Novel Waveform Design and Scheduling For High-Resolution Radar and Interleaving

Size: px

Start display at page:

Download "Novel Waveform Design and Scheduling For High-Resolution Radar and Interleaving"

Rosalind Roberts
5 years ago
Views:

1 Novel Waveform Design and Scheduling For High-Resolution Radar and Interleaving

2 Phase Phase

3 Basic Signal Processing for Radar 101 x = α 1 s[n D 1 ] + α 2 s[n D 2 ] +... s signal h = filter if h = s * "matched filter" y = x* h = x* s * = α 1 r ss [n D 1 ] + α 2 r ss [n D 2 ] +... where: r ss = s* s * "autocorrelation" Transmit Signal Design: desire autocorrelation to be spiky, i.e., to look like a large narrow spike (to rise above noise and clutter)

6 Waveform Design: 4x4 System The most useful realization of the 4x4 system is four beams formed as shown in the figure. 6

7 Ultra-High-Resolution Delay-Doppler Radar Advances in Radar Waveform Diversity M. Zoltowski, Purdue U. GOAL: Ultra-High-Resolution Delay-Doppler Radar in high noise and clutter environments. APPROACH: Employ waveform diversity in conjunction with 4x4 Transmit-Receive Radar System. 4x4 Tx-Rx realization: 4 overlapping beams formed simultaneously on transmit & receive. Each beam transmitting a different waveform Enhanced waveform diversity improves detection performance as shown in this ROC plot. Waveforms transmitted simultaneously on 2 different polarizations at 2 spatially separated transmit arrays Employ even & odd parts of Golay Pair forming 4- ary complementary set in conjunction with novel unitary waveform scheduling and joint matched filtering over multiple PRIs RESULTS: Enhanced waveform diversity leads to ultra-high time resolution of closely-spaced targets. Multiplying by a different complex sinewave for each 4-PRI further reduces background level while maintaining unimodularity. Other 4x4 realizations: 4 overlapping beams formed simultaneously on transmit & receive, with each beam transmitting a different waveform. Enhanced waveform diversity with unitary scheduling improves target resolution and detection.

8 Matrix Waveform Design Criterion S(n) S H ( n) = MNδ I Waveform Scheduling Matrix: S(n) dictates waveform transmitted from m-th emitter (i.e., beam), m=1,, M during k-th PRI, k=1,, M H Matrix Matched Filter: S (-n) for each receiver, k-th component of m-th column dictates matched filter tuned to what was transmitted from m-th emitter during k-th PRI Note: can repeat or use another scheduling matrix for each successive block of M PRIs Note 2: asterisk * denotes convolution: in matrix-matrix product: in forming dot-product between row of 1 st matrix and column 2 nd matrix pointwise multiplication of two numbers replaced by convolution of two sequences

9 Observations for General Case S(n) S H ( n) = MNδ I Perfect Separation: all off-diagonal elements equal to zero (for all n) Perfect Reconstruction: follows from all of the diagonal elements equal to a Kronecker delta function VIP Note: desire unimodular sequences satisfying these conditions to facilitate power amplifier efficiency Note 1: can t achieve an autocorrelation equal to a delta function for a single unimodular sequence, BUT can design two unimodular sequences for which sum of autocorrelations is a delta function Note 2: can t achieve a cross-correlation between two unimodular sequences equal to 0 for all n, BUT can design so that sums of crosscorrelations are 0 for all n

10 Example: Circulant Bank of DFT Filters w 0 w 3 w 2 w 1 w 1 w 0 w 3 w 2 w 2 w 1 w 0 w 3 w 3 w 2 w 1 w 0 w * 0 w * 1 w * 2 w * 3 w * 3 w * 0 w * 1 w * 2 w * 2 w * 3 w * 0 w * 1 w * 1 w * 2 w * 3 w * 0 = 16δI Where the sequences are the four rows of a 4-pt DFT matrix Each sequence is unimodular and of length 4 w 0 = { 1,1,1,1 } w 1 = { 1, j, 1, j} w 2 = { 1, 1,1, 1 } w 3 = { 1, j, 1, j} Amazingly, a circulant bank of DFT filters satisfies both perfect reconstruction and perfect separation, with no negative signs or conjugate-time reverses true for any dimension DFT matrix

11 4x4 without Doppler Transmit Waveforms Rx 1 Rx 2 Rx 3 Rx 4 y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 PRI 1 PRI 2 PRI 3 PRI 4 = L l=0 Γ l 11 Γ l 21 Γ l 31 Γ l 41 Γ l 12 Γ l 22 Γ l 32 Γ l 42 Γ l 13 Γ l 23 Γ l 33 Γ l 43 Γ l 14 Γ l 24 Γ l 34 Γ l 44 δ[n τ l ] I s 11 s 12 s 13 s 14 s 21 s 22 s 23 s 24 s 31 s 32 s 33 s 34 s 41 s 42 s 43 s 44 PRI 1 PRI 2 PRI 3 PRI 4 Beam1 Beam2 Beam3 Beam4 Receiver Processing per each Rx: Matched Filtering over 4 PRIs: E H = s * 11 s * 21 s * 31 s * 41 s 12 s * 22 s * 32 s * 42 s * 13 s * 23 s * 33 s * 43 s * 14 s * 24 s * 34 s * 44 = f 11 f 21 f 31 f 41 f 12 f 22 f 32 f 42 f 13 f 23 f 33 f 43 f 14 f 24 f 34 f 44 PRI 1 PRI 2 PRI 3 PRI 4 Each of the 16 cross-correlations below ideally has a delta function centered at each true target delay (typically summed incoherently): r 11 r 12 r 13 r 14 r 21 r 22 r 23 r 24 r 31 r 32 r 33 r 34 r 41 r 42 r 43 r 44 = y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 f 11 f 21 f 31 f 41 f 12 f 22 f 32 f 42 f 13 f 23 f 33 f 43 f 14 f 24 f 34 f 44

12 Simple Example for 2x2 Case: 2 pt DFT Sequences Math details for 2x2 case with simple 2-pt DFT sequences: {1,1} {1, 1} {1, 1} {1,1} {1,1} { 1,1} { 1,1} {1,1} = {1,1} *{1,1} + {1, 1} *{ 1,1} {1,1} * { 1,1} + {1, 1} *{1,1} {1, 1} * {1,1} + {1,1} *{ 1,1} {1, 1} *{ 1,1} + {1,1} *{1,1} Beam 1 Beam 2 {1,1} {1, 1} {1, 1} {1,1} PRI 1 PRI 2 {1,1} { 1,1} { 1,1} {1,1} PRI 1 PRI 2 Receiver Processing per each Rx: Matched Filtering over 2 PRIs = {0,4,0} {0,0,0} {0,0,0} {0, 4,0} In 2x2 case, obtain FOUR cross-correlation outputs each of which ideally exhibits a delta function (sharp peak) at each target delay

13 Unitary waveform scheduling for 2x2 Case 2x2 transmit scheduling and receiver processing in time domain: h 11 h 12 h 21 h 22 = 2N h 11 h 21 h 12 h 22 Complementary Tx Waveforms * Tx 1 e 1 e * 2 Tx 2 e 2 e 1 PRI 1 PRI 2!Have isolated each of these four terms individually, where: e * 1 e * 2 PRI 1 e 2 e 1 PRI 2 Receiver Processing per each Rx: Matched Filtering over 2 PRIs L h ik = Γikδ[n l τ l ] l = 1 e 1 * e 1 * + e 2 * e 2 * = 2Nδ In 2x2 case, obtain FOUR cross-correlation outputs each of which ideally exhibits a delta function (sharp peak) at each target delay Golay pair of complementary {+1,-1) sequences (unimodular)

14 Example for 2x2 Case: Length-4 Barker Codes Math details for 2x2 case with length-4 Barker Codes: {1,1, 1,1} {1, 1, 1, 1} {1,1,1, 1} {1, 1,1,1} {1, 1,1,1} { 1,1,1,1} { 1, 1, 1,1} {1,1, 1,1} = {1,1, 1,1} * {1, 1,1,1} + {1, 1, 1, 1} *{ 1, 1, 1,1} {1,1, 1,1} * { 1,1,1,1} + {1, 1, 1, 1} * {1,1, 1,1} {1,1,1, 1} * {1, 1,1,1} + {1, 1,1,1} * { 1, 1, 1,1} {1,1,1, 1} * { 1,1,1,1} + {1, 1,1,1} * {1,1, 1,1} Tx 1 Tx 2 {1,1, 1,1} {1, 1, 1, 1} {1,1,1, 1} {1, 1,1,1} {1, 1,1,1} { 1,1,1,1} { 1, 1, 1,1} {1,1, 1,1} PRI 1 PRI 2 = {0,0,0,8,0,0,0} {0,0,0,0,0,0,0} {0,0,0,0,0,0,0} {0,0,0,8,0,0,0} PRI 1 PRI 2 Receiver Processing per each Rx: Matched Filtering over 2 PRIs In 2x2 case, obtain FOUR cross-correlation outputs each of which ideally exhibits a delta function (sharp peak) at each target delay The 4 cross-correlation outputs are incoherently summed

15 Analysis of 2x2 Case in Frequency Domain 2x2 transmit scheduling and receiver processing: Transmit Waveforms H 11 (ω) H 21 (ω) E (ω) E * (ω) E * H 12 (ω) H 22 (ω) (ω) E * 2 (ω) PRI 1 E 2 (ω) E 1 (ω) E 2 (ω) E 1 (ω) PRI 2 PRI 1 PRI 2 Receiver Processing = H (ω) H (ω) Ε 1 (ω) 2 + Ε 2 (ω) 2 0 H 12 (ω) H 22 (ω) 0 Ε 1 (ω) 2 + Ε 2 (ω) 2 Ε 1 (ω) 2 + Ε 2 (ω) 2 = 2N ω = 2N H 11(ω) H 21 (ω) H 12 (ω) H 22 (ω) Have isolated each of these four terms individually The point-wise Fourier Transform of the 2x2 matrix of transmitted waveforms is a unitary matrix for any and all (digital) frequencies

16 From Sequences to Continuous-Time Matched Filtering Complementary Sequences: e 1 * e 1 * + e 2 * e 2 * = 2Nδ CT Waveforms Transmitted: N 1 e 1 (t) = e 1 [k]p(t kt c ) e 2 (t) = e 2 [k]p(t kt c ) k = 0 N 1 k = 0 Easy to show: r ei e i (t) = e i (t) e i * ( t) = N 1 r ei e i [k]r pp (t kt c ) i = 1,2 k = (Ν 1) r e1 e 1 (t) + r e2 e 2 (t) = N 1 (r e1 e 1 [k] + r e2 e 2 [k])r pp (t kt c ) = 2Nδ[k]r pp (t kt c ) = 2Ν r pp (t) k = (Ν 1) N 1 k = (Ν 1) e.g., if the chip waveform was a rectangle, a triangle would be centered at each target delay (of duration twice the chip duration) Sampling above the Nyquist rate, one can digitally upsample to do nearcontinuous-time matched filtering

17 4x4 without Doppler Transmit Waveforms Rx 1 Rx 2 Rx 3 Rx 4 y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 PRI 1 PRI 2 PRI 3 PRI 4 = L l =0 Γ l 11 Γ l 21 Γ l 31 Γ l 41 Γ l 12 Γ l 22 Γ l 32 Γ l 42 Γ l 13 Γ l 23 Γ l 33 Γ l 43 Γ l 14 Γ l 24 Γ l 34 Γ l 44 δ[n τ l ] I e 1 e * 2 e 3 e * 4 e 2 e * 1 e 4 e * 3 e 3 e * 4 e 1 e * 2 e 4 e * 3 e 2 e * 1 PRI 1 PRI 2 PRI 3 PRI 4 Tx 1 Tx 2 Tx 3 Tx 4 Receiver Processing per each Rx: Matched Filtering over 4 PRIs E H = e * 1 e * 2 e * 3 e * 4 e 2 e 1 e 4 e 3 e * 3 e * 4 e * 1 e * 2 e 4 e 3 e 2 e 1 = f 11 f 21 f 31 f 41 f 12 f 22 f 32 f 42 f 13 f 23 f 33 f 43 f 14 f 24 f 34 f 44 PRI 1 PRI 2 PRI 3 PRI 4 Each of the 16 cross-correlations below ideally has a delta function centered at each true target delay (typically summed incoherently): r 11 r 12 r 13 r 14 r 21 r 22 r 23 r 24 r 31 r 32 r 33 r 34 r 41 r 42 r 43 r 44 = y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 f 11 f 21 f 31 f 41 f 12 f 22 f 32 f 42 f 13 f 23 f 33 f 43 f 14 f 24 f 34 f 44

18 4x4 Waveform Scheduling Premised on 4x4 OSTBC Consider a set of waveforms satisfying: e 1 * e 1 * + e 2 * e 2 * + e 3 * e 3 * + e 4 * e 4 * = 4Nδ 4x4 waveform scheduling facilitating separation of superimposed returns (under certain conditions): Where: e 1 e 2 * e 3 e 4 * e 2 e 1 * e 4 e 3 * e 3 e 4 * e 1 e 2 * e 4 e 3 * e 2 e 1 * 4Ν * δ 0 φ 0 0 δ 0 φ φ 0 δ 0 0 φ 0 δ e 1 * e 2 * e 3 * e 4 * e 2 e 1 e 4 e 3 e 3 * e 4 * e 1 * e 2 * e 4 e 3 e 2 e 1 φ = e 3 * e 1 * + e 4 * * e 2 + e 1 * e 3 * e 2 * * e 4 =

19 Conditions for Perfect Separation 4x4 waveform yields key matrix: δ 0 φ 0 0 δ 0 φ Φ 4N φ 0 δ 0 0 φ 0 δ φ = e 3 * e 1 * + e 4 * * e 2 + e 1 * e 3 * e 2 * * e 4 Off-diagonal term vanishes if all 4 signals exhibit conjugate symmetry e i * = e i, i = 1,2,3,4 φ = 0 Time-reversed pairs also serve to nullify off-diagonal terms: e 2 * = e 1, e 4 * = e 3, φ = 0

20 Example 4x4 Unitary Waveform Matrix Design φ = (e 1 * e 3 * e 1 * * e 3 ) + (e 2 * e 4 * e 2 * * e 4 ) e 1 * = e 1, e 3 * = e 3, e 2 * = e 2, e 4 * = e 4, φ = 0 Perfect separation is also achieved if signals 1 and 3 are symmetric, while signals 2 and 4 are anti-symmetric Consider a Golay pair of causal sequences of length N, and form even and odd symmetric signals of twice the length, 2N-1, as follows e 1 = g 1 + g 1 e 3 = g 2 + g 2 e 2 = g 1 g 1 e 4 = g 2 g 2 Can show these form a 4-ary complementary waveform set 4 r ei e i = e i e * i = 4Nδ i=1 4 i=1

21 4x4 Case with Doppler Transmit Waveforms Rx 1 Rx 2 Rx 3 Rx 4 y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 PRI 1 PRI 2 PRI 3 PRI 4 = L l =0 h l 11 h l 21 h l 31 h l 41 h l 12 h l 22 h l 32 h l 42 h l 13 h l 23 h l 33 h l 43 h l 14 h l 24 h l 34 h l 44 δ[n τ l ] I e 1 e jυ l e * 2 e j 2υ l e e j 3υ l e * 3 4 e 2 e jυ l e * 1 e j 2υ l e e j 3υ l e * 4 3 e 3 e jυ l e * 4 e j 2υ l e e j 3υ l e * 1 2 e 4 e jυ l e * 3 e j 2υ l e e j 3υ l e * 2 1 PRI 1 PRI 2 PRI 3 PRI 4 Tx 1 Tx 2 Tx 3 Tx 4 Receiver Processing per each Rx: Matched Filtering over 4 PRIs E l H = e 1 * e 2 * e 3 * e 4 * e jυ l e 2 e jυ l e 1 e jυ l e 4 e jυ l e 3 e j 2υ l e * 3 e j 2υ l e * 4 e j 2υ l e * 1 e j 2υ l e * 2 e j 3υ l e 4 e j 3υ l e 3 e j 3υ l e 2 e j 3υ l e 1 = Tx 1 Tx 2 Tx 3 Tx 4 f 11 f 21 f 31 f 41 e jυ l f 12 e jυ l f 22 e jυ l f 32 e jυ l f 42 e j 2υ l f 13 e j 2υ l f 23 e j 2υ l f 33 e j 2υ l f 43 e j 3υ l f 14 e j 3υ l f 24 e j 3υ l f 34 e j 3υ l f 44 PRI 1 PRI 2 PRI 3 PRI 4 PRI 1 PRI 2 PRI 3 PRI 4 r 11 r 12 r 13 r 14 r 21 r 22 r 23 r 24 r 31 r 32 r 33 r 34 r 41 r 42 r 43 r 44 = y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 f 11 f 21 f 31 f 41 e jυ l f 12 e jυ l f 22 e jυ l f 32 e jυ l f 42 e j 2υ l f 13 e j 2υ l f 23 e j 2υ l f 33 e j 2υ l f 43 e j 3υ l f 14 e j 3υ l f 24 e j 3υ l f 34 e j 3υ l f 44 PRI 1 PRI 2 PRI 3 PRI 4

22 4x4 Case with Doppler Consider matched-filtering over 4 PRIs at i-th receive antenna, i=1,2,3,4: PRI 1 PRI 2 PRI 3 PRI 4 r i1 r i2 r i 3 r i 4 = y i1 y i2 y i 3 y i 4 f 11 f 21 f 31 f 41 e jυ l f 12 e jυ l f 22 e jυ l f 32 e jυ l f 42 e j 2υ l f 13 e j 2υ l f 23 e j 2υ l f 33 e j 2υ l f 43 e j 3υ l f 14 e j 3υ l f 24 e j 3υ l f 34 e j 3υ l f 44 PRI 1 PRI 2 PRI 3 PRI 4 r i1 r i2 r i 3 r i 4 = 1 e jυ l e j 2υ l e j 3υ l y i1 f 11 y i1 f 21 y i1 f 31 y i1 f 41 y i2 f 12 y i2 f 22 y i2 f 32 y i2 f 42 y i 3 f 13 y i 3 f 23 y i 3 f 33 y i 3 f 43 y i 4 f 14 y i 4 f 24 y i 4 f 34 y i 4 f 44 Thus, matched-filtering can be done separably: first do matchedfiltering at each PRI (for each Rx) assuming zero Doppler, and then account for the Doppler phase progression across the multiple PRIs prior to summing PRI 1 PRI 2 PRI 3 PRI 4 Can sweep through range of potential Dopplers (via FFT) for each post-mf time-delay to generate delay-doppler (range-velocity) plots

23 Periodically Transmit 4-PRI sets for Higher SNR and Doppler Resolution Y (m) i = y (m) i1 f 11 y (m) i2 f 12 y (m) i1 f 21 y (m) i2 f 22 y (m) i1 f 31 y (m) i2 f 32 y (m) i1 f 41 y (m) i2 f 42 y (m) i 3 f 13 y (m) i 3 f 23 y (m) i 3 f 33 y (m) i 3 f 43 y (m) i 4 f 14 y (m) i 4 f 24 y (m) i 4 f 34 y (m) i 4 f 44 Y (m) = Y (m) 1 Y (m) 2 Y (m) 3 Y (m) 4 In ideal no-noise, single-target case: Y (m +1) = e j 4υ l Y(m) Stack these 4-PRI data matrices for each post-matched filtering timedelay, e.g., for two sets of 4-PRIs, form: Y = Y (m) Y (m +1) Apply FFT column-wise to compensate for Doppler

Delay-Doppler plot on next slide done via

, Doppler compensated for via FFT Processed 6

24 Enhanced Waveform Diversity Unimodular Constraint for Power Amplifier Efficiency Delay-Doppler plot on next slide done via conventional processing, i.e., Doppler compensated for via FFT Processed 6 sets of 4-PRIs with 4x4 scheduling, symmetric/antisymmetric Golay pair, rectangular chip waveform, 4 samples per chip Each successive 4-PRI set multiplied by a different common complex sinewave

25 Two-Target Simulations with Doppler VIP For a fair comparison, the transmitted power for the 1x1 case was boosted by a power of 4 so that it was the same as the total transmitted power in the 4x4 case. Rectangular chip pulse shaping was used. The sampling rate was 4 times per chip. The unit along the delay axis is sub-chip samples. Golay code pair of length 16 was used to form the 4-ary complementary symmetric / anti-symmetric codes of length 32 (per PRI)

26 Comparison between 4x4 and 1x1 (a) 4x4 (b) Conventional 1x1 26

27 Coherent Processing of Multiple Beams (a) 4x4 with coherent combining (b) 1x1 with coherent combining 27

28 Novel Application to Interleaving As opposed to waveforms being pulses being transmitted simultaneously from different beams, pulses are transmitted at equi-spaced intervals over the same PRI. No association problem: returns/echoes are automatically paired with with proper transmitted pulse DESPITE partially overlapping in time VIP: Scheduling and Unitary Waveform Design are KEY to automatic pairing of return echoes with proper transmitted pulse

30 4x4 without Doppler Transmit Waveforms y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 = L l=0 Γ l 11 Γ l 21 Γ l 31 Γ l 41 Γ l 12 Γ l 22 Γ l 32 Γ l 42 Γ l 13 Γ l 23 Γ l 33 Γ l 43 Γ l 14 Γ l 24 Γ l 34 Γ l 44 δ[n τ l ] I e 1 e 2 * e 3 e 4 * e 2 [n D] e 1 * [ (n D)] e 4 [n D] e 3 * [ (n D)] e 3 [n 2D] e 4 * [ (n 2D)] e 1 [n 2D] e 2 * [ (n 2D)] e 4 [n 3D] e 3 * [ (n 3D)] e 2 [n 3D] e 1 * [ (n 3D)] PRI 1 PRI 2 PRI 3 PRI 4 PRI 1 PRI 2 PRI 3 PRI 4 Receiver Processing per each Rx: Matched Filtering over 4 PRIs Rx 1 Rx 2 Rx 3 Rx 4 E H = e * 1 e * 2 e * 3 e * 4 e 2 e 1 e 4 e 3 e * 3 e * 4 e * 1 e * 2 e 4 e 3 e 2 e 1 = f 11 f 21 f 31 f 41 f 12 f 22 f 32 f 42 f 13 f 23 f 33 f 43 f 14 f 24 f 34 f 44 PRI 1 PRI 2 PRI 3 PRI 4 Each of the 16 cross-correlations below ideally has a delta function centered at each true target delay (typically summed incoherently): r 11 r 12 r 13 r 14 r 21 r 22 r 23 r 24 r 31 r 32 r 33 r 34 r 41 r 42 r 43 r 44 = y 11 y 12 y 13 y 14 y 21 y 22 y 23 y 24 y 31 y 32 y 33 y 34 y 41 y 42 y 43 y 44 f 11 f 21 f 31 f 41 f 12 f 22 f 32 f 42 f 13 f 23 f 33 f 43 f 14 f 24 f 34 f 44

Unitary Design of Radar Waveform Diversity Sets

Unitary Design of Radar Waveform Diversity Sets Michael D. Zoltowski, ariq R. Qureshi, Robert Calderbank, and William Moran Abstract In this work, multiple radar waveforms are simultaneously transmitted,