Echo strength in free space
4.6 Calculations and graphs
The range equation is so important that it is worthwhile illustrating how it is used for calculation of echo strength and graphical depiction of variation of echo strength with range.
4.6.1 Fixed range example
Suppose we wish to find the free-space echo from a navigation buoy of RCS 10 m2 at 10 km range received by a typical deep-sea ship's 9 GHz radar having: / = 9400 MHz
giving X = 3.19 cm, 25 kW transmitter pulse power, scanner gain 1260, total transmitter loss 4 dB, total receiver loss 5 dB.
For comparison we set out a range budget, with linear and decibel alternatives, Table 4.1. Splitting the decibels into positive and negative columns helps avoid minus sign blunders, so easy with hand calculation. This method is well suited to initial 'order of magnitude' tests to check feasibility of a proposed system, or as a cross-check for gross errors within more detailed calculations - computers are very accurate but work on the garbage in, garbage out principle, never questioning inputting blunders.
For this radar Fi2 = —85.9dBW. Multiplying and dividing numerical values, -Se(FSi2) = 2.572 x 10~12 W. Adding and subtracting decibel equivalents (expressed to nearest 0.1 dB), Se(FSi2) = —115.9 dBW, which converts by the methods of Chapter 2, Section 2.1.7 to 25.7 x 10~12W. The difference is a trivial rounding error of 0.1 percent.
The calculated echo power of less than a hundred-thousandth of a microwatt would probably be detectable in free-space conditions, but environmental effects such as Earth curvature might prevent detection in practice.
4.6.2 Graphs
Calculations as Table 4.1 for a spot range do not reveal the full story. It is worthwhile preparing a graph of signal strength versus range to indicate whether signal is changing rapidly and so might be unduly sensitive to parameter variations.
The 'obvious' way to do this is as follows.
1. Draw linear range (abscissa, X-axis) and power (ordinate, F-axis) scales on a sheet of ordinary linear graph paper.
2. Calculate (as Table 4.1) echo strength at several spot ranges.
3. Plot the echoes as a series of points at the proper ranges and powers.
4. Insert the minimum detectable signal curve.
5. Draw a smooth curve through the points, freehand or by French curve.
The result is Figure 4.1. The echo curve falls sharply at short range and less sharply as range increases. We know from Eq. (4.6a) that it has the form power oc R~4, but this is by no means obvious from viewing the graph.
The echo power curve in itself is only part of the story, which is why a curve of the radar's minimum detectable signal (MDS), the minimum echo strength needed for detection, has been added. For simplicity, MDS is here assumed 0.5 x 10~12 W irrespective of range; in practice it would vary. At short range, the echo exceeds MDS and the target is detected. At long range, the echo falls below MDS and is undetectable.
The intersect gives maximum free-space detectable range or first detection range, which can be scaled off the graph as 15.1 km.
For most purposes this presentation is tiresome; unless the range bracket of interest is less than about 2 : 1 the curve clings close to the axes and is difficult to read.
Changing to a linear decibel power scale - equivalent to logarithmically in watts -in
Table 4.1 Range budget by linear and decibel methods
Free space
Quantity Numerical value dB (pos) dB (neg) Notes Dimensionless except as stated
Equation Eq. (4.6a) Eq. (4.8) Eq. (4.8) P 25 000 W 44dBW
G2 1587 600 62 dBi 12602 = 1587 600;
2 x 3 1 = 6 2 X2 0.00102m2 - 2 9 . 9 (0.0319)2 = 0.00102 Conversion (l/47r)3/101 2 -153.0 Negative dB as < 1
1/Lt 1/2.51 - 4 . 0
1/Lr 1/3.16 - 5 . 0
Operation Multiply Add Add Risk of decimal point error when using numerical values 106 -191.9 AdddB (+ve) and dB (~ve)
columns
\t
F1 2 2.572 x 1(T9 -85.9 -85.9 dB = 2.5704 x 10"9
There is a small rounding error
o 10 m2 1OdBm2
I//?4 (10km) 1/104 - 4 0
10 -125.9 Add dB (+ve) and dB (-ve) columns
Multiply \
^r(FS12) 2 . 5 7 2 x l O "1 2W -115.9dBW
= - 1 2 0 + 4 . I d B W = 10"1 2 x 2.570
= 2.57 x 10~1 2W
Figure 4.2 opens things up a bit, but some disadvantages remain:
• the curve is tedious to calculate and draw
• the law relating echo strength to R is not revealed
• the curve becomes very cramped at short range
• it is not easy to change a parameter, for instance target RCS or transmitter power.
Figure 4.3 overcomes the problems. As in Figure 4.2, power is scaled linearly in dBW. But range is now also to a logarithmic scale, so each cm now represents a certain range ratio. This linearises the echo strength line, which falls —40 dB (power ratio 1:10~4) when range is increased by a factor of 10, immediately revealing the R~4
law (S ex R~4) followed by the echo.
Figure 4.1 Echo strength versus range, linear power and range scales. Echo strength has to be plotted from a series of values calculated at judi- cious ranges. Note how rapidly power falls at short range. Law is not obvious and short ranges are cramped. The maximum detectable range intersect at 15.1km is difficult to determine accurately
1. Draw scales Range, km
Figure 4.2 Echo power in dBW, linear range scale. Less cramped than Figure 4.1, but still inconvenient. Draughting procedure similar to Figure 4.1.
Range, km Echo, 1OdBm2 target
Scale 5. Read off intercept with MDS
4. Add MDS threshold 3. Draw smooth curve through points Echo, W
2. Plot calculated strengths at spot ranges
Power, WxIO"12Echo, dBW
Figure 4.3 Logarithmic range scale, otherwise as Figure 4.2. Simple to draw and instructive to use. Linearises echo/range law, revealing that echo is changing —40dB/decade and so following R~4 law. Easy to assess changes to parameters, for example changed RCS, light lines
The procedure is easy, accurate and quick, with only a single calculation.
6. If log graph-paper is not at hand, it is easy enough to mark off plain paper with a range axis. If interested in ranges between 0.1 km and 100 km, which is three decades, with graph width 15 cm to give 5 cm/decade, mark 0.1 km at the start, 0.2km at 1.5cm (5log2 ~ 1.5), 0.5km at 3.5cm (5log5 ~ 3.5), 1.0km at 5 cm, and so on to 100 km at 15 cm as shown in italics.
As with Figure 4.2, mark the vertical axis as echo strength linearly in decibels to a convenient scale, say 1 cm per 10 dB between say —160 dBW at the foot of the page
Centimetres Range, km, log scale
2:1 range change Sketching steps 6
Scale
Entry p< >int: RCS 0 dl I m2 at 1.0 km
Slope -40 dB/decade
= -12dB/octave Echo, 4 dB m2 target
(Light line) Echo, 1OdBm2 target
(Heavy line) Octave
decade
Echo, 3OdBm2 target (Light line) down
Power, dBW
and O dBW near the top. Few radars will detect signals weaker than —130 dBW, but it is convenient to take the graph down further to aid construction of the curve.
7. Draw construction line AB in a corner at slope —40 dB/decade (R~4 law).
8. Calculate Fi2 per Eq. (4.12) or Table 4.1 with R = 1000m and or = OdBm2. This gives the reference echo from a 0 dB m2 (Im2) target at 1 km.
9. Enter on graph (point C).
10. Enter point D, a dB above C (here 10 dB), to represent the echo at 1 km from the 1OdBm2 target.
11. Draw a straight line parallel to AB (a navigator's parallel rule is handy) through D, produced to the sheet borders. The curve, of the form y = JCZ, becomes linearised with slope z9 so log y = z log x. This is the echo from a 10 dB m2 target in free space.
The graph is no longer cramped. To examine the effect of changing a parameter, we merely slide the curve vertically up or down by the appropriate number of dB.
For example, an echo of a ship of RCS 3OdBm2 (1000 m2) lies a further 2OdB up and a dinghy of RCS 4 dB m2 (2.5 m2) lies 6 dB down from our 10 dB m2 buoy (light lines). The intercepts with MDS indicate the ship has maximum free-space detectable range 47.5 km and the dinghy 10.6 km.
It is equally easy to judge the effect on detection range of changing MDS, e.g. when using another pulselength/receiver bandwidth combination. One merely slides the MDS line up or down by the appropriate number of dB.
4.6.3 Computer spreadsheet and charting
Using a personal computer (PC), life is easy. All we do is compile a simple spreadsheet with cells for each of the terms of whichever of Eqs (4.8)-(4.11) suits the system, using either numerical or dB forms. A column of a hundred or so ranges, incremented from the lowest to the highest of interest, is followed by a column containing the chosen equation. The PC can then automatically 'chart' the relationship. Most Windows com- puters have EXCEL spreadsheet facility, which permits computation of logarithms, allowing quick numerical/dB swapping. And the chart can scale itself logarithmically, so if we produce a Figure 4.1 lookalike which turns out to be cramped, a Figure 4.3 version can quickly be generated. To try the effect of changing a parameter, we merely enter the revised value in the appropriate cell. The signal strength graphs in this book were generated in this way. The IEE website (www.iee.org) includes comprehensive spreadsheets which include environmental effects. Chapter 14 gives full details. It is all too easy to enter an incorrect value in one of the many cells necessary to quantify all the environmental, radar and target variables, and it is good practice to make rough sketches similar to Figure 4.3 as a check for gross error.