# Chapter 10. Detection of Signals in Noise

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2 ise Prbability Density Functins Cnsider a typical radar frnt-end that cnsists f an antenna fllwed by a wide band amplifier, a mixer that dwn cnverts the signal t an intermediate frequency (IF) where it is further amplified and filtered (bandwidth β IF ). This is fllwed by an envelpe detectr and further filtering (bandwidth β V = β IF /). The nise entering the IF filter is assumed t be Gaussian (as it is thermal in nature) with a prbability density functin (PDF) given by 1 v p( v) = exp, (10.3) πψ ψ where p(v)dv - prbability f finding the nise vltage v between v and v+dv, ψ - variance f the nise vltage. If Gaussian nise is passed thrugh a narrw band filter (ne whse bandwidth is small cmpared t the centre frequency), then the PDF f the pst-detectin envelpe f the nise vltage utput can be shwn t be R R p( R) = exp, (10.4) ψ ψ where R is the amplitude f the envelpe f the filter utput. This has the frm f the Rayleigh prbability density functin Figure 10.1: Amplitude distributins f thermal nise pre and pst detectin

3 Prbability f False Alarm A false alarm ccurs whenever the nise vltage exceeds a defined threshld vltage, V t, as illustrated in the figure belw. The prbability f this ccurring is determined by integrating the PDF as shwn Pr b( V t R R V < R < ) = exp dr = exp ψ ψ ψ Vt t = P fa. (10.5) It can be seen that the average time interval between crssings f the threshld, called the false alarm time, T fa, can be written as T fa 1 = lim Tk, (10.6) k = 1 where T k Time between crssings f the threshld V t by the nise envelpe (when the slpe f the crssing is psitive). The false alarm prbability culd als have been defined as the rati f the time that the envelpe is abve the threshld t the ttal time as shwn graphically in the figure belw P fa = t t = 1, (10.7) T k k 1 k ave = = Tk T ave k k = 1 faβ where t k and T k are defined in the figure, and the average duratin f a nise pulse is the reciprcal f the bandwidth β. Envelpe f ise Vltage Vt Ψο tk Tk Threshld Tk+1 tk+ tk+1 RMS ise Vltage Time Figure 10.: Receiver utput vltage illustrating false alarms due t nise Fr a bandwidth β = β IF, the false alarm time is just T fa 1 Vt = exp. (10.8) β ψ IF The false alarm times f practical radars must be very large (usually a cuple f hurs), s the prbability f false alarm must be very small, typically P fa < 10-6.

4 Prbability f Detectin Cnsider that a sine wave with amplitude, A, is present alng with the nise at the input t the IF filter. The frequency f the sine wave is equal t the centre frequency f the IF filter. It is shwn by Rice that the signal at the utput f the envelpe detectr will have the fllwing PDF (knwn as a Rician distributin) R R + A p = s( R) exp ψ ψ I RA. (10.9) ψ I (Z) mdified Bessel functin f rder zer and argument Z. It can be shwn that fr large Z, an asympttic expansin fr I (Z) is Z e 1 I( Z) (10.10) πz 8Z The prbability that the signal will be detected is the same as the prbability that the envelpe, R, will exceed the threshld, V t is p d = p ( R) dr = R R + A exp ψ ψ I s Vt V t RA dr. (10.11) ψ Unfrtunately, this cannt be evaluated in a clsed frm and s numerical techniques r a series apprximatin must be used. Hwever, this has already been dne, and tables and a series f curves have been prduced. In terms f the PDFs fr the nise and the signal plus nise vltages, the detectin and false alarm prcess is shwn graphically in the figure belw. The lightly shaded area represents the P fa and the dark shaded area (including the tail cvered by the light shading) represents the P d. 0.8 Prbability Density Rayleigh ise Detectin Threshld Ricean Signal + ise SR = 15dB Amplitude (V) Figure 10.3: PDF s f nise and signal plus nise fr P fa = 10 -

5 87 Figure 10.4: Detectin prbability as a functin f signal t nise rati with false alarm prbability as a parameter A typical radar system will perate with a detectin prbability f 0.9 and a prbability f false alarm f The required signal t nise rati can be read directly ff the graph as 13.dB. te that this is fr a single pulse f a steady sinusidal signal in Gaussian nise with n detectin lsses. % Pd and Pfa Determined umerically by running a time dmain simulatin % Generate nise and signals fr SR=13dB a = (1: ); x = randn(size(a)); sigi = 6.31*sin(a/1000); y = randn(size(a)); sigq = 6.31*cs(a/1000); % Determine the envelpe c = sqrt(x.^+y.^); csig = sqrt((x+sigi).*(x+sigi)+(y+sigq).*(y+sigq)); %Plt the distributins edges=(0:0.1:1); n=histc(c,edges); ns=hist(csig,edges); plt(edges,n,edges,ns); grid % Determine the prbabilities fr different threshlds vt=5; % Lk fr nise peaks abve the threshld nfa = find(c>vt);

6 88 pfa = length(nfa)/ % Lk fr S+ peaks abve the threshld nd = find(csig>vt); pd = length(nd)/ Detectr Lss Relative t an Ideal System An envelpe detectr is used by a radar system when the phase f the received pulse is unknwn. This is called nn-cherent detectin, and it results in a slightly higher SR requirement than the curves abve shw. Figure 10.5: Envelpe detectr lss as a functin f signal t nise rati This lss factr C x is apprximately C x SR(1).3 (1), (10.1) SR(1) where C x (1) Lss in SR, SR(1) The pre detectr single pulse SR required t achieve a particular P d and P fa. The graph shws that fr gd SR, the detectr lss is very small. In the case where the P d = 0.9 and the P fa = 10-6 it is nly abut 0.4dB. This effect is knwn as small signal suppressin as it becmes much mre prnunced as the SR decreases. One f the advantages f cherent detectin is that it has zer respnse t the quadrature nise cmpnent, whereas, this cmpnent is translated int phase mdulatin after envelpe detectin with the result that the effective SR in the latter case is degraded. Hwever the phase mdulatin is very small fr high SR, and few advantages can be gained by cherent detectin fr SR > 10dB (C x = 1dB) ver envelpe detectin.

7 The Matched Filter T achieve the best pssible SR, the characteristics f the IF filter must be matched t thse f the signal pulse. The peak signal t (average) nise pwer rati f the utput respnse f the matched filter is equal t twice the received signal energy, E, divided by the single-sided nise pwer per Hz, S ˆ E =, (10.13) ut where Ŝ Peak instantaneus signal pwer seen during the matched filter respnse t a pulse (W), Average nise pwer (W), E Received signal energy (J), Single sided nise pwer density (W/Hz). The received energy is the prduct f the received pwer, S, as determined by the range equatin and the pulse duratin, τ E = Sτ, (10.14) And the nise pwer density is the received nise pwer,, divided by the bandwidth, β IF =. (10.15) β IF Substituting int (10.13) Sˆ ut = Sτ B IF = S in B τ. (10.16) IF When the bandwidth f the signal at IF is small cmpared t the centre frequency then the peak pwer is apprximately twice the average pwer in the received pulse. S the utput SR is S ut S in B IFτ. (10.17) The matched filter shuld nt be cnfused with the circuit thery cncept f matching that maximises pwer transfer rather than SR. In practise a matched filter implementatin is ften hard t achieve exactly, s cmprmises are made as shwn in the fllwing table:

8 90 Input Signal Shape Table 10.1: Efficiency f nn-matched filters Matched Filter Characteristic Optimum B.τ Lss in SR cmpared t Matched Filter (db) Rectangular Pulse Rectangular Rectangular Pulse Gaussian Gaussian Pulse Rectangular Gaussian Pulse Gaussian (matched) Rectangular Pulse Single tuned circuit Rectangular Pulse Rectangular Pulse Tw cascaded tuned circuits Five cascaded tuned circuits Integratin f Pulse Trains The relatinships develped earlier between SR, P d and P fa apply t a single pulse nly. Hwever, it is pssible t imprve the radar perfrmance by averaging a number f returns. Fr example, as a search-radar beam scans, the target will remain in the beam sufficiently lng fr mre than ne pulse t hit it. This number, knwn as hits per scan, can be calculated as fllws n b θ f b p b p = & =, (10.18) θ s θ f 6ω m where n b Hits per scan, θ b Azimuth beamwidth (deg), & θ - Azimuth scan rate (deg/s), s ω m Azimuth scan rate (rpm). Fr a typical grund based radar with an azimuth beamwidth r 1.5, a scan rate f 5rpm and a pulse repetitin frequency f 30Hz, the number f pulses returned frm a single pint target is 15. The prcess f summing all these hits is called integratin, and it can be achieved in many ways sme f which were discussed in Chapter 5. If integratin is perfrmed prir t the envelpe detectr, it is called pre-detectin r cherent integratin, while if integratin ccurs after the detectr, it is called pst-detectin r nn-cherent integratin. Pre-detectin integratin requires that the phase f the signal be preserved if the full benefit f the summing prcess is t be achieved and because phase infrmatin is destryed by the envelpe detectr, pst-detectin integratin, thugh easier t achieve, is nt as efficient. If n pulses are perfectly integrated by a cherent integratin prcess, the integrated SR will be exactly n times that f a single pulse in white nise. Hwever, in the nn-cherent case, thugh the integratin prcess is as efficient, there are the detectr

9 91 lsses discussed earlier that reduce the effective SR at the utput f the envelpe detectr. Integratin imprves the P d and reduces the P fa by reducing the nise variance and thus narrwing the ise and Signal + ise PDFs as shwn in the figure belw Figure 10.6: Effect f integratin n signal and nise PDFs befre and after integratin Fr n pulses integrated, the single pulse SR required t achieve a given P d and P fa will be reduced. Hwever, this results in increased detectr lsses, and hence a reduced effective integratin efficiency. The integratin efficiency may be defined as SR(1) E i ( n) =, (10.19) nsr( n) where: E I (n) Integratin efficiency, SR(1) Single pulse SR required t prduce a specific P d if there is n integratin, SR(n) Single pulse SR required t prduce a specific P d if n pulses are integrated perfectly. The imprvement in SR if n pulses are integrated, pst detectin is thus ne I (n). This is the integratin imprvement factr, r the effective number f pulses integrated.

10 9 Figure 10.7: Integratin imprvement factr as a functin f pulses integrated The integratin lss in db is defined as fllws 1 L = i ( n) 10lg10. (10.0) Ei ( n) The integratin imprvement factr is nt a sensitive functin f either the P d r the P fa as can be seen by the clustering f the curves in the figure abve. Figure 10.8: Integratin lss as a functin f the number f pulses integrated

11 Detectin f Fluctuating Signals The discussin in the previus sectin assumes that the signal amplitude des nt vary frm pulse t pulse during the integratin perid. Hwever, frm the discussin f target crss sectin in Chapter 9, it is bvius that the RCS f any mving target (with the exceptin f a sphere) will fluctuate with time as the target aspect as seen by the radar changes. T prperly accunt fr these fluctuatins, bth the prbability density functin and the crrelatin prperties with time must be knwn fr a particular target and trajectry. Ideally, these characteristics shuld be measured fr a target, but this is ften impractical. An alternative is t pstulate a reasnable mdel fr the target fluctuatins and t analyse the effects mathematically. Fur fluctuatin mdels prpsed by Swerling are used: Swerling 1:Ech pulses received frm the target n any ne scan are f cnstant amplitude thrughut the scan, but uncrrelated frm scan t scan. The PDF is given by 1 σ p( σ ) = exp, (10.1) σ σ where σ av is the average crss sectin ver all target fluctuatins. av Swerling : The PDF is as fr case 1, but the fluctuatins are taken t be independent frm pulse t pulse. Swerling 3: The fluctuatins are independent frm scan t scan, but the PDF is given by 4σ σ p( σ ) = exp. (10.) σ σ av Swerling: The PDF is as fr case 3, but the fluctuatins are independent frm pulse t pulse. Swerling 5: n fluctuating av av Figure 10.9: Radar returns fr different Swerling fluctuatins

12 94 The PDF fr cases 1 and is indicative f a target with many (>5) scatterers f equal amplitude. These are typical fr cmplicated targets like aircraft, while the PDF fr cases 3 and 4 is indicative f a target with ne large scatterer and many small scatterers. As ne wuld expect, the single pulse SR required t achieve a particular P d (fr P d > 0.4) will be higher fr a fluctuating target than fr a cnstant amplitude signal. Hwever, fr P d < 0.4, the system takes advantage f the fact that a fluctuating target will ccasinally present ech signals larger than the average, and s the required SR is lwer. Figure 10.10: Effect f target fluctuatin n required signal t nise rati T cater fr fluctuating targets and integratin, a further set f curves describing the integratin imprvement factr have been develped. Figure 10.11: Integratin imprvement factr as a functin f the number f pulses integrated fr fluctuating targets

13 95 If these curves are examined in islatin, it wuld appear that the integratin efficiency E I (n) > 1 under certain cnditins. One is nt getting smething fr nthing as the single pulse SR is much higher in these cases than it wuld be fr the single pulse case. The prcedure fr using the range equatin when ne f these Swerling targets is as fllws: 1. Find the SR fr the single pulse, nn-fluctuating case that crrespnds t the P d and P fa required using the curves in Figure Fr the specific Swerling target, find the additinal SR needed fr the required P d. using the curves in Figure If n pulses are t be integrated, the integratin imprvement factr I I (n)=ne I (n) is then fund using the curves in Figure The SR(n) and ne I (n) are substituted int the range equatin alng with σ av and the detectin range fund Cnstant False Alarm Rate (CFAR) Prcessrs As can be seen frm the tail in the Rayleigh distributed nise pwer, the false alarm rate is very sensitive t the setting f the detectin threshld vltage. Changes in radar characteristics with time (ageing) and changes in the target backgrund characteristics mean that a fixed detectin threshld is nt practical and s adaptive techniques are required t maintain a cnstant false alarm rate irrespective f the circumstances. These are called Cnstant False Alarm Rate (CFAR) prcessrs. The implementatin f CFAR is nt difficult, but in general the actual perfrmance will be a functin f the target, clutter and nise statistics. Fr aircraft these are nt a prblem as the area arund the craft is generally empty s gd backgrund statistics can be btained. Hwever fr grund targets, the CFAR threshld is determined frm the clutter statistics, which may nt be hmgeneus. This is made wrse by the fact that targets (tanks etc) ften hide at the edges f clutter bundaries t reduce the prbability f detectin. Envelpe Detectr k Cmparatr + - Decisin Average Threshld Figure 10.1: Multiple cell averaging CFAR eedless t say, target detectin using the CFAR prcess intrduces additinal lsses as the statistics are incmpletely characterised. Fr example a cell averaging prcess will exhibit a 3.5dB lss (cmpared t an ideal single pulse detectr) if 10 cells are used in bradband nise r clutter with a Rayleigh PDF. This decreases t 1.5dB fr 0 cells and 0.7dB fr 40 cells. The lss decreases with increasing numbers f pulses integrated. Fr a 10 cell CFAR, with 10 pulses integrated the lss is nly 0.7dB and it reduced t nly 0.3dB fr 100 pulses integrated.

14 96 CFAR can perate acrss range cells, crss-range cells r bth, as shwn in the figure belw. Figure 10.13: Cnstant false alarm rate ptins (a) area CFAR, (b) range Only CFAR and (c) azimuth angle nly CFAR In the fllwing examples where a 5+5 pint mving average is used t determine the mean value f the signal, it can be seen that the CFAR prcess detects the tw targets withut detecting the nise. Hwever, a fixed threshld f 1.8 wuld perfrm just as well.

15 97 Figure 10.14: CFAR perfrmance fr a cnstant nise flr In many radar systems, the nise flr varies with range, as shwn in the fllwing example. In this instance a fixed threshld des nt perfrm adequately, but the CFAR prcess still des. Figure 10.15: CFAR perfrmance fr a slping nise flr

16 Air Traffic Cntrl Radar Perfrmance Type: Band: Frequency: Peak Pwer: Antenna size: Antenna Gain: D air surveillance radar L 150 t 1350MHz 5MW m 36dB (lwer beam) 34.5dB (upper beam) Beam shape: Csec Elev beamwidth: 4 Csec t 40 Azim beamwidth: 1.5 Scan: Scan Rate: PRF: PRF Stagger: Pulse width: ise Figure: Mechanical 6rpm 360pps Quadruple μs 4dB Calculate the theretical detectin range fr a 1m aircraft target if the detectin prbability P d = 0.9, and the mean time between false alarms is 9 hurs. Matched Filter Assumptins: Rectangular Pulse Secnd rder bandpass filter Lss 0.56dB B.τ = 0.613

17 99 θb f p Hits per Scan: Use the frmula nb = = = ϖ 6 6 m [use n b = 10] False Alarm Prbability: Use the frmula P fa 1 = T faβ Frm the matched filter assumptins B.τ = and the pulsewidth τ = μs, the IF bandwidth β = 306kHz. T fa = 9 hrs = 3400s P fa = 1 10 = Single Pulse SR: Use the curves in Figure.10.4 fr P d = 0.9 and P fa = SR(1) = 15.dB Fluctuating Target: Use the curves in Figure fr Swerling fr an aircraft Additinal SR required 8dB Pulse Integratin: Use the curves in Figure fr 10 pulses integrated, a Swerling target and P d = 90% t btain an imprvement factr f 15dB Ttal n Pulse SR Required: Add up the requirements SR(10) = = 8.dB Applying the Radar Range Equatin Lsses: We assume the fllwing lsses: Transmitter Line = L tx = db (incrprated int Tx pwer) Receiver Line = L rec = db (incrprated int receiver nise figure) 1D Scanning Lss = 1.6dB Matched Filter = 0.56dB CFAR Lss = 0.7dB Misc. Lss = 1.3dB Ttal L = = 4.16dB P = PG λ σ t r 4 3 ( 4π ) R L

18 300 Transmitter Pwer: 10lg 10 ( ) = 67dBW Less L tx = db Radiated peak pwer P t = 65dBW Antenna gain: (Lwer beam) 36dB Radar crss sectin: σ = 10lg 10 (1) = 0dBm Cnstant: 10lg 10 λ ( 4π ) 3 = -45.7dB Received Pwer: Calculated frm the range equatin P r = lg 10 R = lg 10 R dbw Receiver ise: Use the equatin = P F = kt sys BF K = Bltzmann s Cnstant 1.38x10-3 J/deg T sys = Temperature 90K B = Hz We are given the receiver nise figure in db 10lg 10 (F n ) = 4dB = ( kt B) + 10lg ( F ) + L = = dbw 10lg10 sys 10 n rec 143 Received Signal t ise Rati SR = Pr = lg10 R = lg rec 10 This must equal the required single pulse SR if 10 pulses are integrated SR(10) t achieve the specified P d and P fa. 8. = lg 10 R Slving fr the Detectin Range Slve fr R R = 35.8km R

19 301 Atmspheric Attenuatin The atmspheric attenuatin α db at L-and is abut 0.003dB/km (ne way) S the ttal tw way attenuatin ver 353km is.1db which will result in a significant reductin f the detectin range. These equatins are best slved graphically using MATLAB. The graph belw shws tw received pwer curves, ne that des nt take int accunt the atmspheric attenuatin which intersects the SR(10) curve at 353km. The ther graph which takes attenuatin int accunt intersects the SR(10) curve at 30km. Figure 10.16: Graphical slutin t the radar range equatin including the atmspheric lss Analysis Sftware Radar perfrmance analysis sftware based n Blake has been available fr many years. In this case a package called RGCALC is used. Fr the D radar it prvides the fllwing results.

20 30 te that the predicted perfrmance using RGCALC fr a Swerling target is nly 85km, this lwer range is because it makes the assumptin that the ptimum matched filter has B.τ = 1, and hence calculates a wider IF bandwidth by a factr f 1/0.613 than was made abve. This adds.1db t the nise flr which accunts fr the difference References [1] D.Bartn, Mdern Radar Systems Analysis, Artech, 1988 [] M.Sklnik, Intrductin t Radar Systems, McGraw Hill, 1980 [3].Currie (ed), Principles and Applicatins f Millimeter-Wave Radar, Artech, 1987 [4] Radar Detectin, 09/03/001. [5] RGCALC Range Analysis Sftware [6] B.Mahafza, Radar Systems Analysis and Design Using MATLAB, ChapMan & Hall, 000

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