FDR PROGRAM

**1. Introduction**

The FDR program computes frequency dependent rejection (FDR) and
optionally frequency-distance relationships between a transmitter and
receiver. FDR is calculated using the Gauss-Legendre Quadrature
integration method. An optional

F-D relationship is calculated using either a Smooth Curve-Smooth Earth
or Free-space propagation model, for each frequency for which the FDR is
calculated.

**2. Analysis**

FDR is the amount of attenuation offered by an electronic receiver to a
transmitted signal. This attenuation is composed of two parts: on-tune
rejection (OTR) and off-frequency rejection (OFR).

FDR (f) = OFR (f) + OTR (2.1)

where,

> FDR is the rejection provided by a receiver to transmitted signal as a
> result of both the limited bandwidth of the receiver with respect to
> the emission spectrum and the specified detuning, in dB
>
> OTR is the rejection provided by a receiver selectivity characteristic
> to a co-tuned transmitter as a result of an emission spectrum
> exceeding the receiver bandwidth, in dB
>
> OFR is the additional rejection, caused by specified detuning of the
> receiver with respect to the transmitter, in dB

f is the tuned transmitter frequency minus the tuned receiver frequency.

FDR, OFR, and OTR are positive numbers.

The precise mathematical definitions of FDR, OFR, and OTR (average power
only) are as follows:

FDR (f) = 10 log~10~
$\left\lbrack \frac{\int_{- \infty}^{\infty}{S(f)\text{df}}}{\int_{- \infty}^{\infty}S(f)R(f + \Delta f)\text{df}} \right\rbrack$
(2.2)

where,

S (f) is the transmitter power spectral density, in Watts/kHz, assuming
the signal is uninterrupted, and

R (f) is the receiver selectivity with the receiver tuned to the
transmitter frequency.

OFR (f) = 10 log~10~
$\left\lbrack \frac{\int_{- \infty}^{\infty}{S(f)R(f)\text{df}}}{\int_{- \infty}^{\infty}{S(f)R(f + \Delta f)\text{df}}} \right\rbrack$
(2.3)

OTR = 10 log~10~
$\left\lbrack \frac{\int_{- \infty}^{\infty}{S(f)\text{df}}}{\int_{- \infty}^{\infty}{S(f)R(f)\text{df}}} \right\rbrack$
(2.4)

Once the FDR value has been determined for a transmitter-receiver pair,
a frequency-distance separation relationship can be calculated from the
following formula:

*L~P~ (f) = L~R~ -- FDR (f)* (2.5)

where,

> L~P~(f) is the loss to be used in a propagation model to calculate a
> separation distance in dB
>
> L~R~ is the required loss in dB. It may be calculated from either of
> the following formulas:

L~R~ = P~I~ + G~TI~ + G~RI~ -- N -- INR~T~ (2.6)

where,

P~I~ is the power of the interfering transmitter, in dBm

G~TI~ is the gain of the interfering transmitter antenna, in dBi

> G~RI~ is the gain of the victim receiving antenna in the direction of
> the interfering transmitter, in dBi
>
> N is the noise power, in dBm (N = 10 log~10~ B~r~ + F -- 144)
>
> B~r~ is the receiver bandwidth, in kHz
>
> F is the noise figure of the receiver, in dB
>
> INR~T~ is the threshold value of the interference-to-noise ratio (INR)
> at the receiver, in dB (when an INR at the receiver is greater than
> INR~T~, an incompatible situation is predicted; and if INR is less
> that INR~T~, a compatible situation is predicted)

Or,

> L~R~ = P~I~ + G~TI~ + G~RI~ - P~S~ -- G~TS~ -- G~RS~ + L~S~ + SIR~T~
> (2.7)

where,

> P~S~ is the power of the desired-signal transmitter, in dBm
>
> G~TS~ is the gain of the desired-signal transmitter antenna, in dBi
>
> G~RS~ is the gain of the receiver antenna in the direction of the
> desired signal, in dBi
>
> L~S~ is the median propagation and coupling losses of the desired
> signal, in dB
>
> SIR~T~ is the threshold value of the signal-to-interference ratio
> (SIR) at the receiver for a particular type of interference, in dB.
> SIR~T~ depends on the type of desired signal and type of interference.
> (When SIR is less than SIR~T~, an incompatible situation is predicted
> and when SIR is greater than SIR~T~, a compatible situation is
> predicted)

The user must establish the SIR~T~ or INR~T~ for the particular
modulation type of the transmitter and receiver under consideration.

OFR is defined in Equation 2.3 as:

OFR (f) = 10 log~10~
$\left\lbrack \frac{\int_{- \infty}^{\infty}{S(f)R(f)\text{df}}}{\int_{- \infty}^{\infty}{S(f)R(f + \Delta f)\text{df}}} \right\rbrack$

R (f + f) is the receiver selectivity off tuned from the transmitter by
f kHz.

The transmitter spectrum is usually given relative to peak values.
Therefore, we can write:

S (F) = P p (f) (2.8)

where,

P is the transmitter peak power density, in Watts/kHz

p (f) is the transmitter relative power density spectrum.

By substituting P p (f) for S (f), equation 7.3 becomes:

> OFR (f) = 10
> log~10~$$$\left\lbrack \frac{\int_{- \infty}^{\infty}{p(f)R(f)\text{df}}}{\int_{- \infty}^{\infty}{p(f)R(f + \Delta f)\text{df}}} \right\rbrack$
> (2.9)

However, for practical applications, the transmission emission spectrum
and receiver selectivity are given in dB as follows:

> P (f) = 10 log~10~ p (f) (2.10)
>
> G (f) = -10 log~10~ R (f)

where,

P (f) is the normalized input emission spectrum, in dB

G (f) is the insertion loss associated with R (f).

From Equation 2.10,

p (f) = 10 ${}^{\frac{P(f)}{\text{10}}}$

(2.11)

R (f) = 10 ${}^{\frac{- G(f)}{\text{10}}}$

Therefore, Equation 2.9 becomes

OFR = 10 log~10~
$\left\lbrack \frac{\int_{- \infty}^{\infty}\text{10}^{\frac{P(f)}{\text{10}}}\text{10}^{\frac{- G(f)}{\text{10}}}\text{df}}{\int_{- \infty}^{\infty}{\text{10}^{\frac{P(f)}{\text{10}}}\text{10}^{\frac{- G(f + \Delta f)}{\text{10}}}}\text{df}} \right\rbrack$

> = 10 log10
> $\left\lbrack \frac{\int_{- \infty}^{\infty}\text{10}^{\frac{P(f) - G(f)}{\text{10}}}\text{df}}{\int_{- \infty}^{\infty}\text{10}^{\frac{P(f) - G(f + \Delta f)}{\text{10}}}\text{df}} \right\rbrack$

OFR = 10 log~10~
$\int_{- \infty}^{\infty}{\text{10}^{\frac{P(f) - G(f)}{\text{10}}}\text{df}}$ -
10 log~10~
$\int_{- \infty}^{\infty}\text{10}^{\frac{P(f) - G(f + \Delta f)}{\text{10}}}\text{df}$
(2.12)

The integrals in the FDR equations are evaluated numerically by the
trapezoidal method using a variable integration step size. Each of the
integrals is given by a sum of the form:

> $\sum_{I - 1}^{K}{\frac{1}{2}\text{AB}}$ (2.13)

where,

> A = \[f (I) -- f (I-1)\]
>
> B =
> $\text{10}^{\frac{P_{\text{dB}}\lbrack f(I)\rbrack - G_{\text{dB}}\lbrack f(I)\rbrack}{\text{10}}}$ +
> 10${}^{\frac{P_{\text{dB}}\lbrack f(I - 1)\rbrack - G_{\text{dB}}\lbrack f(I - 1)\rbrack}{\text{10}}}$

The fineness of the step size f (I) -- f (I --1) is determined by the
degree with which the emission spectrum and selectivity curves are
defined.

The input emission spectrum and selectivity curves are defined at a
discrete number of frequencies lying between finite limits (Figure 1).
The discrete set of frequencies for which emission levels are given does
not have to coincide with the discrete sets of frequencies for which the
selectivity levels are given. In the general off-tuned case, these two
ordered sets of frequencies are merged into one ordered set of
frequencies, which becomes the set of abscissas f (I) of the function to
be integrated. In order to evaluate the sum representing the integral,
it is necessary to have an emission and selectivity dB level at each f
(I). Therefore, interpolation or extrapolation is necessary at each f
(I) to provide the other dB level.

Except for one case, interpolation is log-linear, which assumes a
straight line on semi-log paper. Log-linear interpolation is not
possible near f (I) = 0. In this case, linear interpolation is done. Two
steps can be taken to minimize the region of linear interpolation.
First, the point representing the tuned frequency on the input emission
spectrum and selectivity curve should be between two non-zero points
which are as close to the zero points as possible. Second, the
selectivity curve should be very well defined since each portion of the
curve may eventually be subject to the linear interpolation at

f (I) = 0 as the curve is shifted by the required sequence of DEL F's.

Extrapolation is always log-linear and is done relative to the input
emission and selectivity curves (not the shifted selectivity curve).
Provision is made in the program to allow for the input of extrapolation
slopes (dB/decade) to be used to generate emission and selectivity
levels beyond the endpoints of each input curve.

If it is required to maintain some total loss between a transmitter and
receiver at some degree of detuning, the OFR calculated by OFRCAL can be
used to determine the remaining loss that must be made up by a distance
separation between the transmitter and the receiver. The remaining loss
to be obtained by the distance separation is given by:

Loss = total loss -- FDR

This loss is input to the Inverse IPS, which outputs the required
distance.

**3. Input Parameters**

The input parameters for OFR calculations are the following:

1.  Transmitter specifications, i.e., transmitter description, -3 dB
    bandwidth, left and right extrapolation slopes and the transmitter
    emission curve points (delta frequency and emission spectrum level).

2.  Receiver specifications, i.e., receiver description, effective
    antenna height, left and right extrapolation slopes and the receiver
    selectivity curve points (delta frequency and power level).

3.  Output stepping options, i.e., initial delta F for frequency
    stepping, step size, and number of points.

Additional parameters required for inverse IPS are:

1.  System parameters, i.e., transmitter primary frequency, transmitter
    effective antenna height, transmitter polarization (V or H), total
    required loss.

2.  Environmental parameters, i.e., refractivity (200 to 450),
    permittivity (1 to 81) and conductivity (.0001 to 5)

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1.  Computation parameters, i.e., maximum number of iterations and
    maximum relative error (0 to 1)

**4. Output**

The following output graphs are produced:

-   input transmitter emission spectrum curve

-   input receiver selectivity curve

-   OFR vs. delta F

-   FDR vs. delta F

-   F-D vs. delta F (optional)

Optionally, the x-y data for the above curves can be written to a disk
file or can be displayed and printed as output reports.

![](media/image1.png){width="5.864583333333333in"
height="7.489583333333333in"}

Figure 1. Typical Off-Tuned Relationship between Tx and Rx (DEL F\<0)

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