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 TT
Ambiguity Function of Pulse MIMO Radar 32Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT
Ambiguity Function of Pulse MIMO Radar 33Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT Doppler processing is separable
Ambiguity Function of Pulse MIMO Radar 34Chun-Yang Chen, Caltech DSP Lab | Asilomar Conference 2007 TT 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