Radar Signal Processing

PTM Pulse TrainsReed-Muller Pulse Trains

PTM Sequencing Across Frequency

Radar Signal Processing

Doppler Resilient Pulse Trains

p-Pulse Train: Transmission of a Golay pair x and y iscoordinated according to a unimodular sequence p = {pn},n = 0, . . . , 2M − 1 over N = 2M PRIs.

At nth PRI: 12(1 + (−1)pn)x+ 1

2(1− (−1)pn)y

Composite ambiguity function:

G(k, θ) =1

2[Cx(k) + Cy(k)]

2M−1∑n=0

ejnθ

︸ ︷︷ ︸+

1

2[Cx(k)− Cy(k)]

2M−1∑n=0

(−1)pne

jnθ

︸ ︷︷ ︸Sidelobe free Range sidelobes

Key observation: Magnitude of range sidelobes areproportional to the magnitude of the spectrum of thesequence (−1)pn :

Sp(θ) =2M−1∑n=0

(−1)pnejnθ

Approach: Design p = {pn} to shape the spectrum Sp(θ).

Radar Signal Processing

PTM Pulse Train

The PTM pulse train clears out the range sidelobes in aDoppler interval around the zero-Doppler axis.

Length-2M PTM sequence zero forces low-order terms of theTaylor expansion of Sp(θ) around θ = 0:

S(m)p (0) = 0, m = 1, . . . ,M − 1

Theorem: The PTM pulse train of length 2M has thefollowing composite ambiguity function:

G(k, θ) = ej(2M−1−1)θ sin(2Mθ/2)

sin(θ/2)Lδk,0 +

12

(M−1∏m=0

(1− ej2mθ)

)[Cx(k)− Cy(k)]

= ej(2M−1−1)θ sin(2Mθ/2)

sin(θ/2)Lδk,0

+12

[(−j)M2M(M−1)/2θM +O(θM+1)][Cx(k)− Cy(k)]

Radar Signal Processing

PTM Pulse Train

Proof:

Sp(θ) =2M−1∑n=0

(−1)pnejnθ

=1∑

n0=0

· · ·1∑

nM−1=0

(−1)n0+...+nM−1ej(n020+n12

1+...+nM−12M−1)θ

=M−1∏m=0

(1− ej2mθ) =

M−1∏m=0

Λm(θ)

where Λm(θ) = 1− ej2mθ. Since Λm(0) = 0, we have

S(`)p (0) = 0, if ` < M,

and

S(M)p (0) = M !

M−1∏m=0

Λ(1)m (0) = (−j)M2M(M−1)/2M !

Radar Signal Processing

Sidelobe Suppression at Higher Doppler Frequencies

Question: Can we clear out range sidelobes in other Dopplerintervals?

Theorem: The k-oversampled PTM sequence of length 2Mkproduces an M th order null at θ = 2π`/k for all co-prime ` and k.

Corollary: Oversampled PTM produces an (M − 1)th order null atθ = 0 and (M − h− 1)th order nulls at all θ = 2π`/2hk.

Example: M = 3, k = 3 −→ {pn} = 000111111000 · · ·

Radar Signal Processing

Reed-Muller Pulse Trains

First order Reed-Muller code RM(1,M) consists of 2M codewords of the form

rb(n) =2M−1∑m=0

bmnm for n = 0, · · · , 2M − 1

where nm denotes the mth binary digit of n.

Walsh functions are the exponentiated Reed-Muller codes

wb(n) = (−1)rb(n), for n = 0, · · · , 2M − 1

The length-2M PTM sequence is equal to rb(n) withb = (1, 1, . . . , 1), and (−1)rb(n) is its corresponding Walshfunction.

Radar Signal Processing

Reed-Muller Pulse Trains

Theorem: For a Reed-Muller code pn = rb(n) of length 2M

the magnitude spectrum |Sp(θ)| is given by

|Sp(θ)| =

∣∣∣∣∣∣2M−1∑n=0

(−1)rb(n)ejnθ

∣∣∣∣∣∣= 2M

M−1∏m=0bm=0

| cos(2m−1θ)|M−1∏m=0bm=1

| sin(2m−1θ)|

where bm, m = 0, . . . ,M − 1 is the mth entry in the binaryM -tuple b.

Proof: Homework

Question: Given a Doppler interval, which first-order RM (orWalsh function) minimizes |Sp(θ)|?

Radar Signal Processing

Reed-Muller Pulse Trains

|Sp(θ)| is 2π−periodic. Only need to look at 0 ≤ θ ≤ 2π.

|Sp(θ)| = 2MM−1∏m=0bm=0

| cos(2m−1θ)|M−1∏m=0bm=1

| sin(2m−1θ)|

Theorem: Among first order Reed-Muller codes of length 2M

there is a single code which minimizes |Sp(θ)| across theentire Doppler interval [πk/2M , π(k + 1)/2M ], where k is aninteger.

This allows us to clear out the range sidelobes along aparticular Doppler bin.

Radar Signal Processing

Reed-Muller Pulse Trains

Proof:

|Sp(θ)| = 2MM−1∏m=0bm=0

| cos(2m−1θ)|M−1∏m=0bm=1

| sin(2m−1θ)|

= 2MM−1∏m=0

| cos(2m−1θ +π

2bm)|

Minimize one by one to find the optimal RM code:

bm =

{1, 2mθ ∈ [0, π2 ] ∪ [3π2 , 2π] mod(2π)

0, otherwise

For all θ inside a given Doppler interval [πk/2M , π(k + 1)/2M ] theminimizers bm, m = 0, 1, . . . ,M − 1 stay unchanged.

Radar Signal Processing

Reed-Muller Pulse Train in Action

Suppose we want to minimize sidelobes in the region ofθ = 0.25 using an RM pulse train of length 256.

This means minimizing |Sp(θ)| in the interval[20π/256, 21π/256].

The right sequence is rb(n), with b being the binaryrepresentation of 135.

-70

-60

-50

-40

-30

-20

-10

0

10

20

0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

|D(θ

)|dB

θ

(20π/256, 21π/256)

Radar Signal Processing

Clearing Doppler Sidelobes

Time-Domain PAM:

z(t) =M−1∑m=0

ams(t−mTc)

Impulse in Delay:

χz(τ, ν) =

∞∫−∞

z(t)z(t− τ)e−jνt

≈ δ(τ)α(ν), ∀ν ∈ ∆ν

Frequency-Domain PAM:

z(ω) =M−1∑m=0

ams(ω −mWc)

Impulse in Doppler:

χz(ν, τ) =

∞∫−∞

z(ω)z(ω − ν)e−jτω

≈ δ(ν)α(τ), ∀τ ∈ ∆τ

Radar Signal Processing

PTM Sequencing Across Frequency

Time-domain OFDM ⇐⇒ Frequency-domain PAM

Z(t) =(M−1∑m=0

ame−jmωct

)S(t) ⇐⇒ z(ω) =

M−1∑m=0

ams(ω −mWc)

Radar Signal Processing

Clearing in a Desired Range/Doppler Interval

Sequencing in Time:

Stacking in Frequency:

Game: Shaping the spectrum of the coordinating sequence

Radar Signal Processing

References

1 A. Pezeshki, A. R. Calderbank, and L. L. Scharf, “Sidelobe suppression is a desired range/Dopplerinterval,” Proc. IEEE Radar Conf., Pasadena, CA, May 4-8, 2009.

2 Y. Chi, A. Pezeshki, and A. R. Calderbank, “Complementary waveforms for sidelobe suppression and radarpolarimetry,” in Applications and Methods of Waveform Diversity, V. Amuso, S. Blunt, E. Mokole, R.Schneible, and M. Wicks, Eds., SciTech Publishing, Inc., to appear 2009.

3 Y. Chi, A. Pezeshki, and A. R. Calderbank, “Range sidelobe suppression in a desired Doppler interval,”Proc. Waveform Diversity and Design Conference, Orlando, FL, Feb. 8-13, 2009.

4 R. Calderbank, S. D. Hoawrd, and W. Moran, “Waveform diversity in radar signal processing: A focus onthe use and control of degrees of freedom,” IEEE Signal Processing Magazine, vol. 26, no. 1, pp. 32-41,Jan. 2009.

5 A. Pezeshki, A. R. Calderbank, W. Moran, and S. D. Howard, “Doppler resilient Golay complementarywaveforms,” IEEE Trans. Information Theory, vol. 54, no. 9, pp. 4254-4266, Sep. 2008.

6 S. Suvorova, S. D. Howard, W. Moran, R. Calderbank, and A. Pezeshki, “Doppler resilience, Reed-Mullercodes, and complementary waveforms,” Conf. Rec. Forty-first Asilomar Conf. Signals, Syst., Comput.,Pacific Grove, CA, Nov. 4-7, 2007.

7 A. Pezeshki, A. R. Calderbank, S. D. Howard, and W. Moran, “Doppler resilient Golay complementarypairs for radar,” Proc. IEEE Workshop on Statistical Signal Processing, Madison, WI, Aug. 26-29, 2007.

8 S. D. Howard, A. R. Calderbank, and W. Moran, “The finite Heisenberg-Weyl groups in radar andcommunications,” EURASIP Journal on Applied Signal Processing, Article ID 85685, 2006.

Radar Signal Processing