stopband constraint case and the ambiguity function daniel jansson

Stopband constraint case and the ambiguity function

Daniel Jansson

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Institutionen för informationsteknologi | www.it.uu.se

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)

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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 FÑ corresponding to the frequencies in Ω

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

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Stopband CAN (SCAN)

• The problem on the previous slide is equivalent to

where G are the remaining columns of FÑ .

• Suppressing the correlation sidelobes is done using the CAN formulation

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Stopband CAN (SCAN)

• Combining the frequency band suppression and the correlation sidelobe suppression problems we get

where 0 ≤ λ ≤ 1 controls the relative weight on the two penalty functions.

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

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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

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Weighted SCAN (WeSCAN)

• Minimization of J2 is a way of minimizing the ISL

• The more general WISL (weighted ISL) is given by

where are weights

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Weighted SCAN (WeSCAN)

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

solving

where

and

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

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Numerical examples

The spectral power of a SCAN sequence generated with parameters N = 100,Ñ = 1000, λ = 0.7 and Ω = [0.2,0.3] Hz. Pstop = -8.3 dB (peak stopband power)

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Numerical examples

The autocorrelation of a SCAN sequence generated with parameters N = 100,Ñ = 1000, λ = 0.7 and Ω = [0.2,0.3] Hz, Pcorr = -19.2 dB (peak sidelobe level)

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Numerical examplesPstop and Pcorr vs λ

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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)

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Numerical examples

The autocorrelation of the WeSCAN sequence

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Numerical examples

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

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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 is

where u(t) is a probing signal which is assumed to be zero outside [0,T], τ is the time delay and f is the Doppler frequency shift.

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The Ambiguity Function

Three properties worth noting

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

2. d|χ(τ,f)|= |χ(-τ,-f)|

3. D

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The Ambiguity FunctionProofs1. Cauchy-Schwartz gives

and since | χ(0,0)| = E, property 1 follows.

2. Use the variable change t -> t+ τ

which implies property 2.

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The Ambiguity FunctionProofs3. The volume of |χ(τ,f)|2 is given by

Let Wτ(f) be the Fourier transform of u(t)u*(t- τ). Parseval gives

therefore

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The Ambiguity FunctionAmbiguity function of a chirp

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The Ambiguity FunctionAmbiguity function of a Golomb sequence

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The Ambiguity FunctionAmbiguity function of CAN generated sequences

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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 get

and the sinc-function decreases quickly as f increases.

• No universal method that can synthesize an arbirtrary ambiguity function.

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The Discrete AF• Assume u(t) is on the form

where pn(t) is an ideal rectangular pulse of length tp

• The ambiguity function can be written as

• Inserting τ = ktp and f = p/(Ntp) gives

where

is called the discrete AF.

• If |p|<<N then

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The Discrete AF• Minimizing the sidelobes of the discrete AF in a certain region

where and are the index sets specifying the region.

• Define the set of sequences as

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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.

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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

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The Discrete AF

stopband constraint case and the ambiguity function daniel jansson

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