Stopband constraint case and the ambiguity function Daniel Jansson.
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Transcript of Stopband constraint case and the ambiguity function Daniel Jansson.
Stopband constraint case and the ambiguity function
Daniel Jansson
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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
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The Discrete AF