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A New Discrete-Time Analytic Signal for Reducing Aliasing in Discrete Time–Frequency Distributions

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A New Discrete-Time Analytic Signal for Reducing Aliasing in Discrete Time–Frequency Distributions 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP A NEW DISCRETE-TIME ANALYTIC SIGNAL FOR REDUCING ALIASING IN DISCRETE TIME–FREQUENCY DISTRIBUTIONS John M. O' Toole, Mostefa Mesbah and Boualem Boashash Perinatal Research Centre, University of Queensland, Royal Brisbane & Women's Hospital, Herston, QLD 4029, Australia. e-mail: [email protected] web: www.som.uq.edu.au/research/sprcg/ ABSTRACT tained from the finite discrete-time signal s(nT) of length N, The commonly used discrete-time analytic signal for discrete where T represents the sampling period. time–frequency distributions (DTFDs) contains spectral en- The requirements for an alias-free discrete-time, discrete- ergy at negative frequencies which results in aliasing in the frequency TFDs are as follows: a discrete-time TFD requires DTFD. A new discrete-time analytic signal is proposed that that a signal with half the Nyquist bandwidth is used, which approximately halves this spectral energy at the appropri- is the case for the analytic signal [2]. A discrete-frequency ate discrete negative frequencies. An empirical comparison TFD requires that a signal with half the time duration is shows that aliasing is reduced in the DTFD using the pro- used, that is s(nT) is zero for N < n ≤ 2N − 1 [3]. Thus posed analytic signal rather than the conventional analytic these two conditions must be combined to produce a discrete- signal. The time domain characteristics of the two analytic time, discrete-frequency TFD (simply referred to as a dis- signals are compared using an impulse signal as an exam- crete TFD, (DTFD)). As the (t; f ) kernel g is independent of ple, where the DTFD of the conventional signal produces the signal, the formation of the discrete WVD (DWVD) will more artefacts than the DTFD of the proposed analytic sig- now be examined. nal. Furthermore, the proposed discrete signal satisfies two The DWVD for a signal s(nT) can be represented as [4] important properties, namely the real part of the analytic sig- nT k nal is equal to the original real signal and the real and imag- Wzc ( 2 ; 2NT ) inary parts are orthogonal. N−1 ∗ − j2p(m−n=2)k=N = ∑ zc(mT)zc((n − m)T)e (2) 1. INTRODUCTION m=0 TFDs are two dimensional representations of signal energy for n = 0;1;:::;2N − 1 and k = 0;1;:::;N − 1. This particu- in the joint time–frequency (t; f ) domain [1]. A commonly lar DWVD, based on the one proposed in [3], is presented used class of TFDs is the quadratic shift-invariant TFD class here as it satisfies more desirable mathematical properties (henceforth referred to as simply TFDs). The most basic than the more common DWVD definition proposed in [5]. TFD of this class is the Wigner-Ville distribution (WVD) de- The implications, however, arising from the choice of ana- fined, for a real signal s(t), as lytic signal are the same regardless of the DWVD definition. ∞ The signal zc(nT), periodic in 2NT, is defined as t ∗ t − j2pt f Wz(t; f ) = z(t + 2 )z (t − 2 )e dt Z−∞ z(nT); 0 ≤ n ≤ N − 1 zc(nT) = (3) where z(t) is the analytic associate of s(t) [1, pp. 13]. The (0; N ≤ n ≤ 2N − 1 reason the analytic associate, z(t), is used rather than real signal s(t) is twofold: firstly it eliminates the crossterms be- where z(nT) represents the discrete-time analytic associate tween the positive and negative frequency components in the of s(nT), which will be discussed in more detail in the next (t; f ) domain and secondly it makes an alias-free discrete- section. The same DWVD can also be expressed in terms of k time representation possible [2]. The general form of this the discrete-frequency domain signal Zc( 2NT ), class of TFDs can be expressed using the (t; f ) kernel g(t; f ) as follows: nT k WZc ( 2 ; 2NT ) rz(t; f ) = Wz(t; f ) ∗ ∗ g(t; f ) (1) t f 1 2N−1 = Z ( l )Z∗( 2k−l )e jp(l−k)n=N (4) ∗ ∑ c 2NT c 2NT where represents the convolution operation. 2N l=0 1.1 Discrete Time–Frequency Distributions where WZc = Wzc and Zc is obtained from the discrete Fourier The implementation of TFDs for digital signal processing transform (DFT) of zc. purposes requires a discrete version of the continuous TFD defined in (1). This requires that both the time t and fre- 1.2 Discrete-Time Analytic Signals quency f arguments in rz(t; f ) be discretised. As TFDs are The discrete-time analytic signal z(nT) used in (3) is formed obtained from mapping a time domain signal s(t), then like- using a frequency domain method described in [6]. The wise discrete-time, discrete-frequency TFDs need to be ob- method zeros all the discrete frequency domain samples ©2007 EURASIP 591 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP at negative frequencies. This is achieved by multiplying discrete spectrum (implicit from its definition [6]), zc(nT) is k k k not. This is because the Fourier transform of the the discrete- S( NT ), the DFT of s(nT), by H( NT ). That is, Z( NT ) = k k k time signal z(nT), Z( f ), is only zero at the sample points H( NT )S( NT ), with H( NT ) defined as f = k=NT for N < k ≤ 2N − 1. As zc(nT) is of length 2N, N then Zc( f ) will be nonzero for the odd negative frequency 1; k = 0 and k = 2 ; sample points f = (2k + 1)=2NT. Thus the DWVD in (2) is k N H( NT ) = 82; 1 ≤ k ≤ 2 − 1; (5) not entirely alias-free. In an effort to reduce this aliasing, a N new discrete-time analytic signal will now be introduced. <>0; 2 + 1 ≤ k ≤ N − 1: The analytic signal z(nT:> ) is then obtained by taking the in- 2. PROPOSED ANALYTIC SIGNAL k verse DFT (IDFT) of Z( NT ). The proposed analytic signal is defined as The justification for using this method is now exam- z (nT); 0 ≤ n ≤ N − 1 ined, with alternative methods presented. Preceding this, two z (nT) = a (8) properties inherent to the continuous-time analytic signal are p 0; N ≤ n ≤ 2N − 1 introduced in the discrete-time context. The properties were proposed in [6] in order for z(nT) = for n = 0;1;:::;2N − 1. The signal za(nT) is defined as (r) (i) the analytic associate of zc(nT), obtained using the fre- z (nT) + jz (nT) to be an “analytic-like” discrete-time k signal. First, the real part of the discrete-time analytic sig- quency domain method described in [6]. That is, Za( 2NT ) = k k k nal is exactly equal to the original real signal s(nT), i.e. H( 2NT )Zc( 2NT ), where H( 2NT ) is described in (5) with N k replaced by 2N. Taking the IDFT of Za( ) results in (r) 2NT z (nT) = s(nT) for 0 ≤ n ≤ N − 1 (6) za(nT). The proposed discrete analytic signal is then ob- tained by forcing za(nT) to zero for N ≤ n ≤ 2N − 1. and second, the orthogonality between the real and imaginary The two analytic signals can be described completely in components of the signal is maintained, i.e. the time domain as a function of the real signal s(nT) (of length N) as N−1 z(r)(nT)z(i)(nT) = 0: (7) ∑ ( ) = ( ) ( ) ~ ˆ( ) ( ) n=0 zc nT s˜ nT u1 nT h nT u1 nT (9) h ~ i Alternative methods to creating the discrete-time analytic zp(nT) = s˜(nT)u1(nT) h(nT) u1(nT) (10) signal z(nT) of length N include using dual quadrature FIR where s˜(nT) is equal to s(nT) periodically extended to length filters to jointly produce the real and imaginary components 2N and where ~ represents circular convolution. The func- of z(nT), as described in [7]. This approach preserves the tion u1(nT) is defined as a time-reversed and time-shifted orthogonality property (7) but will not preserve the original step function, described as u (nT) = u((N −1−n)T), where real signal (6) [6]. Another approach is to approximate the 1 H u(nT) represents the unit step function. Thus the term Hilbert transform operation [·] with an FIR filter [7], as s˜(nT)u (nT) is equal to zero padding s(nT) to 2N. h(nT), the analytic signal is related to the real signal by the rela- 1 N H( k ) tion z(nT) = s(nT) + jH [s(nT)]. This approach preserves of length 2 , is the IDFT of the transfer function 2NT the original real signal (6) but not the orthogonality property defined in (5), which equates to (7). Both of these approaches are implemented in the time ( ); ; domain. d nT n even h(nT) = j pn (11) A more recent approach has been proposed in [8], which ( N cot( 2N ); n odd: combines the frequency domain method of [6] with the zero- ing of a single value of the continuous spectrum in the nega- where d(nT) is the Kronecker delta function, i.e. d(nT) = 1 tive frequency range. This method preserves the original real if n = 0 and 0 for n 6= 0. hˆ(nT) can be expressed in terms of signal (6) but not the orthogonality property (7). h(nT) as In comparison, the frequency domain method [6] is par- hˆ(nT) = h(nT) + h((n + N)T): (12) ticularly attractive for three reasons.
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