Consisting of time-frequency shifts of functions in $L^2$(R), the Gabor system (also known as the Weyl-Heisenberg system) provides the ground for time-frequency analysis of data. This thesis is devoted to give a self-contained proofs to the known Gabor frame conditions in multiwindow setting and to propose another set of conditions with certain restrictions. We discuss how to utilize the new conditions as well.
Walnut`s representation of Gabor frame operator \cite{Walnut} reflects the internal structure of the Gabor system into the frame operator. Walnut`s representation assumes the window functions and their dual window functions are not in $L^2$(R) but in the Wiener space, roughly speaking, the space of functions which are locally bounded and are globally in $l^1$. We observe that if we restrict the domain of Walnut`s representation of the Gabor frame operator to the space of compactly supported and bounded functions, then we can assume the window functions and their dual window functions are in $L^2$(R). In this case, the representation holds in a weak sense. This observation enables us to comprehend the rest of the known theory of Gabor frames in a unified manner including Ron-Shen`s necessary and sufficient condition for Gabor frame, Janssen`s representation of Gabor frame operator, Wexler-Raz biorthogonal relation, and Ron-Shen`s duality principle.
Ron-Shen`s necessary and sufficient condition for Gabor frame is in terms of the infinite matrix associated with the Gabor frame operator. By using the above observations, we provide the proof of the condition and two other equivalent conditions in terms of infinite matrices associated with the pre-frame operator and its adjoint respectively. In addition, we present the necessary and sufficient condition for two Gabor systems(which are Bessel sequences) to form a pair of dual Gabor frames. The condition is in terms of infinite matrices associated with the dual frame operators. When there are more than one window functions, the infinite matrix becomes infinite block-matrix. Multiwindow Gabor systems are discussed separately in the following subsections.
When the \textit{correlation function}, a periodic function which appears in the Walnut`s representation of Gabor frame operator, is in $L^2$ on its period interval, there exists another important representation of Gabor frame operator, Janssen`s representation. Janssen`s representation of Gabor frame operator is interesting in that the coefficients in the summation representing the Gabor transform of a given function depend not on the function itself but on the Gabor system with adjoint lattice parameters $(1/b, 1/a)$.
In fact, two Gabor systems with the adjoint lattice parameters have a close relationship, Wexler-Raz biorthogonality relation; if two Gabor systems with lattice parameter $(a,b)$, say $(g; a, b)$ and $(\tilde g; a,b)$, are Bessel sequences, then $(g; a, b)$ and $(\tilde g; a, b)$ form a dual pair of Gabor frames if and only if $(g; 1/b, 1/a)$ and $(\tilde g; 1/b, 1/a)$ form a biorthogonal system. We note that if there are more then one windows in the Gabor system, say $(G_N;a,b)$ and $(\widetilde G_N;a,b)$, then $(\vecg, 1/b, 1/a)$ and $(\tilde \vecg, 1/b, 1/a)$ form a biorthogonal systems in $L^2(R)^N$ where $G_N=\{g_1,...,g_N\}\subset\LtwoR$ and $\vecg=(g_1,...,g_N)^T \in\LtwoR^N$. This biorthogonality eventually leads us to the Ron-Shen duality principle; $(G_N,a,b)$ is a Gabor frame for $L^2$(R) if and only if $(\vecg, 1/b, 1/a)$ is a Riesz sequence in $L^2(R)^N$.
For our main result, we consider multiwindow Gabor systems $(G_N;a,b)$ with $N$ compactly supported windows and rational sampling density $N/ab$. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to Zibulski-Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski-Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle(UEP) condition generate a tight Gabor system when restricted on $[0,2]$ with $a=1$ and $b=1$. As another application, we show that a multiwindow Gabor system $(G_N;1,1)$ forms an orthonormal basis if and only if it has only one window $(N=1)$ which is a sum of characteristic functions whose supports `essentially` form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows.
We then show how to utilize the proposed characterization in order to design tight Gabor frames with favorable features for applications. For example, we introduce some techniques of generating unimodal and symmetric windows, splitting windows, smoothing windows, and generating windows from B-splines.
$L^2$(R) 공간에 속하는 창함수를 시간이동 및 주파수변조를 시켜 만든 가보 시스템은 시간 및 주파수 분석의 기반이된다. 본 논문은 $N$개의 옹근 받침을 갖는 다중창 가보 시스템 $(G_N;a,b)$이 유리밀도 하에서, 즉 $ab$ 값이 유리수일 때, 프레임이 될 새로운 필요충분 조건을 규명하였다. 이 새로운 조건은 기존에 알려진 조건 중 Zak 도메인에서 표현된 Zibulski-Zeevi 조건을 시간 도메인으로 역변환하여 얻을 수 있다. 특히 웨이블릿 이론에 등장하는 UEP(Unitary Extension Principle)조건을 재해석하여 가보시스템 이론과의 관계를 얻어내었다. UEP 조건을 만족하는 mask들을 $[0,2]$ 구간 상으로 제한하면 시간이동 및 주파수변조 파라미터 (즉 $a$와 $b$)가 각각 $1$인 경우에 타이트 가보 프레임을 생성한다. 또한 다중창 가보시스템 $(G_N;1,1)$이 $L^2$(R) 공간의 정규직교기저가 될 필요충분 조건은 창함수가 단 $1$개 이며 그 단 하나의 창함수는 받침이 사실상 단위 구간의 분할이 되는 characteristic 함수라는 점을 밝혀내었다. 새 조건을 이용하면 시간이동 및 주파수 변조 파라미터 값, 창함수의 개수와 받침의 길이 등이 주어졌을 때 다양한 가보프레임을 만들어 낼 수 있다. 좋은 성질을 갖는 다중창 타이트 가보 프레임을 만들어내기 위해서 새 조건을 활용하는 방안도 몇가지 제안하였다. 이상적인 창함수에 가장 근접한 형태인 단봉대칭 모양 창함수를 만들고 그 함수를 원하는 시간구간 길이에 맞추어 쪼개어도 여전히 타이트 가보 프레임을 구성하도록 디자인 하는 방법, 구간별로 $C^\infty$에 속하는 함수를 전구간 $C^\infty$에 속하도록 함수 파라미터를 변환하는 방법, B-spline이 단위분할 성질을 만족하는 점을 이용하여 창함수를 얻는 방법 등을 논하였다.
또한 본 논문에서는 가보 시스템이 프레임이 될 필요충분 조건인 Ron-Shen 조건을 포함한 몇가지 주요 성질들을 증명하였으며, 이 과정에서 가보 프레임 연산자의 Walnut 표현식으로부터 유용한 Corollary를 이끌어내어 사용하였다.
