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Signal reconstruction from the undersampled signal samples
Optics Communications 268 (2006) 245–252 www.elsevier.com/locate/optcom
Signal reconstruction from the undersampled signal samples Kamalesh Kumar Sharma a
a,*,1
, Shiv Dutt Joshi
b
Electronics and communication Engineering Department, Malaviya National Institute of Technology, Jaipur, India b Department of Electrical Engineering, Indian Institute of technology, New Delhi, India Received 16 June 2006; accepted 14 July 2006
Abstract It is well-known from the celebrated Shannon sampling theorem for bandlimited signals that if the sampling rate is below the Nyquist rate, aliasing takes place and the original signal cannot be reconstructed back by simply passing the signal samples through an ideal lowpass filter. However, researchers such as Stern and Gori have shown the existence of some classes of signals for which the signals are sampled below the Nyquist rate but perfect signal reconstruction is still possible from the given signal samples. Here, we present a generalized lowpass sampling theorem and show that Stern’s and Gori’s lowpass sampling theorems are special cases of it. A sampling theorem for the bandpass signals in the linear canonical transform domains is also presented and its special cases are discussed. Using a modification of the conventional natural sampling waveform with a specific width of the pulses, it is shown that the sampling rate in our generalized lowpass sampling theorem and hence in the Stern’s and the classical Shannon sampling theorems can be further reduced by a factor of two, while for the bandpass signals, the reduction in the sampling rate by some factor is possible only under some restricted conditions. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Linear canonical transform; Fractional Fourier transform; Signal reconstruction and filtering; Sampling theorem
1. Introduction Signal reconstruction from its samples is an important signal processing operation, which is needed very frequently in many applications in several diverse areas such as signal processing, communications, geophysics, radar and sonar, and optics including optical signal processing [4,16,17]. A large number of sampling theorems exist in the literature for representation and interpolation of various types of signals [7,11,16,18,21,22]. For bandlimited signals in the conventional Fourier domain (CFD), the classical Shannon sampling theorem gives us the minimum sampling rate, called Nyquist rate (twice the maximum frequency of the signal), required to reconstruct the signal *
Corresponding author. Tel./fax: +91 141 2551693. E-mail addresses: [email protected] (K.K. Sharma), [email protected] (S.D. Joshi). 1 Presently, he is on deputation to the Department of Electrical Engineering, Indian Institute of technology, New Delhi, India. 0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.07.045
uniquely from its samples. If the sampling rate is below the Nyquist rate or the signal is not bandlimited in the CFD, aliasing takes place in the CFD and it is not possible to recover the original signal from the samples of this undersampled signal by performing the filtering with the linear time-invariant (LTI) systems. It is possible, however, for the signals not bandlimited in the CFD to be bandlimited in some other fractional Fourier transform (FRFT) domain. This is evident from [11], where it has been proved that if a nonzero signal f(t) is bandlimited with angle a then it cannot bandlimited with another angle b, where b 5 a + np for any integer n. The other possibility is that the signals not bandlimited in the CFD can be bandlimited in some linear canonical transform (LCT) domain. This fact has been exploited for signal reconstruction purposes with samples taken below the Nyquist rate by Stern in [21]. He presented a lowpass sampling theorem for this class of signals. The classical Shannon sampling theorem originally established by Whittaker [16,17] and the previously developed sampling
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theorems for the Fresnel transformed and the FRFTed signals [11,22] were shown to be special cases of it. However, Stern presented the results for the class of lowpass signals bandlimited in some LCT domain only. He also discussed that perfect signal reconstruction is possible for the class of signals bandlimited in some LCT domain even if the sampling rate is below the Nyquist rate. Gori [22] has also established similar results earlier for the class of signals bandlimited in the Fresenl transform domains. Here, we consider a larger class of the signals (not bandlimited in the CFD) having their FRFT with angle a to be bandlimited in some LCT domain. A generalized lowpass sampling theorem for this class of signals is presented and it is shown that the Stern’s and Gori’s theorems are special cases of it. We also present a sampling theorem for the bandpass signals in the LCT domains. Bandpass signals in the FRFT and the conventional Fourier transform (CFT) domains are of course special cases of it. It is established in this paper that there exists some bigger class of signals than considered by Stern in [21] for which the sampling need not be done above the Nyquist rate but perfect signal reconstruction from the signal samples is still possible by performing some filtering/processing with the linear time-variant (LTV) systems using the CFT, the FRFT and/or the LCT. All the above theorems consider ideal impulse-train sampling, which is unrealizable in practice. The natural sampling employing pulses of finite duration is often used in practice and this is further replaced by flat-top sampling for the ease of realization [19]. Using a slight modification of the conventional natural sampling waveform as shown in Fig. 3, and using a specific width of these pulses, it is shown here that the sampling rate in our generalized sampling theorem (and hence in the Stern’s and the classical Shannon sampling theorems) can be further reduced by a factor of two. This may result in the more efficient representation of these classes of signals and consequently it may further lead to more applications in the signal compression area. The paper is organized as follows. We briefly review the LCT and the FRFT in Section 2. In Section 3, we propose some schemes for signal reconstruction from the undersampled signal samples using the LTV systems involving the CFT, the FRFT, and the LCT. We consider three classes of signals here, each assumed to be bandlimited in the CFD or in the FRFT domains or in the LCT domains. Some techniques of sampling rate reduction in the above theorems are discussed in Section 4. The conclusions are presented in Section 5. 2. Review of the LCT and the FRFT The LCT RM[f](u) of a signal f(u) with parameter matrix M is given by [4] Z RM ½f ðuÞ ¼ C M ðu; u0 Þf ðu0 Þ du0 ; ð1Þ
where
D 2 1 0 A 02 u 2 uu þ u ; C M ðu; u Þ ¼ AM exp jp B B B pffiffiffiffiffiffiffiffi A B AM ¼ 1=B expðjp=4Þ, and M with determiC D nant AD BC = 1. Various signal transforms such as the Fresnel transform and the FRFT are simply special cases of the LCT. The FRFT is in fact a one-parameter subgroup of the group of LCTs. To be specific, if we substitute A = D = cos a, and B = C = sin a in the matrix M and evaluate (1), we obtain the FRFT of the signal within a unit magnitude complex constant. One important property of the LCT is its delay property, which is reproduced below [4, p. 97]: 0
LCT
f ðu sÞ $ exp½jpð2usC s2 ACÞ RM ½f ðu AsÞ:
ð2Þ
It is clear from this property that the LCT of a delayed signal is a shifted version of the LCT of the original signal multiplied by a chirp factor. This can be exploited in the signal reconstruction problems as discussed below in Section 3. The other details and more discussion on the LCT and its eigenfunctions can be seen in [4,14] and the references therein. The FRFT [1–6,8–13], which is a generalization of the CFT, generalizes the usual time and frequency domain representations of the signals to the continuum of infinite FRFT domains. The FRFT of a signal f(t), denoted as Fa(u), is defined as: Z 1 F a ðuÞ ¼ f ðtÞK a ðt; uÞ dt; and Z 11 ð3Þ f ðtÞ ¼ F a ðuÞK a ðt; uÞ du; 1
where the transform kernel Ka(t, u) of the FRFT is given by [3]: 8 qffiffiffiffiffiffiffiffiffiffiffiffi 1j cot a > t2 þu2 > exp j cot a jtu cosec a ; > 2 2p > > > > > if a is not a multiple of p > > < dðt uÞ; ð4Þ K a ðt; uÞ ¼ > > if a is a multiple of 2p > > > > > dðt þ uÞ; > > > : if a þ p is a multiple of 2p Here a indicates the rotation angle of the transformed signal for FRFT and * denotes complex conjugation. In short notation, we can write as: a
f ðtÞ $ F a ðuÞ:
ð5Þ
Thus the FRFT reduces to the CFT for a = p/2. Several useful properties are currently under study in signal processing community [1–3,5–8]. The FRFT has also been proved to relate to other signal analysis tools such as Wigner distribution, neural network, wavelet transform and various chirp related operations [2]. It may be mentioned here that the FRFT may be more useful in the cases, where
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the signal has some desirable property in the FRFT domain, such as compact support, and lesser number of coefficients for it’s adequate representation (this applies specially for nonstationary/chirp type of signals). We now mention the delay property of the FRFT, which will be used in this correspondence for signal reconstruction problems for ready reference. This property relates the FRFT of the delayed signal to that of the original signal. a
f ðt sÞ $ expðjps2 sin a cos aÞ expðj2pus sin aÞFa ðu s cos aÞ:
ð6Þ
It is clear from (6) that as opposed to the case of the CFT, the support of the FRFT of the delayed signal f(t s) is different from the support of the FRFT of the signal f(t). In fact the Fa(u) gets shifted by s cos a in the FRFT of the delayed signal f(t s). Similarly, multiplication by an exponential signal exp (j2pst) to a signal f(t) results in a similar shift of Fa(u) by s sin a. An excellent and comprehensive discussion of the FRFT can be found in [4]. The MATLAB code for fast computation of the FRFT is also available at the Mathworks website [15]. 3. Signal reconstruction techniques from the undersampled signal samples Signal reconstruction from the given signal samples is an important signal processing operation, which is needed quite frequently in many applications in several diverse areas such as signal processing, communications, geophysics, radar and sonar, and optics including optical signal processing [4,16,17]. A large number of sampling theorems for the representation of various types of signals have also been derived in the literature including in the FRFT domains [7,11,16,18,21,22]. We consider here three different classes of signals, each being assumed to be bandlimited either in the CFD, or in the FRFT domains, or in the LCT domains. As is well-known, signals with compact support in the LCT domain need to have infinite support in the time domain. As a result, no practical signal of finite support is strictly bandlimited or bandpass in the LCT domains. However, the sampling theorems can be applied to signals whose energy is mostly concentrated in a finite interval in the LCT domains with a negligible nonzero reconstruction error. This applies specially to the chirp signals/nonstationary signals encountered in radar and sonar, etc. The optical signal processing involving the FRFT and the LCT may also be benefited from this. We discuss the lowpass and bandpass signal cases separately. 3.1. Signal class whose FRFT with angle a is bandlimited in some LCT domain with parameter M Lowpass signals: Here, we are considering the class of signals with the property that their FRFTs with angle a are bandlimited to uM in some LCT domain with parameter M.
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We start with the Shannon sampling theorem [17,19] which states that a continuous-time CFD bandlimited signal can be reconstructed exactly from its samples taken at a rate, which is greater than or equal to the Nyquist rate by passing the signal samples through an ideal lowpass filter with a suitable cut-off frequency in the CFD. If the sampling frequency is less than the Nyquist rate, then aliasing will take place and it is not possible to reconstruct the original signal back by simply passing the samples through an ideal lowpass filter. To be specific, the CFT of the sampled signal, F sp=2 ðuÞ, can be written as F sp=2 ðuÞ ¼
1 1 X F p=2 ðu nus Þ; T s n¼1
ð7Þ
where Ts is the sampling interval and Fp/2(u) is the CFT of the signal f(t) assumed to be sampled by an ideal impulsetrain given by 1 X 2p dðt nT s Þ with us ¼ ¼ 2pfs : ð8Þ dT s ðtÞ ¼ Ts n¼1 On taking the inverse FRFT with angle (p/2 a) of (7) and using the delay property of the FRFT given in (6), we get 1 1 X expðjpn2 u2s sin a cos aÞ F sa ðuÞ ¼ T s n¼1 expðj2punus cos aÞF a ðu nus sin aÞ:
ð9Þ
We now take the LCT with parameter M of (9) on both sides to get RM ½F sa ðuÞ 1 1 X ¼ expðjpn2 u2s sin a cos aÞ T s n¼1 exp½jpnus cos aDð2u nus cos aBÞ exp½jp2ðu Bnus cos aÞnus sin aC n2 u2s sin2 aAC RM ½F a ðu nus sin aA Bnus cos aÞ: ð10Þ It is clear from (10) that the original signal f(u) can be obtained from the output of the lowpass filter by taking the inverse LCT with parameter M followed by inverse FRFT with angle a as the term corresponding to n = 0 in (10) can be easily filtered from the nearest term corresponding to n = 1 using the ideal lowpass filter in that particular LCT domain for the class of signals under consideration. Mathematically, the signal reconstruction formula can be expressed as
1 1 1 u s f ðtÞ ¼ F a RM rect ð11Þ RM ½F a ðuÞ ; Ts 2uM 1 where F 1 a and RM denotes inverse FRFT and inverse LCT operators. The sampling rate for perfect signal reconstruction must satisfy the condition given below
us ðA sin a þ B cos aÞ P 2uM :
ð12Þ
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Thus using the filtering in the LCT domain, we are able to reconstruct a signal belonging to a class which has infinite bandwidth in the CFT domain, from its samples by sampling it at a finite rate given by (12) and the sampling frequency can be less than twice the bandwidth of the signal in the CFT domain, which in fact is infinite here. The reconstruction formula given in (10) of [21] is a special case of (11) for a = 0. Thus the class of signals covered by (11) is bigger than the classes considered in [21]. Special cases: 1. Signals whose CFT is bandlimited in some LCT domain with parameter M This class of signals is obtained for the value of the parameter a = p/2. If we substitute a = p/2 in (10), we get 1 1 X RM ½S p=2 ðuÞ ¼ exp½ipð2unus C n2 u2s ACÞ T s n¼1 RM ½F p=2 ðu Anus Þ:
ð13Þ
The term in (13) corresponding to n = 0 can be easily filtered from the term corresponding to n = 1 using the ideal lowpass filter in that particular LCT domain provided Aus P 2uM :
ð14Þ
By choosing the value of parameter A to be very large, the sampling frequency can be less than the Nyquist rate, which in fact is infinite here. 2. Signals bandlimited in some LCT domain with parameter M This class of signals is obtained for the value of the parameter a = 0. If we substitute a = 0 in (10), we get the condition (9) of [21], i.e., Bus P 2uM :
ð15Þ
3. Signals which are bandlimited in some FRFT domain with a 5 np/2 This class of signals will not be bandlimited in the CFT domain as discussed in [11]. If we substitute B = sina in (15), we get the condition (15) of [21], i.e., us sin a P 2uM :
or uL is a harmonic of (A sin a + B cos a)us. If neither uM or uL is a harmonic of (A sin a + B cos a)us, it can be shown that the sampling rate must satisfy k1 ðA sin a þ B cos aÞus 6 2Bw ; and N 1 k ; ð17Þ ðA sin a þ B cos aÞus P 2Bw N where Bw = uM uL and k = uM/Bw. Here also k P N and N is a natural number. The signal reconstruction formula for this case can also be written as
1 u uC 1 s ðtÞ ¼ F 1 R rect ½F ðuÞ R M a Ts a M uM uL
1 u þ uC 1 s þ F 1 R rect ½F ðuÞ ; ð18Þ R M a Ts a M uM uL where uC = (uM + uL)/2. Special cases: 1. Signals whose CFT is a band-pass signal in some LCT domain with parameter M If we substitute a = p/2 in (17), we obtain the sampling rate criterion as given by k1 k Aus 6 2Bw ; and Aus P 2Bw : ð19Þ N 1 N 2. Signals whose LCT is a band-pass signal in some LCT domain with parameter M Similarly, if we substitute a = 0 in (17), we get the condition k1 k Bus 6 2Bw ; and Bus P 2Bw : ð20Þ N 1 N 3. Signals whose FRFT is a band-pass signal in some FRFT domain with angle a If we substitute B = sin a in (20), we get the condition for this class of signals as given by
ð16Þ
Ts
-Ts
The signal reconstruction formulas for these special cases can be easily obtained from (11) by substituting the corresponding values of a. We also illustrate the Eq. (16) in Figs. 1.1 and 1.2. Here the effect of chirp multiplication factors in (10) is neglected for clarity of illustration.
0
-2Ts
2Ts
Fig. 1.1. Ideal impulse-train sampling waveform.
3.2. Signal class whose FRF T with angle a is a band-pass signal in some LCT domain with parameter M Bandpass signals: If the LCT with parameter M of the signal Fa(u) is a bandpass signal with bandwidth extending from uL to uM, then along the lines of [19], it can be easily shown that the minimum sampling frequency allowable is (A sin a + B cos a)us = 2(uM uL) provided that either uM
• • •
• • • -2 us •sinα
•
- uM 0• uM
- us sinα
•
• 2 us sinα
us sinα
Fig. 1.2. Spectrum of the sampled signal in FRFT domain.
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249
4. Reduction of sampling rate using modified sampling scheme
ð23Þ
All the above sampling theorems are based on the ideal impulse-train sampling, which is unrealizable in practice. The natural sampling employing pulses of finite duration is often used in practice and this is further replaced by flat-top sampling for the ease of circuit realization [19]. Using a slight modification of the conventional natural sampling waveform as shown in Fig. 3, and using a specific width of the pulses, it is shown here that the sampling rate in our generalized sampling theorem (and hence in the Stern’s and the classical Shannon sampling theorem) can be further reduced by a factor of two. We discuss the modified sampling scheme for the Signal class, which is bandlimited in the CFT domain for the ease of illustration, although the proposed scheme is applicable to all the lowpass sampling theorems discussed above. The basic idea is to eliminate all the even harmonics and some selected odd harmonic of the sampling frequency from the spectrum of the sampling waveform. This increases the gap between the two successive terms in the spectrum of the sampled signal. This increased gap in the spectrum of the sampled signal can be exploited for either increasing the maximum frequency of the signal to be sampled or for reducing the sampling rate of the pulses used in the natural sampling by a factor of two. Here also we discuss the lowpass and bandpass signal cases separately. Lowpass signals: The basic technique proposed here can be considered as an extension of the natural sampling of signals by finite duration pulses as described in [19]. It is based upon the use of sampling waveform, which results in the elimination of all the even harmonics of the sampling frequency and selective elimination of the third or fifth or any other odd harmonic from the conventional spectrum of the sampled signal [20]. This allows us to sample the original signal at a rate below the Nyquist rate. For the sampling waveform shown in Fig. 3, accounting both the positive and negative pulses in a cycle, the conventional Fourier series of this can be written as [20] X 4 nd us SðtÞ ¼ sin sin n t: ð26Þ np 2 2 n¼1;3;5;...
Sampling frequencies for inverse placement of the spectrum [23] should similarly satisfy
Here due to the symmetry of the output voltage along the x-axis, all the even harmonics (n = 2, 4, 6, . . .) are absent.
Fig. 2. Shaded regions showing the allowed range of sampling frequencies for FRFT limited signals given in (21) for a = p/4.
us sin a 6 2Bw
k1 k ; and us sin a P 2Bw : N 1 N
ð21Þ
The allowed range of sampling frequencies given by (21) for a = p/4 is depicted as shaded regions in Fig. 2. The signal reconstruction formulas for these special cases can be easily obtained from (18) by substituting the required values of a. Although the aliasing problem is solved perfectly in (17), to demodulate correctly from RM[Sp/2](u) in the baseband spectrum as done in [23], sampling frequencies for normal placement of the spectrum [23] should satisfy 2uM uL 6 ðA sin a þ B cos aÞus 6 ; ð22Þ 2n þ 1 j kn where 0 6 n 6 m, m ¼ 2ðuMuLuL Þ and n is odd. Here the symbol bxc denotes the largest integer smaller than or equal to x. The robust sampling frequency [23] for normal placement of the spectrum can also be easily written as ðA sin a þ B cos aÞus ¼
2ðuM þ uL Þ : 4n þ 1
uM 2uL 6 ðA sin a þ B cos aÞus 6 ; ð24Þ n 2n j k 1 where 1 6 n 6 m, m ¼ 2ðuMuMuL Þ , and n is even. The robust sampling frequency for inverse placement of the spectrum can also be easily written as ðA sin a þ B cos aÞus ¼
2ðuM þ uL Þ : 4n 1
δ Ts 2π
• •
ð25Þ
It is clear from (17)–(25) that for proper values of the parameters A, B and a, the sampling rate can be less than the 2Bw.
0
• Ts/4
Fig. 3. Sampling waveform.
• • 3Ts/4
•
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Now for a specific choice of d, any of the third, fifth, seventh or any other odd harmonic can be eliminated from the conventional spectrum of the sampled signal. The higher the order of the harmonic, the lower will be the value of d allowing time-division multiplexing of many other signals in this scheme. To be specific, for d = 2p/3, the third harmonic will be absent from the spectrum and it can be easily seen that the sampling rate must satisfy the condition given below for exact signal reconstruction: ð27Þ
us P uM :
This sampling rate is precisely one-half the Nyquist rate given by Shannon sampling theorem. The original signal can be reconstructed from the given signal samples by passing the samples through a pair of BPFs (one having the bandwidth from us/2 to us/2 + uM and the other having the bandwidth from us/2 + uM to 5us/2 in CFT domain) and appropriate frequency translation of the spectrum to the lowpass frequency range as explained in Fig. 4. The signal reconstruction formula for the general case of elimination of the kth harmonic from the spectrum of the sampling waveform can be written as 1 ðk 2Þp 1 h n us oi f ðtÞ ¼ F F u þ ðk 2Þ 1 p=2 4 sin ðk2Þd 2 2 1 ðk þ 2Þp 1 h n us o i þ F p=2 F 2 u þ ðk þ 2Þ ðkþ2Þd 4 sin 2 2 h n 1 ðk 2Þp 1 us o i F F u ðk 2Þ þ 3 4 sin ðk2Þd p=2 2 2 h n 1 ðk þ 2Þp 1 us o i þ F p=2 F 4 u ðk þ 2Þ ; ðkþ2Þd 4 sin 2 2 ð28Þ u u1 fF p=2 ½f ðtÞSðtÞðuÞg; where F 1 ðuÞ ¼ rect uM u u2 F 2 ðuÞ ¼ rect fF p=2 ½f ðtÞSðtÞðuÞg; uM u þ u1 F 3 ðuÞ ¼ rect fF p=2 ½f ðtÞSðtÞðuÞg; uM u þ u2 F 4 ðuÞ ¼ rect fF p=2 ½f ðtÞSðtÞðuÞg; uM us uM us uM ; and u2 ¼ k þ : u1 ¼ ðk 2Þ þ 2 2 2 2 It is clear from this that it is possible to reconstruct the original signal from the signal samples taken below Nyquist rate. If flat-top sampling is utilized here for practical pur-
•• •
•
-5 us /2
• - u /2 s
0
•us / 2
• 5 u /2
poses, similar to the flat-top sampling considered in [19], then it will result in a distortion of the signal at the out put of the BPF but it can be eliminated at least theoretically by an equalizer having a x/sin x type of frequency response [19]. The second technique proposed below, illustrated in Fig. 5, is valid for the class of bandlimited signals in the CFT domain having a real and even symmetric spectrum around the folding frequency within a frequency range of ±dM, i.e., within us/2 ± dM, where us/2 + dM = uM. The typical spectrum of such signals is shown in Fig. 6. Let f(t) be a signal belonging to this class with bandwidth in the CFT domain uM sampled by an ideal impulse-train shown in Fig. 7. The spectrum of the sampled signal is shown in Fig. 7. We know that sampling a signal below the Nyquist rate results in the problem, called aliasing [17,19] which manifests itself in the corruption of the high frequency portion of the CFT of the signal. However, we have some portion in the spectrum of the signal f(t) up to the frequency of (us uM), which is still intact and unaffected by the aliasing. This part of the spectrum can be easily filtered in the CFT domain using an ideal lowpass filter (LPF) with cut-off frequency (us uM). We can further select the aliased portion using an ideal bandpass filter with cut-off frequencies (us uM) to uM and pass it through an LTI system with impulse response satisfying the condition Hp/2(u) = 1/2. The output of such system can be added to the signal at the output of the LPF. This total signal obtained is nothing but the CFT of the original signal from which the original signal can be easily recovered. 1 f ðtÞ ¼ F 1 ½F 1 ðuÞ þ F 2 ðuÞ; where T s p=2 " # )
( 1 X u F 1 ðuÞ ¼ rect dðt nT s Þ ðuÞ ; F p=2 f ðtÞ 2ðus uM Þ n¼1 " # )
( 1 X u us =2 dðt nT s Þ ðuÞ F 2 ðuÞ ¼ rect F p=2 f ðtÞ ð2uM us Þ n¼1 " # )
( 1 X u þ us =2 þ rect dðt nT s Þ ðuÞ : F p=2 f ðtÞ ð2uM us Þ n¼1 ð29Þ f(t) f(t)
Sampler
Low pass Filter
•••
s
us /2+ uM Fig. 4. Spectrum of the sampled signal in CFT domain.
Band-pass Filter
H
/2
(u ) = 1/ 2
Fig. 5. Signal reconstruction for CFT limited signals with real and even spectrum around folding frequency.
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251
A1 A2 0
uM
us /2
us /2−δM
Fig. 6. Spectrum of the signal having even symmetry around folding frequency within ±dM.
BPF
LPF A1
•••
•••
2A2
0
us us / 2
us / 2 − δM
uM
Fig. 7. Spectrum of the sampled signal by sampling the signal with spectrum shown in Fig. 6.
Bandpass signals: We consider the sampling of the bandpass signals using the sampling waveform shown in Fig. 3. Here, as the sampling waveform has only odd harmonics present in it, the approach for finding the range of the sampling frequencies is slightly different from the approach in [19]. Suppose we want to avoid the aliasing in the range of frequencies occupied by the spectrum of the signal, i.e., between uL and uM in the positive portion of the spectrum of the sampled signal and consider its possible overlap due to (N 1)th and (N + 3)th shifts of the negative portion of the spectrum, assuming (N + 1)th harmonic is absent from the spectrum because of the proper choice of the width of the pulses used in the sampling, then it can be easily shown that the sampling rate (accounting both the positive and negative pulses in the sampling waveform) must satisfy 4kBw us 6 4ðk 1ÞBw =ðN 1Þ; and us P ; k P N: N þ3 ð30Þ Here the value of be largest odd integer less than j N must k L M, where M ¼ uM2uu . If we compare it with the samL pling rate given by (5.1.7) and (5.1.8) in [19], it is clear that the reduction in sampling rate will happen if k P N2kþ3 for a given value of Bw. This implies that N 3 P N. Thus the reduction in the sampling rate in case of bandpass signals will happen under this restricted condition only.
We also mention here another class of bandpass periodic signals with a given period, which can also be undersampled but the perfect signal reconstruction is still possible, at least theoretically. This is based on the fact that the spectrum of a periodic signal consists of impulse functions, located at the fundamental frequency and harmonics of it. Hence if the sampling frequency is not an integer multiple of one-half the fundamental frequency, then aliasing will result in the appearance of the frequencies which are not an integer multiple of the fundamental frequency and can be easily identified in the spectrum of the reconstructed signal and these can be filtered using ideal bandpass filters and these can be translated back to their original place by frequency translation methods and perfect signal reconstruction is possible. Thus we have shown the existence of some classes of signals bandlimited in the CFD, the LCT and the FRFT domains, for which the sampling below the Nyquist rate is done but perfect signal reconstruction from the signal samples is still possible. We would also like to mention that the equations similar to (7) are also encountered in a communication scenario related to the multi-path fading problem and this can also be taken care of, if these multipath components can be separated using the techniques discussed here for different classes of signals. The extension of the one-dimensional sampling waveforms discussed here to two-dimensional sampling waveforms/ grids is straightforward.
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5. Conclusions It was shown in this paper that there exists some classes of signals bandlimited as well as bandpass in the CFD, the FRFT, and the LCT domains, for which the sampling rate can be below the Nyquist rate but perfect signal reconstruction from the signal samples is still possible by performing some filtering/processing with the linear time-variant (LTV) systems using the FRFT, CFT and/ or the LCT. This may result in the more efficient representation for theses classes of signals and consequently it may further find its applications in the signal compression area. This may also be helpful in tackling the fading problem encountered in the multi-path environments in wireless communications. References [1] [2] [3] [4]
V. Namias, J. Inst. Maths. Appl. 25 (1980) 241. H.M. Ozaktas, B. Barshan, J. Opt. Soc. Am. A 11 (1994) 547. L.B. Almeida, IEEE Trans. Sig. Proc. 42 (11) (1994) 3084. H.M. Ozaktas, Z. Zalevsky, M.A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, Wiley, New York, 2001. [5] L.B. Almeida, IEEE Sig. Proc. Lett. 4 (1) (1997) 15. [6] A.I. Zayed, IEEE Sig. Proc. Lett. 5 (4) (1998) 101.
[7] K.K. Sharma, S.D. Joshi, Opt. Commun. 257 (1) (2006) 27. [8] K.K. Sharma, S.D. Joshi, Filtering in fractional Fourier transform domains, in: Proceedings of NCC New Delhi, India, 2006, p. 151, Jan. 2006. [9] M.F. Erden, H.M. Ozaktas, D. Mendlovic, Opt. Commun. 125 (Apr) (1996) 288. [10] B. Barshan, M.A. Kutay, H.M. Ozaktas, Opt. Commun. 135 (1–3) (1997) 32. [11] Xiang-Gen Xia, IEEE Sig. Proc. Lett. 3 (3) (1996) 72. [12] G. Cariolaro, T. Erseghe, N. Laurenti, IEEE Trans. Sig. Proc. 48 (1) (2000) 227. [13] M. Alper Kutay, Haldun M. Ozaktas, IEEE Trans. Sig. Proc. 45 (5) (1997) 1129. [14] S.-C. Pei, J.-J. Ding, IEEE Trans. Sig. Proc. 50 (1) (2002) 11. [15] J.O’ Neill, Discrete TFDs: A collection of Matlab files for timefrequency analysis, 1999. Available from:. [16] Robert J. Marks II, Advanced topics in Shannon Sampling and Interpolation Theory, Springer-Verlag, New York, 1993. [17] Alan V. Oppenheim, Ronald W. Schafer, Digital Signal Processing, Prentice Hall of India, 1994. [18] K.K. Sharma, S.D. Joshi, Opt. Commun. 256 (4–6) (2005) 272. [19] H. Taub, D.L. Schilling, Principles of Communication Systems, second ed., McGraw-Hill, New York, 1989, p. 183. [20] M.H. Rashid, Power Electronics: Circuits, Devices, and Applications, third ed., Pearson Education, 2004, pp. 249, 281. [21] A. Stern, Sig. Proc. 86 (7) (2006) 1421. [22] F. Gori, Opt. Commun. 39 (5) (1981) 293. [23] J. Liu, X. Zhou, Y. Peng, IEEE Trans. Sig. Proc. 49 (6) (2001) 1260.
Signal reconstruction from the undersampled signal samples Kamalesh Kumar Sharma a
a,*,1
, Shiv Dutt Joshi
b
Electronics and communication Engineering Department, Malaviya National Institute of Technology, Jaipur, India b Department of Electrical Engineering, Indian Institute of technology, New Delhi, India Received 16 June 2006; accepted 14 July 2006
Abstract It is well-known from the celebrated Shannon sampling theorem for bandlimited signals that if the sampling rate is below the Nyquist rate, aliasing takes place and the original signal cannot be reconstructed back by simply passing the signal samples through an ideal lowpass filter. However, researchers such as Stern and Gori have shown the existence of some classes of signals for which the signals are sampled below the Nyquist rate but perfect signal reconstruction is still possible from the given signal samples. Here, we present a generalized lowpass sampling theorem and show that Stern’s and Gori’s lowpass sampling theorems are special cases of it. A sampling theorem for the bandpass signals in the linear canonical transform domains is also presented and its special cases are discussed. Using a modification of the conventional natural sampling waveform with a specific width of the pulses, it is shown that the sampling rate in our generalized lowpass sampling theorem and hence in the Stern’s and the classical Shannon sampling theorems can be further reduced by a factor of two, while for the bandpass signals, the reduction in the sampling rate by some factor is possible only under some restricted conditions. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Linear canonical transform; Fractional Fourier transform; Signal reconstruction and filtering; Sampling theorem
1. Introduction Signal reconstruction from its samples is an important signal processing operation, which is needed very frequently in many applications in several diverse areas such as signal processing, communications, geophysics, radar and sonar, and optics including optical signal processing [4,16,17]. A large number of sampling theorems exist in the literature for representation and interpolation of various types of signals [7,11,16,18,21,22]. For bandlimited signals in the conventional Fourier domain (CFD), the classical Shannon sampling theorem gives us the minimum sampling rate, called Nyquist rate (twice the maximum frequency of the signal), required to reconstruct the signal *
Corresponding author. Tel./fax: +91 141 2551693. E-mail addresses: [email protected] (K.K. Sharma), [email protected] (S.D. Joshi). 1 Presently, he is on deputation to the Department of Electrical Engineering, Indian Institute of technology, New Delhi, India. 0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.07.045
uniquely from its samples. If the sampling rate is below the Nyquist rate or the signal is not bandlimited in the CFD, aliasing takes place in the CFD and it is not possible to recover the original signal from the samples of this undersampled signal by performing the filtering with the linear time-invariant (LTI) systems. It is possible, however, for the signals not bandlimited in the CFD to be bandlimited in some other fractional Fourier transform (FRFT) domain. This is evident from [11], where it has been proved that if a nonzero signal f(t) is bandlimited with angle a then it cannot bandlimited with another angle b, where b 5 a + np for any integer n. The other possibility is that the signals not bandlimited in the CFD can be bandlimited in some linear canonical transform (LCT) domain. This fact has been exploited for signal reconstruction purposes with samples taken below the Nyquist rate by Stern in [21]. He presented a lowpass sampling theorem for this class of signals. The classical Shannon sampling theorem originally established by Whittaker [16,17] and the previously developed sampling
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theorems for the Fresnel transformed and the FRFTed signals [11,22] were shown to be special cases of it. However, Stern presented the results for the class of lowpass signals bandlimited in some LCT domain only. He also discussed that perfect signal reconstruction is possible for the class of signals bandlimited in some LCT domain even if the sampling rate is below the Nyquist rate. Gori [22] has also established similar results earlier for the class of signals bandlimited in the Fresenl transform domains. Here, we consider a larger class of the signals (not bandlimited in the CFD) having their FRFT with angle a to be bandlimited in some LCT domain. A generalized lowpass sampling theorem for this class of signals is presented and it is shown that the Stern’s and Gori’s theorems are special cases of it. We also present a sampling theorem for the bandpass signals in the LCT domains. Bandpass signals in the FRFT and the conventional Fourier transform (CFT) domains are of course special cases of it. It is established in this paper that there exists some bigger class of signals than considered by Stern in [21] for which the sampling need not be done above the Nyquist rate but perfect signal reconstruction from the signal samples is still possible by performing some filtering/processing with the linear time-variant (LTV) systems using the CFT, the FRFT and/or the LCT. All the above theorems consider ideal impulse-train sampling, which is unrealizable in practice. The natural sampling employing pulses of finite duration is often used in practice and this is further replaced by flat-top sampling for the ease of realization [19]. Using a slight modification of the conventional natural sampling waveform as shown in Fig. 3, and using a specific width of these pulses, it is shown here that the sampling rate in our generalized sampling theorem (and hence in the Stern’s and the classical Shannon sampling theorems) can be further reduced by a factor of two. This may result in the more efficient representation of these classes of signals and consequently it may further lead to more applications in the signal compression area. The paper is organized as follows. We briefly review the LCT and the FRFT in Section 2. In Section 3, we propose some schemes for signal reconstruction from the undersampled signal samples using the LTV systems involving the CFT, the FRFT, and the LCT. We consider three classes of signals here, each assumed to be bandlimited in the CFD or in the FRFT domains or in the LCT domains. Some techniques of sampling rate reduction in the above theorems are discussed in Section 4. The conclusions are presented in Section 5. 2. Review of the LCT and the FRFT The LCT RM[f](u) of a signal f(u) with parameter matrix M is given by [4] Z RM ½f ðuÞ ¼ C M ðu; u0 Þf ðu0 Þ du0 ; ð1Þ
where
D 2 1 0 A 02 u 2 uu þ u ; C M ðu; u Þ ¼ AM exp jp B B B pffiffiffiffiffiffiffiffi A B AM ¼ 1=B expðjp=4Þ, and M with determiC D nant AD BC = 1. Various signal transforms such as the Fresnel transform and the FRFT are simply special cases of the LCT. The FRFT is in fact a one-parameter subgroup of the group of LCTs. To be specific, if we substitute A = D = cos a, and B = C = sin a in the matrix M and evaluate (1), we obtain the FRFT of the signal within a unit magnitude complex constant. One important property of the LCT is its delay property, which is reproduced below [4, p. 97]: 0
LCT
f ðu sÞ $ exp½jpð2usC s2 ACÞ RM ½f ðu AsÞ:
ð2Þ
It is clear from this property that the LCT of a delayed signal is a shifted version of the LCT of the original signal multiplied by a chirp factor. This can be exploited in the signal reconstruction problems as discussed below in Section 3. The other details and more discussion on the LCT and its eigenfunctions can be seen in [4,14] and the references therein. The FRFT [1–6,8–13], which is a generalization of the CFT, generalizes the usual time and frequency domain representations of the signals to the continuum of infinite FRFT domains. The FRFT of a signal f(t), denoted as Fa(u), is defined as: Z 1 F a ðuÞ ¼ f ðtÞK a ðt; uÞ dt; and Z 11 ð3Þ f ðtÞ ¼ F a ðuÞK a ðt; uÞ du; 1
where the transform kernel Ka(t, u) of the FRFT is given by [3]: 8 qffiffiffiffiffiffiffiffiffiffiffiffi 1j cot a > t2 þu2 > exp j cot a jtu cosec a ; > 2 2p > > > > > if a is not a multiple of p > > < dðt uÞ; ð4Þ K a ðt; uÞ ¼ > > if a is a multiple of 2p > > > > > dðt þ uÞ; > > > : if a þ p is a multiple of 2p Here a indicates the rotation angle of the transformed signal for FRFT and * denotes complex conjugation. In short notation, we can write as: a
f ðtÞ $ F a ðuÞ:
ð5Þ
Thus the FRFT reduces to the CFT for a = p/2. Several useful properties are currently under study in signal processing community [1–3,5–8]. The FRFT has also been proved to relate to other signal analysis tools such as Wigner distribution, neural network, wavelet transform and various chirp related operations [2]. It may be mentioned here that the FRFT may be more useful in the cases, where
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the signal has some desirable property in the FRFT domain, such as compact support, and lesser number of coefficients for it’s adequate representation (this applies specially for nonstationary/chirp type of signals). We now mention the delay property of the FRFT, which will be used in this correspondence for signal reconstruction problems for ready reference. This property relates the FRFT of the delayed signal to that of the original signal. a
f ðt sÞ $ expðjps2 sin a cos aÞ expðj2pus sin aÞFa ðu s cos aÞ:
ð6Þ
It is clear from (6) that as opposed to the case of the CFT, the support of the FRFT of the delayed signal f(t s) is different from the support of the FRFT of the signal f(t). In fact the Fa(u) gets shifted by s cos a in the FRFT of the delayed signal f(t s). Similarly, multiplication by an exponential signal exp (j2pst) to a signal f(t) results in a similar shift of Fa(u) by s sin a. An excellent and comprehensive discussion of the FRFT can be found in [4]. The MATLAB code for fast computation of the FRFT is also available at the Mathworks website [15]. 3. Signal reconstruction techniques from the undersampled signal samples Signal reconstruction from the given signal samples is an important signal processing operation, which is needed quite frequently in many applications in several diverse areas such as signal processing, communications, geophysics, radar and sonar, and optics including optical signal processing [4,16,17]. A large number of sampling theorems for the representation of various types of signals have also been derived in the literature including in the FRFT domains [7,11,16,18,21,22]. We consider here three different classes of signals, each being assumed to be bandlimited either in the CFD, or in the FRFT domains, or in the LCT domains. As is well-known, signals with compact support in the LCT domain need to have infinite support in the time domain. As a result, no practical signal of finite support is strictly bandlimited or bandpass in the LCT domains. However, the sampling theorems can be applied to signals whose energy is mostly concentrated in a finite interval in the LCT domains with a negligible nonzero reconstruction error. This applies specially to the chirp signals/nonstationary signals encountered in radar and sonar, etc. The optical signal processing involving the FRFT and the LCT may also be benefited from this. We discuss the lowpass and bandpass signal cases separately. 3.1. Signal class whose FRFT with angle a is bandlimited in some LCT domain with parameter M Lowpass signals: Here, we are considering the class of signals with the property that their FRFTs with angle a are bandlimited to uM in some LCT domain with parameter M.
247
We start with the Shannon sampling theorem [17,19] which states that a continuous-time CFD bandlimited signal can be reconstructed exactly from its samples taken at a rate, which is greater than or equal to the Nyquist rate by passing the signal samples through an ideal lowpass filter with a suitable cut-off frequency in the CFD. If the sampling frequency is less than the Nyquist rate, then aliasing will take place and it is not possible to reconstruct the original signal back by simply passing the samples through an ideal lowpass filter. To be specific, the CFT of the sampled signal, F sp=2 ðuÞ, can be written as F sp=2 ðuÞ ¼
1 1 X F p=2 ðu nus Þ; T s n¼1
ð7Þ
where Ts is the sampling interval and Fp/2(u) is the CFT of the signal f(t) assumed to be sampled by an ideal impulsetrain given by 1 X 2p dðt nT s Þ with us ¼ ¼ 2pfs : ð8Þ dT s ðtÞ ¼ Ts n¼1 On taking the inverse FRFT with angle (p/2 a) of (7) and using the delay property of the FRFT given in (6), we get 1 1 X expðjpn2 u2s sin a cos aÞ F sa ðuÞ ¼ T s n¼1 expðj2punus cos aÞF a ðu nus sin aÞ:
ð9Þ
We now take the LCT with parameter M of (9) on both sides to get RM ½F sa ðuÞ 1 1 X ¼ expðjpn2 u2s sin a cos aÞ T s n¼1 exp½jpnus cos aDð2u nus cos aBÞ exp½jp2ðu Bnus cos aÞnus sin aC n2 u2s sin2 aAC RM ½F a ðu nus sin aA Bnus cos aÞ: ð10Þ It is clear from (10) that the original signal f(u) can be obtained from the output of the lowpass filter by taking the inverse LCT with parameter M followed by inverse FRFT with angle a as the term corresponding to n = 0 in (10) can be easily filtered from the nearest term corresponding to n = 1 using the ideal lowpass filter in that particular LCT domain for the class of signals under consideration. Mathematically, the signal reconstruction formula can be expressed as
1 1 1 u s f ðtÞ ¼ F a RM rect ð11Þ RM ½F a ðuÞ ; Ts 2uM 1 where F 1 a and RM denotes inverse FRFT and inverse LCT operators. The sampling rate for perfect signal reconstruction must satisfy the condition given below
us ðA sin a þ B cos aÞ P 2uM :
ð12Þ
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Thus using the filtering in the LCT domain, we are able to reconstruct a signal belonging to a class which has infinite bandwidth in the CFT domain, from its samples by sampling it at a finite rate given by (12) and the sampling frequency can be less than twice the bandwidth of the signal in the CFT domain, which in fact is infinite here. The reconstruction formula given in (10) of [21] is a special case of (11) for a = 0. Thus the class of signals covered by (11) is bigger than the classes considered in [21]. Special cases: 1. Signals whose CFT is bandlimited in some LCT domain with parameter M This class of signals is obtained for the value of the parameter a = p/2. If we substitute a = p/2 in (10), we get 1 1 X RM ½S p=2 ðuÞ ¼ exp½ipð2unus C n2 u2s ACÞ T s n¼1 RM ½F p=2 ðu Anus Þ:
ð13Þ
The term in (13) corresponding to n = 0 can be easily filtered from the term corresponding to n = 1 using the ideal lowpass filter in that particular LCT domain provided Aus P 2uM :
ð14Þ
By choosing the value of parameter A to be very large, the sampling frequency can be less than the Nyquist rate, which in fact is infinite here. 2. Signals bandlimited in some LCT domain with parameter M This class of signals is obtained for the value of the parameter a = 0. If we substitute a = 0 in (10), we get the condition (9) of [21], i.e., Bus P 2uM :
ð15Þ
3. Signals which are bandlimited in some FRFT domain with a 5 np/2 This class of signals will not be bandlimited in the CFT domain as discussed in [11]. If we substitute B = sina in (15), we get the condition (15) of [21], i.e., us sin a P 2uM :
or uL is a harmonic of (A sin a + B cos a)us. If neither uM or uL is a harmonic of (A sin a + B cos a)us, it can be shown that the sampling rate must satisfy k1 ðA sin a þ B cos aÞus 6 2Bw ; and N 1 k ; ð17Þ ðA sin a þ B cos aÞus P 2Bw N where Bw = uM uL and k = uM/Bw. Here also k P N and N is a natural number. The signal reconstruction formula for this case can also be written as
1 u uC 1 s ðtÞ ¼ F 1 R rect ½F ðuÞ R M a Ts a M uM uL
1 u þ uC 1 s þ F 1 R rect ½F ðuÞ ; ð18Þ R M a Ts a M uM uL where uC = (uM + uL)/2. Special cases: 1. Signals whose CFT is a band-pass signal in some LCT domain with parameter M If we substitute a = p/2 in (17), we obtain the sampling rate criterion as given by k1 k Aus 6 2Bw ; and Aus P 2Bw : ð19Þ N 1 N 2. Signals whose LCT is a band-pass signal in some LCT domain with parameter M Similarly, if we substitute a = 0 in (17), we get the condition k1 k Bus 6 2Bw ; and Bus P 2Bw : ð20Þ N 1 N 3. Signals whose FRFT is a band-pass signal in some FRFT domain with angle a If we substitute B = sin a in (20), we get the condition for this class of signals as given by
ð16Þ
Ts
-Ts
The signal reconstruction formulas for these special cases can be easily obtained from (11) by substituting the corresponding values of a. We also illustrate the Eq. (16) in Figs. 1.1 and 1.2. Here the effect of chirp multiplication factors in (10) is neglected for clarity of illustration.
0
-2Ts
2Ts
Fig. 1.1. Ideal impulse-train sampling waveform.
3.2. Signal class whose FRF T with angle a is a band-pass signal in some LCT domain with parameter M Bandpass signals: If the LCT with parameter M of the signal Fa(u) is a bandpass signal with bandwidth extending from uL to uM, then along the lines of [19], it can be easily shown that the minimum sampling frequency allowable is (A sin a + B cos a)us = 2(uM uL) provided that either uM
• • •
• • • -2 us •sinα
•
- uM 0• uM
- us sinα
•
• 2 us sinα
us sinα
Fig. 1.2. Spectrum of the sampled signal in FRFT domain.
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249
4. Reduction of sampling rate using modified sampling scheme
ð23Þ
All the above sampling theorems are based on the ideal impulse-train sampling, which is unrealizable in practice. The natural sampling employing pulses of finite duration is often used in practice and this is further replaced by flat-top sampling for the ease of circuit realization [19]. Using a slight modification of the conventional natural sampling waveform as shown in Fig. 3, and using a specific width of the pulses, it is shown here that the sampling rate in our generalized sampling theorem (and hence in the Stern’s and the classical Shannon sampling theorem) can be further reduced by a factor of two. We discuss the modified sampling scheme for the Signal class, which is bandlimited in the CFT domain for the ease of illustration, although the proposed scheme is applicable to all the lowpass sampling theorems discussed above. The basic idea is to eliminate all the even harmonics and some selected odd harmonic of the sampling frequency from the spectrum of the sampling waveform. This increases the gap between the two successive terms in the spectrum of the sampled signal. This increased gap in the spectrum of the sampled signal can be exploited for either increasing the maximum frequency of the signal to be sampled or for reducing the sampling rate of the pulses used in the natural sampling by a factor of two. Here also we discuss the lowpass and bandpass signal cases separately. Lowpass signals: The basic technique proposed here can be considered as an extension of the natural sampling of signals by finite duration pulses as described in [19]. It is based upon the use of sampling waveform, which results in the elimination of all the even harmonics of the sampling frequency and selective elimination of the third or fifth or any other odd harmonic from the conventional spectrum of the sampled signal [20]. This allows us to sample the original signal at a rate below the Nyquist rate. For the sampling waveform shown in Fig. 3, accounting both the positive and negative pulses in a cycle, the conventional Fourier series of this can be written as [20] X 4 nd us SðtÞ ¼ sin sin n t: ð26Þ np 2 2 n¼1;3;5;...
Sampling frequencies for inverse placement of the spectrum [23] should similarly satisfy
Here due to the symmetry of the output voltage along the x-axis, all the even harmonics (n = 2, 4, 6, . . .) are absent.
Fig. 2. Shaded regions showing the allowed range of sampling frequencies for FRFT limited signals given in (21) for a = p/4.
us sin a 6 2Bw
k1 k ; and us sin a P 2Bw : N 1 N
ð21Þ
The allowed range of sampling frequencies given by (21) for a = p/4 is depicted as shaded regions in Fig. 2. The signal reconstruction formulas for these special cases can be easily obtained from (18) by substituting the required values of a. Although the aliasing problem is solved perfectly in (17), to demodulate correctly from RM[Sp/2](u) in the baseband spectrum as done in [23], sampling frequencies for normal placement of the spectrum [23] should satisfy 2uM uL 6 ðA sin a þ B cos aÞus 6 ; ð22Þ 2n þ 1 j kn where 0 6 n 6 m, m ¼ 2ðuMuLuL Þ and n is odd. Here the symbol bxc denotes the largest integer smaller than or equal to x. The robust sampling frequency [23] for normal placement of the spectrum can also be easily written as ðA sin a þ B cos aÞus ¼
2ðuM þ uL Þ : 4n þ 1
uM 2uL 6 ðA sin a þ B cos aÞus 6 ; ð24Þ n 2n j k 1 where 1 6 n 6 m, m ¼ 2ðuMuMuL Þ , and n is even. The robust sampling frequency for inverse placement of the spectrum can also be easily written as ðA sin a þ B cos aÞus ¼
2ðuM þ uL Þ : 4n 1
δ Ts 2π
• •
ð25Þ
It is clear from (17)–(25) that for proper values of the parameters A, B and a, the sampling rate can be less than the 2Bw.
0
• Ts/4
Fig. 3. Sampling waveform.
• • 3Ts/4
•
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Now for a specific choice of d, any of the third, fifth, seventh or any other odd harmonic can be eliminated from the conventional spectrum of the sampled signal. The higher the order of the harmonic, the lower will be the value of d allowing time-division multiplexing of many other signals in this scheme. To be specific, for d = 2p/3, the third harmonic will be absent from the spectrum and it can be easily seen that the sampling rate must satisfy the condition given below for exact signal reconstruction: ð27Þ
us P uM :
This sampling rate is precisely one-half the Nyquist rate given by Shannon sampling theorem. The original signal can be reconstructed from the given signal samples by passing the samples through a pair of BPFs (one having the bandwidth from us/2 to us/2 + uM and the other having the bandwidth from us/2 + uM to 5us/2 in CFT domain) and appropriate frequency translation of the spectrum to the lowpass frequency range as explained in Fig. 4. The signal reconstruction formula for the general case of elimination of the kth harmonic from the spectrum of the sampling waveform can be written as 1 ðk 2Þp 1 h n us oi f ðtÞ ¼ F F u þ ðk 2Þ 1 p=2 4 sin ðk2Þd 2 2 1 ðk þ 2Þp 1 h n us o i þ F p=2 F 2 u þ ðk þ 2Þ ðkþ2Þd 4 sin 2 2 h n 1 ðk 2Þp 1 us o i F F u ðk 2Þ þ 3 4 sin ðk2Þd p=2 2 2 h n 1 ðk þ 2Þp 1 us o i þ F p=2 F 4 u ðk þ 2Þ ; ðkþ2Þd 4 sin 2 2 ð28Þ u u1 fF p=2 ½f ðtÞSðtÞðuÞg; where F 1 ðuÞ ¼ rect uM u u2 F 2 ðuÞ ¼ rect fF p=2 ½f ðtÞSðtÞðuÞg; uM u þ u1 F 3 ðuÞ ¼ rect fF p=2 ½f ðtÞSðtÞðuÞg; uM u þ u2 F 4 ðuÞ ¼ rect fF p=2 ½f ðtÞSðtÞðuÞg; uM us uM us uM ; and u2 ¼ k þ : u1 ¼ ðk 2Þ þ 2 2 2 2 It is clear from this that it is possible to reconstruct the original signal from the signal samples taken below Nyquist rate. If flat-top sampling is utilized here for practical pur-
•• •
•
-5 us /2
• - u /2 s
0
•us / 2
• 5 u /2
poses, similar to the flat-top sampling considered in [19], then it will result in a distortion of the signal at the out put of the BPF but it can be eliminated at least theoretically by an equalizer having a x/sin x type of frequency response [19]. The second technique proposed below, illustrated in Fig. 5, is valid for the class of bandlimited signals in the CFT domain having a real and even symmetric spectrum around the folding frequency within a frequency range of ±dM, i.e., within us/2 ± dM, where us/2 + dM = uM. The typical spectrum of such signals is shown in Fig. 6. Let f(t) be a signal belonging to this class with bandwidth in the CFT domain uM sampled by an ideal impulse-train shown in Fig. 7. The spectrum of the sampled signal is shown in Fig. 7. We know that sampling a signal below the Nyquist rate results in the problem, called aliasing [17,19] which manifests itself in the corruption of the high frequency portion of the CFT of the signal. However, we have some portion in the spectrum of the signal f(t) up to the frequency of (us uM), which is still intact and unaffected by the aliasing. This part of the spectrum can be easily filtered in the CFT domain using an ideal lowpass filter (LPF) with cut-off frequency (us uM). We can further select the aliased portion using an ideal bandpass filter with cut-off frequencies (us uM) to uM and pass it through an LTI system with impulse response satisfying the condition Hp/2(u) = 1/2. The output of such system can be added to the signal at the output of the LPF. This total signal obtained is nothing but the CFT of the original signal from which the original signal can be easily recovered. 1 f ðtÞ ¼ F 1 ½F 1 ðuÞ þ F 2 ðuÞ; where T s p=2 " # )
( 1 X u F 1 ðuÞ ¼ rect dðt nT s Þ ðuÞ ; F p=2 f ðtÞ 2ðus uM Þ n¼1 " # )
( 1 X u us =2 dðt nT s Þ ðuÞ F 2 ðuÞ ¼ rect F p=2 f ðtÞ ð2uM us Þ n¼1 " # )
( 1 X u þ us =2 þ rect dðt nT s Þ ðuÞ : F p=2 f ðtÞ ð2uM us Þ n¼1 ð29Þ f(t) f(t)
Sampler
Low pass Filter
•••
s
us /2+ uM Fig. 4. Spectrum of the sampled signal in CFT domain.
Band-pass Filter
H
/2
(u ) = 1/ 2
Fig. 5. Signal reconstruction for CFT limited signals with real and even spectrum around folding frequency.
K.K. Sharma, S.D. Joshi / Optics Communications 268 (2006) 245–252
251
A1 A2 0
uM
us /2
us /2−δM
Fig. 6. Spectrum of the signal having even symmetry around folding frequency within ±dM.
BPF
LPF A1
•••
•••
2A2
0
us us / 2
us / 2 − δM
uM
Fig. 7. Spectrum of the sampled signal by sampling the signal with spectrum shown in Fig. 6.
Bandpass signals: We consider the sampling of the bandpass signals using the sampling waveform shown in Fig. 3. Here, as the sampling waveform has only odd harmonics present in it, the approach for finding the range of the sampling frequencies is slightly different from the approach in [19]. Suppose we want to avoid the aliasing in the range of frequencies occupied by the spectrum of the signal, i.e., between uL and uM in the positive portion of the spectrum of the sampled signal and consider its possible overlap due to (N 1)th and (N + 3)th shifts of the negative portion of the spectrum, assuming (N + 1)th harmonic is absent from the spectrum because of the proper choice of the width of the pulses used in the sampling, then it can be easily shown that the sampling rate (accounting both the positive and negative pulses in the sampling waveform) must satisfy 4kBw us 6 4ðk 1ÞBw =ðN 1Þ; and us P ; k P N: N þ3 ð30Þ Here the value of be largest odd integer less than j N must k L M, where M ¼ uM2uu . If we compare it with the samL pling rate given by (5.1.7) and (5.1.8) in [19], it is clear that the reduction in sampling rate will happen if k P N2kþ3 for a given value of Bw. This implies that N 3 P N. Thus the reduction in the sampling rate in case of bandpass signals will happen under this restricted condition only.
We also mention here another class of bandpass periodic signals with a given period, which can also be undersampled but the perfect signal reconstruction is still possible, at least theoretically. This is based on the fact that the spectrum of a periodic signal consists of impulse functions, located at the fundamental frequency and harmonics of it. Hence if the sampling frequency is not an integer multiple of one-half the fundamental frequency, then aliasing will result in the appearance of the frequencies which are not an integer multiple of the fundamental frequency and can be easily identified in the spectrum of the reconstructed signal and these can be filtered using ideal bandpass filters and these can be translated back to their original place by frequency translation methods and perfect signal reconstruction is possible. Thus we have shown the existence of some classes of signals bandlimited in the CFD, the LCT and the FRFT domains, for which the sampling below the Nyquist rate is done but perfect signal reconstruction from the signal samples is still possible. We would also like to mention that the equations similar to (7) are also encountered in a communication scenario related to the multi-path fading problem and this can also be taken care of, if these multipath components can be separated using the techniques discussed here for different classes of signals. The extension of the one-dimensional sampling waveforms discussed here to two-dimensional sampling waveforms/ grids is straightforward.
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5. Conclusions It was shown in this paper that there exists some classes of signals bandlimited as well as bandpass in the CFD, the FRFT, and the LCT domains, for which the sampling rate can be below the Nyquist rate but perfect signal reconstruction from the signal samples is still possible by performing some filtering/processing with the linear time-variant (LTV) systems using the FRFT, CFT and/ or the LCT. This may result in the more efficient representation for theses classes of signals and consequently it may further find its applications in the signal compression area. This may also be helpful in tackling the fading problem encountered in the multi-path environments in wireless communications. References [1] [2] [3] [4]
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