main forward direction. Therefore, the use of a 90° convention to set the limits between forward and rearward lobes seemed quite natural. The far-field pattern on the left in **Fig. 3** is for a Moxon rectangle in a horizontal orientation. The deep side nulls do not occur at 90° from the main forward bearing, but somewhere between 110° and 120° from that bearing. An automated system for determining the worst-case front-to-back ratio, such as found in NSI software, would identify the worst-case rearward lobe bearing at 91° from the main forward heading. Whether or not this bearing deserves such an identification falls outside of our discussion, but the quandary is clear.

The right side of **Fig. 3** shows a pattern that is typical of many phased vertical arrays. In one sense, there are no rearward lobes, but only a single deep null 180° opposite the direction of maximum gain. From the pattern alone, it is not clear whether any of the font-to-back ratio conventions except the 180° version has appropriate application to such patterns.

**Re-Thinking Directivity**

In the lower HF and the MF portions of the spectrum, noise is a much more important and fundamental factor for receiving antennas than it is at higher frequencies. Noise may come from any direction, ranging from ground-wave paths to very high-angle propagation routes. As well, many more of the antenna used at lower frequencies have cardioidal and similar patterns such as the one on the right in **Fig. 3**. Together, these facts showed some of the shortcomings of the conventional front-to-back ratio ideas as a measure of receiving antenna directivity. Over the years, two efforts emerged to overcome these failings of the 2-dimensional system.

*DMF*: The first of these systems of finding a replacement for the front-to-back ratio emerged from the work of John Devoldere, ON4UN, whose book, *Low-Band DXing* has acquired just fame. John calls his concept the Directivity Merit Figure (DMF). ON4UN calculates the average gain in the entire back azimuth half of the antenna, from 90° to 270° (where the bearing of maximum forward gain is presumed to be 0°), and over the entire elevation range from 2.5° to 87.5°. Doing all of this at 5° increments means that he considers 666 gain values. The average rearward gain now is the average of 666 values. **Fig. 4** shows the rearward areas evaluated as elevation and azimuth slices of a 3-dimensional pattern (for a phased 2-element vertical array). He then defines a figure of merit for the directivity (front response to back half-hemisphere) as being the difference between the forward gain at an optimum wave angle (for example, 20°) and the average rearward gain. (See Chapter 7 of the most recent edition, section 1.8, page 7-8.)

The process requires a separate utility program, since John compensates for the changing equivalent physical distance between angular points on the azimuth rings for different elevation angles. The elevation angles extend from 2.5° to 87.5° because NEC does not calculate a far field at 0°, that is, at ground-wave level using the RP0 command for real lossy, ground. (NEC does allow RP1 ground-wave analysis as a separate command, although this command may not be available on entry-level implementations of NEC.)

DMF has the advantage of allowing a comparison of any bearing with a specific azimuth and elevation setting against the full rear half-hemisphere of the pattern. Hence, it takes into account the sensitivity of the pattern to noises from virtually all angles, as well as the various vertical as well as horizontal lobes and nulls in the rearward pattern. However, the advantage may also be a disadvantage insofar as noise may come from any direction. Hence, DMF provides a rough directivity figure that extends the concept of the averaged front-to-rear idea, but it does not directly provide an indicator of the overall directivity of an antenna with respect to sorting noise from signals in the desired direction.

*RDF*: Several years ago, one Ham suggested an alternative analysis with several simplifying steps for antenna modelers and some inherent advantages over the DMF measure. This Ham's Receiving Directivity Factor (RDF) compares forward gain at a desired direction and elevation angle to average gain over the entire hemisphere. RDF includes all areas around and above the antenna, considering noise to be evenly distributed and aligned with the element polarization. (See Chapter 7 of the most recent edition of *Low-Band DXing*, section 1.9, page 7-9.)

The RDF measure rests in part on the same calculations used to determine the value for the Average Gain Test (AGT). To obtain the average gain test value for a given antenna, the modeler removes all resistive loads, including the material conductivity of the model wires. The one sets up an RP0 command with an even spread of both azimuth and elevation (phi and theta) points. For most purposes, a 5° increment will suffice, but some complex patterns may require a small increment. In free space, the request will include a complete sphere, while over perfect ground, the request will create a hemisphere of sampling points. **Fig. 5** shows the difference in the 3-dimensional pattern produced, in one case a phased 2-element vertical array and in the other a simple vertical dipole.

To obtain the average gain, the RP0 XNDA entry should be either 1001 or 1002. The former prints the radiation pattern values plus the average gain data, while the latter prints only the average gain information. The following line is the NEC output report of the average power value for a simple monopole over perfect ground.

AVERAGE POWER GAIN= 1.99891E+00 SOLID ANGLE USED IN AVERAGING=( 2.0000)*PI STERADIANS.

A free-space pattern would have shown a value of 4 * PI steradians, and the value--assuming a very good model, would have been very close to 1.00000E+00. However, over perfect ground, the solid angle value is 2 * PI steradians, and the value of the very good model is close to 2.00000E+00. To remove any ambiguity, programs like EZNEC perform the necessary division to arrive at an AGT score over perfect ground that is consistent with the free-space value, in this case, 0.99945E+00.

All AGT values are convertible to gain correction values in dB. 10 times the log of the AGT score (relative to 1.00000) yields the correction factor, which the modeler should subtract from the raw gain reported by NEC. In the sample case, no correction is necessary because the value is so close to the ideal. In fact, there is no universal standard of how close the AGT value should be to 1.00000 to be truly adequate. The allowable range of variation depends upon the specific modeling task. However, as we progress toward a hopefully reliable RDF measure, the initial AGT should be as close to 1.00000 as the modeler can make it. The AGT value is a measure of model adequacy and stands as a necessary but not a sufficient condition of true model adequacy.

When we place an antenna over real lossy ground, we may still request the average gain via the RP0 XNDA values of 1001 or 1002. However, the value that we obtain will be significantly lower than the AGT value used to evaluate model adequacy. Consider a vertical monopole with 4 radials only a few feet above average ground (conductivity 0.005 S/m, permittivity 13). A sample model that includes material losses under these conditions returns the following report.

AVERAGE POWER GAIN= 5.72269E-01 SOLID ANGLE USED IN AVERAGING=( 2.0000)*PI STERADIANS.

The average power gain for this example over ground is 1/2 the value shown or 2.86135E-01 relative to a standardized gain of 1.00000E00. One useful interpretation of this value is as a measure of radiation efficiency (in contrast to the power efficiency value provided by the NEC power budget section of the output report). Essentially, the antenna is almost 29% efficient relative to radiation in the far field. Like the AGT value, the average gain report is convertible to a gain value in dB by the same calculation used earlier. In this case, the calculation returns -5.43 dB.

To calculate the RDF, we need one more modeled value: the gain at the elevation angle and azimuth angle selected by the user. The selected heading for the gain value need not necessarily be the elevation and azimuth angle of maximum gain, although we may often find it convenient for a general evaluation to use these values. The antenna model that produced the listed average power gain happens to show an omni-directional pattern with maximum gain at an elevation angle of 19°. The gain is 0.72 dBi. The difference between the overall average gain and gain at the desired direction and elevation angle is the RDF. Hence, the RDF for this antenna is 6.15 dB.

Although we may easily calculate the RDF for an antenna in EZNEC as a 2 step process, some implementations of NEC, such as 4NEC2, have automated the process of obtaining an average gain value and then obtaining the gain at the desired azimuth and elevation angle in order to calculate the RDF.

Unless used wisely, the RDF can mislead us, just as can any of the other measures of directivity that compare forward gain vs. rearward or overall gain. **Fig. 6** provides the elevation pattern and the 3-dimensional pattern of an omni-directional vertical monopole for 3.6 MHz. At the TO angle, the gain is 0.1 dBi, while the average gain is 0.310 or -5.08 dB. Therefore, the RDF is 5.18 dB. As ON4UN points out in his book, omni-directionality in an antenna does not necessarily result in a low or non-existent RDF (or DMF), since the pattern shows relatively low gain at high elevation angles, all of which go into the calculation of average gain.

If we create a simple pair of phased monopoles, we can obtain the pair of patterns shown in **Fig. 7**. These patterns show an average gain close to that of the single monopole (0.314 or -5.03 dB). This result is natural since the array elements use the same height, radial system, and material as the single monopole. However, phasing gives the array a gain of 3.37 dBi at the TO angle. The resulting RDF is 8.40 dB. The difference between the two antennas is 3.22 dB, roughly corresponding to the difference in their maximum gain (3.27 dB).

The close relationship between the gain differential and the RDF differential occurs with these two antennas due to the similarities in the type of antenna and their elevation pattern properties. Had we selected very disparate antenna types for the examples, the two differentials might not have correlated well.

In addition, when noise abatement is a key issue, the RDF measure will not always tell a complete story. As our Ham reports, for best noise attenuation, a narrow half-power beamwidth may be as important as a very high front-to-rear ratio. Moreover, the factor does not itself account for the bandwidth of an antenna. Many noise sources are very broad band. Receiving antennas vary in their bandwidth in terms of signal strength across a span of frequencies corresponding to the input bandwidth of a receiver. In some application, using a narrow bandwidth antenna may yield a better signal-to-noise ratio. These are factors that fall outside the single-frequency requirement for obtaining an RDF calculation.

Nevertheless, the RDF is an adjunct function to NEC that some implementations of the modeling software may offer. Where not offered, we can easily calculate the value. It adds to list of useful measures that we may derive, even from entry-level versions of antenna modeling software.