1

Detection of Signals in Noise

Receiver Thermal NoiseNoise is the unwanted electromagnetic energy that interferes with the ability of a receiver to detect the wanted signalNoise may enter the receiver through the antennaNoise may also be generated by the thermal motion of the conduction electrons in the ohmic portions of the receiver inputstages. This is known as thermal or Johnson noiseThe 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)β – Receiver Noise Bandwidth (Hz)

WβoN kTP =

2

Receiver Noise FigureThe noise power in practical receivers is always greater than that which can be accounted by thermal noise aloneThe 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 β==

dBW )(log10 10 dBodB NFkTN += β

Detected NoiseConsider a typical radar front end that consists of an antenna followed by a wide band amplifier, a mixer that down converts the signal to an intermediate frequency (IF) where it is further amplified and filtered (bandwidth βIF )This is followed by an envelope detector and further filtering (bandwidth βV = βIF/2).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.ψo - variance of the noise voltage.

oo

vvpψπψ 2

exp2

1)(2−

=

3

Gaussian Noise

Rayleigh EnvelopeIf 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 sumR = sqrt(I2 + Q2)

oo

RRRpψψ 2

exp)(2−

=

4

Rayleigh Distributed Envelope

Rayleigh Envelope for Ultrasound Example

5

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 ψo = 1Small changes in the threshold (in the tail) results in large changes in the Pfa 0.0000045

0.00033540.011130.1352

PfaVt

fao

t

oV ot PVdRRRRVob

t

=−

=−

=∞<< ∫∞

ψψψ 2exp

2exp)(Pr

22

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

∑=

∞→=

N

kkNfa T

NT

1

1lim

6

False Alarm ProbabilityThe false alarm probability could also be defined as the ratio of the time that the envelope is above the threshold to the total timeThe average duration of a noise pulse tk ≈ 1/β

For a bandwidth β = βif the false alarm time is therefore

βfaavek

avekN

kk

N

kk

fa TTt

T

tP 1

1

1 ===

∑

∑

=

=

o

t

IFfa

VTψβ 2

exp1 2

=

Duration of a Noise Pulse for Ultrasound Example

βfil = 2kHz thereforeτ ≈ 1/βfil = 0.5ms

7

Probability of DetectionConsider that a sine wave with amplitude A is present along with the noise at the input to the IF filterThe frequency of the sine wave is equal to the centre frequency of the IF filterIt was shown by Rice that the signal at the output of the envelope detector will have the following PDF (a Riceandistribution)

Io(Z) is the modified Bessel function order zero with argument Z. For large Z an asymptotic expansion for Io(Z)

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

oo

oos

RAIARRRpψψψ 2

exp)(22

⎟⎠⎞

⎜⎝⎛ ++≈ ...

811

2)(

ZZeZI

Z

o π

PDFs for Noise and Signal + Noise in Ultrasound Example

8

PDFs for Noise and Signal + Noise

Detection ProbabilityDetection probability is determined by integration

Unfortunately this cannot be evaluated in a closed form so numerical integration techniques or a series expansion must be usedFortunately this has been done for us already by North

where erfc is the complementary error function

∫ ∫∞ ∞

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ +−==

t tV V oo

oosd dRRAIARRdRRpp

ψψψ 2exp)(

22

( )5.0ln21

+−−= SNRPerfcP fad

9

Detection and False Alarm Probabilitiesfor Envelope

Detection

Detection Loss due to Non Coherent Detection

)1(3.2)1()1(

SNRSNRCx

−≈

Cx(1) – Loss in SNR (not dB)SNR(1) – Pre detector SNR needed to achieve the required Pd and Pfa

10

The Matched FilterThis 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 SNRThe 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 NE

NS 2ˆ

=⎟⎟⎠

⎞⎜⎜⎝

⎛

Matched Filter cont….The energy received is the product of the received power S and the pulse duration τ

And the noise power density is the received noise power N divided by the bandwidth βIF

Substituting for the peak signal power to average noise

τSE =

IFo

NNβ

=

τββτ

IFinIFout

NS

NS

NS 22ˆ

⎟⎠⎞

⎜⎝⎛==⎟

⎟⎠

⎞⎜⎜⎝

⎛

11

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

τβ IFinout N

SNS

⎟⎠⎞

⎜⎝⎛≈⎟

⎠⎞

⎜⎝⎛

Matched and Non Matched Filters

0.50.6725 cascaded tuned cctsRectangular

0.560.6132 cascaded tuned cctsRectangular

0.880.4Single tuned circuitRectangular

00.44GaussianGaussian

0.390.72RectangularGaussian

0.490.72GaussianRectangular

0.851.37RectangularRectangular

Loss in SNR (dB)

Optimum β.τ

Filter ShapeInput Pulse Shape

12

Integration of Pulse TrainsThe relationships developed earlier between SNR, Pd and Pfa apply to a single pulse only.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. Thenumber can be calculated using the following formula:

where nb – Hits per scanθb – Azimuth beamwidth (deg)θs – Azimuth scan rate (deg/s)ωm – Azimuth scan rate (rpm)

For a long-range ground based 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, and it can be achieved in many ways some of which were discussed in an earlier lecture.

m

pb

s

pbb

ffn

ωθ

θθ

6==

&

Effect of Integration on PDF’s

Note that though the mean values of both the Noise and Signal+Noiseremain unchanged, the variance decreasesThis results in a reduction of the

required single pulse SNR to achieve a particular Pd and Pfa

Decreasing SNR

13

Integration EfficiencyWith integration, the required SNR decreases as a function of the number of samples integratedHowever as the single pulse SNR decreases, detector losses increase which result in reduced integration efficiency

where: EI(n) – Integration efficiencySNR(1) – Single pulse SNR required to produce a specific Pd if there is no integration.SNR(n) – Single pulse SNR required to produce a specific Pd if n pulses are integrated perfectly.

)()1()(nnSNR

SNRnEi =

The improvement in SNR if n pulses are integrated post detection is nEi(n). This is also the effective number of pulses integrated

nEi(n) ≈ n0.8

14

Integration Loss

Detection of Fluctuating SignalsAll moving targets (with the exception of the sphere) will produce echoes whose RCS changes with timeTo account for these fluctuations, both the PDF and the correlation properties of the target must be knownIdeally these should be measured for each target, however this is often not feasibleA practical alternative is to postulate a reasonable model for target fluctuations and the analyse the effects mathematicallyFour fluctuation models have been proposed by Swerling for this purpose

15

Swerling Fluctuation ModelsSwerling 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 σav – 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

pσσ

σσ −

= exp1)(

avav

pσσ

σσσ 2exp4)( 2

−=

Additional Single Pulse SNR Required

The single pulse SNR to achieve a particular Pd is higher for a fluctuating target if Pd>0.4If 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

16

Effect of Fluctuations on Integration

Constant False Alarm RateAs was shown in the table, the false alarm rate is very sensitive to the detection threshold voltageComponent aging and changes in background mean that a fixed detection threshold is not practicalAdaptive techniques that maintain a constant false alarm rate irrespective of the circumstances are called Constant False Alarm Rate (CFAR) processorsFor aircraft this is not a problem as the area around the target is generally clear, and good background statistics can be obtainedFor ground targets where the background is determined from clutter statistics, the terrain may not be homogeneous, and so additional processing is requiredCFAR losses decrease with the number of cells used from 3.5dB for 10cells to 0.7dB for 40cellsCFAR losses decrease with pulses integrated for a 10cell averagewith 10 pulses integrated it is 0.7dB decreasing to 0.3dB for 100 pulses

17

Cell Averaging CFAR Options

Area CFAR used in imaging or scanning systemsRange CFAR used by pencil beam radarsAzimuth CFAR perimeter protection radar

Cell Averaging (CA) CFAR Processor

1 2 3 48 49 50EnvelopeDetector

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

18

Cell Averaging CFAR Simulation

Constant Noise Floor

Sloping, 1/Rn Noise Floor

Air Traffic Control Radar PerformanceBand: LFrequency: 1250 to 1350MHzPeak Power: 5MWAntenna size: 12.8x6.7mAntenna Gain:36dB (lower beam)

34.5dB (upper)Beam shape: Cosec2

Elev beam: 4° Cosec2 to 40°Azim beam: 1.25°Scan: MechanicalScan Rate: 6rpmPRF: 360ppsPRF Stagger: QuadruplePulse width: 2μsNoise 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

19

Calculate the PfaAssume 2 cascaded bandpass sections for the matched filter. Loss = 0.56dB βτ = 0.613For τ = 2μs the IF bandwidth β = 306kHzTfa = 9hrs = 32400sec

1033 10

10306104.3211 −=

×××==

βfafa T

P

Required SNR

For Pd = 90% and Pfa = 10-10

SNR(1) = 15.2dB

20

Fluctuating Target

For an aircraft target use Swerling 1 or 2 require additional 8dB required

Pulse IntegrationHits per scan

Use n = 10

From the graph the Ii(n) = 15dB

5.126636025.1

6=

××

==m

pbb

fn

ωθ

21

Required Single Pulse SNR

For Pd = 0.9 and Pfa = 10-10 15.2dBAdditional for fluctuating target +8dBIntegration improvement Ii(n) -15dB

Single pulse SNR required 8.2dB

Losses in SNR

Transmitter Line LTX = 2dB incorporated into Tx powerReceiver Line LRX = 2dB incorporated into noise figure

1D scanning loss 1.6dBMatched filter loss 0.56dBCFAR loss 0.7dBMisc. additional losses 1.3dB

Total Loss 4.16dB

22

Transmitted Power

10log10(5x106) 67dBWLine loss 2dB

Transmit Power Pt 65dBW

Receiver Noise

N = 10log10(KTβ)+NF+LRX

10log10(1.38x10-23 x 290 x 306x103) -149dBWNoise Figure NF 4dBReceiver line loss LRX 2dB

Total Receiver Noise -143dBW

23

Apply the Radar Range Equation to Determine the Received Power

Pt 65dBWGdB 36dB (lower beam)LdB 4.16dBσdB 10log10(1) 0dBm2

Const -43.7dB

Pr = 65 + 2x36 - 45.7 - 4.16 + 0 - 40log10R

Pr = 87.14 – 40log10R

RLGPP dBdBdBtr 103

2

10 log40)4(

log102 −+−++= σπλ

Detection Range

For a single pulse SNR of 8.2dB, the received power must be 8.2dB above the noise floorSmin = N + SNR = -143 + 8.2 = -134.8dBW

Smin = Pr

-134.8 = 87.14 – 40log10RSolve for R

R = 10(87.14+134.8)/40 = 353580m (353.6km)

24

Atmospheric AttenuationThe clear air attenuation αdB 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 rangeThe radar range equation that includes this range dependent term is best solved graphically using MATLAB

Graphical Solution

≈310km

25

RGCALC

Radar performance analysis based on work by Blake has been available for many yearsFor the Enroute 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 βτ=1, and so their bandwidth is larger, so their noise floor is higher