Studies on Generalized Fourier Representations and Phase Transforms

# Studies on Generalized Fourier Representations and Phase Transforms

Pushpendra Singh
1 School of Engineering & Applied Sciences, Bennett University – Greater Noida, India
###### Abstract

Fourier representation (FR) is an indispensable mathematical formulation for modeling and analysis of physical phenomenon, engineering systems and signals in numerous applications. In this study, we present the generalized Fourier representation (GFR) that is completely based on the FR of a signal, and introduce the phase transform (PT) which is a special case of the GFR and a true generalization of the Hilbert transform. We derive the PT kernel to obtain any constant phase shift, discuss the various properties of the PT, and demonstrate that (i) a constant phase shift in a signal corresponds to variable time-delays in all harmonics, (ii) to obtain a constant time-delay in a signal, one need to provide variable phase shift in all harmonics, (iii) a constant phase shift is same as the constant time-delay only for single frequency sinusoid. The time derivative and time integral, including fractional order, of a signal can be obtained using the GFR. We propose to use discrete cosine transform (DCT) based implementation to avoid end artifacts due to discontinuities present in both end of the signal. We introduce fractional delay of a discrete time signal using the FR, and present the fast Fourier transform (FFT) implementation of all the above proposed representations.

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Keywords: Generalized Fourier representation (GFR); Hilbert transform (HT); phase transform (PT); Analytic signal representation; discrete cosine transform (DCT).

## 1 Introduction

The Fourier representation (FR) of a signal is the most important mathematical formulation for modeling and analysis of physical phenomena and engineering systems. It has been used to obtain solution of problems in almost all fields of mathematics, science, engineering and technologies. FR is the fundamentals of a signal processing, analysis, information extraction and interpretation. There are many variants of the FR such as continuous-time Fourier series (FS), Fourier transfrom (FT), Fourier sine transform (FST) and Fourier cosine transform (FCT), discrete-time Fourier transform (DTFT), discrete time Fourier series (DTFS), discrete Fourier transform (DFT), discrete sine transform (DST) and discrete cosine transform (DCT) [1, 2]. All these are orthogonal transforms which can be computed by the Cooley–Tukey fast Fourier transform (FFT) algorithms . Recently, many studies [4, 5, 6, 7, 8, 9] have been performed using the Fourier theory and many applications including signal decomposition and time-frequency analysis of a nonlinear and nonstationary time-series are proposed.

The DCT was proposed in the seminal paper  for applications to image processing based pattern recognition and Wiener filtering. The modified DCT (MDCT)  is based on the DCT of overlapping data which uses the concept of time-domain aliasing cancellation . The DCT and MDCT are widely-used due to decorrelation and energy compaction properties in many application like image (e.g. JPEG), video (e.g. Motion JPEG, MPEG, Daala, digital video, Theora) and audio (e.g. MP3, WMA, AC-3, AAC, Vorbis, ATRAC) compression, electrocardiogram data analysis , and for numerical solution of partial differential equations by spectral methods. There are 8-types of DCTs and 8-types of DSTs depending upon the symmetry about a data point and boundary conditions.

The Fourier theory based quadrature method was proposed by Gabor  in 1946 as a practical approach for obtaining the Hilbert Transform (HT), and thus Gabor analytic signal (GAS) representation of a signal. The GAS has been extensively used in communications engineering, physics, time-frequency-energy (TFE) representation, and signal analysis. The TFE representation of a signal is obtained using the concept of instantaneous frequency (IF) [21, 22, 23, 24, 25, 29, 30, 26, 27, 28, 15, 16, 17, 18, 19, 20] which is an important parameter in many applications. Recently, using 8-types of DCTs and 8-types of DTSs, 16-types of quadrature Fourier transforms (QFTs) and corresponding Fourier analytic signal (FSAS) representations are introduced in  for nonlinear and non-stationary time-series analysis. The FQTs and FSAS representations are alternatives to the HT and GAS representation, respectively. The HT and FQTs are phase shifter. However, there is no general method to provide a desired phase shift to signal under analysis. In this study, we propose phase transform (PT), which is based on the proposed generalized Fourier representation (GFR) of a signal, to obtain the desired phase shift and time-delay. We also discuss the various special cases of the GFR namely Fourier representation, PT, time-delay including fractional delay of discrete time signals, time derivative and integral including fractional order, amplitude modulation (AM) and frequency modulation (FM).

The main contributions of this study are summarized as follows:

1. Introduction of the generalized Fourier representation (GFR) which is completely based on the Fourier representation of a signal.

2. Introduction of the phase transform (PT) which is a special case of the GFR, and a true generalization of the Hilbert transform (HT). Using the proposed PT, the desired parse-shift and time-delay can be introduced to a signal under analysis. We derive PT kernel to obtain any constant phase-shift, discuss the various properties of the PT, and demonstrate that the HT is a special case of PT when phase-shift is radian. We also provide an extension of the one-denominational PT for two-dimensional image signals in Appendix A, which can easily be extended for multidimensional signal.

3. Using the PT, we demonstrate that (i) a constant phase shift (e.g. HT as phase shift) in a signal corresponds to variable time-delays in all harmonics, (ii) to obtain a constant time-delay in a signal, one need to provide variable phase shift in all harmonics, (iii) a constant phase shift is same as the constant time-delay only for single frequency sinusoid.

4. The time derivative and time integral, including fractional order, of a signal can be obtained using the GFR. We proposed to use discrete cosine transform (DCT) based implementation to avoid end artifacts due to discontinuities present in both end of the signal.

5. Introduction of the fractional delay of a discrete time signal using the Fourier representations, i.e. DFT, DSTs and DCTs.

6. We present the fast FFT implementation of all the above proposed representations.

This study is organized as follows: A Generalized Fourier Representation using Fourier series and its various special cases are presented in Section 2. PT using Fourier transform is presented in Section 3.1. PT using Fourier sine and cosine transforms is presented in Section 3.2. Implementation of the GFR using DFT and DCT is presented in Section 3.3 and Section 3.4, respectively. Simulation results and discussions are presented in Section 4. Section 5 presents conclusion of the work.

## 2 The Generalized Fourier Representation

In this section we propose the generalized Fourier representation (GFR) and consider its various special cases.

### 2.1 The GFR using Fourier series representation

Let be a real valued periodic signal (i.e., ) which follows the Dirichlet conditions. The Fourier series expansion of is given by

where , and . Using , , where and , one can write

 x\tiny T(t)=a0+∞∑k=1[rkcos(kω0t+ϕk)]. (2)

Using Fourier series representation (2), we hereby propose the GFR as

 x\tiny T(t,ck(t),αk(t))=a0c0(t)cos(α0(t))+∞∑k=1[ck(t)rkcos(kω0t+ϕk−αk(t))], (3)

where and (for ) are introduced as amplitude and phase scaling/modulating functions of frequency () and possibly time (t) as well. Now we consider the various cases of the GFR as follows:

Case 1: The GFR (3) is Fourier series representation of a signal when and , for all and .

Case 2: Using the GFR (3) with and , , we hereby propose the phase transform (PT) as

 x\tiny T(t,αk)=a0cos(α0)+∞∑k=1[rkcos(kω0t+ϕk−αk)], (4)

where is the phase shift in -th harmonics. The Hilbert transform (HT), , a special case of the PT (4) where it has constant phase shift of (i.e. , for all ), can be defined as

 ^x\tiny T(t)=x\tiny T(t,π/2)=∞∑k=1[rksin(kω0t+ϕk)]. (5)

Case 3: The time-delay of a signal is defined as

 x\tiny T(t−tk)=a0+∞∑k=1[rkcos(kω0t+ϕk−kω0tk)]. (6)

From (4), (5) and (6), we observe that (a) HT is a constant phase shift which introduces variable time-delays in all harmonics, i.e., ; (b) to obtain a constant time-delay (say, ) in a signal, we need to provide variable phase shift in all harmonics, i.e., ; (c) constant phase shift is same as constant time-delay only for single frequency sinusoid (say and hence ); (d) variable phase shift is same as variable time-delay only for a zero mean () signal if, , .

It is interesting to observe that we presented the phase shift of a constant signal in (4) as, , which is valid because (i) for the HT, it is zero, (ii) for phase shift of , it is multiplied by minus one, and (iii) there is no change in its value if phase shift is zero. However, time-delay operation in a constant signal [e.g., in (6)] does not change its value.

Case 4: Using (2), one can obtain -th order fractional time derivative of a signal as

 Dμ{x\tiny T(t)}=a0t−μΓ(1−μ)+∞∑k=1(kω0)μ[rkcos(kω0t+ϕk+μπ/2)],μ≥0, (7)

where is gamma function. The GFR (3) is -th order time derivative of the signal when , , and , for all and .

Case 5: Using (2), one can obtain -th order fractional time integral of a signal as

 D−ν{x\tiny T(t)}=a0tνΓ(1+ν)+∞∑k=1(kω0)−ν[rkcos(kω0t+ϕk−νπ/2)],ν≥0. (8)

The GFR (3) is -th order time integral of the signal when , , and , for all and .

Case 6: Using (2), one can obtain amplitude modulated (AM) signal for one arbitrary but fixed value of (say ) and thus carrier frequency , , , , where is a message signal whose maximum frequency .

Case 7: Using (2), one can obtain angle modulated, frequency modulated (FM) and phase modulated (PM), signal for one arbitrary but fixed value of (say ) and thus carrier frequency , , , , where is a message signal whose maximum frequency .

## 3 Phase Transforms using the FT, FCT and FST

In this section, to obtain a desired phase shift in a signal, we present PT using Fourier transform (FT), Fourier cosine transform (FCT), and Fourier sine transform (FST), along with FFT implementation of various cases of the proposed GFR.

### 3.1 PT using Fourier Transform

The Fourier transform (FT) and inverse FT (IFT) pairs of a signal are defined as

 X(ω) =∫∞−∞x(t)exp(−jωt)dt,−∞<ω<∞, (9) x(t) =12π∫∞−∞X(ω)exp(jωt)dω,−∞

subject to the existence of the integrals, and these pairs can be denoted by , where . From the definitions of FT and IFT (9), one can observe that , and , provided that and are zero-mean functions, respectively. Moreover, an odd function is a zero-mean function, however, reverse is not always true.

First, we consider subtle details of the zero-mean function , where and sign function is defined as :

 sgn(t)=⎧⎨⎩1,t>0,0,t=0,−1t<0. (10)

The FT of is evaluated as : , which implies that at . It is pertinent to notice that, , , and which implies does not exist, and therefore is not defined, however, the FT of a zero-mean function constrains that . Thus, we write the FT of the zero-mean sign function, , as

 lima→0X1(f)=S(f)={0,f=0,1jπf,f≠0. (11)

Using the duality principle of the FT, i.e. , one can obtain, , as sign function is an odd function. We denote , and thus obtain

 H(f)=−jsgn(f), and h(t)={0,t=0,1πt,t≠0. (12)

Now, we compute from using IFT (9) as and obtain

 h(t)=1π∫∞0sin(ωt)dω, (13)

and, clearly, , which is also required by the definition (12). The function is the well-known HT kernel which is not defined at origin, moreover, , and , therefore does not exist. Thus, we have presented a trivial but important modification to HT kernel in (12) by defining it at origin, and obtained its integral form in (13). These definitions of the HT kernel is further supported by the discrete time HT kernel , defined as , which is also zero at origin (i.e., ), and it can be obtained by using the discrete counter part of (13), i.e., .

For a real-valued signal , we can write Gabor analytic signal (GAS) as

 z(t)=x(t)+j^x(t)=1π∫∞0X(ω)exp(jωt)dω,−∞

where real part is the original signal and imaginary part is the HT of real part. Thus, we write

 x(t) =1π∫∞0|X(ω)|cos(ωt+ϕ(ω))dω,−∞

where , , and the GFR corresponding to (3) as

 x(t,c(ω,t),α(ω,t))=1π∫∞0c(ω,t)|X(ω)|cos(ωt+ϕ(ω)−α(ω,t))dω. (16)

We hereby define the PT of signal using the IFT as

 x(t,α(ω))=1π∫∞0|X(ω)|cos(ωt+ϕ(ω)−α(ω))dω,−∞

where is the phase shift introduced in the signal . The PT, defined in (17), is the real part of the PT of analytic signal hereby defined as

 z(t,α(ω))=1π∫∞0X(ω)exp(j(ωt−α(ω)))dω, (18)

and therefore obtain transfer function (TF) as

 H(α(ω))={e−jα(ω),ω≥0,ejα(ω),ω<0. (19)

For example, to obtain a constant time-delayed signal from input signal , we set , and therefore from (17) and (9), .

Thus, we have defined a general phase shifter of a signal in (17) which is a generalization of the IFT as well as HT because it is (i) IFT if , (ii) HT if , and (iii) when , we obtain its impulse response, designated as constant phase-transform (PT) kernel, from (17) as

 h(t,α)=cos(α)δ(t)+sin(α)h(t), (20)

where is the Dirac delta function and is the impulse response or HT kernel as defined in (12) or (13). The direct derivation of the PT kernel defined in (20) from (17) is elementary. Here, we present indirect proof of that as follows: one can easily show [using (9) or (14) and setting and ] that, , and . Using these facts, from (17) we obtain, , and thus (20).

The arbitrary but fixed phase shifter defined in (20) is a linear time-invariant (LTI) system model, thus its output can be written as convolution of input with impulse response, i.e.,

 x(t,α)=Hα{x(t)}=x(t)∗h(t,α)=cos(α)x(t)+sin(α)^x(t), (21)

or , where and as defined in (15). Clearly, the PT of a time-domain signal is another time-domain and phase shifted signal . There are some obvious properties of the PT (21) which follow directly from the definition such as one can easily show , where and are identity and HT operators, respectively, i.e., and ; inverse PT (or ); , which implies , , and if . Now, we explore some basic properties of the proposed PT (21) as follows:
(1) Linearity: The PT is a linear operator, i.e., for arbitrary scalars and , functions and .
(2) The PT of constant signal: For any constant , .
(3) Time-shifting and time-dilation: If has PT , then has PT , and has PT where .
(4) Relation with the Fourier Transform: The Fourier transform of PT kernel is or

 H(f,α)=⎧⎨⎩e−jα,f>0,cos(α),f=0,ejα,f<0, (22)

thus PT provides and phase shifts to positive and negative frequencies, respectively, and when, , it becomes HT. If has Fourier transform , then, . It is to be noted that a generalized Hilbert transform to obtain a phase shift to any angle is defined in , where author defined for , and for , is defined same as (22).
(5) Orthogonality: If is a real-valued energy signal (i.e., ), then inner product of and is given by

 ⟨x(t),x(t,α)⟩=cos(α)⟨x(t),x(t)⟩+sin(α)⟨x(t),^x(t)⟩⟹cos(α)=⟨x(t),x(t,α)⟩⟨x(t),x(t)⟩, (23)

as , and thus they are orthogonal only for phase shift .
(6) Energy: If is a real-valued energy signal, then is also a real-valued energy signal and its energy () is computed by inner product of with itself as

 Eα=⟨x(t,α),x(t,α)⟩=cos2(α)⟨x(t),x(t)⟩+sin2(α)⟨^x(t),^x(t)⟩, (24)

and for zero mean signal, energy is preserved in HT, i.e., , so energy is preserved in the proposed PT.
(7) Time-derivative: The PT of the derivative of a signal is the derivative of the PT, i.e.,
.
(8) PT of product of low-pass and high-pass signal: Let be a low-pass signal such that its FT for and let be a high-pass signal with for . Then, PT . One can show it easily using the property of HT as . Thus, to obtain the PT of product of a low-pass signal and a high-pass signal, only the high-pass signal needs to be phase shifted.
(9) PT of analytic signal: From (18) or (21), we obtain which gives . Thus, we can compute by considering the real part of the .

### 3.2 PT using Fourier sine and cosine transforms

The Fourier cosine transform (FCT) and inverse FCT (IFCT) pairs, of a signal, are defined as

 Xc(ω) (25) x(t) =√2π∫∞0Xc(ω)cos(ωt)dω,t≥0,

subject to the existence of the integrals, i.e., is absolutely integrable () and its derivative is piece-wise continuous in each bounded subinterval of .

The Fourier cosine quadrature transform (FCQT), , using the FCT of signal of is defined in  as

 ~xc(t)=√2π∫∞0Xc(ω)sin(ωt)dω, (26)
 ~Xc(ω)=√2π∫∞0~xc(t)sin(ωt)dt, (27)

where

 ~Xc(ω)={0,ω=0,Xc(ω),ω>0. (28)

The FSAS, using the FCQT, is defined in  as

 ~zc(t)=x(t)+j~xc(t)=√2π∫∞0Xc(ω)exp(jωt)dω, (29)

where real part is the original signal and imaginary part is the FQT of real part.

We hereby define the PT of signal using the FCT as

 x(t,α(ω))=√2π∫∞0Xc(ω)cos(ωt−α(ω))dω, (30)

where is the frequency dependent phase shift. If phase shift is constant or independent of frequency, i.e. , then we obtain

 x(t,α)=√2π∫∞0Xc(ω)cos(ωt−α)dω=cos(α)x(t)+sin(α)~xc(t), (31)

where is the phase shift.

Thus, we have defined a general phase shifter of a signal which is a generalization of the IFCT as well as FCQT because it is (i) IFCT if , and (ii) FCQT if .

The Fourier sine transform (FST) and inverse FST (IFST) pairs, of a signal, are defined as

 Xs(ω) (32) x(t) =√2π∫∞0Xs(ω)sin(ωt)dω,

subject to the existence of the integrals. The Fourier sine quadrature transform (FSQT), , using the FST of signal of is defined in  as

 ~xs(t)=√2π∫∞0Xs(ω)cos(ωt)dω, (33)
 ~Xs(ω)=√2π∫∞0~xs(t)cos(ωt)dt, (34)

where one can observe that both representations, defined as FST of in (32) and FCT of in (34), are same for all frequencies, i.e. . The FSAS, using the FSQT, is defined in  as

 ~zs(t)=~xs(t)+jx(t)=√2π∫∞0Xs(ω)exp(jωt)dω, (35)

where imaginary part is the original signal and real part is the FQT of imaginary part.

We hereby define the PT of signal using the FST as

 x(t,α(ω))=√2π∫∞0Xs(ω)sin(ωt−α(ω))dω. (36)

If , then we obtain

 x(t,α)=√2π∫∞0Xs(ω)sin(ωt−α)dω=cos(α)x(t)−sin(α)~xs(t), (37)

where is the phase shift. Thus, we have defined a general phase shifter of a signal which is a generalization of the IFST as well as FSQT because it is (i) IFST if , and (ii) FSQT if .

The FQTs, presented in (26) and (33), are different from the HT (15) by definition itself. The proposed FSTP representations, defined in (30) and (36), are effective phase shifter which can be used in various applications such as envelop detection, IF estimation, time-frequency-energy representation and analysis of nonlinear and nonstationary data.

### 3.3 Implementation of the GFR using DFT

The DFT and inverse DFT (IDFT) of a signal of length are defined as

 X[k] =1NN−1∑n=0x[n]exp(−j2πkn/N),0≤k≤N−1, (38) x[n] =N−1∑k=0X[k]exp(j2πkn/N),0≤k≤N−1.

The DFT and IDFT are computed efficiently using the fast Fourier transform (FFT) algorithm. Unless otherwise mentioned, we consider as a real valued signal. We obtain the discrete-time PT kernel corresponding to continuous-time kernel (20) as

 h[n,α]=cos(α)δ[n]+sin(α)h[n]. (39)

where is the discrete-time HT kernel. Thus we can obtain a constant phase shift in a signal as , where and HT . The PT defined in (4) and (17) can be computed using FFT by considering real part of analytic signal

 z[n,αk]=x[n,αk]+jx[n,αk+π/2]=IFFT{X[k]H[k]}, (40)

i.e., , where is defined (if is even) as

 H[k]=⎧⎨⎩cos(αk),k=0,N/2,2exp(−jαk),1≤k≤N/2−1,0,(N/2+1)≤k≤N−1, (41)

or (if is odd)

 H[k]=⎧⎨⎩cos(αk),k=0,2exp(−jαk),1≤k≤(N−1)/2,0,((N−1)/2+1)≤k≤N−1. (42)

For a constant phase shift, imaginary part is the HT of real part in (40), and if , real part is the original signal . In (41) and (42), we can remove the multiplication factor of 2 and define for positive frequencies (e.g., ) and its complex conjugate for negative frequencies (e.g., ).

Observation: The phase shift of a sinusoidal signal does not change its amplitude (or energy), except for lowest (i.e. DC) and highest frequency components. For example, if , then , and for , it becomes zero which cannot net be recovered by further phase shift. So, to overcome this issue, we can define phase shift of a constant signal and highest frequency component as , which preserves the energy in these cases as well, and in the case of phase shift, complete signal is getting transfered to imaginary part, and we can recover original signal by further phase shift of (), and . This is also consistent with complex plane representation, where multiplication of with a complex number () introduces phase shift (i.e. ) without any change in amplitude. Therefore, we can use for in (41) and (42).

We observe that a simply delayed signal obtained using IDFT (38) as

 x[n−n0]=X+N−1∑k=1X[k]exp(j2πk(n−n0)/N), (43)

is valid only for an integer value of , because complex conjugate of is only for some integer value of , and it is not valid for fractional value of delay. Therefore, in order to obtain both integer and fractional delay in a signal , corresponding to (6) which can be computed by (40), we define as

 H[k]=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩1,k=0,2exp(−j2πknk/N),1≤k≤N/2−1,exp(−jπnk),k=N/2,0,(N/2+1)≤k≤N−1, (44)

where is even (similarly it can also we defined when is odd) and for some constant delay one can set .

The -th order derivative approximation of a signal , denoted as , corresponding to (7), can be estimated by

 xμ[n]=a0(n/Fs)−μΓ(1−μ)+Re[IFFT{X[k]H[k]}] (45)

using which we define as

 H[k]=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩0,k=0,2(2πk/N)μexp(jμπ/2),1≤k≤N/2−1,(π)μexp(jμπ/2),k=N/2,0,(N/2+1)≤k≤N−1, (46)

where mean-value , , and is even (similarly it can also we defined when is odd); and when then it is -th order integral of a signal as defined in (8).

### 3.4 Implementation of GFR using DCT

The DCT-2 of a sequence, of length , is defines as 

 Xc2[k]=√2NσkN−1∑n=0x[n]cos(πk(2n+1)2N),0≤k≤N−1, (47)

and inverse DCT (IDCT) is obtained by

 x[n]=√2NN−1∑k=0σkXc2[k]cos(πk(2n+1)2N),0≤n≤N−1, (48)

where normalization factors for , and for . If consecutive samples of a sequence are correlated, then DCT concentrates energy in a few and decorrelates them. The DCT basis sequences, , which are a class of discrete Chebyshev polynomials , form an orthogonal set as inner product for .

The discrete Fourier cosine quadrature transform (FCQT), , of a signal is defined as 

 ~xc2[n]=√2NN−1∑k=0Xc2[k]sin(πk(2n+1)2N),0≤n≤N−1, (49)

where is the DCT-2 of a signal . We define the PT using the DCT-2 as

 x[n,α(k)]=√2N