The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms.

TLDR: The efficacy of GFT in solving the initial value problems (IVPs) and the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform.

Abstract: This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed.

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Table 1: Some signals and their GFT with information whether FT and/or BLT exists. To obtain FT from the given GFT, first separate out the real and imaginary parts by substituting s = σ + jω and then consider limσ → 0. We observe that the ROC is the entire s-plane, if there are no poles; otherwise, it is always right-side of the rightmost pole. Furthermore, the FT exists, if the ROC includes imaginary axis, s = jω.

Table 2: Some signals and their FST along with the information on whether MT exists. We observe that the ROC is the entire s-plane, if there are no poles; otherwise, it is always right-side of the rightmost pole. Also, the FT exists, if the ROC includes the imaginary axis s = jω.

Table 3: Some signals and their GDTFT along with the information whether DTFT and/or BLzT exist. We observe that the ROC is the entire z-plane, if there are no poles; otherwise, it is always outside of the outermost pole. Also, the DTFT exists, if the ROC includes unit circle |z| = 1.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "The generalized fourier transform: a unified framework for the fourier, laplace, mellin and z transforms" ?

This paper introduces Generalized Fourier transform ( GFT ) that is an extension or the generalization of the Fourier transform ( FT ). GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. For example, the authors have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, the authors demonstrate the efficacy of GFT in solving the initial value problems ( IVPs ). Fourthly, the authors introduce Fourier scale transform ( FST ) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, the authors propose Generalized Discrete-Time Fourier transform ( GDTFT ). 

Q2. What is the GFT of successive time difference?

(118)Important and fundamental contributions of this study are the introduction of the generalized Fourier transform (GFT) in both continuous-time and discrete-time domains, the Fourier scale transform, and the solution of initial value problem using the GFT and Fourier transform. 

Q3. What is the inverse of the MT?

The Mellin transform (MT) and inverse MT (IMT) are derived from the BLT (2) by the change of the variable as e−t = τ =⇒ t = − ln(τ) and is defined asY (s) = ∫ ∞ 0 y(τ) τs−1 dτ,and y(t) = 12πj ∫ σ+j∞ σ−j∞ Y (s) τ−s ds,(69)where x(− ln(τ)) = y(τ). 

Q4. What is the frequency of the time-frequency pair?

This time-frequency pair is also denoted by x(t) X(f), where ω = 2πf , ω denotes the angular frequency in radians/sec, and f denotes the frequency in Hz. 

Q5. What is the moment of a random variable?

The moments of a random variable Y are defined as E{Y m} =Mm = ∫ ∞ −∞ ymf(y) dy, m = 1, 2, . . . ,M, (89)that can be written as Mm =Mnm +Mpm = ∫ ∞0(−y)mf(−y) dy + ∫ ∞0ymf(y) dy, (90)where Mnm = ∫∞ 0 (−y)mf(−y) dy, Mpm = ∫∞ 0 ymf(y) dy, f(y) is the probability density function (pdf), and E is the expectation operator. 

Q6. what is the gft of x(t) divided by t?

The GFT of x(t) multiplied by various functions or scaled in amplitude:G{e−at x(t)} = X(s∗ − a) +X(s+ a); ROC is R− |Re{a}| (29) G{e−a|t| x(t)} = X(s∗ + a) +X(s+ a); ROC is R+ Re{a} (30)G{tm x(t)} = d m ds∗m X(s∗) + (−1)m d m dsm X(s); ROC is R (31)G{|t|m x(t)} = (−1)m d m ds∗m X(s∗) + (−1)m d m dsm X(s) ROC is R. (32)It is also observed thatdmdσm X(s, s∗) =∂m∂s∗m X(s, s∗) + (−1)m ∂m∂sm X(s, s∗) =dmds∗m X(s∗) + (−1)m dmdsm X(s), (33)dmdωm X(s, s∗) = jm∂m∂s∗m X(s, s∗) + (−j)m ∂m∂sm X(s, s∗) = jmdmds∗m X(s∗) + (−j)m dmdsm X(s). (34)5. The GFT of x(t) divided by t, provided limt→0 ( x(t) t ) exists:G { x(t)t} = − ∫ ∞ s∗ X(v) dv + ∫ ∞ s X(u) du, (35)G { x(t)|t|} = ∫ ∞ s∗ X(v) dv + ∫ ∞ s X(u) du. (36)6. 

Q7. what is the definition of the polynomial decaying class of functions?

in order to tackle the polynomial decay in the function, their next goal is to define and explore the generalized Fourier transform (GFT) asX(ω, σ, p) = ∫ ∞ −∞ x(t) |t|p exp(−σ |t|) exp(−jωt) dt, (53)which can be written asX(s, s∗, p) = X(s∗, p) +X(s, p) = ∫ ∞ 0 x(−t) tp exp(−s∗t) dt+ ∫ ∞ 0 x(t) tp exp(−st) dt. (54)Using the proposed definition (53), the authors can easily include the polynomially decaying class of functions. 

Q8. What is the difference between the two functions?

(3)If two functions have identical values except at a countable number of points, their integration is also identical because Lebesgue measure of a set of countable points is always zero.