Interpolation of Discrete Chirp-periodic Signals Based on Fractional Fourier Transform Bing-zhao Li1,2, Ran Tao1 , and Yue Wang1 1. Dept. of Electronic Engineering, Beijing Institute of Technology, Beijing 100081 2. Dept. of Mathematics, Beijing Institute of Technology, Beijing 100081, China E-mail:
[email protected]
Abstract The sampling theorem associated with the fractional Fourier transform can be looked as the convolution of the sinc kernel with infinite sequence of signal points and chirp signal modulations. But in most practical applications we only have finite number of samples, which makes a perfect reconstruction of the original signal impossible. To solve this problem, we obtain a new formula for perfect reconstruction of discrete chirp-periodic signal points based on the fractional Fourier transform in this paper. The method is equivalent to trigonometrically interpolation by fractional Fourier series expansion and can be looked as a generalization of the classical results.
1. Introduction As a generalization of classical Fourier transform, the fractional Fourier transform (FrFT) has received many interests in recent years. It has many applications in several areas, including optics, quantum physics and signal processing society [1]-[6], and it can also be viewed as a rotation of the traditional Fourier transforms in time-frequency plane [7]. The relationship between the Fourier transform and the FrFT is deduced in [8]. The discrete fractional Fourier transform definition and the fast computation methods of FrFT are also proposed by different researchers from different view points. For more properties and applications of FrFT in optics and signal processing community, one can refer to [9], [10]. At the same time, the sampling problem is one of the fundamental works on converting the continuous signals to the discrete signals in signal processing communities; the classical sampling theorem expansion for the fractional Fourier transform domain of band-limited, time-limited and band-pass signals have been derived in [8], [12]-[13], these sampling theories establish that a band-limited or band-pass continuous signal in fractional Fourier domain can be
completely reconstructed by a set of equidistantly spaced signal samples. A commonly used method for reconstructing the original signal is the convolution of sinc function with the product of a chirp signal and sampling points first, and then by multiplying another chirp signal to obtain the original signal. However, numerical implementation of these reconstruction methods on a digital computer is not possible, i.e., there is an infinite number of sampling points. Moreover, in most practical applications we are given only a finite number of samples, which makes a perfect reconstruction of the signal impossible. Although there are some alternative methods proposed to solve these problems [13], but their methods are only suitable for band-limited or time-limited signals in traditional Fourier domain. To overcome these limitations, we derive a new representation of the sampling theorem based on the existing sampling theorems and the properties associated with the fractional Fourier transform. The rest of the paper is organized as follows. Section 2 provides a brief review of fractional Fourier transform and the sampling theorem in fractional Fourier transform domain. In section 3, a new interpolation method is introduced in detail for chirpperiodic signals in fractional Fourier domain. An experimental result of the proposed method for a real signal is described in section 4. Finally, in section 5, the conclusion and the future research topics along this way are given.
2. Preliminary 2.1. FrFT The FrFT with angle by the integral [9], [10]
α of a signal f (t ) ∞
X α (u ) = Fα [ x(t ) ] (u ) = ³ Kα (u, t ) x(t )dt −∞
Where
Proceedings of the First International Conference on Innovative Computing, Information and Control (ICICIC'06) 0-7695-2616-0/06 $20.00 © 2006
is defined (1)
2
2
t u j cot α − jut cscα + j cot α 2 , α ≠ kπ ° Aα e 2 ° Kα (u, t ) = ®δ (t − u ), α = 2k π °δ (t + u ), α = (2k + 1) π ° ¯
Aα =
1 − j cot α and k is an integer. 2π
For more details about the properties and applications of FrFT, one can refer to [4]-[6],[9], [10]. From the properties of FrFT, we know that the non band-limited signal in traditional Fourier domain may be a band-limited signal in certain fractional Fourier domain for angle α [11]. So the traditional non bandlimited signal problems in Fourier domain can be resolved by choosing the correct fractional Fourier transform for angle α .
2.2. Sampling theorem in FrFT domain A signal f (t ) is called Ωα band-limited signal in fractional Fourier transform sense, which means that Fα (u ) = 0 for | u |> Ωα (2) Where Ωα is the bandwidth of the signal f (t ) in the
Fractional Fourier domain, when α = π / 2 equation (2) reduces to the classical band-limited definition in Fourier transform domain. We next review the sampling theorem for an Ωα band-limited signal f (t ) with angle α ≠ nπ for any integer n in fractional Fourier transform domain. Let f ( nT ) be the uniformly sampled signal samples of
f (t ) ,
signal
where sampling period T = π sin α / Ωα satisfies the sampling theorem in fractional Fourier domain [11], then the original signal f (t ) can be reconstructed from the infinite
signal samples f ( nT ) by the following equation [8], [12]-[13]. t ( nT ) − j cot α ∞ j cot α sin [ Ωα csc α (t − nT ) ] (3) f (t ) = e 2 e 2 f ( nT ) 2
2
¦
r
Ωα csc α (t − nT )
n =−∞
Substitute T = π sin α / Ωα into equation (3) and obtain: t (nπ sinα / Ω ) − j cotα ∞ j cotα sin[ Ωα cscαt − nπ ] (4) 2 f (t) = e 2 e f (nπ sinα / Ω ) 2
r
¦
n=−∞
α
2
α
Ωα cscαt − nπ
From the above sampling equations we know that the reconstruction formula of a band-limited signal based on fractional Fourier domain equals to the convolution of the sinc kernel with the product of
infinite signal samples and a chirp signal, and finally get the original signal by other chirp signal modulation. In the following section we express the above equation as a finite summation provided that the sampling points of the signal satisfy the chirp-periodic property of fractional Fourier transform domain [14].
3. Interpolation based on FrFT 3.1. Chirp-periodicity From equation (3) we know that if we want to perfectly reconstruct the original signal, we must use the infinite signal samples, but it is not practical to get infinite signal samples in practical applications. To overcome this problem, we first introduce the definition of chirp-periodicity of a signal associated with fractional Fourier transform. A signal f (t ) is called chirp-periodicity with period Tα for order equation [15]: 2
j ( nT2 ) cot α
α
( nT +Tα ) 2 2
cot α
f (nT + Tα ) (5) It should be noted that if we let α = π / 2 in
e
f (nT ) = e
if it satisfies the following j
equation (5), the chirp-periodicity of signal reduces to the classical periodic definition in conventional sense, and from this point we conclude that the chirpperiodicity is an extension of traditional periodic definition associated with fractional Fourier domain.
3.2. Interpolation based on FrFT In this subsection we deduce the interpolation of signal samples based on the sampling theorem given in literatures [8], [12]-[13]. To simplify the problem we normalized T = π sin α / Ωα = 1 in equation (4), and obtain the following reconstructed signal from infinite sampling points.
fr (t ) = e
−j
t2 cot α 2
∞
¦e
j
n2 cotα 2
n=−∞
f (n)
sin [π t − nπ ] π t − nπ
(6)
At the same time we know that the following equation is true for any t .
sin(π t − nπ ) = (−1)n sin(π t )
(7) Substituting equation (7) to equation (6), and obtain t n j cotα − j cot α sin(π t ) +∞ (−1)n (8) fr (t ) = e 2 e 2 f (n) ¦ π n=−∞ t −n If signal f (t ) satisfies the chirp-periodicity with 2
2
period Tα = NT for order α , that is to say the following equation is true for the signal samples:
Proceedings of the First International Conference on Innovative Computing, Information and Control (ICICIC'06) 0-7695-2616-0/06 $20.00 © 2006
e
n2 cot α 2
j
f (n) = e
2
j ( n +2N ) cot α
f (n + N )
So equation (8) can be rearranged as t M −1 − j c o t α s in ( π t ) + ∞ j f r (t ) = e 2 ¦ ¦ e π k = −∞ n =− L 2
⋅ f ( n + kN )
( n + k N )2 2
(9) cotα
(10)
n + kN
( − 1) t − n − kN
Where L, M and N are arbitrary integers that obey the relation L + M = N . The inner summation in equation (10) can be separated for even and odd N ; in detail, equation (10) can be rewritten as: t2 − j cotα 2
fr (t) = e
sin(πt)
π
+∞ M −1
¦ ¦e
j
n2 cotα 2
(−1)n t − n − kN for N even
k =−∞ n=−L
f (n)
(11a)
And f r (t ) = e
−j
t2 cot α 2
sin(π t )
+∞
M −1
¦ ¦e
π
j
n2 cot α 2
k =−∞ n =− L
f ( n)
(−1)n (−1)k t − n − kN
(11b)
for N odd Exchanging the summation in the above equation we obtain: t n − j cotα sin(π t ) M −1 j cot α fr (t ) = e 2 e ¦ 2 f (n)(−1)n (12a) π n=− L 2
+∞
k =−∞
−j
t2 cotα 2
sin(π t )
π
1 t − n − kN
M −1
¦e
j
n2 cot α 2
for N even
f (n)(−1) n
(12b)
n =− L
(−1) ⋅¦ − t n − kN k =−∞ +∞
k
for N odd
We note that the two inner summations in equation (12) have been decomposed into partial fractions of cotangent and cosecant, respectively. Therefore, it is simple to rewrite the above equation as, t n − j cot α sin(π t ) M −1 j cot α t −n fr (t) = e 2 ¦ e 2 f (n)(−1)n cot(π ) (13a) 2
2
N
N
n=−L
for N even and t2 − j cot α 2
fr (t ) = e
2
Equation (14) can be looked as the interpolation formula of the original chirp-periodic sampling signal points in time domain for band-limited signal in fractional Fourier domain for angle α .When α = π / 2 , equation (14) reduces to the classical result in traditional Fourier transform domain [14], in this sense, the results in [14] can be looked as a special case of our results. At the same time, it must be emphasized that the representation of periodic continuous time signals based on the classical Fourier transform are only valid for band-limited signals that are equidistantly sampled in accordance with the sampling theorem. And because non bandlimited signal in Fourier transform domain may be band-limited in fractional Fourier domain for certain angle α , our results are more suitable for non-bandlimited signal in Fourier domain, including the traditional results as a special case.
2
⋅¦
f r (t ) = e
t2 cot α 2
sin(π t ) M −1 j n2 cot α f (t ) = e f (n)(−1)n ¦e 2 N n =− L (14) − − t n t n ⋅ [(−1) N +1 tan(π ) + cot(π )] 2N 2N −j
n2
sin(πt) M −1 j 2 cotα t −n ¦ e f (n)(−1)n csc(π N ) N n=−L
(13b)
for N odd These equations provide two interpolation formulae for recovering chirp-periodicity analogue signals from their even or odd number of samples. If we use some trigonometric relations, it is possible to show that the above equations can be integrated into one equation as:
4. Simulation results In order to verify the results we obtained in previous sections, we present some numerical results and simulations. The signal we used in simulation is a chirp signal:
f (t ) = e − j 2 t
2
(15) The observation time is [0, 2s], sampling period T = 1/ 50 s , the original signal is plotted in figure 1. We know that this chirp will produce certain impulse in fractional Fourier domain of order α = −arc cot(2) . So in the fractional Fourier domain for certain α values (15) yields a band-limited signal. And from the chirp-periodic property we can apply the following chirp-periodic property of period NT to extend the definition of signal f (nT + NT ) for the signal time outside the observation time interval [0, 2s]:
f (nT + NT ) = f ( nT )e
1 1 j ( nT )2 cot α − j ( nT + NT ) 2 cot α 2 2
(16)
So in this case the signal f (t ) satisfy the chirpperiodicity condition of Tα = NT = 100T .
Proceedings of the First International Conference on Innovative Computing, Information and Control (ICICIC'06) 0-7695-2616-0/06 $20.00 © 2006
5. Acknowledgement This work is supported by the National Natural Science Foundation of China (No.60232010, No. 60572094). The authors wish to thank Dr. George Kwamina Aggrey of Beijing Institute of Technology for many fruitful discussions and the proof reading of the paper.
6. References
Fig.1. original signal Figure 2 plots the reconstructed signal from N = 100 sampling points of one period.
Fig.2. reconstructed signal
5. Conclusions and future works Based on the existing sampling theorems associated with the fractional Fourier transform and the properties of FrFT, we show in this paper that the convolution of the sinc kernel with the product of infinite signal samples and a chirp signal can be expressed as a finite summation provided that the sampling points of the signal satisfy the chirp-periodic property of fractional Fourier transform sense, and the simulation results show that the new sampling theorem obtained in this paper is correct. At the same time, the classical results in traditional Fourier transform domain can be looked as a special case of our results. The future works along this way is to deduce and obtain the new interpolation formulae of sampling theorem for high-dimensional signals from the finite non-uniformly sampled band-limited signals with certain fractional Fourier order and the applications of the results in signal processing.
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