Aliased polyphase sampling theorem for the offset linear canonical transform

Optik - International Journal for Light and Electron Optics 200 (2020) 163410

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Original research article

Aliased polyphase sampling theorem for the offset linear canonical transform ⁎

Shuiqing Xua,b, Zhiwei Chenb, , Ke Zhangc, Yigang Hea, a b c

T

⁎

College of Electrical Engineering and Automation, Wuhan University, Wuhan 430072, China College of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China College of Automation, Chongqing University, Chongqing 400044, China

A R T IC LE I N F O

ABS TRA CT

Keywords: Offset Linear canonical transform Aliased polyphase sampling Sampling rate

Sampling theorems for the offset linear canonical transform (OLCT) have been widely studied in the literature. However, the sampling frequency of these theorems all have same condition restrictions, which the sampling frequency in the OLCT domain should be twice greater that the maximum OLCT frequency of the signal. In this paper, the aliased polyphase sampling theorem in the OLCT domain has been well derived, which can use lower sampling rate to obtain the higher sampling rate to eliminate restrictions. First, the model of the aliased polyphase sampling has been briefly introduced. Subsequently, the OLCT spectrum analysis of the aliased polyphase sampled signals has been proposed based on this model. In addition, the aliased polyphase sampling theorem associated with the OLCT also has been derived. Lastly, the simulation results have been provided to show the useful and effective of the derived theorems.

1. Introduction The offset linear canonical transform(OLCT) is an integral transform with six parameters. In some literature, it is also called as the inhomogeneous canonical transform [1] or the special affine Fourier transformation [2]. Many well-known transforms in optics and signal processing, for instance the Fourier transform(FT), the fractional Fourier transform(FRFT), the linear canonical transform (LCT), the Fresnel transform and many others are all the special cases of the OLCT [3–8]. Owing to it has much more extra free parameters, the OLCT is more suitable for processing the nonstationary signal compared to other transforms. For example, many signals are nonbandlimited in the FT, FRFT and LCT domain, whereas can be bandlimited in the OLCT domain. Hence, the OLCT is an effective tool and has much widely applications in optics, communications, signal processing and many others [9–16]. The OLCT with parameters A = (a, b, c, d, u 0 , w0) of a signal f(t) is represented as [1]

⎧ FA (u) =

OLA [f

(t )](u) =

∞

∫−∞ f (t ) KA (t, u)dt,

jcd(u − u0) ⎨ 2 ⎩ de

b≠0

2

+ jw0 u

f [d (u − u 0 )], b = 0

where

⁎

Corresponding authors at. E-mail addresses: [email protected] (Z. Chen), [email protected] (Y. He).

https://doi.org/10.1016/j.ijleo.2019.163410 Received 11 June 2019; Accepted 11 September 2019 0030-4026/ © 2019 Published by Elsevier GmbH.

(1)

Optik - International Journal for Light and Electron Optics 200 (2020) 163410

S. Xu, et al.

2 2 d 2 j 1 e j 2b u0 e 2b [at + 2t (u0 − u) − 2u (du0 − bw0) + du ] j2πb

KA (t , u) =

(2)

and the parameter a, b, c, d, u 0 , w0 are real numbers satisfied ad − bc = 1. In addition, it is easily to know that if b = 0, the OLCT of a signal corresponding to a chirp multiplication. So, in the following sections, we only focus on the case of b ≠ 0 and b > 0 in order to without loss of generality. The inverse OLCT is obtained by a OLCT with the parameters A−1 = (d, − b, − c, a, bw0 − du0, cu0 − aw0), that is

f (t ) = C

∞

∫−∞ OLA [f (t )](u) K A

−1 (u ,

t ) du

(3)

j 2 2 e 2 (cdu0 − 2adu0 w0+ abw 0) .

where C = As the OLCT has proven to be an effective and important tool in optics and signal processing, the properties and theorems of the OLCT has been well concerned and published in the literature. Such as the eigenfunctions of the OLCT [17], the uncertainty principle of the OLCT [18], the convolution and correlation theorems for the OLCT [19–21], the Wigner-Ville distribution associated with the OLCT [22], the Hilbert transform in the OLCT domain [23] and many kinds of sampling theorems for the OLCT [24–30]. Among all the properties and theorems, the sampling theorem is one of the most important theorems in optics and signal processing. That is due to the sampling theorem acts as a bridge between the continuous physical signals and the discrete time domain. However, all the derived theorems associated with the OLCT presented in the literature are satisfied the condition that the sampling frequency in the OLCT domain should be twice greater that the maximum OLCT frequency of the signal. If the sampling rate does not meet this condition, the aliasing will happen. On the other hand, with the fast development of digital signal processing, we often need the sampling rate as low as possible in practical engineering. Unfortunately, to the best of our knowledge, the sampling theorem in the OLCT domain with low sampling rate has never been derived before. Therefore, it is interesting and useful to study the new sampling theory associated with the OLCT. Simultaneously, the aliased polyphase sampling is an effective sampling method, which use L parallel sampler with a sampling rate of Ws to sample, then by combining these sampling signals, the sampling rate of LWs can be obtained [31,32]. In this way, it can use lower sampling rate to obtain the higher sampling rate to meet the sampling condition. Therefore, in this paper, the aliased polyphase sampling theorem for the OLCT is obtained for the first time to resolve this issue. Firstly, the structure of the aliased ployphase sampling model is briefly introduced. Subsequently, based on this model, the OLCT spectrum analysis of the aliased polyphase sampled signals is presented. In addition, the reconstruction for the signal from its aliased polyphase samples is also provided. Finally, the simulation results are presented to show the effective and useful of the derived theorems. The rest of this paper is structured as follows. The uniform sampling theorem for the OLCT and the model of the aliased polyphase sampling have been introduced in section 2. In section 3, the relationship between the OLCT spectrum of the original signal and the OLCT spectrum of the aliased polyphase samples has been presented. In section 4, the reconstruction of the aliased polyphase sampled signals in the OLCT domain has been derived. In section 5, the simulation results has been provided to show the theorems are effective and correct. Section 6 concludes this paper. 2. Preliminaries 2.1. The uniform sampling for the OLCT A signal f(t) with a uniform sampling period T can be expressed as: k

fs (t ) = f (t )

∑

δ (t − kT) (4)

k =−∞

Then based on equation (4), we can obtain the OLCT spectrum of the uniformly sampled signal fs(t) as [26]: k

OLA [fs (t )](u) = OLA [f (t )

∑

δ (t − kT)](u)

k =−∞ ∞

=

1 j d u2 2π kb e 2b ∑ OLA [f (t )](u − T ) T k =−∞

× e j [−

2π kb (du − bw )/ b] 2 0 0 T e−j [d (u − 2π kb/ T ) /2b]

(5)

The detailed proof of the equation (5) can be seen in [26]. Meanwhile, by using the results in equation (5), the uniform sampling theorem for bandlimted signals in the OLCT domain can be obtained as follows [24,26]. at2

f (t ) = e−j 2b

∞

∑

f (kT) e j

u (t − kT) a (kT)2 −j 0 2b e b sinc[UA (t

− kT)/ b] (6)

k =−∞

where f(kT) is the uniformly sampled signal and T is the sampling period satisfied T = πb/UA. 2

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S. Xu, et al.

2.2. The structure of the aliased polyphase sampling A typical sampling process of the continuous signal f(t) in the time domain can be expressed by the following equation

f (k ) = f (t ) δk (t )

(7)

If we write the dirac function as an impulse function, thus the equation (7) can be represented as ∞

∑

f (k ) =

f (kTs) δ (t − kTs) (8)

k =−∞

Meanwhile, if we inserting L − 1 impulses between any adjacent impulse pair of the dirac function, then another impulse train can be obtained as follows, i.e, ∞

L

∑ ∑ δ (t − kTs − (l − 1) Ts/L)

δk, l (t ) =

(9)

k =−∞ l = 1

From the equation (9), we can know that the period of δk,l(t) is Ts/L. Thus, based on the above analysis, the sampling process with the sampling frequency LWs can be further written as [31]

f (kL + l) = f (t ) δk, l (t ) ∞

L

∑ ∑

=

f (t ) δ (t − kTs − (l − 1) Ts / L)

l = 1 k =−∞ L

∑ gk (l)

=

l=1

(10)

where Ws = 1/Ts and ∞

gk (l) =

∑

f (kTs + (l − 1) Ts / L) (11)

k =−∞

Equation (10) indicates that a sampling process with the sampling rate LWs can be implemented by L parallel samplers with the sampling rate Ws. The model of the aliased polyphase sampling is depicted in Fig. 1 [31]. In this Fig. 1, the up-sampler is necessary in each sampling branch, thus the sampling result is the addition of the outputs from all branches. Based on the model of the aliased polyphase sampling presented in Fig. 1, the aliased polyphase sampling theorem in the FT domain can be obtained as follows [31] If the signal f(t) is bandlimited in the FT domain, then the signal f(t) can be reconstructed from its aliased polyphase sampled signals by using the following equation: L

f (t ) =

∞

∑ ∑ l = 1 k =−∞

f (kTs +

l−1 1 l−1 Ts )sinc[ (t − kTs − Ts )] L Ts L

(12)

where Ts denotes as the sampling period in this sampling model. 3. The spectrum analysis of the aliased polyphase sampled signals The structure of the aliased polyphase sampling has been well elaborated in subsection 2.1. Based on this model, the detailed

Fig. 1. The model of the aliased polyphase sampling 3

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S. Xu, et al.

spectrum analysis of the aliased polyphase sampled signals will be considered in this section. First, if f(t) be a analog signal, OLA [f (t )](u) is the OLCT of the signal f(t) with parameters A, gk(l), l = 0, ⋯, L denote the aliased polyphase sampled signals obtained from the Fig. 1. Then, the relationship between the spectrum of the aliased polyphase sampled signals and the spectrum of the analog signal in the OLCT domain can be obtained in the following theorem. Theorem 1. Suppose gk(l) is the aliased polyphase sampled signals of the continuous signal f(t), then the spectrum of these aliased polyphase samples associated with the OLCT can be expressed as L

(13)

∑l = 1 OLA [gk (l)](u) =

j 2 L e 2b [−2u (du0 − bw0) + du ] Ws 2πb j

∞

∑p =−∞ OLA [f (t )](u − bpLW) s

2 j

where L is the numbers of samplers and Ws be the

× e− 2b d (u − bpLWs) e b (du0 − bw0)(u − bpLWs) sampling frequency of these samples. Proof. According to the model shown in Fig. 1, it can easily get that (14)∑lL= 1 gk (l) = f (t ) δk, l (t ) Then, by applying the OLCT for both sides of the equation (14), it can have that L

OLA [ ∑ gk (l)](u) = OLA [f (t ) δk, l (t )](u)

(15)

l=1

Based on the convolution theorem in the OLCT domain [19], we can rewrite the equation (15) as L

j j 2 2 1 u e 2b [−2u (du0 − bw0) + du ] ⎛OLA [f (t )](u) e− 2b [−2u (du0 − bw0) + du ]*Δ( )⎞ 2πb b ⎠ ⎝

∑ OLA [gk (l)](u) = l=1

(16)

u Δ( b )

is the FT of the function δk,l(t). where On the other hand, by using the results in [32], we can further write the function Δ(x) as ∞

Δ(x ) = Ws

L

∑ ∑ e−j

2πn (l − 1) δ (w L

− nW) s (17)

n =−∞ l = 1

Now, by combining (16) and (17), we can get L

∑ OLA [gk (l)](u)

=

l=1

j j 2 2 1 e 2b [−2u (du0 − bw0) + du ] OLA [f (t )](u) e− 2b [−2u (du0 − bw0) + du ] 2πb ∞

× Ws

L

∑ ∑ e−j

2πn (l − 1) u L δ(

n =−∞ l = 1

If we let

u b

b

− nW) s (18)

= v in the equation (18), thus we can reorganize the equation (18) as

L

∑ OLA [gk (l)](bv)

=

l=1

×

1 j bdv 2 −jv(du0 − bw0) e2 e Ws 2πb

∞

L

∑ ∑ OLA [f (t )](b (v − nW)) s

n =−∞ l = 1 j 2πk (l − 1) − bd(v − nWs)2 j (v − nWs)(du0 − bw0) −j e 2 e e L

(19)

Then, rearranged the equation (19), we can have L

∑ OLA [gk (l)](bv)

=

l=1

×

1 j bdv 2 −jv(du0 − bw0) e2 e Ws 2πb

∞

∑

OLA [f (t )](b (v − nW)) s

n =−∞ L j 2 e− 2 bd(v − nWs) e j (v − nWs)(du0 − bw0)

∑ e−j

2πk (l − 1) L

l=1

(20)

From equation (20), it is easy to get that if n ≠ pL, where p is a positive integer, we can have that L

∑ e−j

2πk (l − 1) L

=0 (21)

l=1

If n = pL, then we can have L

∑ e−j

2πk (l − 1) L

=M (22)

l=1

Now, substituting the equation (21) and (22) into the equation (20), we can obtain that 4

Optik - International Journal for Light and Electron Optics 200 (2020) 163410

S. Xu, et al. L

∑ OLA [gk (l)](bv)

=

l=1

L j bdv 2 −jv(du0 − bw0) e2 e Ws 2πb j

∞

∑

OLA [f (t )](b (v − pLW)) s

p =−∞

2

× e− 2 bd(v − pLWs) e j (v − pLWs)(du0 − bw0)

(23)

Finally, let bv = u in the equation (23), we can rewrite the equation (23) as L

∑ OLA [gk (l)](u)

=

l=1

L j [−2u (du0 − bw0) + du2] e 2b Ws 2πb j

2

∞

∑

OLA [f (t )](u − bpLW)) s

p =−∞

j

× e− 2b d (u − bpLWs) e b (du0 − bw0)(u − bpLWs)

(24)

Thus, the proof of theorem 1 is completed. From the results in theorem 1, we can easily get that by using the phase factors, the aliased spectrum shown in Fig. 2(b) will be eliminated and the spectrum without any aliasing can be obtained in Fig. 2(c). In addition, theorem 1 also indicates that the spectrum of the sampled sequence f(kL + l) can be acquired by periodic replication the spectrum of f(t), which is scaled by a factor of bLWs in magnitude and along with some linear phase modulation. Meanwhile, due to the OLCT is a generalization of the FT, FRFT and LCT, thus by substituting the special parameters into the theorem 1, the spectrum analysis of the aliased polyphase sampled signals associated with the FT, FRFT and LCT also can be obtained. 4. The reconstruction of the aliased polyphase sampled signals in the OLCT domain In section 3, the spectrum of the aliased polyphase sampled signals in the OLCT domain has been well studied. Based on the theorem 1, the reconstruction of the aliased polyphase sampled signals associated with the OLCT is derived in the following. Theorem 2. If the signal f(t) is bandlimited signal with the bandwidth UA, and f(kL + l) be its aliased polyphase sampled signals, then the signal f(t) can be reconstructed by its aliased polyphase sampled signals in the following equation: a 2

(25)

∞

L

f (t ) = e−j 2b t ∑l = 1 ∑k =−∞ f (kTs + U

× sinc[ πb (t − kTs −

u0 (t − kTs + (l − 1) Ts / L) 2 a l−1 l−1 b Ts ) e j 2b (kTs − L Ts ) e−j L

l−1 Ts )] L

where Ts is the sampling period and Ts = πb/UA.

Proof. First, we let (26) g (t ) =

d

2

U ∫−UAA OLA [f (t )](u) e−j 2b u e jut du

According to the analysis in [29], it can know that the signal g(t) is bandlimited to UA/b in the FT domain. Thus, the signal f(t) can be further expressed as

f (t ) =

u0 a 2 1 e−j 2b t e−j b t g (t ) −j2πb

(27)

Now, by using the aliased polyphase sampling theorem for g(t) in the FT domain, we substituting the equation (12) into the equation (27) and rearranged it, one can have a 2

f (t ) = e−j 2b t

L

∞

∑ ∑ l = 1 k =−∞

× sinc[

f (kTs +

u0 (t − kTs + (l − 1) Ts / L) a l−1 2 l−1 b Ts ) e j 2b (kTs − L Ts ) e−j L

U l−1 (t − kTs − Ts )] πb L

(28)

Hence, the proof of theorem 2 is completed. According to theorem 2, it is shown that the continuous signal in the OLCT domain can be well reconstructed by its aliased polyphase samples. Particularly, when selecting special parameters into the theorem 2, the aliased polyphase sampling theorem for the FT, FRFT and LCT also can be derived. Furthermore, since the OLCT is a more effective tool for optics and signal processing, so the aliased polyphase sampling for the OLCT is more general and has much more widely applications. 5. Simulation results The spectrum analysis of the aliased polyphase sampled signals and the reconstruction of the aliased polyphase sampled signals in the OLCT domain have been well studied in previous sections. In the section, to prove the validity of the derived theorems, the simulation of the aliased polyphase sampling for bandlimited signals in the OLCT domain will be preformed in this section. In this simulations, the signal f(t) shown in Fig. 3(a) is considered. The OLCT of the signal f(t) with parameter (1, 0.5, 0, 1, 0, 1) is obtained in Fig. 3(b). From Fig. 3(a) and (b), we can obviously know that the signal f(t) is bandlimited in the OLCT domain. The real and imaginary parts of the signal f(t) are shown in Fig. 4(a) and (b), respectively. Meanwhile, in this simulations, the number of the parallel samplers is chosen as L = 5. Then, by applying the aliased polyphase sampling theorem derived in section 4, we can obtain the real and imaginary parts of the reconstructed signal in Fig. 5(a) and (b), 5

Optik - International Journal for Light and Electron Optics 200 (2020) 163410

S. Xu, et al.

Fig. 2. Spectrum analysis of the aliased polyphase sampling: (a) Spectrum in the OLCT domain, (b) Spectrum of gk(l) in the OLCT domain, (c) Final spectrum of the aliased polyphase sampling in the OLCT domain

respectively. In Fig. 5(a) and (b), the red dotted line represents the reconstructed signal, whereas the blue solid line denotes the original signal. It can clearly see that the red dotted line overlaps almost exactly the blue solid line. Therefore, the theorems derived in this paper has been proved to be accurate and effective. 6. Conclusion This paper has analyzed the aliased polyphase sampling theorem in the OLCT domain, which can use lower sampling rate to obtain the higher sampling rate to meet the sampling condition. First, we have briefly introduced the structure of the aliased polyphase sampling model. Then, the spectrum analysis of the aliased polyphase sampled signals in the OLCT domain has been presented. Furthermore, the reconstruction theorem for the signal from the aliased polyphase samples associated with the OLCT also has been derived. Finally, the simulation results have been proposed to show the correctness and effective of the derived theorems. 6

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S. Xu, et al.

Fig. 3. (a) The original signal f(t), (b) The OLCT of the signal f(t).

Fig. 4. (a) The real part of the signal f(t), (b) The imaginary part of the signal f(t).

Fig. 5. (a) The reconstruction of the real part, (b) The reconstruction of the imaginary part.

The future research and work will be in the direction of real applications of these results in nonstationary signals sampling.

Acknowledgment This work was supported by the National Natural Science Foundation of China (61803140, 51807044,51637004,51577046, 51977153), the Opening foundation of the state key laboratory of traction power (TPL1908), the Fundamental Research Funds for the Central Universities (JZ2019HGTB0090, JZ2019HGTB0073), the National Key Research and Development Plan: Important Scientific Instruments and Equipment Development (2016YFF0102200) and Equipment research project in advance (41402040301). 7

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