Solo Hermelin, profile picture
Uploaded bySolo Hermelin
PPT, PDF7,074 views

5 pulse compression waveform

The document discusses pulse compression waveforms in radar systems, emphasizing their role in improving range resolution and waveform energy decoupling. It details various coding techniques, such as linear FM modulation and matched filtering, to enhance signal clarity and distinguishability between targets. Furthermore, it covers the mathematical foundation and implications of these waveforms in radar performance.

Science◦
In this document
Powered by AI

Introduces pulse compression waveforms and their hierarchy, including types of modulation (Chirp, Barker codes, etc.).

Explains resolution in RADAR systems, showing how to distinguish between targets based on signal strength and timing.

Discusses energy characteristics of unmodulated pulses in radar and introduces pulse compression waveforms.

Details various radar waveforms including CW radars and outlines the hierarchy based on modulation techniques.

Focuses on techniques used in pulse compression, emphasizing coding, modulation, and implementation hardware.

Details the linear FM modulated pulse technique, its Fourier transform, and characteristics of chirp and its applications.

Explains phase-coded waveforms, including bi-phase codes and Barker codes, and their sidelobe characteristics.

Describes poly-phase codes, Frank codes, and their applications in pulse compression with a summary of their benefits.

Discusses pseudo-random codes, their generation, properties, and limitations compared to other coding techniques.

Explains frequency codes and Costas codes, their structures, and performance characteristics.

Introduces complementary pulse codes, providing references for further reading on radar principles and techniques.

Discusses the Fourier transform's role in analyzing signals and waves in radar applications, including key formulations.

Embed presentation

Downloaded 574 times
Pulse Compression Waveform SOLO HERMELIN Updated: 27.10.08http://www.solohermelin.com
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
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
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
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
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
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
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
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
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
SOLO • Pulse Compression Techniques
SOLO • Pulse Compression Techniques
SOLO Waveform Hierarchy • Pulse Compression Techniques
SOLO Coherent Pulse Doppler Radar Return to Table of Content
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
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 −∗ +∞ ∞− +∞ ∞− ⋅=⋅= +−=−= ∫∫
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
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
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       −                 −         −≅ =
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
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
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.
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
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
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
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
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
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
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
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
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
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
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.
-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
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
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
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
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
SOLO Pulse Compression Techniques Bi-Phase Codes
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
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
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.
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
SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 3) Fig. 8.8 Levanon pg.157
SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 4) Fig. 8.8 Levanon pg.158,159
SOLO Pulse Compression Techniques Poly-Phase Codes Frank Codes (continue – 5) Return to Table of Content
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
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
SOLO Pulse Compression Techniques P1, P2, P3, P4 Poly-Phase Codes P4, N = 25 Elements
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
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
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
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.
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
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.
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
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
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
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).
SOLO
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
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
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
( ) ( )∫ +∞ ∞− − = 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
( ) ( )∫ +∞ ∞− − = 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
( ) ( )∫ +∞ ∞− − = 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
( ) ( )∫ +∞ ∞− − = 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
( ) ( )∫ +∞ ∞− − = 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
( ) ( )∫ +∞ ∞− − = 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
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
5 pulse compression waveform
5 pulse compression waveform

More Related Content

PPT
10 range and doppler measurements in radar systems
bySolo Hermelin
 
PPT
1 radar signal processing
bySolo Hermelin
 
PDF
Radar 2009 a 11 waveforms and pulse compression
byForward2025
 
PDF
Analysis for Radar and Electronic Warfare
byReza Taryghat
 
PPTX
Radar fundamentals
byAli Sufyan
 
PPSX
Chapter 3- pulsed radar system and MTI
byRima Assaf
 
PPT
Radar Basics
byAziz Zoaib
 
PPT
Introduction to ELINT Analyses
byJoseph Hennawy
 
10 range and doppler measurements in radar systems
bySolo Hermelin
 
1 radar signal processing
bySolo Hermelin
 
Radar 2009 a 11 waveforms and pulse compression
byForward2025
 
Analysis for Radar and Electronic Warfare
byReza Taryghat
 
Radar fundamentals
byAli Sufyan
 
Chapter 3- pulsed radar system and MTI
byRima Assaf
 
Radar Basics
byAziz Zoaib
 
Introduction to ELINT Analyses
byJoseph Hennawy
 

What's hot

PDF
Dr. Wiley - PRI Analysis and Deinterleaving
byRohde & Schwarz North America
 
PPT
1 radar basic -part i 1
bySolo Hermelin
 
PPT
Components of a Pulse Radar System
byÜlger Ahmet
 
PPTX
Radar Systems- Unit- I : Basics of Radar
byVenkataRatnam14
 
PPTX
A Tutorial on Radar System Engineering
byTBSS Group
 
PPTX
Pulse Modulation ppt
bysanjeev2419
 
PPTX
Cw and fm cw radar
byVijendrasingh Rathor
 
PPT
Principle of FMCW radar
bytobiasotto
 
PDF
Elint Interception & Analysis
byJim Jenkins
 
PPT
Chapter 5
bywafaa_A7
 
PPTX
Study of Radar System PPT
byAtul Sharma
 
PDF
Active Phased Array Radar Systems
byReza Taryghat
 
PPTX
satellite communication- UNIT-III.pptx
byEY GDS
 
PPTX
Adaptive linear equalizer
bySophia Jeanne
 
PPT
RF Basics & Getting Started Guide by Anaren
byAnaren, Inc.
 
PPT
Radar Cross Section
byNann Wai
 
PPT
wave-propagation
byATTO RATHORE
 
PPT
Propagation Models
byAyushi Gagneja
 
PPTX
Link budget satalite for calculation system
byTaif Alawsi
 
PPTX
Traveling Wave Antenna
byAbdelaziz Said
 
Dr. Wiley - PRI Analysis and Deinterleaving
byRohde & Schwarz North America
 
1 radar basic -part i 1
bySolo Hermelin
 
Components of a Pulse Radar System
byÜlger Ahmet
 
Radar Systems- Unit- I : Basics of Radar
byVenkataRatnam14
 
A Tutorial on Radar System Engineering
byTBSS Group
 
Pulse Modulation ppt
bysanjeev2419
 
Cw and fm cw radar
byVijendrasingh Rathor
 
Principle of FMCW radar
bytobiasotto
 
Elint Interception & Analysis
byJim Jenkins
 
Chapter 5
bywafaa_A7
 
Study of Radar System PPT
byAtul Sharma
 
Active Phased Array Radar Systems
byReza Taryghat
 
satellite communication- UNIT-III.pptx
byEY GDS
 
Adaptive linear equalizer
bySophia Jeanne
 
RF Basics & Getting Started Guide by Anaren
byAnaren, Inc.
 
Radar Cross Section
byNann Wai
 
wave-propagation
byATTO RATHORE
 
Propagation Models
byAyushi Gagneja
 
Link budget satalite for calculation system
byTaif Alawsi
 
Traveling Wave Antenna
byAbdelaziz Said
 

Viewers also liked

PPT
4 matched filters and ambiguity functions for radar signals
bySolo Hermelin
 
PPT
Matched filter
bysrkrishna341
 
PDF
Radar ppt
bypratibha007
 
PPT
1 radar basic - part ii
bySolo Hermelin
 
PDF
Radar 2009 a 14 airborne pulse doppler radar
byForward2025
 
PPT
4 matched filters and ambiguity functions for radar signals-2
bySolo Hermelin
 
PPTX
CFAR
byUmair Arain
 
PPTX
Matched filter detection
bySURYA DEEPAK
 
PPT
Isi and nyquist criterion
bysrkrishna341
 
PDF
Radar 2009 a 10 radar clutter.2pdf
byForward2025
 
PDF
Radar 2009 a 7 radar cross section 2
byForward2025
 
PPTX
Radar Powerpoint
byRyan rice
 
PPT
Doppler's effect
bySyeda Zermeen Fatima
 
PPT
The Doppler Effect
byTyler Cash
 
PPT
6 radar range-doppler-angular loops
bySolo Hermelin
 
PPT
DOPPLER EFFECT
byKANNAN
 
PPTX
Doppler Effect
bymychiejw
 
PPT
Doppler effect
byMaria Juana Vicktoria
 
PPT
7 air-to-air combat
bySolo Hermelin
 
PPTX
Understanding the FM Slide Chirp Advantages in Hydroacoustics for Fisheries A...
byHTI Hydroacoustic Technology, Inc.
 
4 matched filters and ambiguity functions for radar signals
bySolo Hermelin
 
Matched filter
bysrkrishna341
 
Radar ppt
bypratibha007
 
1 radar basic - part ii
bySolo Hermelin
 
Radar 2009 a 14 airborne pulse doppler radar
byForward2025
 
4 matched filters and ambiguity functions for radar signals-2
bySolo Hermelin
 
CFAR
byUmair Arain
 
Matched filter detection
bySURYA DEEPAK
 
Isi and nyquist criterion
bysrkrishna341
 
Radar 2009 a 10 radar clutter.2pdf
byForward2025
 
Radar 2009 a 7 radar cross section 2
byForward2025
 
Radar Powerpoint
byRyan rice
 
Doppler's effect
bySyeda Zermeen Fatima
 
The Doppler Effect
byTyler Cash
 
6 radar range-doppler-angular loops
bySolo Hermelin
 
DOPPLER EFFECT
byKANNAN
 
Doppler Effect
bymychiejw
 
Doppler effect
byMaria Juana Vicktoria
 
7 air-to-air combat
bySolo Hermelin
 
Understanding the FM Slide Chirp Advantages in Hydroacoustics for Fisheries A...
byHTI Hydroacoustic Technology, Inc.
 

Similar to 5 pulse compression waveform

PPT
synthetic aperture radar
byAmit Rastogi
 
PPT
Lecture13
byINDHULEKSHMI M.C
 
PDF
1.Basics of Signals
byINDIAN NAVY
 
PPTX
Digital communications
bymohammedalimahdi
 
PPT
Lecture13
bySunrise Sky
 
PDF
Pulse Compression Method for Radar Signal Processing
byEditor IJCATR
 
PPTX
RADAR & NAVIGATION (Lecture 5).pptx
byErniDwi3
 
PPT
SAR Radar Basics and Spatial Resolution.ppt
byAizazHussainShah
 
PPT
6Aesa7.ppt
byssuser42ed4e
 
PPTX
chap3.pptx
byssuser478d0e
 
PPT
PAM
byravirec
 
PPT
12936608 (2).ppt
byABHISHEKJHA176786
 
PDF
Communication Theory II_Lecture 1 for engineers.pdf
byPabasaraEranga
 
PDF
Lecture 4-6.pdf
byFaruque13
 
PDF
Solution Manual for Fundamentals of Communication Systems, 2/E J G. Proakis, ...
bypeperxcoderu6
 
PPTX
DC Unit 1 PPT has useful topics like.pptx
bysurya30504
 
PDF
Lecture1
bysrirenga
 
PDF
M0088
byAhmet Sefer
 
PDF
Comparative analysis on an exponential form of pulse with an integer and non-...
byIJERA Editor
 
PDF
Af32216223
byIJERA Editor
 
synthetic aperture radar
byAmit Rastogi
 
Lecture13
byINDHULEKSHMI M.C
 
1.Basics of Signals
byINDIAN NAVY
 
Digital communications
bymohammedalimahdi
 
Lecture13
bySunrise Sky
 
Pulse Compression Method for Radar Signal Processing
byEditor IJCATR
 
RADAR & NAVIGATION (Lecture 5).pptx
byErniDwi3
 
SAR Radar Basics and Spatial Resolution.ppt
byAizazHussainShah
 
6Aesa7.ppt
byssuser42ed4e
 
chap3.pptx
byssuser478d0e
 
PAM
byravirec
 
12936608 (2).ppt
byABHISHEKJHA176786
 
Communication Theory II_Lecture 1 for engineers.pdf
byPabasaraEranga
 
Lecture 4-6.pdf
byFaruque13
 
Solution Manual for Fundamentals of Communication Systems, 2/E J G. Proakis, ...
bypeperxcoderu6
 
DC Unit 1 PPT has useful topics like.pptx
bysurya30504
 
Lecture1
bysrirenga
 
M0088
byAhmet Sefer
 
Comparative analysis on an exponential form of pulse with an integer and non-...
byIJERA Editor
 
Af32216223
byIJERA Editor
 

More from Solo Hermelin

PPT
3 earth atmosphere
bySolo Hermelin
 
PPT
Inner outer and spectral factorizations
bySolo Hermelin
 
PPT
Keplerian trajectories
bySolo Hermelin
 
PPT
Stabilization of linear time invariant systems, Factorization Approach
bySolo Hermelin
 
PPT
2 aircraft flight instruments
bySolo Hermelin
 
PPT
Reduced order observers
bySolo Hermelin
 
PPT
4 navigation systems
bySolo Hermelin
 
PPT
10 fighter aircraft avionics - part iii
bySolo Hermelin
 
PPT
8 fighter aircraft avionics-part i
bySolo Hermelin
 
PPT
Analytic dynamics
bySolo Hermelin
 
PPT
9 fighter aircraft avionics-part ii
bySolo Hermelin
 
PPT
12 performance of an aircraft with parabolic polar
bySolo Hermelin
 
PPT
Slide Mode Control (S.M.C.)
bySolo Hermelin
 
PPT
11 fighter aircraft avionics - part iv
bySolo Hermelin
 
PPT
6 computing gunsight, hud and hms
bySolo Hermelin
 
PPT
5 introduction to quantum mechanics
bySolo Hermelin
 
PPT
Sliding Mode Observers
bySolo Hermelin
 
PPT
Anti ballistic missiles i
bySolo Hermelin
 
PPT
Anti ballistic missiles ii
bySolo Hermelin
 
PPT
3 modern aircraft cutaway
bySolo Hermelin
 
3 earth atmosphere
bySolo Hermelin
 
Inner outer and spectral factorizations
bySolo Hermelin
 
Keplerian trajectories
bySolo Hermelin
 
Stabilization of linear time invariant systems, Factorization Approach
bySolo Hermelin
 
2 aircraft flight instruments
bySolo Hermelin
 
Reduced order observers
bySolo Hermelin
 
4 navigation systems
bySolo Hermelin
 
10 fighter aircraft avionics - part iii
bySolo Hermelin
 
8 fighter aircraft avionics-part i
bySolo Hermelin
 
Analytic dynamics
bySolo Hermelin
 
9 fighter aircraft avionics-part ii
bySolo Hermelin
 
12 performance of an aircraft with parabolic polar
bySolo Hermelin
 
Slide Mode Control (S.M.C.)
bySolo Hermelin
 
11 fighter aircraft avionics - part iv
bySolo Hermelin
 
6 computing gunsight, hud and hms
bySolo Hermelin
 
5 introduction to quantum mechanics
bySolo Hermelin
 
Sliding Mode Observers
bySolo Hermelin
 
Anti ballistic missiles i
bySolo Hermelin
 
Anti ballistic missiles ii
bySolo Hermelin
 
3 modern aircraft cutaway
bySolo Hermelin
 

Recently uploaded

PPTX
Wuchereria Bancrofti Morphology , Pathogenesis , Lab diagnosis , Treatment & ...
byRemya M S
 
PDF
Types-of-culture-media; basal,enriched,enrichment,differential and selective ...
bycassiuskocha
 
PPTX
Forest Farming - Agroforestry Techniques and More
byjosh540578
 
PPT
Postmortem changes veterinary pathology 3
byahmadrahmani7862
 
PDF
Vertebrate Animals, reptiles, birds, mammals, amphibian, fish
bySza Batac
 
PPTX
Surgical Affecation of Horn by Dr H M barot .pptx
bylpschauhan03
 
PDF
Neuroscience 6th Edition PDF Textbook by Dale Purves
byhifateb609
 
PDF
International Chemistry Scientist Awards
bychemistryaward
 
PPTX
OMNICARE PITCH DECK FINAL Design_Thinking_for a robot to function the tasks
byLOHITH886892
 
PPTX
Physical Science SHS 24.2 Expanding Universe.pptx
byJessonSuaga1
 
PDF
Agency phenomenon. Complexity Explorers Krakow #8
byMarcin Stepien
 
PDF
Unit-1 Parsitology, Introduction to Parasitology..pdf
bydemekedaka
 
PPTX
Sustainable Landscaping with Native Ornamental Plants.pptx
bylalaghasahak4149
 
PDF
Muscular System 2025 Batch - Irfanullah Mehar -- World of Wisdom.pdf
byWorld of Wisdom
 
PPTX
CELL ORGANELLES AND PROTEIN TARGETING.pptx
byRashmiMG2
 
PDF
Kaplan and Sadock's Comprehensive Text of Psychiatry 11th Edition PDF
bymokotey464
 
PDF
Multiple outflows and delayed ejections revealed by early imaging of novae
bySérgio Sacani
 
PDF
Influence of grazing intensity and regime on soil nitrogen fixation dynamics ...
byMuhammad Khalid Afzal
 
PDF
Population Estimation and Behavioral Study of the Spotted Owl (Athene brama) ...
byMuhammad Khalid Afzal
 
PDF
A Brief Review on Antibacterial Activity of Citrullus colocynthis
byMuhammad Khalid Afzal
 
Wuchereria Bancrofti Morphology , Pathogenesis , Lab diagnosis , Treatment & ...
byRemya M S
 
Types-of-culture-media; basal,enriched,enrichment,differential and selective ...
bycassiuskocha
 
Forest Farming - Agroforestry Techniques and More
byjosh540578
 
Postmortem changes veterinary pathology 3
byahmadrahmani7862
 
Vertebrate Animals, reptiles, birds, mammals, amphibian, fish
bySza Batac
 
Surgical Affecation of Horn by Dr H M barot .pptx
bylpschauhan03
 
Neuroscience 6th Edition PDF Textbook by Dale Purves
byhifateb609
 
International Chemistry Scientist Awards
bychemistryaward
 
OMNICARE PITCH DECK FINAL Design_Thinking_for a robot to function the tasks
byLOHITH886892
 
Physical Science SHS 24.2 Expanding Universe.pptx
byJessonSuaga1
 
Agency phenomenon. Complexity Explorers Krakow #8
byMarcin Stepien
 
Unit-1 Parsitology, Introduction to Parasitology..pdf
bydemekedaka
 
Sustainable Landscaping with Native Ornamental Plants.pptx
bylalaghasahak4149
 
Muscular System 2025 Batch - Irfanullah Mehar -- World of Wisdom.pdf
byWorld of Wisdom
 
CELL ORGANELLES AND PROTEIN TARGETING.pptx
byRashmiMG2
 
Kaplan and Sadock's Comprehensive Text of Psychiatry 11th Edition PDF
bymokotey464
 
Multiple outflows and delayed ejections revealed by early imaging of novae
bySérgio Sacani
 
Influence of grazing intensity and regime on soil nitrogen fixation dynamics ...
byMuhammad Khalid Afzal
 
Population Estimation and Behavioral Study of the Spotted Owl (Athene brama) ...
byMuhammad Khalid Afzal
 
A Brief Review on Antibacterial Activity of Citrullus colocynthis
byMuhammad Khalid Afzal
 

5 pulse compression waveform

  • 1.
    Pulse Compression Waveform SOLOHERMELIN Updated: 27.10.08http://www.solohermelin.com
  • 2.
    Table of Content SOLO PulseCompression 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 & DopplerMeasurements 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 RangeResolution 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 PulseCompression 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 CWRadars 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 • PulseCompression 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.
  • 12.
  • 13.
    SOLO Waveform Hierarchy • PulseCompression Techniques
  • 14.
    SOLO Coherent PulseDoppler Radar Return to Table of Content
  • 15.
    SOLO Linear FM ModulatedPulse (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 forRADAR 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 ModulatedPulse (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 ModulatedPulse (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 ModulatedPulse (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 JeanFresnel 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 ModulatedPulse (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 PhaseCoded 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 PhaseCoding 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 forRADAR 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 forRADAR 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 forRADAR 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 forRADAR 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 codedof 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-PhaseCodes • 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 CompressionTechniques 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 Barkercoded 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 Barkercoded 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 Barkercoded 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 codedof 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 CompressionTechniques 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 CompressionTechniques Bi-Phase Codes
  • 40.
    SOLO Pulse CompressionTechniques 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 CompressionTechniques 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 CompressionTechniques 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 CompressionTechniques 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 CompressionTechniques Poly-Phase Codes Frank Codes (continue – 3) Fig. 8.8 Levanon pg.157
  • 45.
    SOLO Pulse CompressionTechniques Poly-Phase Codes Frank Codes (continue – 4) Fig. 8.8 Levanon pg.158,159
  • 46.
    SOLO Pulse CompressionTechniques Poly-Phase Codes Frank Codes (continue – 5) Return to Table of Content
  • 47.
    SOLO Pulse CompressionTechniques 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 CompressionTechniques ( ) ( ) ( ) ( )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 CompressionTechniques P1, P2, P3, P4 Poly-Phase Codes P4, N = 25 Elements
  • 50.
    SOLO Pulse CompressionTechniques 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 Codesare 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 CompressionTechniques 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 CompressionTechniques 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 CompressionTechniques 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 CompressionTechniques 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 CompressionTechniques 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.
  • 61.
    Bogler, P.L., “RadarPrinciples 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, 201562 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 ofa 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

Editor's Notes

  • #10 “Principles of Modern Radar” Georgia Tech, 2004, Samuel O.Piper
  • #11 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #12 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #13 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #14 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #16 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
  • #17 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”,
  • #18 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
  • #19 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
  • #20 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
  • #21 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
  • #22 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
  • #23 Reference in J. Meyer-Arendt, “Introduction to Classical &amp; Modern Optics”, 3th Ed.,Prentince Hall, 1989, pg. 258 http://en.wikipedia.org/wiki/Fresnel_integral http://mathworld.wolfram.com/FresnelIntegrals.html
  • #24 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
  • #25 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pg. 102
  • #26 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #27 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #28 N. Levanon, “Radar Principles”, John Wiley &amp; Sons, 1988, pp.113-117
  • #29 N. Levanon, “Radar Principles”, John Wiley &amp; Sons, 1988, pp.113-117
  • #30 N. Levanon, “Radar Principles”, John Wiley &amp; Sons, 1988, pp.113-117
  • #31 N. Levanon, “Radar Principles”, John Wiley &amp; Sons, 1988, pp.113-117
  • #33 “Principles of Modern Radar” Georgia Tech, 2004, Jim Scheer, “Advanced Radar Waveforms”
  • #34 “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
  • #39 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
  • #40 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
  • #41 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pg. 105
  • #42 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.274-275 Levanon, N., “Radar Principles”, John Wiley &amp; 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 &amp; Sons, 1989, pp. 110-115
  • #43 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.274-275 Levanon, N., “Radar Principles”, John Wiley &amp; 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
  • #44 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.274-275 Levanon, N., “Radar Principles”, John Wiley &amp; 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
  • #45 Levanon, N., “Radar Principles”, John Wiley &amp; 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
  • #46 Levanon, N., “Radar Principles”, John Wiley &amp; 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 &amp; Sons, 1989, pp. 110-115
  • #47 Levanon, N., “Waveform Analysis and Design”, 2008 IEEE Radar Conference, May 26 – 30, Rome, Italy
  • #48 http://www.cis.syr.edu/~tksarkar/pdf/2006_Elsevier_DSP_YDSPR680.pdf Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.152-153
  • #49 http://www.cis.syr.edu/~tksarkar/pdf/2006_Elsevier_DSP_YDSPR680.pdf Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.152-153 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 110-115
  • #50 Levanon, N., “Waveform Analysis and Design”, 2008 IEEE Radar Conference, May 26 – 30, Rome, Italy
  • #51 http://www.cis.syr.edu/~tksarkar/pdf/2006_Elsevier_DSP_YDSPR680.pdf Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.152-153 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 110-115
  • #52 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 &amp; 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 &amp; Sons, 1965, Ch.4, Ristenbatt, M.P., “Pseudo-Random Coded Waveforms”
  • #53 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 &amp; 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 &amp; Sons, 1965, Ch.4, Ristenbatt, M.P., “Pseudo-Random Coded Waveforms”
  • #54 Skolnik, M.I., “Radar Handbook”, § 20.5 “Phase-Code Waveform”, pp.20-19,20-22,
  • #55 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 107-108
  • #56 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 &amp; Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.267
  • #57 Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.267
  • #58 Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.267
  • #59 Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.267
  • #60 Levanon, N., “Radar Principles”, John Wiley &amp; Sons, 1988, pp.145-152 Bogler, P.L., “Radar Principles with Applications to Tracking Systems”, John Wiley &amp; Sons, 1989, pp. 319-321 Mahafza, B.R., “Radar System Analysis and Design Using MATLAB”, Chapman &amp; Hall/CRC, 2000, pp.267