it is a question about Sensors and Signals.
1
Detection of Signals in Noise (Ch 9)
Receiver Thermal Noise • Noise is the unwanted electromagnetic energy that interferes with
the ability of a receiver to detect the wanted signal • Noise may enter the receiver through the antenna • Noise may also be generated by the thermal motion of the
conduction electrons in the ohmic portions of the receiver input stages. This is known as thermal or Johnson noise
• The noise power is expressed in terms of the temperature, To of a matched resistor at the input to the receiver
where k – Boltzmann’s Constant (1.38x10-23 J/K) To – System Temperature (usually 290K) – Receiver Noise Bandwidth (Hz)
WoN kTP
2
Receiver Noise Figure • The noise power in practical receivers is always greater
than that which can be accounted by thermal noise alone • The total noise, N, at the output of the receiver can be
considered to be equal to the noise power from an ideal receiver scaled by a factor called the noise figure, NF
• The noise power can also be written as a log ratio relative to 1W
• The receiver noise power can also be described in terms of a temperature Tsys=To(NF-1)
W.NFkTNFPN oN
dBW )(log10 10 dBodB NFkTN
Noise in a Conventional Radar Front End • A typical radar front end that consists of:
• Antenna • Wide band amplifier (defines overall Noise Figure) • Mixer that down converts the signal to an intermediate
frequency (IF) • Second amplifier • Matched filter (bandwidth IF ) • Envelope detector • Possibly further filtering (bandwidth V = IF/2).
3
Calculation of the Noise Figure for a Cascaded Network
• The total noise figure is made up of contributions from each of the stages weighted by the gain of the previous stage
21
3
1
2 1
11
GG
NF
G
NF NFNF
NF1dB=0.5dB G1dB=-0.5dB
NF2dB=1.5dB G2dB=20dB
NF3dB=6dB G3dB=-6dB
Calculation of the Noise Figure for a Cascaded Network 2
• NF1 = 10 0.5/10 = 1.122
• G1 = 10 -0.5/10 = 0.8913
• NF2 = 10 1.5/10 = 1.4125
• G2 = 10 20/10 = 100
• NF3 = 10 6/10 = 3.98
1.4125-1 3.98 1.122 ... 1.122 0.4628 0.0447 ... 1.6295
0.8913 0.8913 100 NF
21
3
1
2 1
11
GG
NF
G
NF NFNF
NF1 G1
NF2 G2
NF3
10=10log 1.6295=2.12dBdBNF
4
Noise Figure for a Cascaded mmWave Radar Receiver Network
• Determine the total noise figure for the mmWave radar where there is no LNA prior to the mixer
2 3 4 1
1 1 2 1 2 3
1 1 1NF NF NF NF NF
G G G G G G
NF1dB=0.5dB G1dB=-0.5dB
NF3dB=1.5dB G3dB=20dB
NF2dB=6dB G2dB=-6dB
NF4dB=0.5dB G4dB=-0.5dB
Thermal Noise Input • The noise entering the IF filter is assumed to be
Gaussian (as it is thermal in nature) with a probability density function (PDF) given by
where p(v)dv - probability of finding the noise voltage V between v and v+dv.
o - variance of the noise voltage.
oo
v vp
2 exp
2
1 )(
2
5
Gaussian Noise
Detected Noise Envelope (Rayleigh)
• If the Gaussian noise is passed through a narrow band filter (one whose bandwidth is small compared to the centre frequency), then the PDF of the envelope of the noise voltage output can be shown to be
Where R – Amplitude of the envelope at the filter output
Note: A good way to generate the Rayleigh distribution is to generate two independent Gaussian distributions I,Q and take the vector sum
R = sqrt(I2 + Q2)
oo
RR Rp
2 exp)(
2
6
Rayleigh Distributed Envelope
Rayleigh Envelope for Ultrasound Example
7
Lidar Noise for Direct Detection
• Noise contributions • Thermal
• Shot
• Avalanche
• 1/f
Total
shunt therm
R
fkT i
4
fiiei darkphotoshot )(2
*
2/1
D
fA NEP d
Sonar Noise
fN therm 10log2015
8
Humpback Whale Spectrogram
Effect of Noise on Probability of Detection and Probability of False
Alarm
9
Probability of False Alarm Pfa
• A false alarm occurs whenever the noise voltage exceeds a threshold Vt
• The probability of this occurring is determined by integrating the PDF as shown
• Values for this integral are as shown in the table for o = 1
• Small changes in the threshold (in the tail) results in large changes in the Pfa
Vt Pfa 2 0.135
3 0.0111
4 0.000335
5 0.000004
fa o
t
oV o t P
V dR
RR RVob
t
2 exp
2 exp)(Pr
22
False Alarm Time
• The average time interval between threshold crossings (+ve crossing slope only) is called the false alarm time, Tfa
N
k k
N fa T
N T
1
1 lim
10
Duration of a Noise Pulse: Ultrasound Example
• Average duration of a noise pulse tk = ≈ 1/
fil = 2kHz therefore theoretically = 0.5ms
False Alarm Probability
• The false alarm probability could also be defined as the ratio of the time that the envelope is above the threshold to the total time
• The average duration of a noise pulse tk ≈ 1/
• For a bandwidth = if the false alarm time is therefore
1
1
1 / 1
N
k kk ave
fa N k fa faave
k k
t t
P T T T
T
o
t
IF fa
V T
2 exp
1 2
11
Example: False Alarm Time
• Determine the false alarm time for a surveillance radar with • noise variance o = 1V2
• pulse width = 0.3s • Pfa = 10
-10
• Determine the “matched” bandwidth
• Using equation (9.13)
3.33MHz 103.0
11 6
if
3 10 6
1 1 3 10 s [50 min]
10 3.33 10 fa
fa if
T P
Example: Threshold Voltage
• For the previous example, find the threshold voltage, Vt required to achieve a Pfa = 10
-10
• From equation (9.14)
• Therefore o
t fa
V P
2 exp
2
V79.610ln12ln2 10 faot PV
12
Probability of Detection • A sine wave (signal) with amplitude A is present along with
the noise at the input to the matched filter • Signal frequency is equal to the centre frequency of the
matched filter • Rice showed that the signal at the output of the envelope
detector will have the following PDF (a Ricean distribution)
• Io(Z) is the modified Bessel function order zero with argument Z. For large Z an asymptotic expansion for Io(Z)
o o
oo s
RA I
ARR RP
2 exp)(
22
... 8
1 1
2 )(
ZZ
e ZI
Z
o
PDFs for Noise and Signal + Noise in Ultrasound Example
13
PDFs for Noise and Signal + Noise
Vt for Pfa = 10 -10
Detection Probability
• Detection probability is determined by integration
• Unfortunately this cannot be evaluated in a closed form so numerical integration techniques or a series expansion must be used
• Fortunately this has been done for us already by North
where erfc is the complementary error function
t tV V o o
oo sd dR
RA I
ARR dRRpP
2 exp)(
22
5.0ln 2
1 SNRPerfcP fad
14
Detection and False Alarm Probabilities for Envelope
Detection
N. Currie and C. Brown, Principles and Applications of Millimeter-Wave Radar: Artech House, 1987.
Detection Probability Example
Vt for Pfa = 10 -10
Exercise: Estimate the Pd from the area to the right of the S+N curve, then confirm using the graphs in Figure 9.8
15
Activity: Pd and Pfa with Threshold
• Setup the pulsed radar to obtain noisy returns from a target across the room
• Set the detection threshold and note the number of detections and false alarms during a given interval
• Adjust the threshold and repeat a number of times
• Calculate and plot the Pd and Pfa curve for this SNR
• Alter the SNR by changing the range or adjusting the target characteristics
• Repeat the exercise
Detection Loss due to Non Coherent (Envelope) Detection
SNR
SNR C
3.2
C – Loss in SNR (not dB) SNR – Pre detector SNR needed to achieve the required Pd and Pfa
C is small for reasonable SNR
M. Skolnik, Introduction to Radar Systems, 2nd ed.: McGraw-Hill Kogakusha, 1980.
C = 10log10((2.3-0.1)/0.1) = 13.4dB
16
The Matched Filter • This SNR can achieve its maximum value when the IF
filter is matched to the signal. • This should not be confused with matching in circuit
theory which maximises power transfer not SNR • The peak signal to (average) noise power ratio of the
output response of the matched filter is equal to twice the received signal energy E divided by the single-sided noise power per Hz, No
where S – Peak instantaneous signal power seen during the matched filter response to a pulse (W) N – Average noise power (W) E – Received signal energy (J) No – Single sided noise power density (W/Hz)
oout N
E
N
S 2ˆ
Matched Filter cont….
• The energy received is the product of the received power S and the pulse duration
• And the noise power density is the received noise power N divided by the bandwidth IF
• Substituting for the peak signal power to average noise
SE
IF o
N N
IF inIFout
N
S
N
S
N
S 2
2ˆ
17
Matched filter cont…..
• When the bandwidth of the signal at the IF is small compared to the centre frequency then the peak power is approximately twice the average power in the received pulse (triangular pulse approximation)
• The output SNR (average power to average noise) is therefore
IF inout N
S
N
S
Matched and Non Matched Filters
Input Pulse Shape
Filter Shape Optimum .
Loss in SNR (dB)
Rectangular Rectangular 1.37 0.85
Rectangular Gaussian 0.72 0.49
Gaussian Rectangular 0.72 0.39
Gaussian Gaussian 0.44 0
Rectangular Single tuned circuit 0.4 0.88
Rectangular 2 cascaded tuned ccts 0.613 0.56
Rectangular 5 cascaded tuned ccts 0.672 0.5
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Coherent (Synchronous) Detection
• Relies on a phase detector to compare the echo signal with a reference generated by the transmitter
• Amplitude of the output product is proportional to the cosine of the phase difference between the signals
• The use of a second phase detector operating in quadrature eliminates blind phases (where cos = 0)
Single Channel Coherent Detection of a Moving Target
19
Integration of Pulse Trains • As a search radar scans past a target, it will remain in
the beam sufficiently long for more than one pulse to hit the target. The number can be calculated using the following formula:
where nb – Hits per scan b – Azimuth beamwidth (deg) s – Azimuth scan rate (deg/s) m – Azimuth scan rate (rpm)
• For a radar with an azimuth beamwidth of 1.5, a scan rate of 5rpm and a pulse repetition frequency of 30Hz, the number of pulses returned from a single point target is 15.
• The process of summing all these hits is called integration
m
pb
s
pb b
ff n
6
Effect of Integration on PDF’s
• Note that though the mean values of both the Noise and Signal+Noise remain unchanged, the variance decreases
• This results in a reduction of the required single pulse SNR to achieve a particular Pd and Pfa
20
Integration Efficiency
• With integration, the required SNR decreases as a function of the number of samples integrated
• However as the single pulse SNR decreases, detector losses increase which result in reduced integration efficiency
where: En – Integration efficiency
SNR1 – Single pulse SNR required to produce a specific Pd if there is no integration.
SNRn – Single pulse SNR required to produce a specific Pd if n pulses are integrated perfectly.
n n
nSNR
SNR E 1
The improvement in SNR if n pulses are integrated post detection is nEn. This is also the effective number of pulses integrated
nEn ≈ n 0.8
39
M. Skolnik, Introduction to Radar Systems, 2nd
ed.: McGraw-Hill Kogakusha, 1980.
21
Integration Loss
n n
E L
1 log10 10
Valid for Pd > 0.7 and Pfa < 10 -4
10log10(39/100) = 4.1dB
M. Skolnik, Introduction to Radar Systems, 2nd
ed.: McGraw-Hill Kogakusha, 1980.
Detection of Fluctuating Signals • All moving targets (with the exception of the sphere) will produce echoes
whose RCS changes with time
• To account for these fluctuations, both the PDF and the correlation properties of the target must be known
• Ideally these should be measured for each target, however this is often not feasible
• A practical alternative is to postulate a reasonable model for target fluctuations and the analyse the effects mathematically
• Four fluctuation models have been proposed by Swerling for this purpose
Mahafza, B. (2000). Radar Systems Analysis and Design Using MATLAB. Boca Raton, London, New York, Washington DC, Chapman & Hall/CRC
22
Swerling Fluctuation Models • Swerling 1&2 are indicative of a complex target made up of many
(>5) scatterers of equal amplitude, such as aircraft • Swerling 1: Echo pulses constant for the target over one scan (search
radar) but uncorrelated from scan to scan. With av – average RCS over all scans
• Swerling 2: The PDF is as for case 1, but the fluctuations are independent from pulse to pulse
• Swerling 3&4 are indicative of a target with one large scatterer and many small scatterers
• Swerling 3: The fluctuations are independent from scan to scan, but the PDF has changed
• Swerling 4: The fluctuations are independent from pulse to pulse with a PDF as for case 3
avav
p
exp
1 )(
avav
p
2
exp 4
)( 2
Additional Single Pulse SNR Required
The single pulse SNR to achieve a particular Pd is higher for a fluctuating target if Pd>0.4 If Pd<0.4 the system takes advantage of the fact that the fluctuating target will occasionally present an echo higher than average, so the required SNR is lower
M. Skolnik, Introduction to Radar Systems, 2nd ed.: McGraw-Hill Kogakusha, 1980.
23
Effect of Fluctuations on Integration
M. Skolnik, Introduction to Radar Systems, 2nd
ed.: McGraw-Hill Kogakusha, 1980.
Constant False Alarm Rate • As was shown in the table, the false alarm rate is very sensitive to
the detection threshold voltage • Component aging and changes in background mean that a fixed
detection threshold is not practical • Adaptive techniques that maintain a constant false alarm rate
irrespective of the circumstances are called Constant False Alarm Rate (CFAR) processors
• For aircraft this is not a problem as the area around the target is generally clear, and good background statistics can be obtained
• For ground targets where the background is determined from clutter statistics, the terrain may not be homogeneous, and so additional processing is required
• CFAR losses decrease with the number of cells used from 3.5dB for 10cells to 0.7dB for 40cells
• CFAR losses decrease with pulses integrated for a 10cell average with 10 pulses integrated it is 0.7dB decreasing to 0.3dB for 100 pulses
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Cell Averaging CFAR Options
Area Range Angle
Cell Averaging (CA) CFAR Processor
1 2 3 48 49 50 Envelope Detector
Average
+
- k
Threshold
Decision +
-
Comparator
•Moving average around the cell-under-test determines the local statistics which are then used to determine whether a target is present or not.
•A number of “guard” cells around the test cell accommodate any leakage from that cell
25
Cell Averaging CFAR Simulation
Constant Noise Floor
Sloping, 1/Rn Noise Floor
Air Traffic Control Radar Performance • Band: L • Frequency: 1250 to 1350MHz • Peak Power: 5MW • Antenna size: 12.8x6.7m • Antenna Gain: 36dB (lower beam) • 34.5dB
(upper) • Beam shape: Cosec2
• Elev beam: 4 Cosec2 to 40 • Azim beam: 1.25 • Scan: Mechanical • Scan Rate: 6rpm • PRF: 360pps • PRF Stagger: Quadruple • Pulse width: 2s • Noise Figure: 4dB
Calculate the detection range for a 1m2 target if the detection probability Pd=0.9 and the mean time between false alarms is 9 hours
http://www.123rf.com/photo_2043149_air-traffic-control- radar-tower-against-blue-sky-with-clouds.html
26
Calculate the Pfa • Assume 2 cascaded bandpass sections for the matched
filter. Loss = 0.56dB = 0.613 • For = 2s the IF bandwidth = 306kHz • Tfa = 9hrs = 32400sec
10 33
10 10306104.32
11
fa
fa T
P
Required SNR
For Pd = 90% and Pfa = 10
-10
SNR1 = 15.2dB
N. Currie and C. Brown, Principles and Applications of Millimeter-Wave Radar: Artech House, 1987.
27
Fluctuating Target
For an aircraft target use Swerling 1 or 2 require additional 8dB required
Pulse IntegrationHits per scan
Use nb = 10
From the graph the I10 ~ 15dB
5.12 66
36025.1
6
m
pb b
f n
28
Required Single Pulse SNR
• For Pd = 0.9 and Pfa = 10 -10 15.2dB
• Additional for fluctuating target +8dB
• Integration improvement I10 -15dB
• Single pulse SNR required 8.2dB
Losses in SNR
• Transmitter Line LTX = 2dB incorporated into Tx power • Receiver Line LRX = 2dB incorporated into noise figure
• 1D scanning loss 1.6dB • Matched filter loss 0.56dB • CFAR loss 0.7dB • Misc. additional losses 1.3dB
• Total Loss 4.16dB
29
Transmitted Power
• 10log10(5106) 67dBW • Line loss 2dB
• Transmit Power Pt 65dBW
Receiver Noise
• N = 10log10(KT)+NF+LRX • 10log10(1.3810-23 290 306103) -149dBW • Noise Figure NF 4dB
• Receiver line loss LRX 2dB
• Total Receiver Noise -143dBW
30
Apply the Radar Range Equation to Determine the Received Power
• Pt 65dBW
• GdB 36dB (lower beam)
• LdB 4.16dB
• dB 10log10(1) 0dBm2
• Const -45.7dB
Pr = 65 + 2 36 - 45.7 - 4.16 + 0 - 40log10R
Pr = 87.14 – 40log10R
RLGPP dBdBdBtr 103
2
10 log40 )4(
log102
Detection Range
• The received SNR is SNRrec = Pr + N = 87.14 – 40log10R + 143
= 230.1 – 40log10R
• This must equal the single pulse SNR
8.2 = 230.1 – 40log10R
• Solve for R
R = 10(230.1-8.2)/40 = 352,777m (352.8km)
31
Atmospheric Attenuation
• The clear air attenuation dB at L-Band is about 0.003dB/km (one way)
• Over 350km the total attenuation will be 2.1dB which will make a significant difference to the detection range
• The radar range equation that includes this range dependent term is best solved graphically using MATLAB
k mtr RRLGPP
2log40 )4(
log102 103
2
10
Graphical Solution
353km
320km
32
RGCALC
• Radar performance analysis based on work by Blake has been available for many years
• For the radar it produces the following results
Note that the predicted range using RGCALC for Swerling 2, and that derived previously, differ slightly because RGCALC assumes that =1, and so their bandwidth is larger, so their noise floor is higher