A Design Method for MIMO Radar Frequency Hopping Codes Chun-Yang Chen and P. P. Vaidyanathan California Institute of Technology Electrical Engineering/DSP Lab Asilomar Conference 2007

Outline  Review of the background –Ambiguity function –Ambiguity function in MIMO radar  The proposed waveform design method –Ambiguity function for MIMO pulse radar –Frequency hopping signals –Optimization of the frequency hopping codes –Examples  Conclusion and future work 2Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

1 Review: Ambiguity function in MIMO radar 3

Ambiguity Function in SIMO Radar  Ambiguity function characterizes the Doppler and range resolution. 4Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

Ambiguity Function in SIMO Radar  Ambiguity function characterizes the Doppler and range resolution. 5Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u(t) ( , ) target TX  delay  Doppler

Ambiguity Function in SIMO Radar  Ambiguity function characterizes the Doppler and range resolution. 6Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u(t) y ( , ) (t) ( , ) target TX RX  delay  Doppler

Ambiguity Function in SIMO Radar  Ambiguity function characterizes the Doppler and range resolution. 7Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u(t) y ( , ) (t) ( , ) target TX RX  delay  Doppler Matched filter output

Ambiguity Function in SIMO Radar  Ambiguity function characterizes the Doppler and range resolution. 8Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u(t) y ( , ) (t) ( , ) target TX RX  delay  Doppler Matched filter output

Ambiguity Function in SIMO Radar  Ambiguity function characterizes the Doppler and range resolution. 9Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u(t) y ( , ) (t) ( , ) target TX RX  delay  Doppler Matched filter output Ambiguity function

 Ambiguity function characterizes the Doppler and range resolution. Ambiguity Function in SIMO Radar 10Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 2 (  ,  ) target 1 (  ,  )

 Ambiguity function characterizes the Doppler and range resolution. Ambiguity Function in SIMO Radar 11Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 2 (  ,  ) target 1 (  ,  )

 Ambiguity function characterizes the Doppler and range resolution. Ambiguity Function in SIMO Radar 12Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 2 (  ,  ) target 1 (  ,  ) Ambiguity function

 Ambiguity function characterizes the Doppler and range resolution. Ambiguity Function in SIMO Radar 13Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 2 (  ,  ) target 1 (  ,  ) Ambiguity function

MIMO Radar 14Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u 0 (t) x T0 u 1 (t) x T1 u M-1 (t) x T,M-1 … Transmitter emits incoherent waveforms. Transmitter emits incoherent waveforms. Transmitter: M antenna elements

MIMO Radar 15Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u 0 (t) x T0 u 1 (t) x T1 u M-1 (t) x T,M-1 … … x R0 x R1 x R,M-1 MF … … … Transmitter emits incoherent waveforms. Transmitter emits incoherent waveforms. Matched filters extract the M orthogonal waveforms. Overall number of signals: NM Matched filters extract the M orthogonal waveforms. Overall number of signals: NM Receiver: N antenna elementsTransmitter: M antenna elements

Ambiguity Function in MIMO Radar 16Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 u 0 (t) x T0 u 1 (t) x T1 u M-1 (t) x T,M-1 … (,f)(,f) TX  delay  Doppler f  Spatial freq.

Ambiguity Function in MIMO Radar 17Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 x T0 x T1 x T,M-1 … … x R0 x R1 x R,M-1 MF … … … (,f)(,f) (,f)(,f) TXRX  delay  Doppler f  Spatial freq. u 0 (t)u 1 (t)u M-1 (t)

Ambiguity Function in MIMO Radar 18Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 x T0 x T1 x T,M-1 … … x R0 x R1 x R,M-1 MF … … … (,f)(,f) (,f)(,f) TXRX  delay  Doppler f  Spatial freq. u 0 (t)u 1 (t)u M-1 (t)

Ambiguity Function in MIMO Radar 19Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 x T0 x T1 x T,M-1 … … x R0 x R1 x R,M-1 MF … … … (,f)(,f) (,f)(,f) Matched filter output TXRX  delay  Doppler f  Spatial freq. u 0 (t)u 1 (t)u M-1 (t)

Ambiguity Function in MIMO Radar 20Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Matched filter output Receiver beamforming  delay  Doppler f  Spatial freq. u m (t): m-th waveform x m : m-th antenna location n: receiving antenna index

Ambiguity Function in MIMO Radar 21Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Matched filter output Receiver beamforming  delay  Doppler f  Spatial freq. u m (t): m-th waveform x m : m-th antenna location n: receiving antenna index Cross ambiguity function

Ambiguity Function in MIMO Radar 22Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Matched filter output Receiver beamforming [San Antonio et al. 07]  delay  Doppler f  Spatial freq. u m (t): m-th waveform x m : m-th antenna location n: receiving antenna index MIMO ambiguity function

Ambiguity Function in MIMO Radar 23Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 2 (  , , f 2 ) target 1 (  , ,f 1 ) f  Ambiguity function characterizes the Doppler, range, and angular resolution.

Ambiguity Function in MIMO Radar 24Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 2 (  , , f 2 ) target 1 (  , ,f 1 ) Ambiguity function f  Ambiguity function characterizes the Doppler, range, and angular resolution.

2 Proposed Waveform Design Method 25

MIMO Radar Waveform Design Problem  Choose a set of waveforms {u m (t)} so that the ambiguity function  f,f’  can be sharp around {(0,0,f,f)}. 26Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 1 (  , ,f 1 ) f

MIMO Radar Waveform Design Problem  Choose a set of waveforms {u m (t)} so that the ambiguity function  f,f’  can be sharp around {(0,0,f,f)}. 27Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  target 1 (  , ,f 1 ) f Ambiguity function

Imposing Waveform Structures  Pulse radar –MTI (Moving Target Indicator) –Doppler pulse radar 28Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m-th waveform

Imposing Waveform Structures  Pulse radar –MTI (Moving Target Indicator) –Doppler pulse radar  Frequency hopping signals –Constant modulus –Can be viewed as generalized LFM (Linear Frequency Modulation) 29Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m-th waveform

Imposing Waveform Structures  Pulse radar –MTI (Moving Target Indicator) –Doppler pulse radar  Frequency hopping signals –Constant modulus –Can be viewed as generalized LFM (Linear Frequency Modulation)  Orthogonal waveforms –Virtual array 30Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m-th waveform

Ambiguity Function of Pulse MIMO Radar 31Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT

Ambiguity Function of Pulse MIMO Radar 32Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT

Ambiguity Function of Pulse MIMO Radar 33Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT Doppler processing is separable

Ambiguity Function of Pulse MIMO Radar 34Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT Define as Doppler processing is separable

Waveform Design Problem in Pulse MIMO Radar 35Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

Waveform Design Problem in Pulse MIMO Radar 36Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  Choose a set of pulses {  m (t)} such that  ( ,f,f’) can be sharp around {(0,f,f)}.

Waveform Design Problem in Pulse MIMO Radar 37Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007  Choose a set of pulses {  m (t)} such that  ( ,f,f’) can be sharp around {(0,f,f)}.  Ex: SIMO case: M=1

Waveform Design Problem in Pulse MIMO Radar 38Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Choose a pulse with a sharp correlation function (e.g. LFM)  Choose a set of pulses {  m (t)} such that  ( ,f,f’) can be sharp around {(0,f,f)}.  Ex: SIMO case: M=1

Orthogonality of the Frequency Hopping Signals 39Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m m' Frequency Time

Orthogonality of the Frequency Hopping Signals 40Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m m'

Orthogonality of the Frequency Hopping Signals 41Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m m'

Orthogonality of the Frequency Hopping Signals 42Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 m m'   is a constant along {(0,f,f)}, no matter what codes are chosen.

 Define a vector Optimization of the Codes 43Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Code C is better than code C’.

 Define a vector  Def: a code C is efficient if there exists no other code C’ such that Optimization of the Codes 44Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Code C is better than code C’.

 Define a vector  Def: a code C is efficient if there exists no other code C’ such that  For any where g i are increasing convex functions Optimization of the Codes 45Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Code C is better than code C’.

 Define a vector  Def: a code C is efficient if there exists no other code C’ such that  For any where g i are increasing convex functions  So a code C is efficient if Optimization of the Codes 46Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Code C is better than code C’. for all C’.

 Define a vector  Def: a code C is efficient if there exists no other code C’ such that  For any where g i are increasing convex functions  So a code C is efficient if for all C’.  Example: Optimization of the Codes 47Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Code C is better than code C’.

Optimization of the Codes 48Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 M:# of waveforms Q: # of freq. hops K: # of freq. Time-bandwidth product: K  fQ  t

Simulated Annealing Algorithm  Simulated annealing –Create a Markov chain on the set A 49Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 subject to … C C’ … [S. Kirkpatrick et al. 85]

Simulated Annealing Algorithm  Simulated annealing –Create a Markov chain on the set A with the equilibrium distribution 50Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 subject to … C C’ … [S. Kirkpatrick et al. 85]

Simulated Annealing Algorithm  Simulated annealing –Create a Markov chain on the set A with the equilibrium distribution –Run the Markov chain Monte Carlo (MCMC) 51Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 subject to … C C’ … [S. Kirkpatrick et al. 85]

 Simulated annealing –Create a Markov chain on the set A with the equilibrium distribution –Run the Markov chain Monte Carlo (MCMC) –Decrease the temperature T from time to time Simulated Annealing Algorithm 52Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 subject to … C C’ … [S. Kirkpatrick et al. 85]

Examples 53Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Parameters: Uniform linear array # of waveforms M =4 # of hops Q=10 # of freq. K=15 norm type p=3 Proposed Freq. Hopping Signals

Examples 54Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Parameters: Uniform linear array # of waveforms M =4 # of hops Q=10 # of freq. K=15 norm type p=3 Orthogonal LFM Proposed Freq. Hopping Signals Parameters: –The same array –The same duration and bandwidth –Initial frequencies

Examples – Ambiguity Function 55Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 |  ( ,f,f’)| Orthogonal LFMProposed Freq. Hopping Signal

Examples – Ambiguity Function 56Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference log10|  ( ,f,f’)| Orthogonal LFMProposed Freq. Hopping Signal

57Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference Sorted samples (%) Examples – Sorted Samples of Ambiguity Functions 10log10(|  ( ,f,f’)|) LFM Randomly selected code Proposed method Sorted samples (%) 10log10(|  ( ,f,f’)|)

Examples – Correlation Function Matrix 58Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 Orthogonal LFMProposed Freq. Hopping Signal

Conclusion  MIMO radar frequency hopping waveform design method –Sharper ambiguity function (Better resolution) –Applicable in the case of pulse radar orthogonal waveforms  Future work –Other optimization tools –Phase coded signals 59Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007

Q&A Thank You! Any questions? 60Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007