Passive Radar Systems Chapter 3

Chapter 3

Passive Radar Systems

In this chapter an introduction to passive radar system is presented.

Passive radar systems, also referred to as Passive Coherent Location systems (PCL), exploit reflection from illuminators of opportunity (IOs) in order to detect and track objects. In a passive radar system there is no dedicated transmitter [14] and the receiver uses third-party non cooperative transmitters in the environment. PCL systems exploit the transmitters in the environment hence it is possible to declare that they are in nature knowledge-based systems.

Passive radar systems are bistatic and thus measure the difference in time of arrival between the signal directly from the transmitter and the signal reflected off the target. This allows the bistatic range of the object to be determined. In addition to the bistatic range, also the bistatic Doppler shift of the reflected signal is calculated.

3.1 Passive Radar System description

A basic PCL radar system requires two antennas. The first antenna, often called the reference antenna, is used to capture a direct version of the signal being utilised and should point in the direction of the transmitter. This captured direct signal is needed for two purposes. Firstly, it provides a reference signal against which the signal from the surveillance channel is correlated in the matched filter processing stage. Secondly, a direct version of the opportunity signal is required in the DPI suppression stage to adaptively delete the direct signal component in the surveillance channel. The second antenna, usually called the surveillance antenna, is used to capture signals containing

time-delayed and Doppler-shifted versions (i.e. reflections from moving targets) of the exploited signal. In practice, the surveillance antenna also receives a strong direct signal component from the transmitter, as well as strong returns from stationary clutter. This section gives an overview of the passive radar signal processing chain. A general block scheme of a complete system is sketched in Figure 3.1 [12]. A complete passive radar system typically consist of the following processing steps[12]:

• Reception of the direct signal from the transmitter and from the surveillance region on dedicated low-noise, linear, digital receivers.

• Digital beam forming to determine the Direction of Arrival of signals and spatial rejection and partially reduction of strong interference as direct signal.

• Adaptive filtering to cancel any residual of unwanted direct signal returns in the surveillance channel(s).

• Transmitter-specific signal conditioning

• Cross-correlation of Doppler-shifted copies of the reference channel with the surveillance channel(s) to determine target bistatic range and Doppler.

• Detection using a constant false alarm rate (CFAR) scheme

• Association and tracking of object returns in range/doppler space, typically employing a Kalman filter.

• Association and fusion of line tracks from each transmitter to form the final estimate of an object’s location, heading and speed.

3.1.1Receiver System

The first section of a PCL radar system contains all the analogue components required to filter, amplify and downconvert the signals received by the reference and surveillance antennas. This is the stage where the analogue signals are digitised for processing. The main requirements are : high dynamic range and highly linear receiving equipment with a low noise figure. One of the factors severely influencing the performance of a PCL radar system is the DPI component received by the surveillance antenna. This DPI correlates perfectly with the reference signal and produces range and Doppler sidelobes that are several orders of magnitude larger than the reflected signals from targets[15]. A number of different techniques can be employed to reduce the DPI component received by the surveillance antenna. Terrain shielding is a technique whereby the surveillance antenna is shielded from the transmitter by placing it behind a building or other physical structure. The Manastash Ridge radar is a good example of physical shielding[14][16], where the surveillance antenna is shielded from the transmitter by a mountain. Spatial suppression may also be achieved by steering the surveillance antenna so that the transmitter falls inside a null or a low sidelobe. Another approach is to use an analogue canceller before digitisation of the surveillance signal.

3.1.2Digital beamforming

Passive radar systems use antenna arrays with several antenna elements. This allows the direction of arrival of echoes to be calculated using standard radar beamforming techniques, such as amplitude monopulse using a series of fixed, overlapping beams or adaptive beamforming. Alternatively, some research systems have used only a pair of antenna elements and the phase-difference of arrival to calculate the direction of arrival of the echoes (known as phase interferometry) [13][16][17].

3.1.3Signal conditioning

When exploiting certain illuminators of opportunity in a PCL radar system, the reference signal first needs to be preprocessed in order to improve its quality for detection porpuses. For systems utilising FM radio transmissions, this stage is not necessary. An example occurs when utilising DAB signals in a single frequency network (SFN). In this case, the raw reference signal would consist of several identical, time-shifted copies of the reference signal from each transmitter within line-of-sight of the receiver. If the matched filter processing is performed on the surveillance signal and the raw reference signal, a single target would show up as multiple time-shifted targets on the amplitude-range-Doppler (ARD) surface. To avoid this, the multipath-resistant features of the DAB waveform need to be utilised in order to construct a pure reference signal[18].

3.1.4Adaptive cancellation

As mentioned earlier, the surveillance channel of a PCL radar system contains a strong DPI component, as well as time-delayed reflections from clutter sources. These unwanted signals correlate highly with the reference signal, causing the weak reflections from moving targets to be masked by the resulting sidelobes.

In addition to analogue, physical and spatial techniques, an adaptive filtering algorithm should also be used for DPI suppression. This involves using a noise canceller structure to estimate the direct signal and clutter components in the surveillance channel.

This estimate is then subtracted from the surveillance signal. Ideally, the surveillance signal should now contain only target reflections. In practice, the amount of suppression that can be achieved is limited due to estimation errors.

3.1.5 Cross-Correlation processing

After suppression of the DPI and clutter components in the surveillance channel, it is necessary to look for time-delayed and Doppler-shifted versions of the reference signal. This is achieved by correlating the surveillance signal with Doppler-shifted versions of the reference signal to form a bank of filters matched to every possible Doppler frequency of interest. This operation is equal to calculate the ambiguity function and can be written as follows:

2 2 0 int d T j f t * d surv ref ( , f ) s ( t )s ( t )e π dt χ τ =

_∫

−τ (3.1) where:

•

χ τ

( , f )d represents the ambiguity function

• s_surv( t ) is the surveillance signal

• s ( t )ref is the reference signal

•

τ

is the time delay of interest

• fd is the Doppler frequency of interest

• Tint is the integration time

The ambiguity function provides a mathematical tool for radar designers to identify resolution and ambiguity in both delay-time and Doppler. It is important to note that the ambiguity function of analogue sources, (e.g. FM radio or analogue TV), is unpredictable as a result of the time-varying structure of the signal, which also means a content dependent bandwidth. On the contrary, digital waveforms exhibit an ambiguity function with a thumb-tack shape and a bandwidth that is constant in time. In a

monostatic configuration the time and Doppler resolution are proportional to the range a velocity resolutions. In Table 3.1 monostatic range resolution attainable with different waveform of opportunity is reported. Among digital waveforms the best range resolution can be achieved by using DVB-T and UMTS signals.

The calculation of ambiguity function is one of the most computational expensive in a passive radar signal processing chain. In the following paragraphs some algorithms used to calculate the ambiguity function are presented.

Signal of opportunity Frequency (MHz) Range Resolution (Monostatic resolution) (Km) FM Radio: speech 93.5 16.5

FM Radio: classical music 100.6 5.8

FM Radio: rock music 104.9 6.55

FM radio: reggae 107.1 1.8 DAB 219.4 0.2 Digital TV (DVB-T) 817 MHz 0.018 GSM 900 944.6 1.8 GSM 1800 1833.6 2.62 UMTS FDD 2110-2170 0.032

3.1.6Target detection

Targets are detected on the cross-correlation surface by applying an adaptive threshold, and declaring all returns above this surface to be targets.

A conventional constant false alarm rate (CFAR) algorithm can be used for this task[12].

3.1.7 Target association

The output of the CFAR algorithm produces all the cells on the ARD surface that contain target detections. It is now necessary to associate this range and Doppler data with individual targets[12]. A standard Kalman filter, provided with measurements of range, Doppler and angle of arrival, and estimates of range, Doppler, Doppler rate, bearing and bearing rate can be used to effectively track targets in the range-Doppler space. Most false alarms are rejected during this stage of the processing [12].

After having associated plots-to-targets, the range and Doppler data for each target are processed by a non-linear estimator to determine the location of the target, speed and heading. Use of a nonlinear estimator allows optimum use of the Doppler information in this tracking process.

3.2Properties of a Passive Radar

3.2.1 The Geometry of a PR

The geometry of a PCL radar system is identical to that of a bistatic radar system. Jackson [19] analysed the geometry of bistatic radar systems and its notation and North- referenced coordinate system has been widely adopted. The North-referenced system and its parameters are shown in Figure 3.2.

Figure 3.2 The North-referenced coordinate system

The coordinate sets

(

x , y , z_{T T T}

)

and

(

x , y , z_{R R R}

)

represent the coordinates of the transmitter, receiver and target respectively, while V represents the velocity of the target. The triangle formed by the transmitter, receiver and target is known as the bistatic triangle. The transmitter and receiver are connected by the bistatic baseline with length L. The transmitter and receiver look angles are given by θT and θR respectively

and are measured clockwise from the north of the coordinate system. The bistatic angle,

β, is given by β = θT − θR . The bistatic bisector bisects the bistatic angle. δ is the target angle relative to the bistatic bisector. RT represents the range from the transmitter to the target and RR the range from the target to the receiver.

There are essentially three parameters that the bistatic receiver may measure:

1. the difference in range (RT + RR – L) between the direct signal and the transmitter-target-receiver path

2. the angle of arrival θR of the received echo, 3. the Doppler shift fD of the received echo

Contours of constant bistatic range (RT + RR) define an ellipse, with the transmitter and

receiver at the two foci. If (RT + RR) and θR are measured and L is known, the range of

the target from the receiver may be found from:

(

)

(

)

2 ₂ 2 sin R T R T R R R R L R R R L θ + − = + +

(3.2)

Instead the Doppler shift is given by:

(

)

2 2 d V f cos cosδ β / λ =

(3.3)

It can be seen that, if the target is crossing the bistatic baseline then β =0 and

fD = 0. Physically this can be understood because at this point the transmitter-to-target

range is changing in an equal and opposite way to the target-to-receiver range.

3.2.2Radar equation

The starting point for an analysis of the performance of a passive bistatic radar system is the basic form of the bistatic radar equation[15]:

2 2 2 0

1

1

4

4

4

t t r r b n T R n

PG

P

G

P

R L

R

kT B F

λ

σ

π

π

π

=

(3.4)

where:

• Pr is the received power (W)

• P_n is the receiver noise power (W)

• Pt is the transmit power (W)

• Gt is the transmit antenna gain

• Gr is the receiver antenna gain

• RT is the transmitter-to-target range (m)

• RR is the receiver-to-target range (m)

•

σ

b is the bistatic radar cross-section

•

λ

is the signal wavelength

•

k

is the Boltzman constant 1.38e-23

• T₀ is the noise reference temperature, 290 K

• B is the receiver effective bandwidth (Hz)

Contours of constant values of

2 1 T R R R    

  and hence of signal-to-noise ratio, define

geometric figures known as Ovals of Cassini [15].

It should be noted the noise figures of receivers at VHF and UHF will be of the order of a few dB at most, so the noise level will be dominated by external noise, most likely in the form of the direct signal, multipath, and other co-channel signals.

3.2.3The bistatic ambiguity function

In a bistatic configuration, the relationship between time-range, Doppler-velocity are strictly dependent on the relative position of target, transmitter and receiver.

A more appropriate formulation of the ambiguity function for bistatic radar is given by[15]:

(

)

(

(

)

)

(

(

)

)

(

)

(

)

(

)

0 2 x 2 2 int H a a H H H a a T * R R H a R a R R R R D R H R D R a R R ,R ,V ,V , ,L s t R , ,L s t R , ,L exp j f R ,V , ,L f R ,V , ,L dt χ θ τ θ τ θ π θ π θ = − − _{− −}   

∫

(3.5)

where

• RRH and RRa are the hypothesised and actual ranges (delays) from the receiver to the target.

• VH and Va are the hypothesised and actual target radial velocities with respect to the receiver.

• fD_H and fDa are the hypothesised and actual Doppler frequencies.

• and

θ

_RR and L are as defined in Figure 3.2

As example, Figure 3.3 shows how the range resolution and such the ambiguity function shape can change with the geometric configuration. The example is referring to a generic waveform.

Ambiguity Function for a target in

Position 1 (800m

from the baseline)

Range resolution is 33.5m

Ambiguity Function for a target in

Position 2 (400m

from the baseline)

Range resolution is 35m

Ambiguity Function for a target in

Position 3 (200m

from the baseline)

Range resolution is 41m

Ambiguity Function for a target in

Position 4 (100m

from the baseline)

Range resolution is 70 m

Figure 3.3 Bistatic geometry effect on ambiguity function shape

-20 -15 -10 -5 0 5 10 15 20 650 700 750 800 850 900 950

F.A. nel punto a 806.2258 m dal Rx in direzione -7.125 °N DR= 33.5219

-20 -15 -10 -5 0 5 10 15 20 250 300 350 400 450 500 550 -20 -15 -10 -5 0 5 10 15 20 50 100 150 200 250 300 350 400 -20 -15 -10 -5 0 5 10 15 20 50 100 150 200 250 300 350

3.3Cross Ambiguity function algorithms

The sampled version of eq. (3.1) is :

[

]

1

[

] [ ]

₂ 0 p N _{j n} * N ref surv n m, p s n m s n e π χ − − = =

∑

− (3.6)

where N =_T_int f_s_{ is the number of samples,} m represents the time bin (

τ

_m=m / f_s), and pis the Doppler bin corresponding to fd_n p f_s

N

= _.

In terms of computational burden, the CAF is one of the most heavy signal processing operation in a passive radar signal chain [9]. In fact, a long integration time is required to obtain a proper level of signal to noise ratio (SNR). Moreover, the dimensions of the time- Doppler map depend on maximum time delay ( or maximum range ) and maximum Doppler frequency ( or velocity ). As a reference, if N_delayis the number of delay bins and N_f is number of Doppler bins, eq. (3.6) requires 2NN_delayN complex _f multiplications and

(

N −1

)

N_delayN_f complex additions.

In the following subsections different implementations of eq. (3.6) are considered, computational burden [21]. Subsections 3.3.1-3.3.2 show algorithms obtained by rearranging the factors in eq. (3.6) according to different criteria, both the algorithms are computational expensive and they calculate the CAF without any approximation. In the following sections we consider some methods for CAF with the aim of having a reduction of computational burden.

3.3.1 Correlation-FFT

In this method, at the p-th Doppler bin, the samples along time bins of the CAF correspond to the cross-correlation samples between the reference signal, s_ref

[ ]

n and the Doppler-shifted echo signal,

[ ]

2

p j n N p surv s [ n ] =s n e− π :

[

]

1

[ ] [

]

0 N * p ref p n m, p s n s n m C [ m ] χ − = =

∑

− =

(3.7)

It is possible to save computational burden by evaluating such cross-correlation in the frequency domain as:

[ ]

{

}

(

{

[ ]

}

)

{

}

{

l

}

* * p p ref p ref C [ m ] =IDFT DFT s n DFT s n =IDFT S [ k ] S [ k ]

(3.8)

For each p-th Doppler bin an 2N-IDFT is applied and 2N-Ndelay output samples are

discarded.

Moreover, it should be noticed that both the DFTs in eq. (3.8) can be executed just once. In fact, it is possible to obtain S [ k ]_p as a circular shifting of the 2N-DFT of

[ ]

surv

s n .At each iteration 2N complex multiplications and one IDFT are performed. This algorithm can be implemented in parallel for the Nf Doppler bins of interest Figure

Figure

3.3.2Direct-FFT

This algorithm considers that at the

frequency domain correspond to the DFT of the sequence

as complex multiplication between complex conjugate of delayed referen surveillance signal:

[

]

[

{

1 0 N * ref surv n m, p s n m s n e DFT s n m s n DFT { x [ n ]} χ − = =

∑

− =

In this case, for each m

iterations of this algorithm are limited to the maximum number of delays ( this means that is possible to parallelize the algorithm over the range bins. shows the Direct-FFT flow

burden has been done.

Figure 3.4 Flowchart for Correlation-FFT algorithm

This algorithm considers that at the m-th time bin, the samples along the Doppler frequency domain correspond to the DFT of the sequence *

[

ref surv

s n−m s n as complex multiplication between complex conjugate of delayed referen

] [ ]

[

] [ ]

{

}

2 p j n N ref surv * ref surv m m, p s n m s n e DFT s n m s n DFT { x [ n ]} π − = − = − =

(

m-th delay bin a DFT is performed and N-Nf

iterations of this algorithm are limited to the maximum number of delays ( this means that is possible to parallelize the algorithm over the range bins.

FFT flow- graph and in Table 3.2 an evaluation of computational th time bin, the samples along the Doppler

]

[ ]

ref surv

s n−m s n obtained as complex multiplication between complex conjugate of delayed reference signal and

(3.9)

f are discarded. The

iterations of this algorithm are limited to the maximum number of delays ( Ndelay ) and

this means that is possible to parallelize the algorithm over the range bins. Figure 3.5 an evaluation of computational

Figure 3

3.3.3Direct-FFT with Decimation

The problem of the Direct-FFT approach is the excessive processing load due to calculations of the FFT for long input signals. An efficient implementation is based on a decimation technique that allows to discard data at Dop

targets do not exist.

This modified integration algorithm utilises some extra processing steps to decimate the signal but greatly reduces the overall computation complexity with almost no loss in signal processing gain. This approach is based on downsample the data after the dot product in order to perform a smaller FFT

of points in the FFT after decimation is summarized as in Figure 3.

reference signal with the echo signal is calculated.

3.5 Flow-graph of Direct FFT algorithm

FFT with Decimation

FFT approach is the excessive processing load due to calculations of the FFT for long input signals. An efficient implementation is based on a decimation technique that allows to discard data at Doppler frequencies that we know

fied integration algorithm utilises some extra processing steps to decimate the signal but greatly reduces the overall computation complexity with almost no loss in signal processing gain. This approach is based on downsample the data after the dot product in order to perform a smaller FFT [12]. If D is the decimation factor the number of points in the FFT after decimation is N_D = _N D_{ . The algorithm can be} .6. First, the product of the conjugated, time delayed reference signal with the echo signal is calculated.

FFT approach is the excessive processing load due to calculations of the FFT for long input signals. An efficient implementation is based on a pler frequencies that we know

fied integration algorithm utilises some extra processing steps to decimate the signal but greatly reduces the overall computation complexity with almost no loss in signal processing gain. This approach is based on downsample the data after the

dot-. If D is the decimation factor the number . The algorithm can be . First, the product of the conjugated, time delayed

Figure

Then the product signal

stage integrator, decimated by factor

section. Figure 3.7 shows the structure of a CIC filter. Becaus magnitude- response for a CIC is

by a FIR filter whose task is to compensate for the CIC non

of FIR filter depends on the decimation factor and on the maxim

considered. Moreover, the FIR filter can be avoided if the maximum Doppler stays within the 3-dB bandwidth of CIC filter frequency response.

Z-1

fs

Figure 3.6 Flow-graph of Direct-FFT with CIC filter

Then the product signal xl[n] goes into the CIC filter where it is integrated using a n

stage integrator, decimated by factor D, and then differentiated in a n shows the structure of a CIC filter. Becaus

for a CIC is sin(x)/x- like, this kind of filter is typically followed by a FIR filter whose task is to compensate for the CIC non-flat-passband. The length of FIR filter depends on the decimation factor and on the maxim

considered. Moreover, the FIR filter can be avoided if the maximum Doppler stays dB bandwidth of CIC filter frequency response.

Z-1 . . . Z-1 -1 . . . N integrators N combs fs fs/D

Figure 3.7 CIC filter structure N-sections

goes into the CIC filter where it is integrated using a n-, and then differentiated in a n-stage comb shows the structure of a CIC filter. Because the frequency- like, this kind of filter is typically followed passband. The length of FIR filter depends on the decimation factor and on the maximum Doppler considered. Moreover, the FIR filter can be avoided if the maximum Doppler stays

Z-1 -1 N combs

The integrator operates at the original sampling rate fs. After decimation, the comb

section operates at the reduced sampling rate of fs/D. Therefore, the length of the output vector is a factor R smaller. Next, the decimated signal dl[n] is low-pass filtered in

order to remove out of band frequencies and also to compensate the CIC frequency response, if the maximum Doppler frequency is . Finally, calculating the FFT algorithm on the filtered data vector fl[n] results in all Doppler velocities for all targets from a

specific range of frequencies at that range. The computational cost for each delay is as follows:

• O(CIC) = N complex additions for the one integrator stage, ND=N/R additions for

the one comb stage

• O(FFT) = (ND/2) log2(ND) complex multiplications and NDlog2(ND) complex additions

• O(FIR) = NFIR complex multiplications and NFIR additions ( NFIR depends on the decimation rate and max frequency Doppler of interest )

• N complex multiplication due to the dot products

Now we explain one of possible ways to choose the decimation factor.

Considering the frequency response of a CIC filter evaluated to the input sampling frequency: 2 2 1 2 1 s s N D N j f f s N N N S I C _N f f s D e _{sin f} f f H( f ) H ( f )H ( f ) D f f sin z _f π π

π

π

− −   _{  } −   _{  }        = = = _{ } ≤     _{ }   −             (3.10)

Assuming a CIC with 1 section ( N=1) , it is possible to choose a maximum Doppler frequency in order to have an attenuation less than 1 dB :

1 0 25 s dB f f . D = (3.11)

In fact, considering a maximum target velocity of 100 m/s, the maximum Doppler frequency is equal to:

6 0 0 2 0 6 10 max D max v f f ( . ) f c − = ≈ (3.12)

where f0 is the carrier frequency and c=3e8 m/s. Combining eq.(3.11) and eq.(3.12)

we are going to obtain an upper limit decimation factor:

0 0 0 25 3 75 5 2 s s max max f f c D . . v f Ε f = = (3.13)

D values less than Dmax provide an attenuation less than 1 dB on ambiguity function peak. Instead, if the algorithm is implemented on FPGA the upper limit will be bounded from Bit Grow effect that is due to the gain at the output of the comb section [20]. The losses for this method are almost negligible in terms of ambiguity function shape and SNR.

This methods has been implemented in Matlab and C++ and it is resulted ten time faster than direct FFT method.

3.3.4Batches Algorithm

The aim of this algorithm is to calculate the CAF in a more efficient way respect to the previous paragraphs. The first step is to simply re-index the sum in (3.6) using

B

n=bL +q where b=0,…,NB , q=0,…,LB-1.This re-indexing operation creates NB

blocks of LB samples. To ensure that NB blocks are created, the length of the signals

should have a length divisible by LB. The value of LB is a parameter of the algorithm

that can be chosen. The eq. (3.6) now becomes:

[ ]

1 1

[

] [

]

2 0 0 B B B p N L j ( bL q ) * _N surv B ref B b q

m, p

s

bL

q s

bL

q m e

π

χ

− − − + = =

=

∑ ∑

+

+ −

(3.14)

Separating the complex exponential in (3.14) and rearranging terms gives

[ ]

1 2 1

[

] [

]

2 0 0 B B B l p p N L j bL j q * N N surv B ref B b q

m, p

e

π

s

bL

q s

bL

q m e

π

χ

− − − − = =

=

∑

∑

+

+ −

(3.15)

To simplify eq. (3.15) further, it is possible to neglect the exponential term in the inner summation: 2 1 0 1 p j q N B e− π ≅ q= ....L −

(3.16)

In order to reduce the losses due at eq.(3.16) we can impose

1 2 0 then 2 f B B f N N L L N N

π

π

γ

γ

≈ ≤ <

(3.17)

where γ is an integer greater than 10. Considering that N=LBNB from eq. (3.17) we

obtain: 2

B f

N > N γ

(3.18)

Then eq.(3.17) and eq. (3.18) guarantee that the exponential has a phase less than π/10. Neglecting the exponential term in the inner summation we obtain:

[

]

[

] [

]

[ ]

1 ₂ 1 0 0 1 2 0 B B B B B p N _{j bL} L * N surv B ref B b q p N j bL N m B b m, p e s bL q s bL q m e y bL π π

χ

− − − = = − ₋ = = + + − = =

∑

∑

∑

(3.19)

which represents the DFT of decimated sequence y_m

[ ]

b ( b=0,…NB ) , obtained by

summing the product sequence

[

] [

]

l *

surv B ref B

s bL +q s bL + −q m for each batch. The resulting algorithm scheme is shown in Figure 3.8 and its computational load is reported in Table 1 as a function of the number of batches assuming that LBNB=N.

There is a different way to implement the Batch

1. Create the matrices

•

[

[

[

0 0 0 1 0 1 ref B ref B ref B ref B ref B B ref B s L s L q s L s L q s ( N )L s ( N ) q       =_{ }       REF •

[

surv k B surv k B surv k B B

surv k B surv k B surv k B B

surv k B B surv k B surv k B B B

s , L s , L q s , L L s , L s , L q s , L L s ,( N )L s ,( N ) q s ,( N )L L       =_{ }       ECHO r r r

2.For each delay:

a. Calculate the cross

x NDelay

[ m ] [ m ]

CrossM REF , ECHO

Figure 3.8 Batch algorithm scheme

There is a different way to implement the Batch algorithm (Batch2):

Create the matrices

]

]

]

[

]

[

]

[

]

[

[

[

0 1 0 0 0 1 1 1 0 1 1 0 1 l ref B B ref B ref B * ref B B ref B ref B

ref B B ref B ref B B B

s L L s L s L q s L L s L s L q s ( N )L s ( N ) q s ( N )L L  _{+ +}    + +        _{− + − + − + −}    ⋯ ⋯ ⋮ ⋮

[

]

[

]

]

[

]

[

]

[

]

0 0 0 0 1 1 0 1 1 1 1 0 1 1 1

surv k B surv k B surv k B B

surv k B surv k B surv k B B

surv k B B surv k B surv k B B B

s , L s , L q s , L L s , L s , L q s , L L s ,( N )L s ,( N ) q s ,( N )L L  + + + −    + + + −        _{− + − + − + −}   

⋯

⋯

⋮

⋮

⋮

r r r r r r r r r

Calculate the cross-correlation along the row, and we obtain a matrix

(

)

xcorr

[ m ]= [ m ]

CrossM REF , ECHO

]

]

]

0 1 1 1 1 1 ref B B ref B B ref B B B s L L s L L s ( N )L L  _{+ −}    + −        _{− + −}    ⋮

[

]

[

]

[

]

0 0 0 0 1 1 0 1 1 1 1 0 1 1 1

surv k B surv k B surv k B B

surv k B surv k B surv k B B

surv k B B surv k B surv k B B B

s , L s , L q s , L L s , L s , L q s , L L s ,( N )L s ,( N ) q s ,( N )L L  + + + −    + + + −        _{− + − + − + −}   

⋮

⋮

⋮

r r r r r r r r r

3.Apply the FFT along the columns and we obtain cc

[

]

l k

s r ,m , p

It should be noted that if we calculate the CrossM matrix with the FFT algorithm step 2 is really fast respect to the scheme in Figure 3.8. Moreover, with a correct choice of LB

the CrossM represents the matrix in the domain slow-time/range that is a typical matrix in a monostatic radar scenario.

For a constant modulus signal the SNR loss for a target at range bin m0 and Doppler bin

p0 can be written as[21]:

[

]

0 0 0 0 0 0 sinc 20 20 sinc B B B p p N sin N N L m , p log log p p N sin N N

π

π

            = =            

(3.20)

As showed on eq.(3.20), the CAF loss only depends on the Doppler bin and

reaches its maximum at the highest Doppler value considered in the CAF, while

no loss is experienced at zero Doppler.

Algorithm Complex Multiplications Complex Additions

Correlatio n-FFT N log ( N )2 2

(

1+Nf

)

+2NNf 2N log ( N )2 2

(

4N+Nf

)

Direct-FFT delay 2 2 N N _N + log N_   NdelayN log ( N )2 Direct-FFT with decimatio n

(

)

2 2 delay FIR N / D N _N+ log N / D +N _

Batch 2

( )

2 B delay B N N _N+ log N _   Ndelay

(

N +N logB 2

( )

NB

)

Batch2 Ndelay(NB(3*Lblog2(2Lb))+(NB/2)*.lo

g2(NB)

Ndelay(NB(3*2Lblog2(2Lb))+(N B)* log2(NB)

3.4Direct Path filtering

Two of serious problems encountered in passive radar systems are: the existence of a reference waveform in the receiver target channel, named “direct path interference”, and the clutter/multipath echo. These unwanted signals correlate perfectly with the reference signal and this can lead to:

• strong clutter echoes masking targets with high Doppler frequencies,

• a fraction of the direct signal being received via the side/backlobe of the surveillance antenna which masks target echo signals,

• strong target echoes masking other echoes from other targets of a lower level, even in the presence of large range-Doppler separations.

In passive radar systems, there are various methods for removing, or at least reducing, those unwanted signal components.

First, and most attractive due to effectiveness, simplicity and cost, is to site the receive antenna so that it is physically shielded from the direct path signal, using topography, buildings. The Manastash Ridge Radar at the University of Washington, is an examples

Errore. L'origine riferimento non è stata trovata.. Spatial cancellation of the direct

path signal is a second principal method. An array antenna at the receiver can be configured to steer a null at the direct path signal [22]. The Adaptive filtering is the third method. The adaptive filtering on target channel is used to remove the direct path and multipath/clutter echoes before to do the matching filter operation.

In order to analyze the adaptive filtering technique a signal model on receiver channel is presented[25][26]:

( )

(

)

2

(

)

( )

0 1 c T dk c N N j f nT R T k T k i T i R k i s n A( n )s ( n ) a s n τ e π c s n τ v n = = = +

∑

− +

∑

− + (3.21) where

• sT

( )

n is a sample of the complex envelope of the reference waveform

• A(n) is the complex amplitude of direct signal received on surveillance channel

• ci and τi are the complex amplitude and the delay ( with respect to the direct signal) of i-th stationary scatter ( i=1,…,Nc)

• ak and τk are the complex amplitude and the delay of k-th target with fdk Doppler

frequency

• vR(n) is the thermal noise contribution at the receiver antenna

The goal of pre-processing techniques is to delete the zero Doppler component (including Multipath/Clutter echoes and direct signal ) of the receiver signal:

( )

(

)

(

)

1 0 c c N N MP i T i T i T i i i s t c s t τ A( t )s ( t ) c s t τ = = =

∑

− + =

∑

− (3.22)

The signal s_MP

( )

t is a linear combination of delayed replies of sT(t) and complex

coefficients ci. To operate using adaptive filters, the knowledge of transmitted

waveform is required. In fact, the reference signal is exploited by assuming it is a good replica of transmitted signal. Specifically, since the direct signal is captured by the main lobe of the reference antenna, the echoes and multipath signals, that could be received from antenna sidelobes, are assumed negligible. Moreover, the reference signal is supposed to be multipath-free signal, this assumption is generally true if we are not in a highly dense urban area and the reference antenna has a gain more than 10 dB.

An adaptive filter consists of two main processes. First, a transversal filter is responsible to perform the filtering process. Second, the adaptive control mechanism exploits the input data and the filtered data to adjust instantaneously the taps of the transversal filter. The transversal filter collects M samples of sref(n) (reference signal captured on reference channel), they form the elements of the M by 1 tap input vector Sref(n) as it has been shown in Figure 3.9.

In a passive radar context, M represents the number of consecutive range bins, namely the maximum distance from the receiver, where we desire to erase the multipath

contributions. Moreover, the M tap weights ŵ0(n), ŵ1(n), …, ŵM-1(n) form the elements of the M by 1 vector ŵ(n).

During the filtering process, the received signal sR(n) is supplied for processing, alongside the M- vector Sref(n). An estimation of the multipath component of sR(n), named

d n

ˆ

( )

, is produced by the transversal filter. Accordingly, we may define an estimation error e(n) as the difference between the desired response, sR(n), and the actual filter output, as indicated in Figure 3.9 Detailed structure of NLMS and LMS filters. The estimation error e(n) and the tap input vector are applied to the adaptive control mechanism.

The adaptive control mechanisms for LMS or NLMS algorithm have the same structure. Specifically, a scalar version of inner product of the estimation error e(n) and the tap input sREF(n-k) is computed for k=0,1,2,…,M-1. The result defines the correction factor

3.4.1 LMS and NLMS filtering [24]

The adaptive algorithms, both LMS and NLMS, compute w(n) in order to come close to the optimum Wiener solution. Rather than terminating on the Wiener solutions, the tap weight vector computed by the LMS/NLMS algorithms executes a random motion around the minimum error performance surface.

Applying the steepest descent algorithm to the Wiener filter theory, we obtain a simple recursive solution:

[

]

1

( n+ =) ( n )+µ − ( n )

w w p Rw (3.23)

where w(n) are the tap weights of a corresponding transversal filter u(n). In addition to these inputs the filter is supplied with a desired response d(n). And p represents the cross-correlation vector between the tap input vector u(n) and the desired response d(n). Instead R is the correlation matrix of tap input vector u(n). The LMS filter derives from an estimation of eq.(3.23).

The simplest choice of estimators is to use instantaneous estimates for R and p that are based on sample value of the tap input vector and desired response defined respectively by: ɵ

_{( )}

H

_{( )}

_{( )}

ref d n =w n S n (3.24) and

( )

( ) ( )

* ref R ˆ n = n s n p S (3.25)

Substituting eq.(3.24) -(3.25) in eq.(3.23), we get a new recursive relation for steepest-descent algorithm:



₍

₎



_{( ) ( )}

_{( )}



1 _{ref R ref}

n+ = ( n )+

µ

n _s n − n ( n )_

w w S S w (3.26)

Filter output: ɵ

_{( )}

H

_{( )}

_{( )}

ref d n =w n S n (3.27) Estimation error:

( )

( )

ɵ

_{( )}

R e n =s n −d n (3.28) Tap-weight adaptation:



₍

₎



_{( )}

_{( ) ( )}

1 _ref * n+ = n +

µ

n e n w w S (3.29)

Equations (3.27)-(3.28) establish the estimation error e(n),based on the current estimate of the tape weight vector



w

( )

n . The term

( ) ( )

*

ref n e n

µS on eq.(3.29) represents the adjustment that is applied to the current estimate of tap weight vector. The algorithm described by eq. (3.27) trough (3.29) is the complex form of the adaptive least mean square (LMS) algorithm. At each iteration this algorithm requires the actual values of

Sref(n), sR(n) and



_{( )}

n

w . The computational complexity of LMS algorithm consists of 2M+1 complex multiplications and 2M complex additions per iteration, where M is the number of range bins.

The practical importance of LMS filters is due to mainly to:

• Simplicity of implementation

• A model independent, and therefore robust, performance

The main limitation of LMS filters is their relatively slow rate of convergence, which is attributed to the exclusive use of first order information.

Two principal factors influence the convergence behaviour: the step size parameter and the eigenvalues of correlation matrix R of the input vector Sref(n). Theoretical studies summarize the proprieties of these factors as follows:

If a small value is given toµthen the adaptation is slow, which means that the LMS filter has a long memory. Moreover the mean-square error has a little variance after the

adaptation On the contrary, when a large value is assigned to the step-size the adaptation is really fast and the variance of mean square error increase respect to the mean value. In the last case the filter is in a short memory condition

When the eigenvalue of correlation matrix are widely spread, the excess mean square error caused by LMS filter is determined primarily by the largest eigenvalue, and time taken by the average tap weight vector E_



w

( )

n _is limited by the smallest eigenvalue. When the correlation matrix of the taps input is ill conditioned ( i.e., when the eigenvalues spread is large ), the convergence of LMS filter may slow down. The necessary condition to the LMS filter convergence on the step-size parameter is defined by: 2 0 max MS µ < < (3.30)

where Smax is the maximum value of the power spectral density of the taps inputs.

The main problem of LMS filter is that suffers from gradient noise amplification. To overcome this difficulty, the normalized LMS is used. The NLMS algorithm alleviates this problem by normalizing the coefficient update equation with respect to the squared-norm of the input data. In structural terms, the squared-normalized LMS filter is exactly the same as the standard LMS filter. In fact, both the filters differ only in the way in which the tap weights are calculated.

The M by 1 tap input vector Sref(n) produces an output

ɵ

_{( )}

d n that is subtracted from the received signal (i.e. the desired response) sR(n) to produce an estimation error signal e(n) without zero-Doppler components. The input vector Sref(n) and the error signal e(n) are combined by the weight controller system that produces adjustments to the transversal filter taps. This sequence of events is repeated until the filter reaches a steady state.

There are two ways to derive the NLMS filter: it is possible to formulate the normalized LMS filter as a natural modification of the ordinary or we may derive the normalized LMS filter on its own right. The design of NLMS filter can be made under the next constraints: Given the tap-input vector Sref(n) and the received signal sR(n), determine

the updated tap weight vector



w

(

n+1

)

so as to minimize the squared Euclidean norm of the change:



₍

₎



₍

₎



_{( )}

1 1 n n n

δ

w + =w + −w (3.31)

under the constraint

( )



₍

₎

_{( )}

1 H

R ref

s n =w n+ S n (3.32)

To solve this problem the method of Lagrange Multipliers has been applied and after some mathematical steps we find:



₍

₎



_{( )}

( )

2

( ) ( )

1 _ref * ref n n n e n n µ + = + w w S S ɶ (3.33)

As it is clearly shows in eq.(3.33) the vector

( ) ( )

*

ref n e n

S is normalized with respect to squared Euclidean norm of the tap input vector S_ref

( )

n . We may make the following observations about a comparison between the LMS and NLMS filters:

Setting

( )

( )

2 ref n n µ µ = S ɶ ɶ _(3.34)

the normalized LMS filter can be seen as an LMS filter with time-varying step-size parameter

The normalized LMS filter exhibits a rate of convergence that is faster than that of the standard LMS for both correlated and uncorrelated data. The NLMS filter introduces a problem when the tap-input vector Sref(n) is small because the inverse of squared norm can arise numerical difficulties and the filter is not able to find the steady state.



₍

₎



_{( )}

( )

( ) ( )



_{( )}



₍

₎

2 1 ref * 1 ref n n n e n n n n µ _δ γ + = + = + + + w w S w w S ɶ (3.35) where γ >0.

As shown in eq.(3.35), the underlying idea is to minimize the incremental change



₍

₎

1

n

δ

w + in the tap weight vector, from iteration n to iteration n+1, under the constraint in eq.(3.32).

The normalized LMS filter has mainly two advantages respect to the LMS filter:

• The NLMS filter attenuates the gradient noise amplification problem.

• The rate of convergence of NLMS filter is potentially faster than that of the conventional LMS filter for both uncorrelated and correlate input data.

Returning to eq.(3.35), we may make two observations:

• The direction of the adjustment

δ



w

(

n+1

)

is the same of the input vector Sref(n) • The size of the adjustment is dependent on the sample correlation coefficient

between the input vectors Sref(n) and Sref(n-1)

From a geometric perspective when Sref(n) and Sref(n-1) are orthogonal vectors the rate of convergence of the NLMS filter is fast, on the contrary when Sref(n) and Sref(n-1) are parallel vectors the rate of convergence of the NLMS filter is slow.

To overcome the problem of slow convergence, we may use a generalization of NLMS filter known as the affine projection filter.

In mathematical terms, we may formulate the criterion for designing an affine projection filter as a generalization of the adopted criterion for NLMS filter ( eq.(3.31)-(3.32) ):

Given the tap-input vector Sref(n) and the received signal sR(n), determine the updated tap weight vector



w

(

n+1

)

so as to minimize the squared Euclidean norm of the change:



₍

₎



₍

₎



_{( )}

1 1

n n n

δ

w + =w + −w (3.36)

( )



₍

₎

₍

₎

1 0 1

H

R ref A

s n =w n+ S n−k k = ,…, N − _(3.37)

where NA is smaller than the dimensionality M of the input data space Sref(n)=[sref(n)… sref(n-M+1)]T. As we readily see the NLMS is special case ( NA=1 ) of the optimization criterion afore mentioned. We may view NA, as the order of the affine projection adaptive filter. For convenience of presentation , we introduce the following definitions:

An NA by M data matrix A(n) whose Hermitian transpose is defined by:

( )

( )

(

1

)

(

1

)

H

ref ref ref A

n =_ n , n− ,..., n−N + _

A S S S (3.38)

An N by 1 desired response vector whose Hermitian transpose is defined by:

( )

( ) (

1

)

(

1

)

H

A

n =_d n ,d n− ,...,d n−N + _

d (3.39)

The update formula for the taps filter is given by:



₍

₎



_{( )}

_{( ) ( )}

(

_{( )}

)

1

_{( )}

1 H H

n+ = n +µ n n n − n

w w ɶ A A A e (3.40)

where

µ

ɶ

is an adaption constant and e(n) is defined as:

( ) ( )

( )



_{( )}

n = n − n n

e d A w (3.41)

Because of the use of (NA-1) past values of both the Sref(n) and d(n), the affine projection filter may be viewed as an adaptive filter that is intermediate between the normalized LMS filter and the recursive least squares filter. The affine projection filter provides a significant improvement in convergence, in front of an increased computational cost.

3.4.2LS/ECA filter

In addition to the mean least square approach exists also the method of least squares. In computational terms the method of Least Square is a batch-processing approach. The input data stream is arranged in block of equal length, and the filtering of input data proceeds on a block by block basis. The filter is adapted to non-stationary data by repeating the computation on a block by block basis, which makes it computationally demanding.

To derive the basic block LS algorithm, we consider the transversal filter structure, the

sR composed by N samples of the surveillance signal sR(n), and the sref vector composed

by N+M-1 samples of the reference signal sref(n). The goal of LS method is to find the tap weights vector in order to minimize the sum of error squares:

( )

1

( )

2 0 min E min N i e i − = =

∑

w w w (3.42)

where wis the tap weights vector. In eq.(3.42) e(i) is defined as follows:

( )

( )

1

(

)

( )

1

( )

0 0 1 M i M * * R k ref R ref i k k k i e i s i w s i k s i s k w i ,..., N − − + − = = = −

∑

− = −

∑

= − (3.43)

Reformulating eq.(3.43) in terms of data matrix we may have:

* R

= −

e s X w (3.44)

The eq.(3.42) becomes:

( )

2

{

2

}

min E min min *

R

= = −

where

( )

(

)

(

)

(

)

2 1 0 1 1 1 ref ref M

ref ref ref ref

ref ref s s M ... s N s N M −  _{− +}      =_{ }= _{ }  _{− − +}    X B s Ds D s D s … ⋮ ⋱ ⋮ ⋯

,D is a matrix that applies a delay of a single sample, and B is an incidence matrix that selects only the last N rows of the following matrix. The columns of X define a basis for M-dimensional clutter subspace, where M is the number of the first range bins.

We may solve eq.(3.45) by expressing the tap weight vector as:



*

(

)

1 H H R −

=

w

X X

X s

(3.46)

Using eq.(3.46) into eq.(3.44):

(

_H

)

1 _H

(

(

_H

)

1 _H

)

R R R R

− −

= − = − = 0

e s X X X X s I X X X X s P s (3.47)

We may view the multiple matrix product _{X X X}

(

H

)

−1_XH _{as a projection operator onto} a linear space spanned by the rows of data matrix X that represents the clutter subspace. On the contrary P0 projects the received vector sR in the subspace orthogonal to the

clutter subspace. The error vector e is the received signal with zero-Doppler components attenuated. The action of LS filter is like a notch filter in zero Doppler. We may want to delete also clutter components near the Zero-Doppler and it is possible to achieve this by extending the disturbance subspace dimension represented by the columns of matrix X by including Doppler shifted replicas of the reference signal. In this case the algorithm is named ECA ( Extensive Cancellation Algorithm ) and the X matrix becomes[25][26]: 1 1 P ... ... P − − = _ _ X _{B Λ S} Λ S S Λ S Λ S (3.48)

where S=B s_{ ref} Ds_ref D s2 _ref ... DM−1s_ref_, D is a matrix that applies a delay of a single sample, B is an incidence matrix that selects only the last N rows of the following

matrix and Λ is a diagonal matrix that applies the phase shift corresponding to the p-p th Doppler bin.

However this approach is computationally intensive, since it corresponds to increasing the dimension of the weight vector



w whose evaluation requires the computation and the inversion of the matrix XHX with dimensions MxM wich corresponds to

O[NM2+M2logM] complex products.

Obviously respect to the adaptive algorithm the LS algorithm or ECA algorithm does not require a variable to control the rate of adaptation. This should help the algorithm to be more robust when used in different signal environments.