1 A New Discrete Analytic Signal for Reducing in the Discrete Wigner–Ville Distribution John M. O’ Toole, Mostefa Mesbah, Boualem Boashash

Abstract— It is not possible to generate an alias-free discrete the original signal is recoverable from the analytic signal as Wigner–Ville distribution (DWVD) from a discrete analytic ℜ[z(t)] = s(t). The second, which we call the orthogonality signal. This is because the discrete analytic signal must satisfy property, states that the real and imaginary components of the two mutually exclusive constraints. We present, in this article, a new discrete analytic signal that improves on the commonly used analytic signal are orthogonal. That is, the inner product of discrete analytic signal’s approximation of these two constraints. the real and imaginary parts is zero, hℜ[z(t)], ℑ[z(t)]i = 0 Our analysis shows that—relative to the commonly used signal— The discrete WVD (DWVD) requires a discrete analytic the proposed signal reduces aliasing in the DWVD by approx- signal. (We use the term discrete analytic signal to refer to a imately 50%. Furthermore, the proposed signal has a simple discrete version of the continuous analytic signal, even though implementation and satisfies two important properties, namely, that its real component is equal to the original real signal and this discrete signal is not an of a continuous that its real and imaginary components are orthogonal. complex variable [4].) Various methods for forming discrete analytic signals exist. We classify these methods as either Index Terms— discrete Wigner–Ville distribution (DWVD), analytic signal, discrete Hilbert transforms, antialiasing, time– time-domain [5], [6] or frequency-domain [4], [7] based frequency analysis methods. The most commonly used approach [3], [8]–[10] for generating discrete analytic signals for the DWVD is a frequency-domain method [4]. This method, which can be I. INTRODUCTION simply implemented, results in a discrete analytic signal that Neither time- nor frequency-domain based methods are satisfies the recovery and orthogonality properties. suitable for analysing nonstationary signals. Time–frequency methods, which represent the signal in the joint time– frequency domain, provide appropriate analysis tools—they B. An Alias-free Discrete Wigner–Ville Distribution are able to display the time-varying frequency content which The DWVD is discrete in both the time and frequency characterise nonstationary signals. directions—a requirement for digital appli- The Wigner–Ville distribution (WVD) is an example of cations [9]. The DWVD is formed from either a discrete-time a time–frequency domain representation. The distribution is signal z[n] or its discrete (DFT) counterpart one of the more widely studied types of time–frequency Z[k]. To obtain an alias-free DWVD, the N-point discrete representations, and is known as the fundamental distribution signal must satisfy two constraints [11]. First, Z[k] must in the class of quadratic time–frequency distributions [1]. The be zero at the Nyquist frequency term and for all negative WVD uses the analytic associate of the real-valued signal, frequencies—that is, Z[k] = 0 for N/2 ≤ k ≤ N − 1. (The rather than the real-valued signal itself [2]. The WVD is free segment N/2

C. A New Discrete Analytic Signal and orthogonality property is We present, in this article, a new discrete analytic signal N−1 based on the commonly used discrete analytic signal [4]. ℜ z[n] ℑ z[n] = 0. (6) Like the commonly used signal, which we refer to as the n=0 X   conventional discrete analytic signal, the proposed signal (We shall refer to the discrete analytic signal simply as an satisfies the time-constraint of (1) but only approximates the analytic signal when the context is clear.) frequency-constraint of (2). We measure this approximation by 2) Review: We classify existing methods for forming ana- quantifying the amount of energy in the ideally-zero region lytic signals as either time- or frequency-domain based meth- in (2). We use this measure to compare the performance of ods. First, we look at two time-domain based methods. One the proposed and conventional discrete analytic signals. In method uses dual quadrature FIR filters to jointly produce the addition, we numerically compare, using a number of signals, real and imaginary components of z[n], as described in [6]. the amount of aliasing energy in the DWVD of the two discrete The resultant analytic signal satisfies the orthogonality prop- analytic signals. erty but not the recovery property [4]. The other method forms We found that the proposed signal, relative to the conven- the analytic signal using the relation z[n] = s[n]+ jH(s[n]), tional signal, reduces aliasing in the DWVD by approximately by approximating the operation H(·) with one-half. This result agrees with our initial result—that the an FIR filter [6]. The resultant analytic signal satisfies the proposed signal, relative to the conventional signal, better ap- recovery property but not the orthogonality property. proximates the frequency-constraint of (2). Also, the proposed Next, we look at two frequency-domain based procedures. signal satisfies the recovery and orthogonality properties and One method forms the analytic signal by setting the negative can be computed simply using DFTs. frequency samples to zero [4]. This method, originally pro- posed in discrete Hilbert transform form by Cˇ ´ızekˇ [13] and Bonzanigo [14], uses the DFT and inverse DFT (IDFT) to II. DISCRETE ANALYTIC SIGNALSFORTHE DISCRETE switch between the time and frequency domains. The method, WIGNER–VILLE DISTRIBUTION which we shall refer to as the Cˇ ´ızek–Bonzanigoˇ method, A. Discrete Wigner–Ville Distribution satisfies both properties. The other frequency-domain based method [7] is a modified version of the Cˇ ´ızek–Bonzanigoˇ To compute the DWVD of an N-point real-valued discrete method; it has the additional step of zeroing an extra single signal s[n], we first form the 2N-point discrete analytic signal value of the continuous spectrum in the negative frequency zc[n] from s[n]. Then, we use zc[n], or its DFT associate Zc[k], to define the DWVD [11] as range. The method satisfies the recovery property but not the orthogonality property. 2N−1 Comparative to the other methods, the analytic signal pro- ∗ −jπ(m−n/2)k/N W [n, k]= zc[m]zc [n − m]e (3) duced by the Cˇ ´ızek–Bonzanigoˇ method is particularly attrac- m=0 tive for the following reasons: X2N−1 1 ∗ π − • its negative frequency samples are exactly zero; = Z [l]Z [k − l]e j (l k/2)n/N . (4) 2N c c • it preserves the recovery and orthogonality properties; l=0 X • it has a simple implementation [4]—no filter design [6] or The signals zc[n] and Zc[k] are periodic in 2N. We present this selection of an arbitrary frequency point [7] is necessary. particular DWVD definition here as it satisfies more desirable The commonly used procedure for obtaining an analytic mathematical properties than other DWVD definitions [5], signal for a DWVD uses the Cˇ ´ızek–Bonzanigoˇ method. The [10], [11]. The implications, however, arising from the choice complete procedure, for the N-point real-valued signal s[n], of analytic signal are the same regardless of the DWVD is as follows [3], [8]–[10]: definition. To generate z [n] from s[n] involves a number of c 1) take the DFT of signal s[n] to obtain S[k]; steps, which we describe in the next subsection. 2) let Zc[k]= Hc[k]S[k], with Hc[k] defined as

and N B. Discrete Analytic Signals b b 1, k =b 0 k = 2 , N Hc[k]= 2, 1 ≤ k ≤ 2 − 1, 1) Desirable Properties: There is no unique definition  0, N + 1 ≤ k ≤ N − 1; for a discrete analytic signal. The discrete definition should,  2 b however, conserve as many properties inherent to the contin-  uous analytic signal as possible. Marple [4] proposed that a 3) take the IDFT of Zc[k] to obtain zˆc[n] (of length N); discrete analytic signal should at least satisfy the recovery and 4) let zc[n] equal zˆc[n] zero-padded to length 2N; we call orthogonality properties. zc[n] the conventionalb analytic signal. We detail these two properties as follows. For a discrete The last step ensures that zc[n] satisfies the time-constraint analytic signal z[n], associated with the N-point real-valued of (1), and therefore not the frequency-constraint of (2). In ˇ signal s[n], the recovery property is addition, the C´ızek–Bonzanigoˇ method does not zero the Nyquist frequency term, which also violates the frequency- ℜ z[n] = s[n], for 0 ≤ n ≤ N − 1 (5) constraint.  3

III. PROPOSED DISCRETE ANALYTIC SIGNAL 0.6 Conventional Proposed The procedure to form the proposed analytic signal zp[n] 0.4 [15], from the N-point real-valued signal s[n], is as follows: 0.2 1) zero-pad s[n] to length 2N; call this sa[n]; 2) take the DFT of sa[n] to obtain Sa[k]; 3) let 0

Za[k]= Ha[k]Sa[k] (7) -0.2 where H [k] is defined as a -0.4

1, k = 0 and k = N, -0.6 Ha[k]= 2, 1 ≤ k ≤ N − 1, (8)  0 20 40 60 80 100 120 0, N + 1 ≤ k ≤ 2N − 1; Time (samples)  4) take the IDFT of Za[k] to obtain za[n]; Fig. 1. Imaginary part of the conventional and proposed analytic signals formed from the N-point impulse signal, where N = 64. 5) and lastly, force the second half of za[n] to zero z [n], 0 ≤ n ≤ N − 1, z [n]= a p 0, N ≤ n ≤ 2N − 1. ( zp[n] are zero for N ≤ n ≤ 2N − 1, because the presence of ut[n] in (9) and (10) guarantee that both signals satisfy Steps 2 to 4 implements the Cˇ ´ızek–Bonzanigoˇ method on the time-constraint. Also, zc[n] has a significant negative the zero-padded signal sa[n] [16]. We, therefore, do the component around n = N − 1, whereas zp[n] does not. The zero-padding process before we generate the signal za[n], relation in (11) explains this difference. unlike the procedure for zc[n], where we do the zero-padding process last. The last step ensures that zp[n] satisfies the time- constraint of (1), although at the expense of the frequency- B. Frequency-Domain Analysis constraint of (2). We can easily show that, like the conventional analytic In the frequency domain, the two analytic signals as a signal, the proposed signal satisfies the recovery and orthog- function of Sa[k] are onality properties presented in Section II-B.1. Zc[k]= Sa[k]Hc[k] ⊛ Ut[k] (12) ⊛ A. Time-Domain Analysis Zp[k]= Sa[k]Ha[k] Ut[k]. (13) The two analytic signals are related to the 2N-point real- where S [k] is the DFT of s [n] and U [k] is the DFT of u [n]. valued signal sa[n] as follows: a a t t The frequency-response function Hc[k] is zc[n]= sa[n] ⊛ hc[n] ut[n] (9) ⊛ zp[n]= sa[n] ha[n] ut[n] (10) 2Ha[k], k even, Hc[k]= (14) where ⊛ represents circular convolution. The time-reversed (0, k odd, and time-shifted step function ut[n] is defined as ut[n] = u[N −1−n], where u[n] represents the unit step function. The with Ha[k] defined in (8). Because of the convolution with Ut[k] in (12) and (13), neither Zc[k] nor Zp[k] satisfy the impulse function ha[n] is the IDFT of the frequency-response frequency-constraint. function Ha[k], defined in (8). We can show that this impulse function equates to To illustrate the difference between the two analytic signals’ spectra, we use the impulse signal once more. These results δ[n], n even, are displayed in Fig. 2. For this signal S [k] = 1 for all values h [n]= a a j πn of k. Neither signal satisfies the frequency-constraint. The ( N cot ( 2N ), n odd, conventional analytic signal’s approximation—comparative to where δ[n] is the Kronecker delta function. The relation the proposed analytic signal—of the frequency-constraint, between the two convolving functions hc[n] and ha[n] is however, is marred by significant oscillation between the odd and even values of . The oscillatory nature of causes hc[n]= ha[n]+ ha[n + N]. (11) k Hc[k] this behaviour, as described by (14). The presence of ut[n] in (9) and (10) guarantees that zc[n] and zp[n] both satisfy the time-constraint. To highlight the differences between the two analytic sig- IV. PERFORMANCE OF PROPOSED ANALYTIC SIGNAL nals, we use the N-point impulse signal s[n] = δ[n] as an example. As both analytic signals preserve the real-valued In this section, we examine the performance of the proposed signal, only the imaginary components for the signals are analytic signal—relative to the conventional analytic signal— plotted in Fig. 1. As expected, the imaginary parts of zc[n] and at approximating the frequency-constraint of (2). 4

TABLE I 1 Conventional PERFORMANCE RATIO MEASURES COMPARING THE Proposed PROPOSED WITH THE CONVENTIONAL ANALYTIC 0.8 SIGNAL. η η 0.6 Signal Type (N even) (N odd) Impulse 0.5078 0.4711 0.4 Step 0.4137 0.4136 Sinusoid 0.4595 0.5709 0.2 NLFM 0.5055 0.4999 WGNac 0.5883 (0.0976) 0.5319 (0.0983) b Magnitude Squared (normalised) EEG 0.4184 (0.0671) 0.4155 (0.0650) 0 -0.5 0 0.5 a 1000 realisations used. b 1000 epochs used. Normalised Frequency a,b Values are in the form, mean (standard deviation).

Fig. 2. Discrete spectra of the two analytic signals formed from the test impulse signal. nonlinear frequency modulated signal (NLFM) signal, white Gaussian noise (WGN), and a real-world signal. This last signal is an electroencephalogram (EEG) recording from a A. Relative Performance newborn baby. The length N for each signal was arbitrarily We use the signals’ spectral energy, at the Nyquist and set to even values between 14 and 2048; 1 was added to this negative frequencies, as a relative performance measure. The value to obtain N odd. following proposition describes this measure. The results, in Table I, for most of the test signals confirm Proposition 1: The spectral energy relation of Zp[k] and the approximation stated in (16). The exceptions to this include Zc[k], at Nyquist and negative frequencies, is the WGN realisations, where the mean ratio value is < 0.6, and the sinusoidal signal when N is odd, where the ratio value 2N−1 2 is also < 0.6. Z [k] p In addition, we plot, in Fig. 3, the spectra of the conventional k=N X − and proposed analytic signals using two of the tests signals: the 2 N 1 2 1 2 1 2 sinusoidal signal with N = 14, and an EEG epoch with N = = Zc[k] + Zp[N] − Z[0] − C (15) 2 2 99. Note, from Fig. 3 and Fig. 2, that the amount of energy in k=N   X the negative spectral region is signal dependent, but the ratio b where Z[k]= Za[k] − Zp[k], with Za[k] defined in (7), and η comparing the analytic signals remains approximately the same. b 0, N even, C = 2 2 Z[N] , N odd. V. REDUCED ALIASED DWVD  Proof: See Appendix I. This section compares the performance of the analytic

From (15), we see that b the energy relation between Zp[k] signals by their contribution to aliasing in the DWVD. and Zc[k] is dependent on the value of Za[k] and Zp[k] at and . If the second term in the right hand side k = 0 k = N A. Aliasing in the DWVD of z [n] of (15) is small, relative to the first term, then we can rewrite p (15) as To begin, we recall how we quantify aliasing in the discrete- 2N−1 2N−1 time domain. Consider a continuous-time signal y(t), bandlim- 2 1 2 Z [k] ≈ Z [k] . (16) ited in the frequency-domain to the region |f| < 1/2T . We p 2 c k=N k=N sample y(t), with sampling period T , to obtain the discrete- X X time signal y[n]. This signal y[n] is alias free because the This equation states that the spectral energy for Zp[k] is periodic copies in the frequency domain for y[n] do not approximately half of the spectral energy for Zc[k] in the specified range. We numerically verify this approximation in overlap. Now consider another discrete signal y1[n], obtained the next subsection using a number of test signals. by sampling y(t) with sampling period T1 = 2T . This discrete signal y1[n] is aliased because the periodic copies in the frequency-domain do overlap. If we know the spectral content B. Numerical Examples for y(t), then we are able to measure the spectral periodic This section provides examples to confirm the approxima- overlap for y1[n], and are therefore able to quantify the aliasing tion in (16). To start, we define the ratio in y1[n]. Similarly, to evaluate aliasing in the DWVD, we 2N−1 2 measure spectral content in a specific region of the doppler– Zp[k] η = k=N . frequency domain. 2N−1 2 The asymmetrical doppler–frequency function of the ana- Pk=N Zc[k] lytic signal Z[k], defined as Next, we compute this ratioP with six different signal types: ∗ an impulse function, a step function, a sinusoidal signal, a K[l,k]= Z[l]Z [k − l] 5

TABLE II 1 Conventional Proposed PERFORMANCE RATIO MEASURES COMPARING ALIASING IN THE DWVD OF THE PROPOSED ANALYTIC 0.8 SIGNAL WITH THE DWVD OF THE CONVENTIONAL 0.013 ANALYTIC SIGNAL. 0.6 µ µ Signal Type (N even) (N odd)

0.4 Impulse 0.4034 0.3750 Step 0.4101 0.4099 Sinusoid 0.4545 0.5600 0.2 NLFM 0.5055 0.4998 0 WGNa 0.5840 (0.0984) 0.5371 (0.0996)

Magnitude Squared (normalised) -0.25 0 EEGb 0.4140 (0.0723) 0.4112 (0.0705) 0 -0.5 0 0.5 a 1000 realisations used. b 1000 epochs used. Normalised Frequency a,b Values are in the form, mean (standard deviation). (a)

N N 1 Conventional 2 Proposed + K[l,k] . k=0 l=k+1 0.8 X X

0.103 As the function K[l,k] is quadratic in Z[k], cross-terms 0.6 between the positive and negatives frequencies will be present in the resultant DWVD. These cross-terms are part of the spectral-leakage in the doppler–frequency function, and are 0.4 therefore incorporated into the α(K) measure. We consider these cross-terms as aliasing as they would not be present if 0.2 the frequency-constraint was satisfied. 0

Magnitude Squared (normalised) -0.25 0

0 -0.5 -0.25 0 0.25 0.5 Normalised Frequency B. Numerical Examples (b) We present the results in Table II for the same set of example Fig. 3. Discrete spectra comparing the two analytic signals for two test signals used in Section IV-B. The results for µ are not equal to signals: (a) a sinusoidal signal, and (b) an EEG epoch. The inset plots show η because the doppler–frequency function K[l,k] is quadratic a portion of the negative frequency axis with a reduced magnitude range. in the signal Z[k]. We see from the results, however, that µ approximates η for all test signals apart from the impulse signal, where µ is less than η. From these results we infer that is used to form the DWVD in (4). If we assume that Z[k] satis- the amount of spectral-leakage for Kp[l,k] is approximately fies the frequency-constraint of (2), then the nonzero content half of the spectral-leakage for Kc[l,k]. Hence, the amount of (or energy) in K[l,k] is contained within a specific region. aliasing present in the DWVD of zp[n] is approximately half Any energy, however, outside this region results in aliasing of the aliasing present in the DWVD of zc[n]. in the DWVD. We refer to this undesirable phenomenon as To show some examples of this reduced aliasing, we plot the spectral-leakage. DWVDs of the two analytic signals using two different signals As Zc[k] does not satisfy the frequency-constraint, its from the test set—namely, the impulse signal and an EEG doppler–frequency function Kc[l,k] contains spectral-leakage. epoch. Fig. 4 shows the two DWVDs of the impulse signal. This is true also for the doppler–frequency function of Zp[k], For this signal, the energy in the DWVD should, ideally, be Kp[l,k]. We quantify this spectral-leakage by summing the concentrated around the time sample n = 0, as δ[n] = 0 squared error over this region, where the ideal here is zero. for n > 0. Fig. 4 shows that the DWVD of the proposed Accordingly, to asses the relative merit of the proposed ana- analytic signal better approximates this ideal compared with lytic signal, we use the ratio squared error measure the DWVD of the conventional analytic signal. In Fig. 5 we show the two DWVDs using the EEG epoch. α(K ) µ = p We previously plotted the spectra of the two analytic signals α(Kc) for this EEG epoch in Fig. 3b. From this frequency-domain plot, we can see that very little relative energy is present where α(K) is a measure of the two-dimensional spectral- above the normalised frequency value of 0.25. Thus, we expect leakage for K[l,k], defined as little energy in the DWVD above the frequency value 0.25. 2N−1 2N−1 2N−1 k−N Accordingly, from Fig. 5, we see that the DWVD of the 2 2 α(K)= K[l,k] + K[l,k] proposed signal has less energy in this region compared with k=0 l=N k=N l=0 that for the DWVD of the conventional analytic signal. X X X X

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Fig. 4. DWVDs of the two analytic signals using the impulse test signal δ[n]: absolute value of the DWVD for the (a) conventional analytic signal, and (b) proposed analytic signal. Both DWVDs are normalised.

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Fig. 5. DWVDs of the two analytic signals using an EEG epoch: absolute value of the DWVD for the (a) conventional analytic signal, and (b) proposed analytic signal. To highlight the differences between the distributions, (c) and (d) display a portion of the distribution where, for this particular signal, we expect little energy. The plot in (c) is one-half of the distribution in (a); likewise, the plot in (d) is one-half of the distribution in (b). Both DWVDs are normalised. 7

VI. CONCLUSION Last, we consider the energy at negative frequencies for The failure of a discrete analytic signal to satisfy both a Zp[k]. By combining (13) and (18), Zp[k] for odd k values is finite-time and finite-frequency bandwidth constraint causes 1 Z [2k +1] = S [2k + 1]H [2k + 1] aliasing in the DWVD. We presented, in this article, a new p 2 a a discrete analytic signal which, compared with the conventional N−1 1 discrete analytic signal, better approximates these constraints + S [2l]H [2l]U [2k + 1 − 2l]. (22) 2N a a t and consequently reduces aliasing in the DWVD. We showed l=0 X that DWVD of our proposed analytic signal has approximately Thus, relating (22) with (20) and (7), we get 50% less aliasing than that for the DWVD of the conven- 1 tional analytic signal. The proposed signal retains two useful Zp[2k +1] = Za[2k +1]+ Zc[2k + 1] . (23) attributes of the conventional signal: it satisfies the recovery 2 and orthogonality properties and has a simple implementation From (8), we know that Za[k] = 0 for k > N; therefore, (23) using DFTs. reduces to Zp[2k +1] = Zc[2k + 1]/2 for (2k + 1) > N. If we combine this relation with (21), then we get the negative APPENDIX I spectral energy relation, PROOF OF PROPOSITION 1 N−1 2N−1 2 1 2 We derive the relation, for the negative spectral energy, Z [2k + 1] = Z [k] . (24) p 4 c between the proposed and conventional analytic signals. To do N k=N+1 k=X⌈ 2 ⌉ X so, we decompose the energy of Zp[k] for N ≤ k ≤ 2N − 1 as follows: B. Case for k even 2N−1 N−1 2 2 2 We start by introducing a new signal zˆ[n], defined as Zp[k] = Zp[N] + Zp[2k] k=N N X k=⌊X2 ⌋+1 0, 0 ≤ n ≤ N − 1, N−1 zˆ[n]= (25) 2 (za[n], N ≤ n ≤ 2N − 1. + Zp[2k + 1] (17) N The signal zˆ[n] is purely imaginary because the real part of k=X⌈ 2 ⌉ za[n] is zero for N ≤ n ≤ 2N − 1. We can also express zˆ[n] where the function ⌊x⌋ returns an integer smaller than or as zˆ[n] = za[n] − zp[n] for all values of n. In the frequency equal to x, and the function ⌈x⌉ returns an integer larger domain, this equates to than or equal to x. We examine each part of the preceding decomposition separately. Z[k]= Za[k] − Zp[k]. (26) where represents the DFT of . We now introduce Z[k] b zˆ[n] A. Case for k odd some properties of Z[k]. First, we write Ut[k], which is the DFT of ut[n], as We canb show easily, because of the form of zˆ[n] in (25), that b Nδ[k], k even, N−1 N−1 U [k]= (18) 2 2 t πk Z[2k] = Z[2k + 1] . (27) (1 − j cot ( 2N ), k odd. k=0 k=0 Next, we consider the energy at negative frequencies for X X Also, because bis purely imaginaryb then following sym- Z [k]. From (12) and (14), we express Z [k] as zˆ[n ] c c metry holds: − ∗ 1 N 1 Z[2N − k]= −Z [k]. (28) Z [k]= S [2l]H [2l]U [k − 2l]. c N a a t l=0 We express the spectral energy for Z[2k], using the sym- X b b If we use (18) in the preceding equation, then we can write metrical relation in (28), as for even–odd values as − − b Zc[k] k N 1 2 2 N 1 2 Z[2k] = Z[0] + 2 Z[2k] + A. (29) Zc[2k]= Sa[2k]Ha[2k] (19) k=0 k= N +1 N−1 X ⌊X2 ⌋ 1 b b b Z [2k +1] = S [2l]H [2l]U [2k + 1 − 2l]. (20) c N a a t where A = |Z[N]|2 when N is even and A = 0 when N is Xl=0 odd. The even k terms for Zc[k] do not contribute to the negative Similarly, theb spectral energy at k odd values of Z[k] is spectral energy because, from (8), Ha[2k] = 0 for 2k > N. N−1 N−1 Thus, the energy in the negative spectral region is solely 2 2 Z[2k + 1] = 2 Z[2k + 1] + B.b (30) caused by the odd k terms in Zc[k], k=0 N k=⌈ 2 ⌉ 2N−1 N−1 X X 2 2 b b Zc[k] = Zc[2k + 1] . (21) where B = 0 when N is even and B = |Z[N]|2 when N is k=N+1 N X k=X⌈ 2 ⌉ odd. This concludes the segment on the properties of Z[k]. b b 8

If we substitute (30) and (29) into (27), then we obtain: [3] B. Boashash, “Note on the use of the Wigner distribution for time– frequency signal analysis,” IEEE Trans. Acoust., Speech, Signal Pro- − − N 1 2 N 1 2 cessing, vol. 36, no. 9, pp. 1518–1521, 1988. Z[2k] = Z[2k + 1] [4] S. L. Marple, Jr., “Computing the discrete-time ‘analytic’ signal via N N FFT,” IEEE Trans. Signal Processing, vol. 47, no. 9, pp. 2600–2603, k=⌊ 2 ⌋+1 k=⌈ 2 ⌉ X X 1999. b 1 b 2 [5] T. Claasen and W. Mecklenbrauker,¨ “The Wigner distribution—a tool for − Z[0] + A − B . time–frequency signal analysis. Part II: discrete-time signals,” Philips J. 2 Research, vol. 35, pp. 276–350, 1980.   [6] A. Reilly, G. Frazer, and B. Boashash, “Analytic signal generation—tips Then we substitute (24), and the relation b Z[k] = −Zp[k] for and traps,” IEEE Trans. Signal Processing, vol. 42, no. 11, pp. 3241– k > N, into the preceding equation to obtain: 3245, 1994. [7] M. Elfataoui and G. Mirchandani, “A frequency domain method for gen- N−1 2N−1 b eration of discrete-time analytic signals,” IEEE Trans. Signal Processing, 2 1 2 Zp[2k] = Zc[k] vol. 54, no. 9, pp. 3343–3352, 2006. 4 [8] F. Peyrin, Y. Zhu, and R. Goutte, “A note on the use of analytic signal N k=N+1 k=⌊X2 ⌋+1 X in the pseudo Wigner distribution,” in Proc. IEEE Int. Symp. on Circuits 1 2 and Systems, vol. 2, May 8–11 1989, pp. 1260–1263. − Z[0] + A − B . (31) [9] B. Boashash and G. R. Putland, “Computation of discrete quadratic 2 TFDs,” in Time–Frequency Signal Analysis and Processing: A Compre-   hensive Reference, B. Boashash, Ed. Oxford, UK: Elsevier, 2003, ch. 6, b pp. 268–278. C. Nyquist Frequency Terms [10] J. O’ Toole, M. Mesbah, and B. Boashash, “A discrete time and The Nyquist term Z [N] is a real number, because of frequency Wigner–Ville distribution: properties and implementation,” in a Proc. Int. Conf. on Digital Signal Processing and Comm. Systems, vol. the definition of Za[k] in (7); the Nyquist term Z[N] is an CD-ROM, Dec. 19–21, 2005. imaginary number, because zˆ[n] is purely imaginary. Thus, [11] F. Peyrin and R. Prost, “A unified definition for the discrete-time, we rewrite (26) as b discrete-frequency, and discrete-time/frequency Wigner distributions,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, no. 4, pp. 858– 2 2 2 866, Aug. 1986. Zp[N] = Za[N] + Z[N] (32) [12] D. Slepian, “On bandwidth,” Proc. IEEE, vol. 64, no. 3, pp. 292–300, 1976. ˇ The remaining relations depend on the parityb of N. [13] V. C´ızek,ˇ “Discrete Hilbert transform,” IEEE Trans. Audio Electroa- 1) Case for N even: We know, from (19) and (7), that when coust., vol. 18, no. 4, pp. 340–343, Dec 1970. [14] F. Bonzanigo, “A note on ‘Discrete Hilbert transform’,” IEEE Trans. N is even, Zc[N]= Za[N]. When we combine this with (32) Audio Electroacoust., vol. 20, no. 1, pp. 99–100, Mar 1972. we get [15] J. M. O’ Toole, M. Mesbah, and B. Boashash, “A new discrete- 2 2 2 time analytic signal for reducing aliasing in discrete time–frequency Zp[N] = Zc[N] + Z[N] . (33) distributions,” in Proc. Fifteenth European Signal Processing Conf. EUSIPCO-07, Poznan,´ Poland, Sept.3–7 2007, pp. 591–595. 2) Case for N odd: We know, from (23), that when N is [16] ——, “A computationally efficient implementation of quadratic time– b frequency distributions,” in Proc. Int. Sym. on Signal Processing and odd, Zc[N] = 2Zp[N] − Za[N]. When we combine this with 2 2 2 its Applications, ISSPA-07, vol. I, Sharjah, United Arab Emirates, Feb. (26) we get |Zc[N]| = |Za[N]| + 4|Z[N]| . If we substitute 12–15 2007, pp. 290–293. this equation into (32), then we get the relation

2 2 b 2 Zp[N] = Zc[N] − 3 Z[N] . (34)

Finally, we are able to assemble the b three parts of the decomposition in (17). If we combine the relation for k even in (31) with the relation for k odd in (24), and add the Nyquist frequency relations in (33) and (34), then we get the following: John M. O’ Toole received the B.E. degree from the 2N−1 2 University College Dublin, Ireland, in 1997, and the Zp[k] M.Eng.Sc degree from the same university in 2000. k=N He is currently working towards a Ph.D. degree at the University of Queensland, Australia. X 2N −1 1 2 1 2 2 From 1997 to 1998 he worked as a research = Zc[k] + Zp[N] − Z[0] − C assistant in the Microwave Research Group at the 2 2 k=N   University College Dublin. He moved to Australia X b in 2001 and worked as a consultant engineer in where C = 2|Z[N]|2 when N is odd and C = 0 when N is control systems from 2001 to 2003. Following this, even. This concludes the proof. he worked as a research assistant in the Signal Processing Research Centre at the Queensland University of Technology from b 2003 to 2005. REFERENCES His research interests include time–frequency signal analysis, discrete-time signal processing, efficient algorithm design, and biosignal analysis. [1] B. Boashash, “Part I: Introduction to the concepts of TFSAP,” in Time– Frequency Signal Analysis and Processing: A Comprehensive Reference, B. Boashash, Ed. Oxford, UK: Elsevier, 2003, ch. 1–3, pp. 3–76. [2] J. Ville, “Theorie´ et applications de la notion de signal analytique,” Cables et Transmissions, vol. 2A, no. 1, pp. 61–74, 1948, in French. English translation: I. Selin, Theory and applications of the notion of complex signal, Rand Corporation Report T-92 (Santa Monica, CA, August 1958). 9

Mostefa Mesbah received his M.S. and Ph.D. de- grees in Electrical Engineering from University of Colorado at Boulder, Colorado, USA, in the area of automatic control systems. He is currently a Senior Research Fellow at the Perinatal Research Centre, the University of Queensland, Queensland, Australia, supervising a number of PhD students and leading a number of biomedical signal processing projects dealing with the automatic characterization and classification of newborn EEG abnormalities and fetal movement monitoring. His research interests include biomedical signal processing, nonstationary signal processing, sensor fusion, and signal detection and classification.

Boualem Boashash obtained a Diplome d’ingenieur-Physique-Electronique from ICIP University of Lyon, France, in 1978, the M.S. and Doctorate (Docteur-Ingenieur) degrees from the Institut National Polytechnique de Grenoble, France in 1979 and 1982 respectively. In 1979, he joined Elf-Aquitaine Geophysical Research Centre, Pau, France. In May 1982, he joined the Institut National des Sciences Applique’es de Lyon, France. In 1984, he joined the Electrical Engineering department of the University of Queensland, Australia, as lecturer. In 1990, he joined Bond University, Graduate School of Science and Technology, as Professor of Electronics. In 1991, he joined Bond University Queensland University of Technology as the foundation Professor of Signal Processing and director of the Signal Processing Research Centre. Currently, Prof. Boashash holds the positions of dean of the college of engineering at the University of Sharjah, United Arab Emirates, and Adjunct Professor at the School of medecine, University of Queensland, Australia. His research interests are time-frequency signal analysis, spectral estimation, signal detection and classification, and higher-order spectra. Prof. Boashash is fellow of IE Aust., fellow of IREE, and fellow of IEEE.