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Stopband constraint case and the ambiguity function

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Daniel Jansson

Goal

Generate discrete, unimodular sequences with frequency notches and good correlation properties

Why?

Avoiding reserved frequency bands is important in many applications (communications, navigation..)

Avoiding other interference

How?

SCAN (Stopband CAN) / WeSCAN (Weighted Stopband CAN)

Let {x(n)}, n = 1...N be the sought sequence

Express the bands to be avoided as

Define the DFT matrix with elements

Form matrix S from the columns of FÑcorresponding to the frequencies in Ω

We suppress the spectral power of {x(n)} in Ω by minimizingwhere

The problem on the previous slide is equivalent towhere G are the remaining columns of FÑ.

Suppressing the correlation sidelobes is done using the CAN formulation

Combining the frequency band suppression and the correlation sidelobe suppression problems we getwhere 0 ≤ λ ≤ 1 controls the relative weight on the two penalty functions.

The problem is solved by using the algorithm on the next slide

If a constrained PAR is preferable to unimodularity the problem can be solved in the same way except x for each iteration is given by the solution to

Minimization of J2 is a way of minimizing the ISL

The more general WISL (weighted ISL) is given bywhere are weights

Let and D be the square root of Γ. Then the WISL can be minimized by

solvingwhereand

Replace in the SCAN problem with and perform the SCAN algorithm, but do necessary changes that are straightforward.

The spectral power of a SCAN sequence generated with parameters N = 100,

Ñ = 1000, λ = 0.7 andΩ = [0.2,0.3] Hz. Pstop = -8.3 dB (peak stopband power)

The autocorrelation of a SCAN sequence generated with parameters N = 100,

Ñ = 1000, λ = 0.7 andΩ = [0.2,0.3] Hz, Pcorr = -19.2 dB (peak sidelobe level)

Pstop and Pcorrvsλ

The spectral power of a WeSCAN sequence generated with γ1=0, γ2=0 and γk=1 for larger k.

Pstop = -34.9 dB (peak stopband power)

The autocorrelation of the WeSCAN sequence

The spectral power of a SCAN sequence generated with PAR ≤ 2

The response of a matched filter to a signal with various time delays and Doppler frequency shifts (extension of the correlation concept).

The (narrowband) ambiguity function iswhere u(t) is a probing signal which is assumed to be zero outside [0,T], τis the time delay and f is the Doppler frequency shift.

Three properties worth noting

The maximum value of |χ(τ,f)| is achieved at | χ(0,0)| and is the energy of the signal, E

d|χ(τ,f)|= |χ(-τ,-f)|

D

Proofs

Cauchy-Schwartz givesand since | χ(0,0)| = E, property 1 follows.

Use the variable change t -> t+ τwhich implies property 2.

Proofs

3. The volume of |χ(τ,f)|2is given byLet Wτ(f) be the Fourier transform of u(t)u*(t- τ). Parseval givestherefore

Ambiguity function of a chirp

Ambiguity function of a Golomb sequence

Ambiguity function of CAN generated sequences

Why is there a vertical stripe at the zero delay cut?

The ZDC is nothing but the Fourier transform of u(t)u*(t). Since u(t) is unimodular we getand the sinc-function decreases quickly as f increases.

No universal method that can synthesize an arbirtrary ambiguity function.

Assume u(t) is on the formwhere pn(t) is an ideal rectangular pulse of lengthtp

The ambiguity function can be written as

Inserting τ = ktpand f = p/(Ntp) giveswhere is called the discrete AF.

If |p|<<N then

Minimizing the sidelobes of the discrete AF in a certain regionwhere and are the index sets specifying the region.

Define the set of sequences as

Denote the correlation between {xm(n)} and {xl(n)} by

All values of are contained in the set

Minimizing the correlations is thus equivalent to minimizing the discrete AF sidelobes.

Define where

All elements of appear in We can thus minimizewhich as we saw before is almost equivalent to

Minimize by using the cyclic algorithm on the next slide