We consider Gabor systems on rectangular lattices generated by a function and lattice parameters :
Here the time-frequency shift is an operator acting on fuctions as
A Gabor system is a frame if and only if there are constants satisfying
We call lower and upper frame bound, respectively. The supremum over all such is also a lower frame bound, referred to as the optimal lower frame bound. Similarly, the infimum over all upper frame bounds is an upper bound – the optimal lower frame bound. In the following, always the denote the optimal frame bounds.
The existence of the frame bounds is equivalent to the stability of the reconstruction of from the samples , i.e., the frame operator
is a bounded bijective operator with a bounded inverse. Moreover, the associated operator norms give the optimal frame bounds.
The volume of the fundamental domain is called the volume of the lattice, , whereas the density of the lattice is .
Given a window function and density , determine the pairs such that
- is a frame;
- is optimal under all pairs which satisfy (a) and (b). Optimality is considered in three versions, namely, with respect to
- the lower frame bound ;
- the upper frame bound ;
- the condition number .
The first condition turns the problem into a univariate optimization problem, where we are free to choose the parametarization. For instance, one option is to use , which explains the optimality with respect to time. Another possibility is , which reflects the symmetry of the parameters with respect to . In this case, the square lattice corresponds to .
- What is the set of admissible parameters ?
- Are there closed forms for and with respect to the lattice parameters?
- Is there an optimizer? If yes, is it unique?
- Do the optimizers of and coincide?
The investigated functions are the hyperbolic secant, cut-off exponentials, one-sided and two-sided exponentials. For these functions, the first two problems were already resolved satisfactorily in . The rest of the questions were answered in .
Parameter transformations and dilations
By the symplectic invariance of Gabor frames, a Gabor system is a frame if and only if is a frame, with same bounds. In particular, dilations as metaplectic operators correspond to dilation matrices
so the pair is optimal (in any of the optimality versions) for fixed and if and only if is optimal for and . In the following we will mostly set additional function parameters to , since we can recover the optimal frames through dilations, anyway.
If and are continuously differentiable and have unique critical points and , then this already implies something for the condition number.
- If tends to or Is unbounded, then has an optimizer. Hence, we only look at the critical points.
- , or equivalently,
- The frame bounds are not negative. But this means that the sign of and is the same at the critical points. Why is this interesting? This simply tells us that the minimum is attained at some is between and . This is a trivial observation if , but otherwise, is strictly between and !
Determining the optimal frame bounds
Janssen’s representation of the frame operator reads
expressing explicitly the dependence on the shifts of . This already motivated the first examples, where the sum is reduced to one and three time-shifts, respectively.
By cut-off exponentials we mean where is a normalizing constant. In order to simplify Jannsen’s representation, we can consider the cut-offs and . Yet, the slight support change results in completely different situations. Letting , we get the well-known box function, whereas the degenerate corresponds to the one-sided exponential explored below.
The frame bounds are
resulting in the condition number . Notice that all expressions depend on the product , so we can optimise interchangeably, by fixing one or the other. Either way, there is no optimal lattice.
The left plot shows the lower bound (red), the upper bound (blue) and the condition number (purple). There is clearly no optimal lattice for any of the optimization problems.
In this case, the frame operator (in Janssen’s representation) depends on three shifts and everything changes.
This time, the limiting cases do not result in frames. Notice that the optimality depends on the product , again. Optimizers exists, but the uniqueness is an issue. The lower frame bound has a unique global maximum. It is attained at the unique positive solution of the equation
The graph shows lower frame bound depending on the lattice parameter .
The upper frame bound is strictly monotonically increasing when or . For all other , has a local maximum and a local minimum, which may or may not be global. If its value at the local minimum is strictly less than 2, then it is a unique minimizer.
The graph shows upper frame bound depending on the lattice parameter .
The hyperbolic secant
The (scaled) hyperbolic secant is a totally positive function in Feichtiger’s algebra, so the lower frame bound vanishes at the critical density . For all higher integer densities , all pairs are admissible, i.e., we are optimizing over . From now on, we only investigate for (see above for the general solution). The optimal frame bounds are given by
The left plot shows the lower bound (red), the upper bound (blue) and the condition number (purple) with their extremal points. The right plot shows their logarithmic counterparts.
- the series converge locally uniformly, hence, are uniformly differentiable;
Therefore, optimal points with respect to exist. All that is left to do is to determine the critical points and compare the values. It would be quite easy if there is only one critical point, then no comparison is needed 😊
For both bounds, we are solving
The next goal is to determine all pairs which solve .
The lower bound
Since the lower bound vanishes for all , the pairs definitely solve the problem. By examining the derivatives of , we showed that there are no further solutions.
Hence, the only critical points are the solutions of and . The second one has no solution when , so we are indeed left with only one critical point, ! This means that the square lattice is the best one for the lower bound for any fixed integer density.
The above plots show (top plot) and plot (bottom plot). The evaluations are at the points and for and for .
The upper bound
Clearly, is a solution. We showed that the objective function is strictly monotonically increasing, so again, a critical point of the upper bound has to satisfy , i.e., the square lattice is again the optimal one.
When the same lattice maximizes the lower and minimizes the upper bound, this is also the unique optimizer of the condition number.
The limiting case of the cut-off exponentials is the one-sided exponential . Due to the dilation invariance, we look at the bounds when . The parameterization of the lattice parameters is , since we initially expected something symmetric.
We first look at the condition number:
The left plot shows the lower bound (red), the upper bound (blue) and the condition number (purple) with their extremal points. If the optimizers coincided, the angle would have been .
The unique minimizer is the point . A not too exhaustive analysis would show that both bounds have unique optimal points, as the critical points are unique and the behaviour in the limiting cases and guarantees an optimizer. That is not to say that this is . Indeed,
If the lower bound is first strictly monotonically increasing, then strictly monotonically decreasing, its maximum has to be less than . Similarly, by evaluating. Or, we recall what we know about coinciding optimizers.
We finish with the two-sided exponential (here: ). This is a totally positive function of type 2, so it belongs to Feichtinger’s algebra and cannot admit a frame at the critical density. For all other integer densities, all pairs are admissable and the frame bounds are given by
Simple enough, right? Besides, we had already dealt with similar products with the other windows. Yet, there seems to be something intricately problematic here.
The left plot shows (scaled) the lower bound (red), the upper bound (blue) and the condition number (purple) with their extremal points. As in the one-sided case, If the optimizers coincided, the angle would have been .
Looking at the graphs, one could almost claim the existence of a unique optimizer will be easy to show. However, this one has proven to be the most challenging of all windows. Our analysis indicated that the visual ”obviousness” is due to a very particular balance of the factors of the bounds. A good global estimate is not possible, because each factor has a varying contribution on different intervals, with high dependancy on the lattice density . Numerical tests hint that the position of the optimizers should be different for each object function and grows logarithmically in . This motivated the following approach:
- Pick a reference point : we chose , which worked well with the first factor.
- Left cut-off: Compare the exact value of the bound, say with a rough estimate for points :
The first inequality is due to the monotonicity of and . To confirm the inequality, one should begin with addition theorems and take it from there.
- Right cut-off: Do the same for points :
To sum up, this shows that
- there is a maximum of ;
- it has to be in .
- Show that the bound is strictly logarithmically concave.
Why logarithmically concave? We expect it to be concave. However, the logarithm is a nice thing to have if we are considering the second and higher derivatives of a product of three functions 😄
Also, we need to keep the end goal in mind: we want a unique maximum. Strict concavity is a typical way to argue this in a homework assignment, but the logarithm is a strictly monotonic function, so it would preserve the position of the optimizers.
We keep in mind that
- (Strictly) concave functions are (strictly) logarithmically concave;
- is strictly concave;
- if is concave, then is strictly logarithmically concave (on the remaining interval, not everywhere). This is the new task.
- Show that strictly convex and unbounded.
- Do the cut-off for .
At this point, I would like to assure you that we are not overdoing it with the cut-off comparisons. Without it, we run into trouble with the condition number . This one is the hardest to understand and we would like to only analyse it on the small interval. We can cut-off everything which is not between the optimizers and . Only then can we focus on the behaviour of on .
- Show the logarithmic concavity of .
Again, this is not an overkill! Unlike the other bounds we have encountered, nothing cancelled out here and no one wants to see the second derivative of a product with so many factors. Besides, we like recycling ♻️
The good news is that the first two summands are taken care of with the lower bound considerations. The bad news is the logarithmic convexity of
This is as far as the reduction goes and at this point the true computations begin. In the end, we managed to prove the strict logarithmic concavity. Similarly to the one-sided exponential, there is no common optimal lattice. This time, .
For small , there were a few adjustments, but this is the overall idea.
 A. J. E. M. Janssen, “Some Weyl-Heisenberg frame bound calculations,” Indagationes
Mathematicae, vol. 7, no. 2, p. 165–183, 1996.
 M. Faulhuber and I. Shafkulovska, “Gabor frame bound optimizations,” arXiv preprint