# Digital Signal Processing For Radar Applications

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1 Digital Signal Processing For Radar Applications Altera Corporation

2 Radar: RAdio Detection And Ranging Need a directional radio beam Measure time between transmit pulse and receive pulse Find Distance: Divide speed of light by interval time 2

3 Radar Band (Frequency) Terminology Radar Band Frequency (GHz) Wavelength (cm) Millimeter 40 to to 0.30 Ka 26.5 to to 0.75 K 18 to to 1.1 Ku 12.5 to to 1.7 X 8 to to 2.4 C 4 to to 3.75 S 2 to 4 15 to 7.5 L 1 to 2 30 to 15 UHF 0.3 to to 30 3 λ = v / f where f = wave frequency (Hz or cycles per second) λ = wavelength (centimeters) v = speed of light (approximately 3 x centimeters/second) Radar Band often dictated by antenna size requirements

4 Radar Range Equation Receiver Power P receive = P t G t A r σ F 4 (t pulse / T) / ((4π) 2 R 4 ) where P t = transmitted power G t = antenna transmit gain A r = Receive antenna aperture area σ = radar cross section (function of target geometric cross section, reflectivity of surface, and directivity of reflections) F = pattern propagation factor (unity in vacuum, accounts for multi-path, shadowing and other factors) t pulse = duration of receive pulse T = duration of transmit interval R = range between radar and target 4

5 Transmit Pulse Repetition Frequency (PRF) From 100s of Hz to 100s of khz Can cause range ambiguities if too fast Aliasing of return to shorter range 0 us 0 km 100 us 15 km 200 us 30 km 100 range bins Increasing range and return echo time Assume PRF of 10 khz (100 us), therefore R maximum = (3x10 8 m/sec) (100x10-6 sec) / 2 = 15 km Target 1 at 5 km range: t delay = 2 R measured / v light = 2 (5x10 3 ) / 3x10 8 = 33 us 5 Target 2 at 21 km range: t delay = 2 R measured / v light = 2 (21x10 3 ) / 3x10 8 = 140 us

6 Doppler concept frequency shift through motion 6

7 Doppler effect Frequency shift in received pulse: f Doppler = 2 v relative / λ Example: assume X band radar operating at 10 GHz (3 cm wavelength) Airborne radar traveling at 500 mph Target 1 traveling away from radar at 800 mph V relative = = -300 mph = -134 meter/s Target 2 traveling towards radar at 400 mph V relative = = 900 mph = 402 meter/s First target Doppler shift = 2 (-134m/s) / (0.03m) = khz Second target Doppler shift = 2 (402m/s) / (0.03m) = 26.8 khz 7

8 Frequency Spectrum of Pulse Spectrum of single pulse Spectrum of slowly repeating pulse (low PRF) Spectrum of rapidly repeating pulse (high PRF) Line spacing equal to PRF 8

9 Doppler Ambiguities Radar bearing aircraft maximum speed: 1200 mph = 536 m/s Target aircraft maximum speed 1200 mph = 536 m/s Maximum positive Doppler = 2 (1072m/s) / (0.03m) = 71.5 khz 9 Radar bearing aircraft minimum speed: 300 mph = 134 m/s Effective radar bearing aircraft minimum speed with θ = 30 degree angle from target track is sin(30) = mph = 67 m/s Target aircraft maximum speed 1200 mph = 536 m/s Maximum negative Doppler = 2 ( m/s) / (0.03m) = khz

10 Critical Choice of PRF Low PRF: generally 1-8 khz Good for maximum range detection Requires long transmit pulse duration to achieve adequte transmit power means more pulse compression Excessive Doppler ambiguity Difficult to reject ground clutter in main antenna lobe Medium PRF: generally 8-30 khz Compromise: get both range and Doppler ambiguities, but less severe Good for scanning use other PRF for precise range or for isolating fast moving targets from clutter High PRF: generally khz Moving target velocity measurement and detection Allows highest transmit power greater detection ranges FFT Spectral masking IFFT detection processing 10

11 Using both Range and Doppler detection Main Lobe Beam High closing rate target Opening rate target Side Lobe Ground Return Ground terrain profile 0 Range Return Assume no range or Doppler ambiguities 11 0 Doppler Return

12 Doppler Pulse Processing Basics fast time L-1 0 Form L range bins in fast time Perform FFT across N pulse intervals for each of L ranges Doppler filter into K frequency bins FFT FFT FFT FFT FFT FFT FFT FFT FFT FFT FFT FFT FFT FFT 0 N-1 0 K-1 slow time Coherent processing interval (CPI) of N radar pulses Target discrimination in both range and Doppler frequency

13 STAP Algorithm

14 MTI Detection among clutter, jamming Pulse Compression matched filtering to optimize SINR MTI filtering for gross clutter removal Pulse Doppler filtering effective to resolve targets with significant motion from clutter STAP temporal and spatial filtering to separate slow moving targets from clutter and null jammers Very high processing requirements Low latency, fast adaptation Dynamic range requires floating point processing 14

15 Target Detection with Spatial Dimension Normalized Doppler Frequency Angle of arrival Jammer is across all Doppler bins Target is same angle as main clutter Target is close to clutter Doppler frequency Spatial and Temporal filtering needed to discriminate

16 Beamforming using ESA A0 T/R A1 T/R A2 T/R A3 A4 A5 A6 A7 T/R T/R T/R T/R T/R d θ A8 A9 T/R T/R λ Each T/R module sample multiplied by complex value A m A m = e -2πdmsin(θ/λ) for m = 1..M-1, for each angle θ Main lobe shape, side lobe height can be adjusted by multiplying angle vector with tapering window similar to FIR filter coefficients All complex receive samples sum/split for single feed to/from processor Angular steering, determined by vector A

17 Add Spatial dimension: Radar Datacube M sub-array weighted samples (no longer summed into single feed) L sampling intervals per pulse interval (fast time) CPI of N radar pulses (slow time) Doppler pulse processing across L & N dimensions STAP processing across M & N dimensions

18 Target steering vector t = f (angle, Doppler) e -2π0F dopp X where A θ = e -2πdmsin(θ/λ) (M long vector) t e -2πnF dopp X e -2π(N-1)F dopp X t is vector of length N M Temporal Steering Vector F d for each Doppler frequency F doppler F d = e -2πnF dopp for n = 1..N-1 A θ = e -2πdmsin(θ/λ) for m = 1..M-1, for given angle of arrival θ t = F d O A θ, Kronecker product of target Doppler and Steering vectors

19 Computing Interference Covariance Matrix y 1 st column j th column 0 Pulse N-1 N th column Slow time (CPI) S I = y* y T M-1 (vector cross product) y is column vector of length NM, so S I is (NM) x (NM) size S Interference = S noise + S jammer + S clutter and is hermitian Range Bin k where S noise = σ 2 I, S jammer is block diagonal matrix, S clutter depends upon environment L-1 Fast time

20 Estimate S I using neighboring bins (NM) x (NM) size matrices M-1 L-1 S I is calculated for each neighboring range bin Do not want target information in covariance matrix, so estimate S I in k th bin by averaging, element by element, nearby S I matrices Orange range cell k contains suspected target under test Red range cells are guard bands Assume nearby bins have same clutter statistics as orange cell Use the nearby cells to for estimate of S I to use in STAP algorithm 0 Pulse Slow time (CPI) N-1 Range Bin k Fast time

21 Calculate weight vector to maximize SIR M-1 h L-1 0 Pulse N-1 Range Bin k Fast time Estimate of S I Slow time (CPI) Calculate optimal weighting vector h = k S I -1 t* k is scalar constant, S I is interference covariance matrix, t is target steering vector Then find z = h T y, where y is output from bin k of radar cube data z(f doppler, θ) is then subject to target threshold detection process

22 Using QRD to find weighting vector h h = k S I -1 t* (h is MN long vector) S I u = t*, where u includes scaling constant k Q R u = t*, where Q Q H = I and Q -1 = Q H Q is constructed of orthogonal normalized vectors R will become upper triangular R u = Q H t* Solve for vector u using back substitution h = u / (t H u*) ( k = t H u* ) z = h T y gives optimal output detection result where z is a complex scalar 22

23 STAP using covariance matrix summary For each range bin of interest: Compute covariance matrix for each range bin Estimate interference covariance matrix S I by averaging the surrounding covariance matrices Perform QRD upon S I Then for each F doppler and A θ of interest: Compute Q H t* R u = Q H t* find u with back substitution Compute h = u / (t H u*) Final result z = h T y This is known as power domain method 23

24 Example GFLOPs estimate (power domain) Process over 12 A θ, 16 F doppler and assume prosecute 32 target steering vectors Use 10 range bins with a PRF = 1 khz Compute Covariance Matrix Average over 10 covariance matrices QR Decomposition Compute Q H t* (each A θ and Doppler) Solve for u using back substitution Compute h = u / (t H u*) and z = h T y Total 1.1 GFLOPS 0.4 GFLOPs 37.7 GFLOPs 9.4 GFLOPs 4.7 GFLOPs 0.2 GFLOPS 53.5 GFLOPs Detection for 8 possible targets over 4 possible velocities over narrow angle and Doppler, low PRF 24

25 Alternate method voltage domain

26 STAP using radar cube data directly y 1 st column j th column 0 Pulse N-1 N th column Slow time (CPI) M-1 Operate directly on y data vectors One vector y per range bin Use L s range bins, where L s > MN range bins Construct matrix Y = [y 0 y 1 y 2.. y L s-1], dimension [MN x L s ] Range Bin k L-1 Fast time

27 STAP using Y data matrix summary S I = Y* Y T = R H Q H Q R (Y T = Q R) = R H R = R 1H R 1 Recall S I u = t*, so substitute R 1H R 1 u = t* Define r R 1 u Then for each F doppler and A θ of interest: Solve for r in R 1H r = t* using forward substitution (R 1H is lower triangular) Solve for u in R 1 u = r using backward substitution (R 1 is upper triangular) Compute h = u / (t H u*) Final result z = h T y This is known as voltage domain method 27

28 Apply QRD to Y data matrix L s -1 L s -1 L s -1 y R 1 MN Y T = Q x 0 0 MN -1 0 L s -1 0 MN -1 Y T = Q x R, where R is composed of R 1 and 0 R 1 is upper triangular, R 1H is lower triangular, [MN x MN]

29 Example GFLOPs estimate (voltage domain) 12 A θ, 16 F doppler and 32 target steering vectors, PRF = 1 khz, with 200 range bins QR Decomposition Solve for r using forward substitution Solve for u using back substitution Compute h = u / (t H u*) and z = h T y Total 40.1 GFLOPs 4.7 GFLOPs 4.7 GFLOPs 0.2 GFLOPs 49.7 GFLOPs Higher rate and resolution system: 48 A θ, 16 F doppler 64 target vectors, PRF = 1 khz, 1000 range bins 29 QR Decomposition Solve for r using forward substitution Solve for u using back substitution Compute h = u / (t H u*) and z = h T y Total 3.51 TFLOPs 37.7 GFLOPs 37.7 GFLOPs 0.8 GFLOPs 3.59 TFLOPs

30 FPGA Processing Flow Implementation (voltage domain) Doppler Vector F d PRF = 1 KHz Compute each PRF over L s range bins Compute each target vector, for each PRF [16 x 1] Angle Vector A θ Kronecker Product Steering Vector t [192 x 1] Complex, Single Precision Floating Point Datapath Y T over L s range bins [192 x 200] [12 x 1] QR Decomposition R 1H (lower triangular) R 1 (upper triangular) Both [192 x 192] Forward Subst. Back Subst. r [192 x 1] u [192 x 1] Dot Product t H u* Norm (/). h [192 x 1] y (range bin of interest) [192 x 1] Output Filter Result z

31 Voltage verses Power domain methods Voltage Domain Operates directly on the data Solve over-determined rectangular matrix, using QRD Each steering vector requires both forward and backward substitution to solve for optimal filter Power Domain Estimate Covariance matrix (Hermitian matrix) Covariance matrix inversion ideal for Choleski algorithm Requires higher dynamic range (none issue with floating point) Choleski more efficient to implement, using Fused Datapath Each steering vector requires only backward substitution to solve for optimal filter 31

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