Radar Frequencies and Waveforms 12th Annual International Symposium on Advanced Radio Technologies Michael Davis Georgia Tech Research Institute Sensors and Electromagnetic Applications Laboratory [email protected]

Based on material created by Byron M. Keel, Ph.D., GTRI Waveforms Extract “Target” Information A radar system probes its environment with specially designed waveforms to identify and characterize targets of interest. Detection For a given range, angle, and/or Doppler, decide if a target is or is not present. Example: Moving target indication (MTI) radar Estimation For a given range, angle, and/or Doppler, estimate Example: Synthetic aperture radar (SAR) imaging

2 Overview Radar frequencies Radar waveform taxonomy CW: Measuring Doppler Single Pulse: Measuring range Ambiguity function Pulse compression waveforms (FM and PM) Coherent pulse trains

3 Radar Frequencies

4

Radar Bands

Range

-

Radar Band Frequency

Long Poor Angular Resolution

Large HF 3 – 30 MHz

VHF 30 – 300 MHz

UHF 300 – 1000 MHz Range Air Air Range

L 1 – 2 GHz -

FOPEN Surveillance

S 2 -4 GHz Long

C 4 – 8 GHz

X 8 – 12 GHz Air-to-Air SAR/

Ku 12 – 18 GHz GMTI

Fire Fire

Control/

Ka 27 – 40 GHz Munitions

mm (V & W) 40 – 300 GHz

Small

Range

-

Resolution

Short Good Good Angular

5 Radar Waveform Taxonomy

6 Continuous Wave (CW) vs. Pulsed CW: Simultaneously transmit and receive Pulsed: Interleave transmit and receive periods

7 Continuous Wave (CW) vs. Pulsed

Continuous Wave Pulsed Requires separate Same antenna is used for transmit and receive transmit and receive. antennas. Isolation requirements Time-multiplexing relaxes limit transmit power. isolation requirements to allow high power. Radar has no blind Radar has blind ranges ranges. due to “eclipsing” during transmit events.

8 Modulated vs. Unmodulated Modulation may be applied to each pulse (intrapulse modulation) or from pulse-to-pulse (interpulse modulation) Classes of Modulation Amplitude “ON-OFF” Amplitude Modulation

Phase Frequency Modulation Frequency Phase Modulation Polarization

9 Measuring Doppler with CW Waveform

10 Measuring Doppler with a CW Radar

Doppler Shift

v Target radial velocity

fD fc Radar frequency

11 CW Doppler Resolution

Df

Velocity resolution improves

as integration time, TCW, and radar frequency, fc, increases.

12 CW Doppler Processing

0 DFT processing Mainlobe -5 Sample CW returns discretely in time

-10 Generate spectrum via Fourier analysis (e.g., FFT) -15 Sidelobes

dB -20 Results in sinc shaped response

-25 Weighting can be applied to reduce Doppler sidelobes -30 SNR loss -35 Resolution degradation -40 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 normalized frequency Sampling of DFT response a function of 1 0.9 Bin spacing 0.8 Frequency 0.7 0.6 Zero padding reduces bin spacing; does not 0.5 improve resolution

Magnitude 0.4

0.3

0.2

0.1

0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Normalized (cycles/sample)

13 Measuring Range with a Single (Unmodulated) Pulse

14 Unmodulated Pulse

PTX Peak transmit power fc Center frequency Tp Pulse width

Baseband

15 The Matched Filter Observe a known signal, s(t), in noise

Apply matched filter to maximize signal-to-noise ratio (SNR)

assuming that signal has unit power, i.e.,

16 The Matched Filter

Matched Filter

17 Waveform Range Response

Matched Filter

The range response, h(t), of a waveform is the auto-correlation function of the transmitted signal.

18 Range Resolution: Unmodulated Pulse

Tp

Range resolution improves as transmitted pulse gets shorter.

19 Ambiguity Function

20 Ambiguity Function

Range Response (No Uncompensated Doppler)

Ambiguity Function

Doppler shift The ambiguity function characterizes the filtered response when the received signal contains an uncompensated Doppler shift

21 Ambiguity Function for a Simple Pulse

1 x tt   0 t Simple Pulse t

t sintf 1 d  t t A t,1 ftd    t t t tfd 1 t Simple Pulse Ambiguity Function Zero Doppler Cut Zero Time-Delay Cut

t Zero Doppler Cut A t,0  1  t t t

sintfd  Zero Time-Delay Cut A0, fd   t t tfd

22 Improving Range Resolution with Pulse Compression

23 Limitations of the Unmodulated Pulse

Decreasing SNR, Radar Performance Increasing

Decreasing Pulse Width Increasing

Increasing Range Resolution Capability Decreasing

For an unmodulated pulse there exists a coupling between range resolution and waveform energy

24 Pulse Compression Range response is the auto-correlation of the transmitted signal. To have “narrow” in range (time) domain, the waveform must have “wide” bandwidth in frequency domain The bandwidth of an unmodulated pulse of duration Tp is 1/ Tp 2/Tp Pulse Compression Use modulated pulses to get better range resolution.

25 Pulse Compression Waveforms Permit a de-coupling between range resolution and waveform energy. Apply modulation to increase bandwidth.

Range resolution, DR, improves as bandwidth, W, increases.

SNR is unchanged if pulse width remains the same.

26 Linear Frequency Modulated (LFM) Waveforms

27 LFM Phase and Frequency Characteristics

Linear Frequency Modulated Waveforms • LFM phase is quadratic • Instantaneous frequency is defined as the time derivate of the phase 2 t t x tt cos   t

t 22 • The instantaneous frequency is linear

t Quadratic Term Linear Term  4 radians 2

d 2 frequency tt 2 dt tt time  t t 1 d  time ft 2 2 2 2tdt

28 Components of LFM Spectrum

X  Xjj  expexp       3 Key Terms 2

0

-2 Magnitude Quadratic Residual -4 -6

Response Phase Phase -8

dB t  20 -10

-12

-14

-16 X  1       For large time-bandwidth products -18 -20 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 frequency, normalized by 

2

0 1 t 2 -2    -4 4 Quadratic phase term -6 -8

dB t  50 -10

-12  -14 Residual phase term -16    -18 -20 4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 frequency, normalized by 

2 1 t 2 0 Xj exp         -2 4 -4  -6 -8

dB t 100 -10 Reference: Cook, Bernfeld, “Radar Signals, An Introduction to Theory and Application”, -12 -14

Artech House, 1993, p. 49 -16

-18

-20 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 frequency, normalized by 

29 LFM Match Filtered Response Bandwidth (MHz) Range Resolution (m) 0.1 1500 1 150

0 2 75 5 30 -4 10 15 20 7.5 -10 50 3 -13 100 1.5 dB 200 0.75 -20 500 0.3 1000 0.15

-30  sin ttt   t t -6 -4 -2 -1 0 0.5 2 4 6 yt 1  multiples of 1/    t ttt   t For t ≥ 20, match filtered response approximates a sinc

1 c ~ -13 dB peak sidelobes t  resolution in time  r  range resolution  2 Rayleigh resolution:

Rayleigh resolution equivalent to 4 dB width

30 LFM Ambiguity Function

t  sint 1ft d t tt y t,1 ftd    t t t  t 1ftd tt

t t sin 1  tf  d   t  t  y t, fd  1  t t t t t 1  t  fd  t  

Sinc functions are time shifted versions of one another t f sin 1  t d   td  t t t  y t,0 1 t t t t 1  t t The triangle shaped response does not shift with the sinc response

31 Amplitude Weighting 5 Amplitude weighting reduces peak sidelobe levels 0 reduces straddle loss -4 Price paid -10 increased mainlobe width (degraded -15

resolution) -20

loss in SNR (loss computable from dB weighting coefficients) -25 -30

-35 N 1 2 -40 wn    -45 n0 SNRLoss  N 1 -50 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 N w n normalized delay n0 Weighting as a part of X() “Fast Convolution”

FFT FFT-1

X*() X W()

32 Taylor Weighting Function

Peak Sidelobe Level (dB) -20 -25 -30 -35 -40 -45 -50 -55 -60 nbar SNR Loss (dB) 2 -0.21 -0.38 -0.51 3 -0.21 -0.45 -0.67 -0.85 4 -0.18 -0.43 -0.69 -0.91 -1.11 -1.27 5 -0.16 -0.41 -0.68 -0.93 -1.14 -1.33 -1.49 6 -0.15 -0.39 -0.66 -0.92 -1.15 -1.35 -1.53 -1.68 7 -0.15 -0.37 -0.65 -0.91 -1.15 -1.36 -1.54 -1.71 -1.85 8 -0.16 -0.36 -0.63 -0.90 -1.14 -1.36 -1.55 -1.72 -1.87 9 -0.16 -0.36 -0.63 -0.90 -1.14 -1.36 -1.55 -1.72 -1.87

Peak Sidelobe Level (dB) -20 -25 -30 -35 -40 -45 -50 -55 -60 nbar 4 dB Resolution Normalized by c/2 2 1.15 1.19 1.21 3 1.14 1.22 1.28 1.33 4 1.12 1.22 1.29 1.36 1.42 1.46 5 1.11 1.20 1.29 1.36 1.43 1.49 1.54 6 1.10 1.19 1.28 1.36 1.43 1.50 1.56 1.61 7 1.09 1.19 1.28 1.36 1.43 1.50 1.56 1.62 1.67 8 1.08 1.18 1.27 1.35 1.43 1.50 1.57 1.63 1.68

33 Exciter / LO

Stretch Processing

Stretch Processing:

a technique for converting  Frequency fb time-delay into frequency ftbd t beat frequency  2DR Time   td BB t c  A/D DFT BB

Filter limits range of frequencies t t+td d Filter defines A/D requirements Target Exciter / LO Filter defines range window extent Return Commonly referred to as

a de-ramp operation Frequency Example Calculation Parameter Value Units Time Pulse Length 50 usec Waveform Bandwidth 500 MHz Caputi, “Stretch: A Time-Transformation Technique”, March 1971 Filter Bandwidth 40 MHz A/D Sampling Rate 40 MHz Range Window Extent 600 m ct BB Nominal Range Resolution 0.3 m DR Range Window Extent 2 

34 Range Resolution and SAR Imagery

1 m resolution (> 150 MHz bandwidth)

Ku band 10 cm resolution (> 1.5 GHz bandwidth)

Source: Sandia National Labs (www.sandia.gov) 35 Phase Coded Waveforms

36 Phase Code Waveforms

Composed of concatenated sub-pulses (or chips) Chip-to-chip phase modulation applied to achieve desired compressed response (e.g., mainlobe, sidelobes, & Doppler tolerance) Phase modulation Bi-phase codes (only 2 phase states) Poly-phase codes (exhibit more than 2 phase states) 7

6

5

4

t chip Chip width 3 amplitude

2 + + + - - + - ct chip  r  2 1 0 -6 -4 -2 0 2 4 6 Nt correlation delay (normazlied by t)

• Consists of N chips each with duration, tchip • For appropriately chosen codes, the Rayleigh range resolution is equal to the chip width • Energy in the waveform is proportional to the number of chips • In general, sidelobe levels are inversely proportional to the number of chips

37 Barker Codes Perfect bi-phase aperiodic codes Correlation Belief that no Barker code exists + + + - - + - above length 13 + + + - - + - Has been proven for odd length sequences Barker codes are applied in radar 12

applications 10

Desire for longer codes however 8 has driven the community to

6 magnitude consider longer sub-optimum codes amplitude 4

2

0 -10 -5 0 5 10 correlation delay (normazlied by t) Code Length Code Sequence Peak Sidelobe Integrated Sidelobe Level, dB Levels, dB 2 ,  -6.0 -3.0 3  -9.5 -6.5 4    ,  -12.0 -6.0 Longer code = Lower PSL 5     -14.0 -8.0 7      -16.9 -9.1 11  -20.8 -10.8 13  -22.3 -11.5

38 Minimum Peak Sidelobe Codes

Binary codes yielding minimum peak sidelobes for a given sequence length Identified through exhaustive searches MPS codes identified through length 69 Peak sidelobe levels = 1 for the Barker length sequences N = 2,3,4,5,7,11, &13 = 2 for N <= 28 (excluding Barker codes & N = 22,23,24,26,27) = 3 for N = (22,23,24,26,27) & 29 <= N <= 48, and N = 51 = 4 for N = 50, and 52 <= N <= 70 Does not ensure optimum integrated sidelobe level Nunn and Coxson (IEEE AES 2008) found codes with peak sidelobe levels

• = 4 for N = 71 through 82

• = 5 for N = 83 through 105 Longer codes with low peak sidelobes have been identified (not necessarily optimum)

39 Doppler Intolerance of Bi-Phase Codes 1 Cycle of Doppler Bi-phase codes are Doppler intolerant

Mainlobe is not preserved 12 Sidelobes increase Waveform designed to limit maximum 10

Doppler shift to ¼ cycle 8 Corresponds to 1 dB loss in peak

amplitude 6 amplitude Poly-phase, quadratic phase 4 response required to achieve Doppler tolerance 2 0 -10 -5 0 5 10 correlation delay (normazlied by t) + + + – – + – Filter ¼ Cycle of Doppler

 12 Doppler Shifted + + + – – + – 10 Received Signal 8

6

 amplitude 4  0 2 0 -10 -5 0 5 10 correlation delay (normazlied by t)

40 Measuring Range and Doppler with Coherent Pulse Train

41 Coherent Pulse Train

Good Doppler resolution No range resolution

Good range resolution Poor Doppler resolution

Good range resolution Good Doppler resolution

42 Coherent Pulse Train

Parameter Symbol

Pulse Width Tp

Pulse Repetition Interval (PRI) Tr

Number of Pulses Np

Tp

Tr

Duty Cycle:

Pulse Repetition Frequency (PRF):

43 Measuring Phase & Radial Velocity 1

2R0 R0 expjf 2 c c 2

R vT 2R0  vT  0 A pulsed Doppler expjf 2 c c waveform measures the phase change between 3 pulses

R0  v2 T 22R0  v T  expjf 2 c 2R0 2vnT c expj 2 fcc exp j 2 f cc n 22vnT   v  expj 2 fc   exp   j 2 nT  c  R0  vnT     2R0  vnT  expjf 2 c 2v c nN0, , 1 f  d  44 Processing Doppler The Discrete Fourier Transform represents a bank of matched filters

The filters are only applied at the zero time-delay lag h n  exp j 2 fk nT  k fFkS N Note:F S  PRF N 1 kF S  y k  x nexp  j 2 nTS k  0, , N  1 , N  N n0 N

FT  1 SS N’ filters Measured signal from N pulses

N 1 k y k  x nexp  j 2 n k  0, , N  1 , N  N n0 N

Measured signal from N pulses N’ filters k fFkS Discrete Fourier Transform N

45 Signal-to-Noise Ratio Prior to applying the Doppler matched filter, the SNR may be less than zero A2 x n exp j 2  0.125 PRF nT n 0, , N 1 SNR  SNR  0 dB    2 I-Channel I-Channel

2 2

0 0 amplitude

amplitude -2 -2

20 40 60 80 100 120 20 40 60 80 100 120 pulse number pulse number Q-Channel Q-Channel

2 2

0 0 amplitude

amplitude -2 -2

20 40 60 80 100 120 20 40 60 80 100 120 pulse number pulse number Blue – noise Blue – noise +signal Green – Doppler shifted signal

46 SNR Gain Associated with Doppler Processing SNR Gain due to Doppler processing Often referred to as coherent processing gain SNR GAIN 10log10  128 21 dB 5 A2 SNR  0  2

-5

10log10 N  -10 N2 A 2 NA 2 SNR  dB

22 SNR Gain N -15

-20

SNR GAIN 10log10  N  -25

-30 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 normalized frequency • Example of radar modes benefiting from coherent integration – SAR: 100s to 1000s of pulses (20 to 30 dB of SNR gain or more) – GMTI: 10s to 100s of pulses (10 to 20 dB of SNR gain or more)

47 Pulse-Doppler Design Considerations Ambiguities Range Doppler Blind Zones Range eclipsing occurs since radar cannot receive while transmitting. Doppler blind zones occur when target is observed with same Doppler as clutter.

48 Pulsed Doppler Waveform Modes Low PRF Range unambiguous Doppler ambiguous High PRF Range ambiguous Doppler unambiguous Medium PRF Range ambiguous Doppler ambiguous Process multiple PRFs to Resolve range and Doppler ambiguities Move range and Doppler blind zones

49 Summary Radar frequencies Radar waveform taxonomy CW: Measuring Doppler Single Pulse: Measuring range Ambiguity function Pulse compression waveforms (FM and PM) Coherent pulse trains

50