Phase retrieval algorithm for generation of constant time envelope with prescribed fourier transform magnitude signal
The invention is an iterative process for performing iteratively the phase retrieval of an adaptive signal x(t) matching two sets of constraint both concerning the time envelope u e (t) of signal x(t) and magnitude distribution U m (f) of its spectral representation. At each iteration k the process computes an estimate {tilde over (x)} k (t) of signal x(t) which is obtained from a first projection P A on a first set of constraint in time domain of a computed value x k (t) of x(t), x k (t) deriving from an estimate {tilde over (X)} k−1 (f) of the spectrum of signal x(t), said estimate {tilde over (X)} k−1 (f) being itself obtained from a second projection P B on a second set of constraints in spectral domain of the Fourier transform X k (f) of the estimate {tilde over (x)} k−1 (t) of x(t) computed at iteration k−1. Iterative computation of estimate {tilde over (x)} k (t) is repeated until {tilde over (x)} k (t) meets a predefined criterion which indicates that estimate {tilde over (x)} k (t) is close enough to expected signal x(t).
1. A process for performing iteratively the phase retrieval of a transmission signal x(t) matching two sets of constraint both concerning the time envelope u e (t) of signal x(t) and magnitude distribution U m (f) of its spectral representation, said process computing at each iteration k an estimate {tilde over (x)}(t) of signal x(t), comprising
said estimate {tilde over (x)} k (t) being obtained from a first projection P A on a first set of constraint in time domain of a computed value x k (t) of x(t) which derives from an estimate {tilde over (X)} k−1 (f) of the spectrum of signal x(t), said estimate {tilde over (X)} k−1 (f) being itself obtained from a second projection P B on a second set of constraints in spectral domain of the Fourier transform X k (f) of the estimate {tilde over (x)} k−1 (t) of x(t) computed at iteration k−1;
estimate {tilde over (x)} k (t) being a weighted sum of projection P A of x k (t) onto the first constraint domain and of x k (t) itself;
estimate {tilde over (X)} k−1 (f) being a weighted sum of projection of P B of X k (f) onto the second constraint domain and of X k (f) itself;
iterative computation of estimate {tilde over (x)} k (t) is repeated until {tilde over (x)} k (t) meets a predefined criterion,
wherein estimates {tilde over (x)} k (t) and {tilde over (X)} k−1 (f) are respectively defined by the following relations:
{tilde over (x)} ( t )=σ P A {x k ( t )}+(1−σ) x k ( t ),and
{tilde over (X)} ( f )=μ P B {X k ( f )}+(1−μ) X k ( f )
where σ and μ are relaxation parameter which values are less than 1 and where P A and P B are respectively defined by the following relations:
P
A
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where A is the expected constant magnitude of x(t), and
P
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where U m (f) is the expected Fourier Transform Magnitude U m (f) of signal x(t), and wherein, at the first iteration the computed value x 1 (t) of x(t) is computed from an estimate {tilde over (X)} 0 (f) of the spectrum of x(t) defined by the following relation:
{tilde over (X)} 0 ( f )= U m ( f ) e iθ 0 (f)
where Um(f) is the expected Fourier Transform Magnitude U m (f) of signal x(t) and where θ 0 is a particular initial phase defined by the following relation:
θ
0
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where Ω is the spectral domain of the second constraints set.
2. The process of claim 1 , wherein the values of relaxation parameters σ and μ are chosen among a set of values extending between 0.7 and 0.9.
3. The process according to claim 1 , wherein a test step takes place at each iteration to determine if the predefined criterion is met or not.
4. The process of claim 3 , wherein the test step consists in comparing the value of the estimate {tilde over (x)} k (t) to the value of the previous estimate {tilde over (x)} k−1 (t) to determine if the modulus |{tilde over (x)} k (t)−{tilde over (x)} k−1 (t)| 2 is less than a determined threshold δ or not.
5. The process of claim 3 , wherein the test step consists in comparing the modulus of the Fourier Transform of estimate {tilde over (x)} k (t) to the prescribed Fourier Transform Magnitude of the expected signal x(t), U m (f), to determine if the modulus ∥F{{tilde over (x)} k (t)}|−U m (f)| 2 is less than a determined threshold δ or not.
6. The process of claim 3 , wherein the test step consists in comparing the number k of iterations already performed to a predefined maximum number N of iterations, to determine if k is greater than N or not.
7. The process of claim 6 , wherein, for the last few iterations the values of relaxation parameters σ and μ are chosen substantially equal to 1.
8. The process according to claim 1 , comprising at least the following steps:
a first initialization step for inputting the data corresponding to the desired time envelope u e (t) as well as those corresponding to the spectral magnitude components U m (f) of its spectrum;
a second initialization step for consists in computing an initial seed value θ 0 (f) for computing the Fourier transform of the desired signal x(t);
a third initialization step for computing a first estimate {tilde over (X)} 0 (f) of the spectral representation of the desired transmission signal x(t), {tilde over (X)} 0 (f) being defined by the following relation:
{tilde over (X)} 0 ( f )= U m ( f ) e i θ 0 (f) ∀f;
a fourth step for computing the current value x k (t), for an iteration k, of the desired transmission signal x(t) using an estimate value {tilde over (X)} k−1 (t) of the spectral representation of x(t) obtained for the previous iteration k−1, x k (t) being defined by the following relation:
x k ( t )= F −1 {{tilde over (X)} k−1 ( f )}
where F represents the Fourier transform operation;
a fifth step for performing the projection P 1 of the current computed value x k (t) of x(t) using projection P A of x k (t) onto a first constraint set constituted by the single value A;
a sixth step, test step, for comparing the value of {tilde over (x)} k (t) with these of some predefined criteria to determine when the considered criterion is matched meaning that the iterative phase of the process comprising steps four to nine can be stopped;
a seventh step for computing the current value X k (f), for an iteration k, of the Fourier transform of the desired transmission signal x(t) using the current value {tilde over (x)} k (t) computed at the fourth step, X k (f) being defined by the following relation:
X k ( f )= F{{tilde over (x)} k ( t )}
where F represents the Fourier transform operation;
an eighth step performing the projection P 2 of the current value X k (f) of the Fourier Transform of x(t) using projection P B of X k (f) onto a second constraint set constituted by the prescribed Fourier Transform Magnitude U m (f) of the expected signal;
a ninth step for incrementing index k indicating the number of the current iteration;
a tenth step implemented only if the sixth step ( 16 ) is successful during which a signal {tilde over (x)} k (t) substantially equal to the expected signal x(t) is delivered.