Volume 49, Number 4, 2008 485 Discrete Hilbert Transform. Numeric Algorithms Gheorghe TODORAN, Rodica HOLONEC and Ciprian IAKAB Abstract - The Hilbert and Fourier transforms are tools used for signal analysis in the time/frequency domains. The Hilbert transform is applied to casual continuous signals. The majority of the practical signals are discrete signals and they are limited in time. It appeared therefore the need to create numeric algorithms for the Hilbert transform. Such an algorithm is a numeric operator, named the Discrete Hilbert Transform. This paper makes a brief presentation of known algorithms and proposes an algorithm derived from the properties of the analytic complex signal. The methods for time and frequency calculus are also presented. 1. INTRODUCTION spectrum in order to avoid the aliasing process – the Nyquist condition. Signals can be classified into two classes: The discrete signal will be analyzed on a analytic signals (for instance = sin)( ωtAtx ), computer system, which implies its and experimental signals (measured signals). digitization (the digital signal is the discrete The last category represents real signals and is signal converted in binary format, accordingly of great importance in applications. to the adopted analog/numeric conversion; in An experimental signal represents a signal most of the cases, the signal acquisition observed during a limited interval of time. It is hardware also does the digitization of the a sample of the original signal, which signal samples). The resulted digital signal has characterizes a physical process of interest. the greatest importance in numeric analysis The experimental signal can be a operations. continuous time signal (analogical), or a Some other remarks need to be made. digital signal (discrete). Since the sampled signal has a limited length, The practical limitations of the systems it needs to either (1) have a infinite frequency used to analyze analogical signals impose that spectrum, or (2) be a periodic signal. In case the experimental analogical signals had a (1), the sampling doesn’t respect the Nyquist limited frequency band [1],[2]. condition. Or, in case (2) we choose to If the original signal doesn’t have a represent the signal as a periodic one, with an limited band, a low-pass filtration needs to be extended period. In both cases, the digital applied in order to obtain the experimental signal cannot exactly represent the original signal which will be analyzed. physical process. The rule also applies to sampled signals, In the case of the Hilbert transform, it’s a which need to have a limited frequency band known fact that the signal x(t) needs to be too. causal (that is x(t)=0, for t < 0). The sampled As a result, before acquisition, the signal x[n] is in this case a non-periodic experimental analogical signal will be low- sequence, real and causal. pass filtered. In such a case, we can talk of a discrete The acquisition frequency needs to be two Hilbert transform applied to the sequence x[n]. times the biggest frequency of the signal’s The complex analytic signal associated to the x[n] sequence has the spectrum different © 2008 – Mediamira Science Publisher. All rights reserved. 486 ACTA ELECTROTEHNICA from zero only for the interval of positive 1 x()() t= x t ∗ . (4) frequencies. π t When x(t) is a periodic signal, x[n] is a The above relation allows the calculus of periodic sequence and we cannot talk of causality (the periodic term implies the the spectral density of x( t) : sequence extension from − ∞ to + ∞ ). A ⎧ 1 ⎫ calculus algorithm for the Discrete Hilbert Xˆ () jω =FFF{}x(t) = {}x (t) ⋅ ⎨ ⎬ (5) π t Transform in this case imposes the condition ⎩ ⎭ that the Discrete Fourier Transform of the Or complex analytic sequence to be equal to zero ⎧ 1 ⎫ in the interval of negative frequencies. And of Xˆ ()() jω= X j ω ⋅F⎨ ⎬ (6) course, for positive frequencies, the spectrum ⎩π t⎭ of the analytic sequence to be two times the Since: spectrum of the signal x[n]. In this case, the ⎧ 1 ⎫ Hilbert transform can be used with all its F⎨ ⎬ = − j sgn (ω ) known advantages regarding the causal ⎩π t⎭ signals. It results: The next paragraphs present the methods for calculating the Discrete Hilbert Transform. X( jω )= X ( j ω )[ − j sgn ω] (7) Or: 2. HILBERT TRANSFORM IN ⎧− j X( jω ), ω > 0 CONTINUOUS TIME X() jω = ⎨ (8) ⎩ j X( jω ), ω < 0 To start, we present first the theory of the As a result, the spectral density function Hilbert transform, definitions, properties [2], of the x(t) signal’s conjugate is obtained by [10]. changing the phase of the spectral density for Let’s consider a real measurement signal: X (jω) by ± π/2. x() t ∈L (2) (1) It results: Where L(2) is the signal class with integral HF{}x(t)==xt ( )-1 { Xˆ ( jω )} (9) square. The Hilbert transform of the signal x(t) is: The inverse Hilbert transform is defined in relation (3). We can write: 1∞ x (τ ) xˆ()() t=H{} x t = v ⋅ p dτ (2) x()() t= H-1 x t = −H xˆ (t) (10) π ∫−∞ t −τ { } { } ∞ x()τ ⎡ t−ε x()τ ∞ x()τ ⎤ Taking into account relation (8) it results: v.p. dt = lim dτ + dτ , ∫−∞ ε →0 ⎢∫∫−∞ t+ε ⎥ -1 t −τ ⎣ t −τ t −τ ⎦ ⎪⎧F{ jXˆ(jω j } ,ω > 0 where v.p. represents the functional named x(t)= − H{} x(t)ˆ = ⎨ (11) ⎪F-1 − jX(jω j ,ω < 0 principal value. ⎩ {} x() t is improper named the conjugate of x(t). The analytic signal We also have x() t ∈L (2) . Having the pairs x (t) and x( t )= H{ x(t)} x(t) is the inverse Hilbert transform of x( t) : we build the analytic signal z (t) : 1 ∞ x() t z()()( t= x t+ j x t) (12) x()() t= H-1{} x t= − v.. p dτ (3) ∫−∞ π t −τ We observe that: Let’s observe that x() t is determined by Z( jω )= F{ z(t)} = X (jω)) + j X (jω (13) 1 the convolution of x(t) with the signal : Referring to relation (7) we obtain: π t Volume 49, Number 4, 2008 487 Zj()ω =+− Xj ()ωωω j [ j sgn]() Xj = (13.a) Which shows that the sample =+=Xj()[1sgn]2()()ω ωωXj uω X[][] N− k= X− k has a correspondent where u (ω) is the unit step function. sample of the spectral density, with the It’s useful to observe that: negative frequency X (−kω 0 ) . 1 For N – even, the samples X() jω = [ Z ( jω )+ Z∗ ( − jω )] (14) 2 ⎧ ⎡ N ⎤⎫ ⎨X[1], X [2],..., x −1 ⎬ 1 ⎢ 2 ⎥ X() jω = []Z()() jω − Z∗ − jω (15) ⎩ ⎣ ⎦⎭ 2 j are named „positive harmonics”, while the samples: 3. DISCRETE HILBERT TRANSFORM. ⎧⎫NN CALCULUS ALGORITHMS. ⎨⎬XX[++ 1], [ 2],..., XNXN [ −−≡ 2], [ 1] ⎩⎭22 ⎧ ⎡⎤⎡⎤⎛⎞NN ⎛ ⎞ ⎫ Definitions ≡−−⎨XX⎢⎥⎢⎥⎜⎟1 , −− ⎜ 2 ⎟ ,..., XX [ − 2], [ − 1]⎬ ⎩⎭⎣⎦⎣⎦⎝⎠22 ⎝ ⎠ Having the signal x(t) defined on the time are named „negative harmonics”. interval [0, tN], using a sampling period Te, we obtain the discrete signal x[n] : For N – odd, the samples xn[]=∈ xnT ( ), n 0, N− 1 (16) ⎧ ⎡⎤N −1 ⎫ e ⎨XX[1], [2],..., x⎢⎥⎬ ⎩⎭⎣⎦2 t N Where: Te = . are called „positive harmonics”, while the N samples: The sampling frequency fe is chosen so ⎧⎫NN++11 f e ⎨⎬XX[ ], [+−−≡ 1],..., XNXN [ 2], [ 1] that the frequency is greater or equal to ⎩⎭22 2 ⎧ ⎡⎤⎡⎤⎛⎞NN−−11 ⎛ ⎞ ⎫ the least significant frequency from the ≡ ⎨XX⎢⎥⎢⎥−−−−⎜⎟, ⎜1 ⎟ ,..., XX [ 2], [− 1]⎬ spectrum of x(t). We consider the discrete ⎩⎭⎣⎦⎣⎦⎝⎠22 ⎝ ⎠ f 2π are called „negative harmonics”. frequency step f = e , ω = f 0 N 0 N e The X[0] component is the continuous ⎡ N ⎤ respectively. component, while the X (N - even) is the The discrete Fourier transform (DFT) is: ⎣⎢ 2 ⎦⎥ 2π N-1 -jnk Nyquist component. It is found when the TFD{} x([n] =X[k] = ∑ x[n]eN , k∈ 0,N − 1(17) number of samples, N, is even – a situation n= 0 frequently found because DFT is implemented And the inverse discrete Fourier transform using an algorithm for which N is even. Also, -1 DFT is: ⎡ N ⎤ 2π The X component is the continuous one. N-1 jnk ⎢ ⎥ −1 1 ⎣ 2 ⎦ TFD{} X[][ [k= x n] = X[ k]eN , k∈ 0,N − 1 N ∑ k= 0 Similarly to relation (10), the discrete (18) Hilbert transform is defined: The sample of the spectral density Hxn{ []} == xn [] TFDXk−1 { []} (20) corresponding to frequency kω is determined 0 Where for N – even: with the relation: ⎧ N X()[ j kω = T X k] ⎪−=−jX[], k k 1, 1,N even 0 e ⎪ 2 Xkˆ []= ⎨ (21a) where X(jω) is the Fourier transform in N ⎪ jX[], k k=+ 1, N − 1, N even continuous time. ⎩⎪ 2 On the other hand: Xk[]*=−=− XN [ k ] X [ k ]. (19) 488 ACTA ELECTROTEHNICA We observe that the continuous and 3.1. The inverse discrete Fourier Nyquist components are excluded (for k = 0 transform algorithm N and k = ). Is based on relations (20) – (21.a) : 2 1) We determine the discrete Fourier While for N - odd: transform of the numeric sequence x [n]: ⎧ N −1 X[][] k= TFD{ x n } ⎪−=jXk[], k 1, , Nodd ⎪ 2 Xkˆ []= ⎨ (21b) 2) We set the continuous component to zero: N +1 ⎪ jX[], k k=− , N 1, N odd X[0]=0 ⎪ ⎩ 2 3) If the length N of sequence X[k] is even, Where the continuous component is excluded. we set the Nyquist component to zero: ⎡ N ⎤ Calculus Algorithms X = 0 ⎣⎢ 2 ⎦⎥ Relation (20) can be written: 4) The sequences X[k], −1 x[]nHxnTFDjSkXk==−{ []} { [] []} N N −1 k ∈,1 −1,N even ; or k = 1, ,.N odd (22) 2 2 Where: (positive harmonics) are multiplied by –j.