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Stopband constraint case and the ambiguity function PowerPoint PPT Presentation


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Daniel Jansson. Stopband constraint case and the ambiguity function. Stopband constraint case. Goal Generate discrete, unimodular sequences with frequency notches and good correlation properties Why?

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Stopband constraint case and the ambiguity function

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Daniel jansson

Daniel Jansson

Stopband constraint case and the ambiguity function


Stopband constraint case

Stopband constraint case

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)


Stopband can scan

Stopband CAN (SCAN)

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


Stopband can scan1

Stopband CAN (SCAN)

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


Stopband can scan2

Stopband CAN (SCAN)

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


Stopband can scan3

Stopband CAN (SCAN)

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


Weighted scan wescan

Weighted SCAN (WeSCAN)

Minimization of J2 is a way of minimizing the ISL

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


Weighted scan wescan1

Weighted SCAN (WeSCAN)

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.


Numerical examples

Numerical examples

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)


Numerical examples1

Numerical examples

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)


Numerical examples2

Numerical examples

Pstop and Pcorrvsλ


Numerical examples3

Numerical examples

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)


Numerical examples4

Numerical examples

The autocorrelation of the WeSCAN sequence


Numerical examples5

Numerical examples

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


The ambiguity function

The Ambiguity Function

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.


The ambiguity function1

The Ambiguity Function

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


The ambiguity function2

The Ambiguity Function

Proofs

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

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


The ambiguity function3

The Ambiguity Function

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


The ambiguity function4

The Ambiguity Function

Ambiguity function of a chirp


The ambiguity function5

The Ambiguity Function

Ambiguity function of a Golomb sequence


The ambiguity function6

The Ambiguity Function

Ambiguity function of CAN generated sequences


The ambiguity function7

The Ambiguity Function

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.


The discrete af

The Discrete AF

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


The discrete af1

The Discrete AF

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


The discrete af2

The Discrete AF

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.


The discrete af3

The Discrete AF

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


The discrete af4

The Discrete AF


The discrete af5

The Discrete AF


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