1. ELEC4600 Radar and Navigation Engineering
Radar Range Equation:
This equation is not very accurate due to several uncertainties in the variables used:
1. Smin is influenced by noise and is determined statistically
2. The radar cross section fluctuates randomly
3. There are losses in the system
4. Propagation effects caused by the earth’ surface and atmosphere
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ELEC4600 Radar and Navigation Engineering

2. ELEC4600 Radar and Navigation Engineering
Probabilities
Due to the statistical nature of the variables in the radar equation we define performance based on two main factors
Probability of Detection (Pd)
The probability that a target will be detected when one is present
Probability of False Alarm (Pfa)
The probability that a target will be detected when one is not present
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ELEC4600 Radar and Navigation Engineering

3. ELEC4600 Radar and Navigation Engineering
Minimum Signal
Detection of Signals in Noise
Typical output from receiver’s video amplifier,
We have to determine how to decide whether a signal is present or not
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ELEC4600 Radar and Navigation Engineering

4. ELEC4600 Radar and Navigation Engineering
Minimum Signal
Threshold Detection
Set a threshold level and decide that any signal above it is a valid reply from a target.
Two problems:
1. If the threshold is set too high Probability of Detection is low
2. If the threshold is set too low Probability of False Alarm is high
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ELEC4600 Radar and Navigation Engineering

5. Receiver Noise and Signal/Noise Ratio
Source of Noise is primarily thermal or Johnson Noise in the receiver itself
Thermal noise Power = kTBn
Where k is Boltzmann’s Constant (1.38 x J/K)
T is the temperature in Kelvins (~Celsius +273)
B is the Noise Bandwidth of the receiver
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ELEC4600 Radar and Navigation Engineering

6. Receiver Noise and Signal/Noise Ratio
Noise Bandwidth
H(f0) Bn
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ELEC4600 Radar and Navigation Engineering

7. Receiver Noise and Signal/Noise Ratio
Noise Bandwidth Bn
H(f0)
In practice, the 3dB bandwidth is used.
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ELEC4600 Radar and Navigation Engineering

8. Receiver Noise and Signal/Noise Ratio
Noise Figure
Amplifiers and other circuits always add some noise to a signal and so the Signal to Noise Ratio is higher at the output than at the input
This is expressed as the Noise Figure of the Amplifier (or Receiver)
Fn= (noise out of a practical reciver) (noise out of an ideal (noiseless) receiver at T0)
Ga is the receiver gain
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ELEC4600 Radar and Navigation Engineering

9. Receiver Noise and Signal/Noise Ratio
Since Ga = So / Si (Output/Input)
and kT0B is the input noise Ni
then
finally
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ELEC4600 Radar and Navigation Engineering

10. Receiver Noise and Signal/Noise Ratio
Since Ga = So / Si (Output/Input)
and kT0B is the input noise Ni
then
finally
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ELEC4600 Radar and Navigation Engineering

11. Receiver Noise and Signal/Noise Ratio
Modified Range Equation
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ELEC4600 Radar and Navigation Engineering

12. Probability Density Functions
Noise is a random phenomenon
e.g. a noise voltage can take on any value at any time
Probability is a measure of the likelihood of discrete event
Continuous random functions such as noise voltage are described by probability density functions (pdf)
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ELEC4600 Radar and Navigation Engineering

13. Probability Density Functions
e.g.
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ELEC4600 Radar and Navigation Engineering

14. Probability Density Functions
e.g. for a continuous function
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ELEC4600 Radar and Navigation Engineering

15. Probability Density Functions
Definitions
Mean
Mean Square
Variance
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ELEC4600 Radar and Navigation Engineering

16. ELEC4600 Radar and Navigation Engineering
Common PDFs
Uniform
This is the pdf for random phase
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ELEC4600 Radar and Navigation Engineering

17. ELEC4600 Radar and Navigation Engineering
Common PDFs
Gaussian or Normal
Very common distribution
Uniquely defined by just the first and second moments
Central Limit Theorem
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ELEC4600 Radar and Navigation Engineering

18. ELEC4600 Radar and Navigation Engineering
Common PDFs
Rayleigh
Variance
Detected envelope of filter output if input is Gaussian
Uniquely defined by either the first or second moment
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ELEC4600 Radar and Navigation Engineering

19. ELEC4600 Radar and Navigation Engineering
Common PDFs
Exponential
Note:
Probable Error in Notes
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ELEC4600 Radar and Navigation Engineering

20. Calculation of Minimum Signal to Noise Ratio
First we will determine the threshold level required to give the specified average time to false alarm (Tfa). This is done assuming no signal input. We shall also get a relationship between Tfa and Pfa .
Then we add the signal and determine what signal to noise ratio we need to give us the specified probability of detection (Pd)
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ELEC4600 Radar and Navigation Engineering

21. Calculation of Minimum Signal to Noise Ratio
Gauss in
Rayleigh out
BVBIF/2
Pfa=
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ELEC4600 Radar and Navigation Engineering

22. Calculation of Minimum Signal to Noise Ratio
assuming tk=1/BIF
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ELEC4600 Radar and Navigation Engineering

23. Calculation of Minimum Signal to Noise Ratio
Now we have a relationship between False alarm time and the threshold to noise ratio
This can be used to set the Threshold level
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ELEC4600 Radar and Navigation Engineering

24. Calculation of Minimum Signal to Noise Ratio
Now we add a signal of amplitude A and the pdf becomes Ricean. i.e. a Rice distribution
This is actually a Rayleigh distribution distorted by the presence of a sine wave
Where I0 is a modified Bessel function of zero order
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ELEC4600 Radar and Navigation Engineering

25. Calculation of Minimum Signal to Noise Ratio
This is plotted in the following graph
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ELEC4600 Radar and Navigation Engineering

26. Calculation of Minimum Signal to Noise Ratio
From this graph, the minimum signal to noise ratio can be derived from:
a. the probability of detection
b. the probability of false alarm
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ELEC4600 Radar and Navigation Engineering

27. Integration of Radar Pulses
Note that the previous calculation for signal to noise ratio is based on the detection of a single pulse
In practice a target produces several pulses each time the antenna beam sweeps through its position
Thus it is possible to enhance the signal to noise ratio by integrating (summing ) the pulse outputs. Note that integration is equivalent to low pass filtering.
The more samples integrated, the narrower the bandwidth and the lower the noise power
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ELEC4600 Radar and Navigation Engineering

28. Integration of Radar Pulses
Note: The antenna beam width nb is arbitrarily defined as the angle between the points at which the pattern is 3dB less than the maximum
Beam Width θB
3dB
If the antenna is rotating at a speed of θS º/s
and the Pulse repetition frequency is fp
the number of pulses on target is nB = θB fp / θS
or if rotation rate is given in rpm (ωm) nB = θB fp / 6 ωm
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ELEC4600 Radar and Navigation Engineering

29. Integration of Radar Pulses
Integration before detection is called predetection or coherent detection
Integration after detection is called post detection or noncoherent detection
If predetection is used SNRintegrated = n SNR1
If postdetection is used, SNRintegrated  n SNR1 due to losses in the detector
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ELEC4600 Radar and Navigation Engineering

30. Integration of Radar Pulses
Predetection integration is difficult because it requires maintaining the phase of the pulse returns
Postdetection is relatively easy especially using digital processing techniques by which digitized versions of all returns can be recorded and manipulated
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ELEC4600 Radar and Navigation Engineering

31. Integration of Radar Pulses
The reduction in required Signal to Noise Ratio achieved by integration can be expressed in several ways:
Integration Efficiency:
Note that Ei(n) is less than 1 (except for predetection)
Where (S/N)1 is the signal to noise ratio required to produce the required Pd for one pulse and
And (S/N)n is the signal to noise ratio required to produce the required Pd for n pulses
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ELEC4600 Radar and Navigation Engineering

32. Integration of Radar Pulses
The improvement in SNR where n pulses are integrated is called
the integration improvement factor Ii(n)
Note that Ii(n) is less than n
Another expression is the equivalent number of pulses neq
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ELEC4600 Radar and Navigation Engineering

33. Integration of Radar Pulses
Integration Improvement Factor
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ELEC4600 Radar and Navigation Engineering

34. Integration of Radar Pulses
False Alarm Number
Note the parameter nf in the graph
This is called the false alarm number and is the average number of “decisions” between false alarms
Decisions are considered as the discrete points at which a target may be detected unambiguously
Recall that the resolution of a radar is half the pulse width multiplied by the speed of light τ τ τ
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ELEC4600 Radar and Navigation Engineering

35. Integration of Radar Pulses
False Alarm Number
Thus the total number of unambiguous targets for each transmitted pulse is T/ τ
where T is the pulse repetition period (1/fP)
We multiply this by the number of pulses per second (fP) to get the number of decisions per second
Finally we multiply by the False alarm rate (Tfa) to get the number of decisions per false alarm.
nf = [T/ τ][fP][Tfa]
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ELEC4600 Radar and Navigation Engineering

36. Integration of Radar Pulses
False Alarm Number
nf = [T/ τ][fP][Tfa]
But T x fP =1
and τ  1/B where B is the IF bandwidth
so nf  Tfa B  1/Pfa
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ELEC4600 Radar and Navigation Engineering

37. Integration of Radar Pulses
Effect on Radar Range Equation
Range Equation with integration
Expressed in terms of SNR for 1 pulse
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ELEC4600 Radar and Navigation Engineering

38. Integration of Radar Pulses
Example:
Radar:
PRF: 500Hz Bandwidth :1MHz
Antenna Beamwidth: 1.5 degrees Gain: 24dB
Transmitter Power 2 MW Noise Figure: 2dB
Pd: 80% PFA: 10-5
σ: 2m2 Freq: 1GHz
Antenna Rotation speed: 30 degrees/s
What is maximum range?
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ELEC4600 Radar and Navigation Engineering

39. ELEC4600 Radar and Navigation Engineering
Radar Cross Section
Radar Cross Section (RCS)
To simplify things the radar range equation assumes that a target with cross sectional area σ absorbs all of the incident power and reradiates it uniformly in all directions.
This, of course, is not true
When the radar pulse hits a target the energy is reflected and refracted in many ways depending on
the material it is made of
Its shape
Its orientation with respect to the radar
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ELEC4600 Radar and Navigation Engineering

40. ELEC4600 Radar and Navigation Engineering
END
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ELEC4600 Radar and Navigation Engineering