1.2 Finite Fourier Transform
1.2.1 Connection between Finite and Centered Discrete Fourier Transform 21
− N
2
+ 1, . . . , N
2
for some threshold valueε >0and large N ∈N. Then we obtain that
f(x)≈
bN2c X
ν=−dN2e+1
cν(f)eiνx ∀x∈[0,2π).
1.2.1 Connection between Finite and Centered Discrete Fourier Transform
For bandlimited 2π-periodic functions the finite Fourier transform and the CDFT are closely related. Let us first formally define the notion of bandlimited functions.
Definition 1.15 (Bandlimited Function)A function f ∈ L22π is called bandlimited on
−N
2
+ 1, . . . ,N
2 for some N ∈Nif
|cν(f)|= 0 ∀ν /∈
− N
2
+ 1, . . . , N
2
. The natural number N is called the bandwidth of N.
Let us now consider a function f ∈L22π with bandwidth N, i.e.,
f(x) =
bN2c X
ν=−dN2e+1
cν(f)eiνx ∀x∈[0,2π). (1.8)
Note that a function f as in (1.8) is a trigonometric polynomial, see, e.g., [PPST19], Section 1.2, and as such contained inC2π. We would like to obtain a connection between the finite spectrum c(f) of f and the CDFT. As the input argument of the CDFT has to be a vector rather than a function, we need to discretize f by constructing a suitable vector of evaluations off. Sincef is bandlimited with bandwidthN, its finite spectrum is determined by N Fourier coefficients. Because of the linearity of the DFT and the CDFT, it is natural to evaluate f at N equidistant points.
We define the following vector ofN equidistant samples of f on [0,2π),
Consequently, by calculating the CDFT of the vectoraN ∈CN ofN equidistant samples of f, we obtain a vector that contains precisely the N Fourier coefficients cν(f) of f for ν ∈
−N
2
+ 1, . . . ,N
2 . This vector is just a restriction of the finite spectrum c(f)∈CZ off to the frequencies contained in
This means that we can compute the finite spectrum of a bandlimited function, which by (1.8) and Theorem 1.12 (i) completely definesf, if we calculate the CDFT of the vector aN ofN equidistant samples of f. Thus, we can recover f fromN discrete, equidistant samples via the FFT in O(NlogN) time.
For functions that are only approximately bandlimited, i.e.,
|cν(f)|< ε ∀ν /∈
for some threshold valueε >0 and large bandwidth N ∈N, with
f(x)≈
bN2c X
ν=−dN2e+1
cν(f)eiνx ∀x∈[0,2π),
the FFT ofaN provides a good approximation of the function.
Remark 1.16 If we apply the CDFT to a vector of equidistant samples off of length s < N, i.e., to
1.2 Finite Fourier Transform
we find that
absη = 1 s
s−1
X
j=0
e−2πijηs
bN2c X
ν=−dN2e+1
cνe2πijνs
= 1 s
bN2c X
ν=−dN2e+1
cν
s−1
X
j=0
ωsj(η−ν)
=
bN2c X
ν=−dN2e+1 ν≡η mod s
cν
for all η∈
−s
2
+ 1, . . . ,s
2 . ♦
Since the O(NlogN) runtime of the FFT is, as noted in Section 1.1.1, optimal for arbitrary N-length vectors, we can only hope to improve the runtime of fast Fourier algorithms for2π-periodic functions if their Fourier coefficients satisfy additional a priori known conditions. The by far most interesting case is the one of functions with sparse frequency support, meaning that most of the corresponding Fourier coefficients are in-significantly small and that only few of them actually contribute to the Fourier series of the function. We will investigate different types of sparsity in Chapters 2 and 3.
2 Sparse FFT for 2π-Periodic Functions with Short Support
Sparse Fourier transforms have many applications in signal processing, for example analog-to-digital conversion, see, e.g., [LKM+06, YRR+12], GPS signal acquisition, see, e.g., [HAKI12], and wideband communication or spectrum sensing, see, e.g., [HSA+14, YG12]. Thus, in the first part of this thesis we are interested in deterministically re-covering 2π-periodic functions f from samples. By Theorem 1.14 it suffices to know the significantly large Fourier coefficients and the frequencies corresponding to them in order to obtain a good approximation off. We can only hope to do this in a more efficient way than by directly applying the CDFT to the vector of N equidistant samples of f as in Section 1.2.1 if the number of Fourier coefficients we need to recover is small compared to the assumed bandwidth N of f.
Most of the existing sparse Fourier transform methods do not assume any further structure of the sparsity. The first sparse methods which achieved runtimes that are sublinear in the bandwidth or vector lengthN were randomized algorithms. This means that with a small, usually tunable probability the returned vector is not a good ap-proximation of the correct solution. Such algorithms have runtimes of O
BlogO(1)N , see, e.g., [AGS03, GMS05, GGI+02, Man92, HIKP12a, HIKP12c, IKP14, IGS07, CLW16, CCW16, LWC13, MZIC18, SI13, CIK18]. More information and implementations can be found in a survey about randomized sparse FFT algorithms, see [GIIS14].
There also exist deterministic sparse Fourier algorithms where the probability of failure is zero. These methods include techniques arising from modifications of Prony’s method with a runtime of O B3
, see, e.g., [HKPV13, PT14, PTV16]. As many Prony-based techniques suffer from numerical instabilities for noisy input data, they cannot be ap-plied to all problems. Other deterministic method utilize arithmetic progressions or the Chinese Remainder Theorem, see, e.g., [Aka10, Aka14, Iwe10, Iwe13], or other properties of the DFT, see, e.g., [Mor16, PWCW18]. All of these non-Prony-based methods have in common that their runtime isO
B2logO(1)N
. Thus, they are sublinear in the vector length or bandwidthN, but quadratic in the sparsityB. For generalB-sparsity it seems to be extremely difficult to reduce the quadratic dependence of the runtime on B, see, e.g., [BDF+11, CI16, FR13].
However, if there is some additional a priori information about the sparsity struc-ture, runtimes scaling subquadratically in the sparsity can indeed be achieved, see, e.g., [PW16a,PW17a] with runtimes ofO(BlogN)andO BlogBlogNB
. Consequently, we will focus on two types of structured sparsity in the first part of this thesis. To be more precise, we will always assume that the frequencies associated with signifi-cantly large Fourier coefficients are contained in a small number, n, of support sets S1, . . . , Sn (
−N
2
+ 1, . . . ,N
2 , and that each of the unknown sets Sj has a “sim-ple” structure.
In this chapter we will investigate the special case that there is only one such set S
and that S is an interval in Z, i.e., thatn= 1 and
S ={ω1, ω1+ 1, . . . , ω1+B−1}
for some starting frequency ω1 ∈
−N
2
+ 1, . . . ,N
2
−B+ 1 . In Chapter 3 we will extend our methods for deterministically recovering2π-periodic functions to the case of frequency supports consisting of several sets with more complex structures.
In the vector setting the special case of n = 1 and S being an interval corresponds to the case that the vector we aim to recover has a short support. This is precisely the sparsity assumption of the deterministic sparse FFT algorithms [PW16a, PW17a]. See Section 5.3 for a more detailed explanation of these two methods.
Sections 2.1 to 2.3 in this chapter are based on my paper [Bit17c] and are in part iden-tical with the representations therein. Section 2.4 presents completely new, previously unpublished results that I developed on my own.
2.1 Sparsity and Short Support
Throughout the next two chapters we will always consider a2π-periodic functionf ∈C2π with finite spectrumc(f)∈CZ. We will assume thatf has approximately a large band-widthN ∈N. This means that the Fourier coefficients corresponding to the frequencies that are not contained in
−N
2
+ 1, . . . ,N
2 have an absolute value which is so small that it can be disregarded. Hence,f is of the form
f(x)≈
bN2c X
ω∈−dN2e+1
cω(f)eiωx.
As in practical applications the given data is usually noisy, we will assume that the function f is perturbed by a 2π-periodic function η ∈ C2π with c(η) ∈ `1 satisfying kc(η)k∞ ≤ ε for some suitably chosen noise threshold ε > 0. Since we aim to recover f from finitely many samples of noisy data, our main object of interest for now are the Fourier coefficients off+η.
Using a threshold parameter ε >0, we can now formally define the notion of signifi-cantly large Fourier coefficients.
Definition 2.1 Letf ∈C2π, let ε >0 be a suitably chosen noise threshold and ω∈Z. A Fourier coefficientcω(f)∈Cis calledsignificantly large if |cω(f)|> ε. A frequencyω is calledenergetic if its corresponding Fourier coefficient cω(f) is significantly large.
As already mentioned above, due to the fact that the runtime of the FFT is opti-mal for arbitrary N-length input vectors, we can only expect to improve its runtime of O(NlogN)if it is known a priori that many of the Fourier coefficients are insignificantly small. This motivates the following formal definition of the concept of sparsity.
Definition 2.2 (Sparsity)Let f ∈ C2π and let ε > 0 be a suitably chosen noise threshold. Thenf is called B-sparse if it has only B energetic frequencies.
In this chapter we are interested in functions whose energetic frequencies are contained in a short interval inZ.