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
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Define the DFT matrix with elements
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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 minimizing
where
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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)
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•
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
changes that are straightforward.
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and perform the SCAN algorithm, but do necessary
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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 examples
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Pstop 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 Function
Proofs
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1.
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 Function
Proofs
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3.
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 Function
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Ambiguity function of a chirp
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The Ambiguity Function
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Ambiguity function of a Golomb sequence
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The Ambiguity Function
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Ambiguity function of CAN generated sequences
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The Ambiguity Function
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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
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•
Assume u(t) is on the form
where pn(t) is an ideal rectangular pulse of length tp
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The ambiguity function can be written as
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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
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•
Minimizing the sidelobes of the discrete AF in a certain region
where
the region.
•
Define the set of sequences
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and
are the index sets specifying
as
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The Discrete AF
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Denote the correlation between {xm(n)} and {xl(n)} by
•
All values of
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Minimizing the correlations is thus equivalent to minimizing the discrete AF
sidelobes.
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are contained in the set
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The Discrete AF
•
Define
•
All elements of
appear in
We can thus minimize
which as we saw before is almost equivalent to
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Minimize by using the cyclic algorithm on the next slide
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where
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The Discrete AF
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The Discrete AF
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