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Describes Pulse Compression in Radar Systems. For comments please contact me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://solohermelin.com. Since some figures were not downloaded, I recommend to see this presentation on my website under RADAR Folder, Signal Processing subfolder.
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5 pulse compression waveform
- 1. Pulse Compression Waveform SOLO HERMELIN Updated: 27.10.08http://www.solohermelin.com
- 2. Table of Content SOLO Pulse Compression Waveform Resolution Pulse Range Resolution Pulse Compression Waveform Introduction Waveform Hierarchy Linear FM Modulated Pulse (Chirp) Barker Codes Combined Barker Codes Poly-Phase Codes Phase Coded Waveforms Matched Filter Response to Phase Coding Bi-Phase Codes
- 3. Table of Content (continue – 1) SOLO Pulse Compression Waveform Poly-Phase Codes Frank Codes P1, P2, P3, P4 Poly-Phase Codes Pseudo-Random Codes Frequency Codes Costas Codes Complementary Pulse Codes Summary of Pulse Compression Codes References
- 4. Range & Doppler Measurements in RADAR SystemsSOLO Resolution Resolution is the spacing (in range, Doppler, angle, etc.) we must have in order to distinguish between two different targets. first target response second target response composite target response greather then 3 db Distinguishable Targets first target response second target response composite target response Undistinguishable Targets less then 3 db The two targets are distinguishable if the composite (sum) of the received signal has a deep (between the two picks) of at least 3 db. Return to Table of Content
- 5. SOLO Unmodulated Pulse Range Resolution Resolution is the spacing (in range, Doppler, angle, etc.) we must have in order to distinguish between two different targets. Range Resolution RADAR τ c R RR ∆+ Target # 1 Target # 2 Assume two targets spaced by a range Δ R and a unmodulated radar pulse of τ seconds. The echoes start to be received at the radar antenna at times: 2 R/c – first target 2 (R+Δ R)/c – second target The echo of the first target ends at 2 R/c + τ τ τ time from pulse transmission c R2 ( ) c RR ∆+2 τ+ c R2 Received Signals Target # 1 Target # 2 The two targets echoes can be resolved if: c RR c R ∆+ =+ 22 τ 2 τc R =∆ Pulse Range Resolution ( ) ( ) ≤≤+ = elsewhere ttA ts 0 0cos : 0 τϕω
- 6. Unmodulated Pulse SOLO Energy ( ) ( ) ( )∫∫∫ +∞ ∞− +∞ ∞− +∞ ∞− === ωω π dSdttsdttsEs 222 2 1 : ( ) ( ) ≤≤+ = elsewhere ttA ts 0 0cos : 0 τϕω ( ) ( ) ( )[ ] ( ) ( ) −+ +=++= +== ∫ ∫∫ +∞ ∞− τ ϕϕτωτ ϕω ϕω τ τ 000 2 0 00 2 0 2 00 2 2cos22cos 1 2 22cos1 2 cos A dtt A dttAdttsEs Unmodulated Pulse
- 7. RADAR SignalsSOLO ( ) ( ) ≤≤+ = elsewhere ttA ts 0 0cos : 0 τϕω Energy ( ) ( ) 2 2cos22cos 1 2 2 000 2 τ τ ϕϕτωτ A E A E ss =⇒ −+ += 2 τc R =∆ Pulse Range Resolution Decreasing Pulse Width Increasing Decreasing SNR, Radar Performance Increasing Increasing Range Resolution Capability Decreasing For the Unmodulated Pulse, there exists a coupling between Range Resolution and Waveform Energy. Return to Table of Content
- 8. Pulse Compression WaveformsSOLO Pulse Compression Waveforms permit a decoupling between Range Resolution and Waveform Energy. - An increased waveform bandwidth (BW) relative to that achievable with an unmodulated pulse of an equal duration τ 1 >>BW 22 τc BW c R <<=∆ - Waveform duration in excess of that achievable with unmodulated pulse of equivalent waveform bandwidth BW 1 >>τ PCWF exhibit the following equivalent properties: This is accomplished by modulating (or coding) the transmit waveform and compressing the resulting received waveform. Pulse Compression Waveform Introduction Return to Table of Content
- 9. SOLO Waveform Hierarchy Radar Waveforms CW Radars Pulsed Radars Frequency Modulated CW Phase Modulated CW bi – phase & poly-phase Linear FMCW Sawtooth, or Triangle Nonlinear FMCW Sinusoidal, Multiple Frequency, Noise, Pseudorandom Intra-pulse Modulation Pulse-to-pulse Modulation, Frequency Agility Stepped Frequency Frequency Modulate Linear FM Nonlinear FM Phase Modulated bi – phase poly-phase Unmodulated CW Multiple Frequency Frequency Shift Keying Fixed Frequency
- 10. SOLO Waveform Hierarchy • Pulse Compression Techniques • Wave Coding • Frequency Modulation (FM) - Linear • Phase Modulation (PM)] - Non-linear - Pseudo-Random Noise (PRN) - Bi-phase (0º/180º) - Quad-phase (0º/90º/180º/270º) • Implementation • Hardware - Surface Acoustic Wave (SAW) expander/compressor • Digital Control - Direct Digital Synthesizer (DDS) - Software compression “filter” Return to Table of Content
- 11. SOLO • Pulse Compression Techniques
- 12. SOLO • Pulse Compression Techniques
- 13. SOLO Waveform Hierarchy • Pulse Compression Techniques
- 14. SOLO Coherent Pulse Doppler Radar Return to Table of Content
- 15. SOLO Linear FM Modulated Pulse (Chirp) ( ) ( )2/cos 2 03 ttAtf ωω ∆+= t A 2/τ− 2/τ ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi Pulse Compression Waveforms Linear Frequency Modulation is a technique used to increase the waveform bandwidth BW while maintaining pulse duration τ, such that BW 1 >>τ 1>>⋅ BWτ 222 0 2 0 ττ µω µ ωω ≤≤−+= += tt t t td d
- 16. Matched Filters for RADAR Signals ( ) ( ) ( ) ( ) ≤≤−= = −∗ Ttttsth eSH i tj i 00 0ω ωω SOLO The Matched Filter (Summary( si (t) - Signal waveform Si (ω) - Signal spectral density h (t) - Filter impulse response H (ω) - Filter transfer function t0 - Time filter output is sampled n (t) - noise N (ω) - Noise spectral density Matched Filter is a linear time-invariant filter hopt (t) that maximizes the output signal-to-noise ratio at a predefined time t0, for a given signal si (t(. The Matched Filter output is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 00 tj iii iii eSSHSS dttssdthsts ω ωωωωω ξξξξξξ −∗ +∞ ∞− +∞ ∞− ⋅=⋅= +−=−= ∫∫
- 17. SOLO Linear FM Modulated Pulse (continue – 1) Pulse Compression Waveforms Concept of Group Delay BW 1 >>τ τ BW 1 ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi ( ) ( ) 222 cos 2 0 00 ττµ ω ≤≤− −=−= = t t tAtsth i t MF Matched Filter ( )tsi ( )tso ( ) ( )tsth i t MF −= =00 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωωω ξξξξξξ ∗ = +∞ ∞− =+∞ ∞− ⋅=⋅= −=−= ∫∫ ii t i ii t i SSHSS dtssdthsts 0 0 0 0 0 0
- 18. SOLO Linear FM Modulated Pulse (continue – 2) Pulse Compression Waveforms Concept of Group Delay (continue -1) BW 1 >>τ τ BW 1 ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi Matched Filter ( )tsi ( )tso ( ) ( )tsth i t MF −= =00 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωωω ξξξξξξ ∗ = +∞ ∞− =+∞ ∞− ⋅=⋅= −=−= ∫∫ ii t i ii t i SSHSS dtssdthsts 0 0 0 0 0 0 ( ) ( ) ( ) ( ) >≤≤+− <+≤≤− − +− +=− 0 22 0 22 2 cos 2 cos 2 0 2 0 2 tt tt t tAtss ii τ ξ τ τ ξ τ ξµ ξω ξµ ξωξξ ( ) ( ) ( ) ( ) ( ) ∫∫ + +− >∞+ ∞− > = − +− +=−= 2/ 2/ 2 0 2 0 2 00 0 0 2 cos 2 cos 0 τ τ ξ ξµ ξω ξµ ξωξξξ t t ii t t d t tAdtssts ( ) ( ) 222 cos 2 0 00 ττµ ω ≤≤− −=−= = t t tAtsth i t MF Ignoring terms of 2 ω0 ( ) ( ) ( ) ( ) ( ) t tttA t tttA t tttA dttt A ts tt t t µ µτµω µ µτµω µ µξµω ξµξµω τ τ τ τ 2/2/sin 2 2/2/sin 2 2/sin 2 2/cos 2 2 0 22 0 2 2/ 2/ 2 0 22/ 2/ 2 0 20 0 0 0 +− − −+ = −+ =−+≅ + +− + +− > = ∫ ( ) ( ) βαβαβα coscos2coscos =−++( )[ ] ( )∫∫ + +− + +− −+++−+−= 2/ 2/ 2 0 22/ 2/ 22 0 2 2/cos 2 2/2/2cos 2 τ τ τ τ ξµξµωξµξµξµξω tt dtt A dtt A
- 19. SOLO Linear FM Modulated Pulse (continue – 3) Pulse Compression Waveforms Concept of Group Delay (continue -2) BW 1 >>τ τ BW 1 ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi Matched Filter ( )tsi ( )tso ( ) ( )tsth i t MF −= =00 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ωωωωω ξξξξξξ ∗ = +∞ ∞− =+∞ ∞− ⋅=⋅= −=−= ∫∫ ii t i ii t i SSHSS dtssdthsts 0 0 0 0 0 0 ( ) ( ) 222 cos 2 0 00 ττµ ω ≤≤− −=−= = t t tAtsth i t MF Ignoring terms of 2 ω0 ( ) ( ) ( ) t tttA t tttA ts t t µ µτµω µ µτµω 2/2/sin 2 2/2/sin 2 2 0 22 0 20 0 0 0 +− − −+ ≅ > = ( ) ( )t tt tt tA ts t t 0 20 0 0 cos 1 2 1 2 sin 1 2 0 ω τ τµ τ τµ τ τ − − −≅ > = ( ) ( ) βαβαβα sincos2sinsin =−−+ −== τ τµ βωα tt t 1 2 ,0 If we re-due for t < 0 and combine, we obtain ( ) ( )t tt tt tA ts t 0 20 0 cos 1 2 1 2 sin 1 2 0 ω τ τµ τ τµ τ τ − − −≅ =
- 20. SOLO ( ) ( )2/cos 2 03 ttAtf ωω ∆+= t A 2/τ− 2/τ Linear FM Modulated Pulse (continue – 4) ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi The Fourier Transform is: ( ) [ ] ( ) ( )∫∫ ∫ −− − ++−+ +−= − += 2/ 2/ 2 0 2/ 2/ 2 0 2/ 2/ 2 0 2 exp 2 1 2 exp 2 1 exp 2 cos τ τ τ τ τ τ µ ωω µ ωω ω µ ωω dt t tjAdt t tjA dttj t tASi ∫∫ −− + +− + + − − − −= 2/ 2/ 2 0 2 0 2/ 2/ 2 0 2 0 2 exp 2 exp 22 exp 2 exp 2 τ τ τ τ µ ωωµ µ ωω µ ωωµ µ ωω dttjj A dttjj A Change variables: xt = − − µ ωω π µ 0 yt = + + µ ωω π µ 0 ( ) ∫∫ −− − + + − −= 2 1 2 1 2 exp 2 exp 22 exp 2 exp 2 2 2 0 2 2 0 Y Y X X i dt y jj A dt x jj A S π µ ωωπ µ ωω ω − −= − += µ ωωτ π µ µ ωωτ π µ 0 2 0 1 2 & 2 XX + −= + += µ ωωτ π µ µ ωωτ π µ 0 2 0 1 2 & 2 YY Define: ( )f n f ∆=−=∆ πωωτµ π 2 2 & 2 1 : 0 Pulse Compression Waveforms
- 21. SOLO ( ) ( )2/cos 2 03 ttAtf ωω ∆+= t A 2/τ− 2/τ Linear FM Modulated Pulse (continue – 5) ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi The Fourier Transform is: ( ) ( ) ( ) ∫∫ −− − + + − −= 2 1 2 1 2 exp 2 exp 22 exp 2 exp 2 22 0 22 0 Y Y X X i dt y jj A dt x jj A S π µ ωωπ µ ωω ω The first part gives the spectrum around ω = ω0, and the second part around ω = -ω0 : where: are Fresnel Integrals, which have the properties: ( ) ( ) ∫∫ == UU dz z USdz z UC 0 2 0 2 2 sin& 2 cos ππ ( ) ( ) ( ) ( )USUSUCUC −=−−=− & ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )−+ ++−=−+− − + +++ − −= ωωωω µ ωω µ π µ ωω µ π ω 002211 2 0 2211 2 0 2 exp 2 2 exp 2 ii i SSYSjYCYSjYCj A XSjXCXSjXCj A S ωωωπωτµ π ∆=−∆=∆=∆ 2 :&2: 2 1 : 0 n ff Pulse Compression Waveforms
- 22. SOLO Fresnel Integrals Augustin Jean Fresnel 1788-1827 Define Fresnel Integrals ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∑ ∑∫ ∞ = + ∞ = + + −= = ++ −= = α α αα π α αα π α 0 0 14 2 0 34 0 2 !214 1 2 sin: !1234 1 2 cos: n n n n n n nn x dS nn x dC ( ) ( )αααα πα SjCdj += ∫0 2 2 exp ( ) ( ) 5.0±=∞±=∞± SC ( ) ( ) ( ) ( )USUSUCUC −=−−=− & The Cornu Spiral is defined as the plot of S (u) versus C (u) duuSd duuCd = = 2 2 2 sin 2 cos π π ( ) ( ) duSdCd =+ 22 Therefore u may be thought as measuring arc length along the spiral.
- 23. SOLO ( ) ( )2/cos 2 03 ttAtf ωω ∆+= t A 2/τ− 2/τ Linear FM Modulated Pulse (continue – 6) ( ) 222 cos 2 0 ττµ ω ≤≤− += t t tAtsi The Fourier Transform is: ωωωπωτµ π ∆=−∆=∆=∆ 2 :&2: 2 1 : 0 n ff Define: ( ) ( ) ( )[ ] ( ) ( )[ ]{ }2 21 2 210 2 XSXSXCXC A Si +++=− + µ π ωωAmplitude Term: Square Law Phase Term: ( ) ( ) µ ωω ω 2 2 0 1 − −=Φ Residual Phase Term: ( ) ( ) ( ) ( ) ( ) 4 1tan 5.05.0 5.05.0 tantan 11 1 21 211 2 π ω τ == + + → + + =Φ −− >>∆ − f XCXC XSXS ( ) ( )nfXnfX −∆=+∆= 1 2 &1 2 21 ττ ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )−+ ++−=−+− − + +++ − −= ωωωω µ ωω µ π µ ωω µ π ω 002211 2 0 2211 2 0 2 exp 2 2 exp 2 ii i SSYSjYCYSjYCj A XSjXCXSjXCj A S ( )ω2Φ( ) + − ωω0iS Pulse Compression Waveforms
- 24. SOLO Linear FM Modulated Pulse (continue – 7) Pulse Compression Waveforms Linear FM Modulated Pulse (Chirp) Summary • Chirp is one of the most common type of pulse compression code • Chirp is simple to generate and compress using IF analog techniques, for example, surface acoustic waves (SAW) devices. • Large pulse compression ratios can be achieved (50 – 300). • Chirp is relative insensitive to uncompressed Doppler shifts and can be easily weighted for side-lobe reduction. • The analog nature of chirp sometimes limits its flexibility. • The very predictibility of chirp mades it asa poor choice for ECCM purpose. Return to Table of Content
- 25. SOLO Pulse Compression Techniques Phase Coded Waveforms • Phase Coded Waveforms consists of N contiguous sub-pulses where the phase of each pulse is chosen to shape the range sidelobe response at the output of the matched filter. - sub-pulse length = τ - total length = N τ Poly-phase codes allow for any of M phase shifts on a sub-pulse basis, where M is called the order of the code and the possible phase states are φi = (2π/M) i, for i = 1,…,M
- 26. SOLO Pulse Compression Techniques Phase Coding A transmitted radar pulse of duration τ is divided in N sub-pulses of equal duration τ’ = τ /N, and each sub-pulse is phase coded in terms of the phase of the carrier. The complex envelope of the phase coded signal is given by: ( ) ( ) ( )∑ − = −= 1 0 2/1 ' ' 1 N n n ntu N tg τ τ where: ( ) ( ) ≤≤ = elsewhere tj tu n n 0 '0exp τϕ Return to Table of Content
- 27. Matched Filters for RADAR SignalsSOLO Matched Filter Response to Phase Coding ( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp 2 1 exp 2 1 ωω −+= ∗ ( ) ( ) ( ) ∆<< =∆−= ∑ − = elsewhere tt tftptfctg M p p 0 011 0 Let the signal be a phase-modulated carrier, in which the modulation is in discrete and equal steps Δt. The complex envelope of the signal can be described by a sequence of complex numbers , such thatkc ( ) [ ] ( ) ( )∫ +∞ ∞− ∗ +−−= dtttgtgtjgo 000exp 2 1 τωτ Constant Phase Matched Filter output envelope (change t ↔τ): ( )ttk ∆<≤+∆→ τττ 0 tMt ∆=0 ( ) [ ] ( ) ( )[ ] [ ] ( )[ ] ( ) ∑ ∫ ∫∑ − = ∆+ ∆ ∗ +∞ ∞− ∗ − = ∆−+−∆−= ∆−+−∆−∆−=+∆ 1 0 1 0 1 0 0 exp 2 1 exp 2 1 M p tp tp p M p po dttkMtgctMj dttkMtgtptfctMjtkg τω τωτ Change variable of integration to t1 = t – τ + (M - k) Δt ( ) [ ] ( ) ( ) ( ) ∑ ∫ − = −∆+−+ −∆−+ ∗ ∆−=+∆ 1 0 1 110exp 2 1 M p tkMp tkMp po dttgctMjtkg τ τ ωτ
- 28. Matched Filters for RADAR SignalsSOLO Matched Filter Response to Phase Coding (continue – 1( Matched Filter output envelope for a Phase Coding is: ( ) [ ] ( )[ ] ( ) ∑ ∫ − = ∆+ ∆ ∗ ∆−+−∆−=+∆ 1 0 1 0exp 2 1 M p tp tp po dttkMtgctMjtkg τωτ Change variable of integration to t1 = t – τ + (M - k) Δt ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ∑ ∫∫∑ ∫ − = −∆+−+ ∆−+ ∗ ∆−+ −∆−+ ∗ − = −∆+−+ −∆−+ ∗ +∆−=∆−=+∆ 1 0 1 11110 1 0 1 110 exp 2 1 exp 2 1 M p tkMp tkMp tkMp tkMp p M p tkMp tkMp po dttgdttgctMjdttgctMjtkg τ τ τ τ ωωτ ( ) ( ) ( ) ( ) ( ) ( ) τ τ −∆+−+<<∆−+= ∆−+<<−∆−+= −+ ∗ −−+ ∗ tkMpttkMpctg tkMpttkMpctg kMp kMp 11 * 1 11 * 1 ( ) [ ] ∑ ∫∫ − = −∆ −+ − −−+ +∆−=+∆ 1 0 0 1 * 0 11 * 0exp 2 1 M p t kMpkMppo dtcdtcctMjtkg τ τ ωτ ( ) [ ] ∑ − = −+−−+ ∆ −+ ∆ ∆− ∆ =+∆ 1 0 * 1 * 0 1exp 2 1 M p kMpkMppo t c t cctMj t tkg ττ ωτ This equation describes straight lines in the complex plane, that can have corners only at τ = 0. At those corners ( ) [ ] ∑ − = −+∆− ∆ =∆ 1 0 * 0exp 2 1 M p kMppo cctMj t tkg ω Constant Phase
- 29. Matched Filters for RADAR SignalsSOLO Matched Filter Response to Phase Coding (continue – 2( Matched Filter output envelope for a Phase Coding is: ( ) [ ] ∑ − = −+−−+ ∆ −+ ∆ ∆− ∆ =+∆ 1 0 * 1 * 0 1exp 2 1 M p kMpkMppo t c t cctMj t tkg ττ ωτ This equation describes straight lines in the complex plane, that can have corners only at τ = 0. At those corners ( ) [ ] ∑ − = −+∆− ∆ =∆ 1 0 * 0exp 2 1 M p kMppo cctMj t tkg ω Constant Phase We can see that is the discrete autocorrelation function for the observation time t0 = M Δt (the time the received Radar signal return is expected) ∑ − = −+ 1 0 * M p kMpp cc
- 30. Matched Filters for RADAR SignalsSOLO Matched Filter Response to Phase Coding (continue – 3( Example: Pulse poly-phase coded of length 4 Given the sequence: { } 1,,,1 −−++= jjck which corresponds to the sequence of phases 0◦, 90◦, 270◦ and 180◦, the matched filter is given in Figure bellow. { } 1,,,1 * −+−+= jjck
- 31. Pulse poly-phase coded of length 4 At the Receiver the coded pulse enters a 4 cells delay lane (from left to right), a bin at each clock. The signals in the cells are multiplied by -1,+j,-j or +1 and summed. clock SOLO Poly-Phase Modulation -1 = -11 1+ -j +j = 02 1+j+ +j -1-j = -13 1+j+j− +1 +1+1+1 = 44 1+j+j−1− -j-1+j = -1 5 j+j−1− +j - j = 0 6 j−1− 7 1− -1 = -1 8 0 Σ { } 1,,,1 −−++= jjck 1− 1+j+ j− {ck*} 0 = 00 0 1 2 3 4 5 6 7 { } 1,,,1* −+−+= jjck Return to Table of Content
- 32. SOLO Pulse Compression Techniques Bi-Phase Codes • easy to implement • significant range sidelobe reduction possible • Doppler intolerant A bi-phase code switches the absolute phase of the RF carrier between two states 180º out of phase. Bandwidth ~ 1/τ Transmitted Pulse Received Pulse • Peak Sidelobe Level PSL = 10 log (maximum side-lobe power/ peak response power) • Integrated Side-lobe Level ISL = 10 log (total power in the side-lobe/ peak response power) Bi-Phase Codes Properties The most known are the Barker Codes sequence of length N, with sidelobes levels, at zero Doppler, not higher than 1/N. Return to Table of Content
- 33. SOLO Pulse Compression Techniques Bi-Phase Codes Length N Barker Code PSL (db) ISL (db) 2 + - - 6.0 - 3.0 2 + + - 6.0 - 3.0 3 + + - - 9.5 - 6.5 3 + - + - 9.5 - 6.5 4 + + - + - 12.0 - 6.0 4 + + + - - 12.0 - 6.0 5 + + + - + - 14.0 - 8.0 7 + + + - - + - - 16.9 - 9.1 11 + + + - - - + - - + - - 20.8 - 10.8 13 + + + + + - - + + - + - + - 22.3 - 11.5 Barker Codes -Perfect codes – Lowest side-lobes for the values of N listed in the Table.
- 34. -1 Pulse bi-phase Barker coded of length 3 Digital Correlation At the Receiver the coded pulse enters a 3 cells delay lane (from left to right), a bin at each clock. The signals in the cells are multiplied according to ck* sign and summed. clock -1 = -11 +1 -1 = 02 -( +1) = -15 0 = 06 +1 +1-( -1) = 33 +1-( +1) = 04 SOLO Pulse Compression Techniques 1 2 3 4 5 6 0 +1+1 0 = 00
- 35. Pulse bi-phase Barker coded of length 5 Digital Correlation At the Receiver the coded pulse enters a 7 cells delay lane (from left to right), a bin at each clock. The signals in the cells are multiplied by ck* and summed. clock SOLO Pulse Compression Techniques +1-1+1+1+1 { }* kc +1 = +11 +1 = 19 0 = 010 2 -1 +1 = 0 +1 +1 -1-( +1) = 04 +1 +1 +1 –(-1)+1 = 55 0 = 00 3 +1-1 +1 = 1 +1 +1 -(+1) -1 = 06 +1-( +1) +1 = 17 –(+1) +1 = 08
- 36. Pulse bi-phase Barker coded of length 7 Digital Correlation At the Receiver the coded pulse enters a 7 cells delay lane (from left to right), a bin at each clock. The signals in the cells are multiplied by ck* and summed. clock -1 = -11 +1 -1 = 02 -1 +1 -1 = -13 -1 -1 +1-( -1) = 04 +1 -1 -1 –(+1)-( -1) = -15 +1 +1 -1-(-1) –(+1)-1= 06 +1+1 +1-( -1)-(-1) +1-(-1)= 77 +1+1 –(+1)-( -1) -1-( +1)= 08 +1-(+1) –(+1) -1-( -1)= -19 -(+1)-(+1) +1 -( -1)= 010 -(+1)+1-(+1) = -111 +1-(+1) = 012 -(+1) = -1 13 0 = 014 SOLO Pulse Compression Techniques -1-1 -1+1+1+1+1 { }* kc
- 37. Pulse bi-phase coded of length 8 Digital Correlation At the Receiver the coded pulse enters a 8 cells delay lane (from left to right), a bin at each clock. The signals in the cells are multiplied by ck* and summed. clock SOLO Pulse Compression Techniques +1 = 11 -1-1 -1+1+1+1+1 { }* kc+1 -1 +1 = 02 -1 -1 +1 = -13 +1 -1 -1-( +1) =-24 -1 +1 -1 –(-1)+1= 15 +1 -1 +1-(-1) -1–(+1)= 06 1+1 -1-( +1)-1 –(-1)-(+1)=- 17 +1+1+1 –(-1)+1-( -1) -( -1)+1= 88 +1+1 –(+1) -1-( +1)-(-1) -1= -19 +1-(+1)+1-(-1) -( +1)-1= 010 -(+1)+1-(+1)-(-1)+1 = 111 +1-(+1)-(+1)-1 = -212 -(+1)-(+1)+1 = -113 -(+1)+1 = 014 +1 = 115
- 38. SOLO Pulse Compression Techniques Bi-Phase Codes Combined Barker Codes One scheme of generating codes longer than 13 bits is the method of forming combined Barker codes using the known Barker codes. For example to obtain a 20:1 pulse compression rate, one may use either a 5x4 or a 4x5 codes. The 5x4 Barker code (see Figure) consists of the 5 Barker code, each bit of which is the 4-bit Barker code. The 5x4 combined code is the 20-bit code. • Barker Code 4 • Barker Code 5
- 39. SOLO Pulse Compression Techniques Bi-Phase Codes
- 40. SOLO Pulse Compression Techniques Bi-Phase Codes Binary Phase Codes Summary • Binary phase codes (Barker, Combined Barker) are used in most radar applications. • Binary phase codes can be digitally implemented. It is applied separately to I and Q channels. • Binary phase codes are Doppler frequency shift sensitive. • Barker codes have good side-lobe for low compression ratios. • At Higher PRFs Doppler frequency shift sensitivity may pose a problem. Return to Table of Content
- 41. SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes In this case the pulse of width τ is divided in N equal groups; each group is subsequently divided into other N sub-pulses each of width τ’. Therefore the total number of sub-pulses is N2 , and the compression ratio is also N2 . A Frank code of N2 sub-pulses is called a N-phase Frank code. The fundamental phase increment of the N-phase Frank code is: N/360 =∆ ϕ For N-phase Frank code the phase of each sub-pulse is computed from: ( ) ( ) ( ) ( ) ( ) ϕ∆ −−−− − − 2 1131210 126420 13210 00000 NNNN N N Each row represents the phases of the sub-pulses of a group
- 42. SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 1) Example: For N=4 Frank code. The fundamental phase increment of the 4-phase Frank code is: 904/360 ==∆ ϕ We have: −− −− −− ⇒ → jj jjj form complex 11 1111 11 1111 901802700 18001800 270180900 0000 90 Therefore the N = 4 Frank code has the following N2 = 16 elements { }jjjjF 11111111111116 −−−−−−= The phase increments within each row represent a stepwise approximation of an up- chirp LFM waveform.
- 43. SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 2) Example: For N=4 Frank code (continue – 1). If we add 2π phase to the third N=4 Frank phase row and 4π phase to the forth (adding a phase that is a multiply of 2π doesn’t change the signal) we obtain a analogy to the discrete FM signal. If we use then the phases of the discrete linear FM and the Frank-coded signals are identical at all multipliers of τ’. '/1 τ=∆ f
- 44. SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 3) Fig. 8.8 Levanon pg.157
- 45. SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 4) Fig. 8.8 Levanon pg.158,159
- 46. SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 5) Return to Table of Content
- 47. SOLO Pulse Compression Techniques P1, P2, P3, P4 Poly-Phase Codes The phase-code pulses envelope is given by: ( ) ( ) ( ) ∑= −− = N m m mt recj NT tg 1 ' 1 exp 1 τ ϕ The phases φm are chosen such that the autocorrelation function has the smallest Peak-to-sidelobe ratio (PSR), for a certain code length. PSR is bounded from bellow by the code length N ( )NPSR log20= Binary phase codes use only φm=0 or π. The main drawback of binary codes, such as Barker codes or m-sequences is their sensitivity to Doppler shift. Poly-phase codes are not restricted on code elements and are generated from phase history of frequency-modulated pulse. The Frank code and the P1 and P2 codes, The modified version of Frank code, are derived from the linear stepped frequency Modulation. These three codes are only applicable for perfect square length (N = L2 ), and can be expressed as: ( ) ( ) ( ) ( )[ ] ( )[ ] ( )[ ] ( )[ ]jLiL L P jLiLj L P ji L Frank ji ji ji −+−+= −−−+−= −−= 2/12/1 2 :2 1211 2 :1 11 2 : , , , π ϕ π ϕ π ϕ
- 48. SOLO Pulse Compression Techniques ( ) ( ) ( ) ( )Nii N P oddNNiii N evenNNii N P i i −−−= =− =− = 11:4 ;,,2,1;1 ;,,2,1;1 :3 2 π ϕ π π ϕ Another two well known poly-phase codes are P3 and P4 derived from linear frequency modulated pulse. Unlike Frank, P1 and P2 codes, P3 and P4 code lengths are arbitrary. P3 and P4 codes can be expressed as: It is known that Frank, P1 and P2 codes are more Doppler shift insensitive than binary codes, but P3 and P4 are even better. P1, P2, P3, P4 Poly-Phase Codes
- 49. SOLO Pulse Compression Techniques P1, P2, P3, P4 Poly-Phase Codes P4, N = 25 Elements
- 50. SOLO Pulse Compression Techniques Frank, P1, P2, P3, P4 Codes Summary • Frank, P1, P2, P3, P4 Codes are digital versions of the chirp • They are insensitive to Doppler frequency shift provided that fmax . τ’ < 0.3 but more sensitive then chirp. • They can have very long length.. P1, P2, P3, P4, P(n,k) Poly-Phase Codes Return to Table of Content
- 51. SOLO Pseudo-Random Codes Pseudo-Random Codes are binary-valued sequences similar to Barker codes. The name pseudo-random (pseudo-noise) stems from the fact that they resemble a random like sequence. The pseudo-random codes can be easily generated using feedback shift-registers. It can be shown that for N shift-registers we can obtain a maximum length sequence of length 2N -1. 0 1 0 0 1 1 1 23 -1=7 Register # 1 Register # 2 Register # 3 XOR clock A B Input A Input B Output XOR 0 0 0 0 1 1 1 0 1 1 1 0 Register # 1 Register # 2 Register # 3 0 1 0 s e q u e n c e I.C. 0 0 11 1 0 02 1 1 03 1 1 14 0 1 15 1 0 16 0 1 07 clock 0 0 18 0 Pulse Compression Techniques
- 52. SOLO Pseudo-Random Codes (continue – 1) To ensure that the output sequence from a shift register with feedback is maximal length, the biths used in the feedback path like in Figure bellow, must be determined by the 1 coefficients of primitive, irreducible polynomials modulo 2. As an example for N = 4, length 2N -1=15, can be written in binary notation as 1 0 0 1 1. The primitive, irreductible polynomial that this denotes is (1)x4 + (0)x3 + (0)x2 + (1)x1 + (1)x0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 24 -1=15 s e q u e n c e 1 0 0 1 I.C.0 The constant (last) 1 term in every such polynomial corresponds to the closing of the loop to the first bit in the register. Register # 1 Register # 2 Register # 3 XOR clock A B Input A Input B Output XOR 0 0 0 0 1 1 1 0 1 1 1 0 Register # 4 Register # 1 Register # 2 Register # 3clock Register # 4 1 0 1 0 0 0 0 1 02 0 0 0 13 1 0 0 04 1 1 0 05 1 1 1 06 1 1 1 17 0 1 1 18 1 0 1 19 0 1 0 110 1 0 1 011 1 1 0 112 0 1 1 013 0 0 1 114 1 0 0 115 0 1 0 016 0 0 1 017 Pulse Compression Techniques
- 53. SOLO Pseudo-Random Codes (continue – 2) Pulse Compression Techniques Input A Input B Output XOR 0 0 0 0 1 1 1 0 1 1 1 0 Register # 1 Register # 2 Register # n XOR clock A B Register # (n-1) Register # m . . .. . . 2 3 1 2,1 3 7 2 3,2 4 15 2 4,3 5 31 6 5,3 6 63 6 6,5 7 127 18 7,6 8 255 16 8,6,5,4 9 511 48 9,5 10 1,023 60 10,7 11 2,047 176 11,9 12 4,095 144 12,11,8,6 13 8,191 630 13,12,10,9 14 16,383 756 14,13,8,4 15 32,767 1,800 15,14 16 65,535 2,048 16,15,13,4 17 131,071 7,710 17,4 18 262,143 7,776 18,11 19 524,287 27,594 19,18,17,14 20 1,048,575 24,000 20,17 Number of Stages n Length of Maximal Sequence N Number of Maximal Sequence M Feedback stage connections Maximum Length Sequence n – stage generator N – length of maximum sequence 12 −= n N M – the total number of maximal-length sequences that may be obtained from a n-stage generator ∏ −= ipN n M 1 1 where pi are the prime factors of N.
- 54. SOLO Pseudo-Random Codes (continue – 3) Pulse Compression Techniques Pseudo-Random Codes Summary • Longer codes can be generated and side-lobes eventually reduced. • Low sensitivity to side-lobe degradation in the presence of Doppler frequency shift. • Pseudo-random codes resemble a noise like sequence. • They can be easily generated using shift registers. • The main drawback of pseudo-random codes is that their compression ratio is not large enough. Return to Table of Content
- 55. SOLO Pulse Compression Techniques Frequency Codes Costas Codes In this case a pulse of duration T is divided in N equal sub-pulses of duration NT /1 =τ In Linear Stepped Frequency Modulation (LSFM) the frequency of each sub-pulse is increased linearly according to: Nififfi ,,2,10 =+= δ where f0 is a constant frequency and f0 >> δ f. The maximum change in frequency is Δ f = N δ f during the time τ. The pulse has a time-bandwidth of: ( ) 2 1 1 2 1 NfNNfNTf ≈⋅=⋅=⋅∆ ≈ τδτδ 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Column number, j (time) Column number, j (time) Rownumber,i(frequency) Rownumber,i(frequency) Frequency-time array for LSFM Frequency-time array for Costas code Costas codes are similar to LSFM, only the frequency steps are chosen randomly.
- 56. SOLO Pulse Compression Techniques Frequency Codes Costas Codes (continue – 1) The normalized complex envelope of a Costas signal is given by: ( ) ( ) ( ) ( ) ≤≤ =−= ∑ − = elsewhere ttfj tgltg N tg l l N l l 0 02exp & 1 1 1 0 1 1 τπ τ τ Costas showed that the output of the matched filter is given by: ( ) ( ) ( ) ( )( )∑ ∑ − = − ≠ = −−Φ+Φ= 1 0 1 0 1,,2exp 1 , N l N lq q DlqDlllD fqlftfj N f τττπτχ ( ) ( ) 1 1 1 2exp sin , τττπβ α α τ τ ττ ≤−− −=Φ qq q q Dlq fjjf ( ) ( ) ( ) ( )ττπβ ττπα +−−= −−−= 1 1 Dqlq Dqlq fff fff
- 57. SOLO Pulse Compression Techniques Frequency Codes Costas Codes (continue – 2) 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Column number, j (time) Rownumber,i(frequency) Frequency-time array for Costas code Fig. 8.3, 8.4 Levanon pg.150,151
- 58. SOLO Pulse Compression Techniques Frequency Codes Costas Codes (continue – 3) • All side-lobes, except for few around the origin, have amplitude 1/N. Few side-lobes close to the origin have amplitude 2/N, which is typical to Costas codes. • The compression ratio of Costas codes is approximately N. • The ambiguity function of Costas codes is approaching the ideal thumbtack shape.. • Costas codes have low sensitivity to coherence requirements. Return to Table of Content
- 59. SOLO Pulse Compression Techniques Complementary Pulse Codes Complementary codes consist of a pair of codes with complementary side-lobes, that is, their side-lobes are equal and opposite Golay, 1961).
- 60. SOLO
- 61. Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989 References SOLO Pulse Compression Techniques Levanon, N., “Radar Principles”, John Wiley & Sons, 1988 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000 Nathanson, F.E., “Radar Design Principles”, McGraw-Hill, 1969 Morris, G.V., “Airborne Pulse Radar”, Artech House, 1988 Berkowitz, R.S. Ed., “Modern Radar – Analysis, Evaluation, and System Design”, John Wiley & Sons, 1965 Richards, M.A., “Fundamentals of Radar Signal Processing”, Georgia Tech Course ECE 6272, Spring 2000 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms” Hermelin, S., “Matched Filters and Ambiguity Functions for RADAR Signals”, Power Point Presentation Return to Table of Content
- 62. January 20, 2015 62 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA
- 63. SOLO Fourier Transform of a Signal The Fourier transform of a signal f (t) can be written as: A sufficient (but not necessary) condition for the existence of the Fourier Transform is: ( ) ( ) ∞<= ∫∫ ∞ ∞− ∞ ∞− ωω π djFdttf 22 2 1 JEAN FOURIER 1768-1830 ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 The Inverse Fourier transform of F (j ω) is given by: ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω
- 64. ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 Signal (1) C.W. ( ) 2 cos 00 0 tjtj ee AtAtf ωω ω − + == 0ω - carrier frequency Frequency ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω Fourier Transform ( ) ( ) ( )00 22 ωωδωωδω ++−= AA jF Fourier Transform SOLO Fourier Transform of a Signal
- 65. ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 Signal (2) Single Pulse ( ) > ≤≤− = 2/0 2/2/ τ ττ t tA tf τ - pulse width Frequency ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω Fourier Transform ( ) ( ) ( ) ( )2/ 2/sin 2/ 2/ τω τω τω τ τ ω AdteAjF tj == ∫− Fourier Transform SOLO Fourier Transform of a Signal
- 66. ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 Signal ( ) ( ) > ≤≤− = 2/0 2/2/cos 0 τ ττω t ttA tf τ - pulse width Frequency ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω Fourier Transform ( ) ( ) ( ) ( ) ( ) ( ) − − + + + = = ∫− 2 2 sin 2 2 sin 2 cos 0 0 0 0 2/ 2/ 0 τωω τωω τωω τωω τ ωω τ τ ω A dtetAjF tj Fourier Transform 0ω - carrier frequency (3) Single Pulse Modulated at a frequency 0ω ω ( )ωjF 0 τ π ω 2 0 + 2 τA 0ω τ π ω 2 0 − τ π ω 2 0 +− 2 τA 0ω− τ π ω 2 0 −− τ π ω 2 20 + τ π ω 2 20 − SOLO Fourier Transform of a Signal
- 67. ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 Signal ( ) ( ) ±±=>− ≤−≤−+ = ,2,1,0,2/0 2/2/cos 0 kkkTt kTttA tf rand τ ττϕω τ - pulse width Frequency ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω Fourier Transform ( ) ( ) ( ) ( ) ( ) ( ) − − + + + = = ∫− 2 2 sin 2 2 sin 2 cos 0 0 0 0 2/ 2/ 0 τωω τωω τωω τωω τ ωω τ τ ω A dtetAjF tj Fourier Transform 0ω - carrier frequency (4) Train of Noncoherent Pulses (random starting pulses), modulated at a frequency 0ω T - Pulse repetition interval (PRI) SOLO Fourier Transform of a Signal
- 68. ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 Signal ( ) ( ) ( ) ( )( ) ( )( )[ ] −++ += ±±=>− ≤−≤− = ∑ ∞ =1 000 0 coscos 2 2 sin cos ,2,1,0,2/0 2/2/cos n PRPR PR PR series Fourier tntn n n t T A kkkTt kTttA tf ωωωω τω τω ω τ τ ττω τ - pulse width Frequency ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω Fourier Transform Fourier Transform 0ω - carrier frequency 5) Train of Coherent Pulses, of infinite length, modulated at a frequency 0ω T - Pulse repetition interval (PRI) ( ) ( ) ( ){ ( ) ( ) ( ) ( )[ ] +−+−+−−++ + −+= ∑ ∞ =1 0000 00 2 2 sin 2 n PRPRPRPR PR PR nnnn n n T A jF ωωδωωδωωδωωδ τω τω ωδωδ τ ω T/1 - Pulse repetition frequency (PRF) TPR /2πω = SOLO Fourier Transform of a Signal
- 69. ( ) ( )∫ +∞ ∞− − = ωω π ω dejF j tf tj 2 1 Signal ( ) ( ) ( ) ( )( ) ( )( )[ ] −++ += ±±=>− ≤−≤− = ∑ ∞ = ≤≤− 1 000 22 0 coscos 2 2 sin cos 2/,,2,1,0,2/0 2/2/cos n PRPR PR PRNT t NT tntn n n t T A NkkkTt kTttA tf ωωωω τω τω ω τ τ ττω τ - pulse width Frequency ( ) ( )∫ +∞ ∞− = dtetfjF tjω ω Fourier Transform Fourier Transform 0ω - carrier frequency 6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω T - Pulse repetition interval (PRI) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) −− −− + +− +− + + + + −+ −+ + ++ ++ + + + = ∑ ∑ ∞ = ∞ = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 sin 2 2 sin 2 2 sin 2 2 sin 2 2 sin 2 2 sin 2 2 sin 2 2 sin 2 n PR PR PR PR PR PR n PR PR PR PR PR PR TN n TN n TN n TN n n n TN TN TN n TN n TN n TN n n n TN TN T A jF ωωω ωωω ωωω ωωω τω τω ωω ωω ωωω ωωω ωωω ωωω τω τω ωω ωω τ ω T/1 - Pulse repetition frequency (PRF) TPR /2πω = SOLO Fourier Transform of a Signal
- 70. Signal ( ) ( ) += ±±=>− ≤−≤− = ∑ ∞ =1 1 cos 2 2 sin 21 ,2,1,0,2/0 2/2/ n PR PR PR Series Fourier tn n n T A kkkTt kTtA tf ω τω τω τ τ ττ τ - pulse width 0ω - carrier frequency 6) Train of Coherent Pulses, of finite length N T, modulated at a frequency 0ω T - Pulse repetition interval (PRI) T/1 - Pulse repetition frequency (PRF) TPR /2πω = ( ) ( )tAtf 03 cos ω= t A A ( )tf1 t 2 τ 2 τ −T A T T 2 2 τ+T 2 2 τ−T T T 2 τ− 2 τ+T ( )tf2 t TN 2/TN2/TN− ( ) ( ) ( ) ( )tftftftf 321 ⋅⋅= ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )[ ] −++ += ±±=>− ≤−≤− =⋅⋅= ∑ ∞ = ≤≤− 1 000 22 0 321 coscos 2 2 sin cos 2/,,2,1,0,2/0 2/2/cos n PRPR PR PRNT t NT tntn n n t T A NkkkTt kTttA tftftftf ωωωω τω τω ω τ τ ττω ( ) > ≤≤− = 2/0 2/2/1 2 TNt TNtTN tf ( ) ( )ttf 03 cos ω= SOLO Fourier Transform of a Signal
Notes de l'éditeur
- “Principles of Modern Radar” Georgia Tech, 2004, Samuel O.Piper
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- J.V.DiFranco, W.I. Rubin, “RADAR Detection”, Artech House, 1981, Ch.5, pp.143-201 D. Curtis Schleher Ed., “Automatic Detection and Radar Data Processing”,
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- Reference in J. Meyer-Arendt, “Introduction to Classical & Modern Optics”, 3th Ed.,Prentince Hall, 1989, pg. 258 http://en.wikipedia.org/wiki/Fresnel_integral http://mathworld.wolfram.com/FresnelIntegrals.html
- Cook, C.E., Barnfeld, M., “Radar Signals: An Introduction to Theory and Applications”, Artech House, 1993, Ch.6, “The Linear FM Waveform and Matched Filter”, pp.130-172
- Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pg. 102
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- N. Levanon, “Radar Principles”, John Wiley & Sons, 1988, pp.113-117
- N. Levanon, “Radar Principles”, John Wiley & Sons, 1988, pp.113-117
- N. Levanon, “Radar Principles”, John Wiley & Sons, 1988, pp.113-117
- N. Levanon, “Radar Principles”, John Wiley & Sons, 1988, pp.113-117
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
- “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms” Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.135
- Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 in Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, or Eaves, J.L., Reedy, E.K., “Principlesof Modern Radar”, Van Nostrand Reinhold Company, 1967, pp. 481-483 Richards, M.A., “Fundamentals of Radar Signal Processing”, GeorgiaTech Course ECE 6272, Spring 2000, Lecture #11, Slide # 30
- Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 Nathanson, F.E., “Radar Design Principles”, 2nd Ed., McGraw Hill, 1991, pg.540
- Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pg. 105
- Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.274-275 Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 Nathanson, F.E., “Radar Design Principles”, 2nd Ed., McGraw Hill, 1991, pg.540 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 110-115
- Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.274-275 Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 Nathanson, F.E., “Radar Design Principles”, 2nd Ed., McGraw Hill, 1991, pg.540
- Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.274-275 Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 Nathanson, F.E., “Radar Design Principles”, 2nd Ed., McGraw Hill, 1991, pg.540
- Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 Nathanson, F.E., “Radar Design Principles”, 2nd Ed., McGraw Hill, 1991, pg.540
- Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Morris, G.V., “Pulsed Doppler Radar”, Artech House, 1988, Ch. 8, Cohen, M.N., “Pulse Compression in Pulsed Doppler Radar System”, pg.136 Nathanson, F.E., “Radar Design Principles”, 2nd Ed., McGraw Hill, 1991, pg.540 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 110-115
- Levanon, N., “Waveform Analysis and Design”, 2008 IEEE Radar Conference, May 26 – 30, Rome, Italy
- http://www.cis.syr.edu/~tksarkar/pdf/2006_Elsevier_DSP_YDSPR680.pdf Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153
- http://www.cis.syr.edu/~tksarkar/pdf/2006_Elsevier_DSP_YDSPR680.pdf Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 110-115
- Levanon, N., “Waveform Analysis and Design”, 2008 IEEE Radar Conference, May 26 – 30, Rome, Italy
- http://www.cis.syr.edu/~tksarkar/pdf/2006_Elsevier_DSP_YDSPR680.pdf Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.152-153 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 110-115
- Nathanson, F.E., “Radar Design Principles”, McGraw-Hill, 1969, pp.457-466, 1991, pp.543-554 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 106-108 Morris, G.V., Ed., “Airborne Pulse Radar”, Artech House, 1988, pp. 138 – 141 Berkowitz, R.S. Ed., “Modern Radar – Analysis, Evaluation, and System Design”, John Wiley & Sons, 1965, Ch.4, Ristenbatt, M.P., “Pseudo-Random Coded Waveforms”
- Nathanson, F.E., “Radar Design Principles”, McGraw-Hill, 1969, pp.457-466, 1991, pp.543-554 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 106-108 Morris, G.V., “Airborne Pulse Radar”, Artech House, 1988, pp. 138 – 141 Berkowitz, R.S. Ed., “Modern Radar – Analysis, Evaluation, and System Design”, John Wiley & Sons, 1965, Ch.4, Ristenbatt, M.P., “Pseudo-Random Coded Waveforms”
- Skolnik, M.I., “Radar Handbook”, § 20.5 “Phase-Code Waveform”, pp.20-19,20-22,
- Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 107-108
- Costas, J., “A study of a class of detection waveforms having nearly ideal range-doppler ambiguity properties”, Proc. IEEE, vol. 72, Aug. 1984, pp. 996 - 1009 Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.267
- Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.267
- Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.267
- Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.267
- Levanon, N., “Radar Principles”, John Wiley & Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley & Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman & Hall/CRC, 2000, pp.267