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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)

WoN kTP 

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).

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

2 1

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         



2 1

NF NFNF

 

 

NF1 G1

NF2 G2

NF3

10=10log 1.6295=2.12dBdBNF

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.

v vp

 2 exp

1 )(

2 

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)

RR Rp

 2 exp)(

2 

Rayleigh Distributed Envelope

Rayleigh Envelope for Ultrasound Example

Lidar Noise for Direct Detection

• Noise contributions • Thermal

• Shot

• Avalanche

• 1/f

 Total

shunt therm

fkT i

 

fiiei darkphotoshot  )(2

  *

2/1

fA NEP d

 

Sonar Noise

fN therm 10log2015

Humpback Whale Spectrogram

Effect of Noise on Probability of Detection and Probability of False

Alarm

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

oV o t P

V dR

RR RVob

 

  

 2 exp

2 exp)(Pr

False Alarm Time

• The average time interval between threshold crossings (+ve crossing slope only) is called the false alarm time, Tfa

 

 

k k

N fa T

N T

1 lim

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

k kk ave

fa N k fa faave

k k

t t

P T T T

 



    



IF fa

V T

 2 exp

1 2 

Example: False Alarm Time

• Determine the false alarm time for a surveillance radar with • noise variance o = 1V2

• pulse width  = 0.3s • 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 

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)(

  

  

 ... 8

1 1

2 )(

e ZI

o 

PDFs for Noise and Signal + Noise in Ultrasound Example

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)(

 5.0ln 2

1  SNRPerfcP fad

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

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

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

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

S 2

2ˆ   

  

  

   



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   

  

  

  

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

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

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

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

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

M. Skolnik, Introduction to Radar Systems, 2nd

ed.: McGraw-Hill Kogakusha, 1980.

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

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.

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

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

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: 2s • 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

Calculate the Pfa • Assume 2 cascaded bandpass sections for the matched

filter. Loss = 0.56dB  = 0.613 • For  = 2s the IF bandwidth  = 306kHz • Tfa = 9hrs = 32400sec

10 33

10 10306104.32

11  

 fa

fa T

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.

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

 

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

Transmitted Power

• 10log10(5106) 67dBW • Line loss 2dB

• Transmit Power Pt 65dBW

Receiver Noise

• N = 10log10(KT)+NF+LRX • 10log10(1.3810-23  290  306103) -149dBW • Noise Figure NF 4dB

• Receiver line loss LRX 2dB

• Total Receiver Noise -143dBW

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

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)

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

10 

Graphical Solution

353km

320km

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