[Wiley Series in Probability and Statistics] Random Data (Analysis and Measurement Procedures) || The Hilbert Transform

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  • C H A P T E R 13

    The Hilbert Transform

    The Hilbert transform of a real-valued time-domain signal x(t) is another real-valued time-domain signal, denoted by x(t), such that z(t) = x{t) + jx(t) is an analytic signal. The Fourier transform of x(t) is a complex-valued frequency domain signal X(f), which is clearly quite different from the Hilbert transform x(t) or the quantity z(t). From z(t), one can define a magnitude function A(t) and a phase function #(/), where A{t) describes the envelope of the original function x{t) versus time, and 0(r) describes the instantaneous phase of x{t) versus time. Section 13.1 gives three equivalent mathematical definitions for the Hilbert transform, followed by examples and basic properties. The intrinsic nature of the Hilbert transform to causal functions and physically realizable systems is also shown. Section 13.2 derives special formulas for the Hilbert transform of correlation functions and their envelopes. Applications are outlined for both nondispersive and dispersive propagation problems. Section 13.3 discusses the computation of two envelope signals followed by correlation of the envelope signals. Further material on the Hilbert transform and its applications appears in Refs 1-5.

    13.1 HILBERT TRANSFORMS FOR GENERAL RECORDS

    The Hilbert transform for general records can be defined in three ways as follows, where all integrals shown will exist in practice:

    1. Definition as Convolution Integral. The Hilbert transform of a real-valued function x(t) extending over the range oo < t < oo is a real-valued function x(t) defined by

    Random Data: Analysis and Measurement Procedures, Fourth Edition. By Julius S. Bendat and Allan G. Piersol Copyright 2010 John Wiley & Sons, Inc.

    473

  • 474 THE HILBERT TRANSFORM

    JE(f) = Jt[x(t)] X [ U ) -du (13.1) "('-")

    Thus x(t) is the convolution integral of x(t) and (l/pi), written as

    (=(*(1/pi (13.2)

    Like Fourier transforms, Hilbert transforms are linear operators where

    3tf[alx\{t)+a2x2{t)]=altf[xl{t)}+a2je{x2{t)} (13.3)

    for any constants a.\, a2 and any functions X](r), x2(f).

    2. Definition as (pi/2) Phase Shift System. Let X(f) be the Fourier transform of x(r), namely,

    x(f) = &[Ht)\ = x(t)e-j2nftdt (13.4)

    Then, from Equation (13.2), it follows that X[f) is the Fourier transform X(f) of x(t), multiplied by the Fourier transform of (l /pi). The Fourier transform of (l/pi) is given by

    J ^ / n / ] = - y s g n / = j " / ^ f

    f> 0 (13.5)

    A t / = 0 , the function s g n / = 0 . Hence, Equation (13.2) is equivalent to the passage of x(t) through a system defined by (j sgn/) to yield

    X(f) = (-jsgnf)X(f) (13.6)

    This complex-valued quantity X(f) is not the Hilbert transform of the complex-valued quantity X(f). Its relation to x(t) is

    x(t) = Xify^df (13.7) J o o

    In words, x(r) is the inverse Fourier transform of X(f). The Fourier transform of ( j sgn f) can be represented by

    fi(f) = - 7 S g n / = | e , W 2 ) ^ < Q (13.8)

    with 5(0) = 0. Also,

    Bif) = \B{f)\e-*M (13.9)

    Hence, B(f) is a (pi/2) phase shift system where

    \B(f)\ = 1 fox all f 0 (13.10)

    pi / 2 f o r / > 0

    l - pi / 2 f o r / < 0 v ;

  • HILBERT TRANSFORMS FOR GENERAL RECORDS 475

    If one lets

    Xif) = \X(f)\e~M (13.12)

    pi/2 pi/2

    0 *

    -it/2

    0 *

    -it/2

    It follows that

    X(f)=X(f)e -]() \X(f)\e -,(/)+)\ (13.13)

    Thus, the Hubert transform consists of passing x(t) through a system which leaves the magnitude of X(f) unchanged, but changes the phase from

  • 476 THE HILBERT TRANSFORM

    The "instantaneous frequency" /n is given by

    , 1 \ de(t)

    /o=Ur (13-19) K2nJ dt

    Let Z(/) be the Fourier transform of z(t), namely,

    Z(f) = 3F\z(t)\ = P[x(t) +jx(t)} = ^[x(t)]+j^[x(t)}=X(f) +jX(f)

    (13.20)

    The inverse Fourier transform of Z(f) yields

    z(t) = [Z(f)] = x(t) +jx(t) (13.21)

    where

    x(t) = Jf[x(t)} = lm[z(t)} (13.22)

    13.1.1 Computation of Hubert Transforms

    To compute Z(f), note from Equation (13.6) that

    X(f) = (-jsgnf)X(f)

    Hence, Equation (13.20) becomes

    Z{f) = [1 + sgn/]X(f) = fl, (f)X(f) (13.23)

    where, as shown in the sketch, B](0) = 1 and

    * "3'24

    This is a very simple transformation to obtain Z(f) from X(f). One should compute X(f) for all/and then define Z(/) by Z(0) =X(0) and

    z ( f ) = { 0 ^ > rj/ v*>

    The inverse Fourier transform of Z(/) then gives z(f) with x(t) = Im[z(r)]. This is the recommended way to compute the Hilbert transform. From Equation (13.25),

  • H I L B E R T T R A N S F O R M S F O R G E N E R A L R E C O R D S 477

    x(t) = Re^2 X(fy2"fidf

    jc(i) = Im^2 J X(f)e>2nftdf

    (13.26)

    Example 13.1. Digital Formulas for xit) and xit). For digital computations, from Equations (11.76) and (11.77), one obtains for = 0, 1, 2 , . . . , ( 1),

    'N/2

    x{nAt) = 2A/Re

    x(nAt) = 2A/Im

    Here, the factor Af= (l/NAt) with

    *=1

    'N/2

    -X0Af

    G X ( W ) e x p ( ^ )

    N-l

    X(kAf) = At y^x(wAf)exp( -j n=0

    2nkn

    ~N~

    Note that values of X(kAf) are needed only from k = 0 up to k = (N/2), where the Nyquist frequency occurs, to obtain the digitized values of x(nAt) and its Hilbert transform x(nAt). The envelope signal of x(t) is given by

    () = [2 (nAt) + 2 ( ) ] 1/2

    13.1.2 Examples of Hilbert Transforms

    Table 13.1 gives several x(t) with their associated x(t) and envelopes A(t). Proofs of these results follow directly from previous definitions. These examples are plotted in Figure 13.1. The last result in Table 13.1 is a special case of a general theorem that

    Jt[u(t) cos 2pi/ 0] = u(t) sin 2nf0t (13.27)

    for any function u(t) which is an even function of t.

    Table 13.1 Examples of Hilbert Transforms

    x(t) x(t) AW

    cos 2pi/0 sin 2pi/ 0 1

    sin 2pi/ 0 cos 2pi/0 1 sinr 1cos t sin(r/2)

    t t it/2) 1 t 1 1/2

    1 + i 2 1+f 2

    e"cl'lcos 2nf0t

  • 478 THE HILBERT TRANSFORM

    *(

    -, 1/2

    Figure 13.1 Examples of Hilbert transforms.

    13.1.3 Properties of Hilbert Transforms

    A number of properties will now be stated for Hilbert transforms, which follow readily from the basic definitions or are proved in the references. Let x(t) = J^[x(t)] and y(t) = Jf\y(t)] be Hilbert transforms of x{t) and y(r), with corresponding Fourier transforms X(f) and Y(f).

    a. Linear Property

    Jt[ax(t) + by(t)] = ax{t) + by(t)

    for any functions x(t), y(t) and any constants a, b.

    (13.28)

  • HILBERT TRANSFORMS FOR GENERAL RECORDS 479

    b. Shift Property

    3/e \x{t-a)] = Jt(f-fl) (13.29)

    c. Hilbert Transform of Hilbert Transform

    J f [jc(f)] = -x{t) (13.30)

    In words, the application of two successive Hilbert transforms gives the negative of the original function.

    d. Inverse Hilbert Transform ofx(t)

    f 0 0 xiu) x(t) = Je-\x(t)\ = - ^ - r d u (13.31)

    Thus x(t) is the convolution of x(t) with ( - l / pi ) . Alternatively, x(t) can be defined by

    x(t) = &-l[(jsgnf)X(f)] (13.32)

    where

    X{f) = P[x(t)] (13.33)

    e. Even and Odd Function Properties If x(t) is an even (odd) function of t, then x(t) is an odd (even) function of t:

    x(t) even i(f) odd

    x(t) odd x(t) even (13.34)

    f. Similarity Property

    g. Energy Property

    je\x(at)\ = x(at) (13.35)

    POO POO

    x2(t)dt = x2(t)dt (13.36) J o o J o o

    This follows from Parseval's theorem because

    POO POO

    x\t)dt= \X(f)\2df J o o J 0 0

    l % ) l 2 #

    (13.37)

    and the fact that

    \x(f)\2 = W)\2 (13.38)

  • 480 THE HILBERT TRANSFORM

    h. Orthogonal Property

    1*00

    x(t)x{t)dt = 0 (13.39) Joo

    This follows from Parseval's theorem because

    x(t)x(t) dt X*(f)X(f)df (13.40)

    and the fact that

    X* (f)X(f) = (-j sgn f)\X(f)\2 (13.41)

    is an odd function of / s o that the right-hand side of Equation (13.40) is zero,

    i. Modulation Property

    JV[x{t) cos 2pi/ 0] = x(t) sin 2pi/ 0 (13.42)

    if x(t) is a signal whose Fourier transform X(f) is bandwidth limited; that is,

    X(,) = {X(f) M F (.3.43) [ 0 otherwise

    provided t ha t / 0 is such t ha t / 0 > F. Also,

    Xf[x(t) sin 2pi/ 0] = -x(t) cos 2pi/ 0 (13.44)

    j . Convolution Property

    jr\x(tyy(t)]=x(tyy(t) =x(tym (13.45)

    This follows from the fact that

    &[x(tyy(t)]=X(f)Y(f) (13.46)

    and

    [(-jSgnf)X(f)}Y(f)=X(f)Y(f)

    = x(f)[(-Jwf)Y(f)]=x(f)Y(f) (13.47)

    k. Lack of Commutation Property

    ^{J#>[x(t)}} 3f{&-[x{t)}} (13.48)

    In words, Fourier transforms and Hilbert transforms do not commute.

    13.1.4 Relation to Physically Realizable Systems

    A physically realizable constant-parameter linear system is defined by a weighting function h(x) satisfying Equation (2.3), that is,

    ( ) = 0 f o r r < 0 (13.49)

  • HILBERT TRANSFORMS FOR GENERAL RECORDS 481

    The corresponding frequency response function H(f) of Equation (2.13) is given by

    f O O

    H(f) = &\h(x)\ = h(x)e-j2nftdx Jo (13.50)

    = HR(f)-jH,(f)

    Here, HR(f) equals the real part of H(f) and Hj(f) equals the imaginary part of H(f)

    as defined by

    HR(f) =

    H,(f) =

    h(x)cos Infxdx

    (13.51)

    h () sin Infxdx

    It will now be proved that for a system to be physically realizable, it is necessary and sufficient that / / / / ) be the Hilbert transform of HR(f).

    This result is a special case of a more general theorem that applies to all causal functions. By definition, a real-valued function y ( f ) is a causal function if

    y ( / ) = 0 f o r i < 0 (13.52)

    Any function y(t) can always be broken down into the sum of an even function ye{t) plus an odd function ya{t) by writing

    y(t)=ye(t)+y0(t) (13.53)

    where

    ye{t)=\[y{t)+y{-t)]

    yo(t)=\\y{t)-y(-t)\

    This gives ye(-t) = ye(t) and ya(-t) - y0(t). The resolution of any causal function into its even and odd components is illustrated in Figure 13.2.

    Assume now that y(t) is a causal function. Then, from Equations (13.52) and (13.54), f o r / > 0 ,

    (13.55) yo(t) = \y{t)=ye{t)

    For t < 0, however,

    * < " - * * - " (,3.56) yo(t) = -h{-t) = -ye{t)

  • 482 THE HILBERT TRANSFORM

    Figure 13.2 Even and odd components of a causal function.

    Hence, for a causal function,

    where

    y 0 { t ) = (sgnr)y e(?)

    sgnf: 1 t > 0

    -1 t < 0

    For any causal function y(r), its Fourier transform Y(f) must satisfy

    Y(f) = P\y(t)\ = P\ye(t) +y0(t)] = YR(f)-jY,(f)

    where

    &\yo{t)] = -jYiif)

    From Equation (13.57),

    &\yo(t)] = &[(sgnt)ye{t)] = ^{sgn t}*aF\ye(t)}

    where

    JHsgnil = - 4

    Hence,

    J - o o pi ( / - ( f - l l )

    (13.57)

    (13.58)

    (13.59)

    (13.60)

  • HILBERT TRANSFORMS FOR GENERAL RECORDS 483

    This proves that

    Yiif) = ^TT^du = X[YR[f)\ (13.61)

    In words, Y^f) is the Hilbert transform of YR(f) when y(f) is a causal function. This completes the proof.

    A direct way to determine whether or not a computed H(f) can represent a physically realized (causal) system is now available. It will be physically realizable if / / / ( / ) is the Hilbert transform of HR(f) since Equation (13.61) is equivalent to Equation (13.57). Thus, Equation (13.61) is both a necessary and a sufficient condition for a system to be physically realizable. In equation form,

    H(f) = HR(f)-jH,(f) with H,(f) = HR(f) (13.62)

    It follows that

    H,(f) = HR(f) = -HR(f) (13.63)

    and

    H(f) = HR(f)-jH,(f) = [(f) +jHR(f) =jH(f) (13.64)

    Deviations from these results can be used to study nonlinear systems.

    Example 13.2. Exponential Causal Function. Consider the causal function

    y(r) = pi~2pi' t > 0 (otherwise zero)

    Here,

    and

    y e ( f ) = J e -2 l ' l a l l ,

    y0(t) =

    Taking Fourier transforms yields

    Yeif) = YHif) 2(1 +p

    and

  • 484 THE HILBERT TRANSFORM

    Thus, YR(f) is an even function of / a n d Y/if) is an odd function off with

    Ylif) = X[YR(f)]

    The total Fourier transform of y(r) is then

    Y(f) = Y(f)-JYI(f)=w^)

    Example 13.3. Exponential-Cosine Causal Function. Consider the causal function

    y(r) = e~ a 'cos 2nbt a > 0, t > 0 (otherwise zero)

    For this case, the Fourier transform of y(r) is given by

    (a+j2%f) + (2nb)

    where the separate formulas for YR(f) and Y]if) are quite complicated. Because y(i) is a causal function, however, one will have

    Yiif) = *W)]

    with Yg(f) an even function of / and Yiif) an odd function of/.

    13.2 HILBERT TRANSFORMS FOR CORRELATION FUNCTIONS

    One of the more important applications for Hilbert transforms involves the calculation of correlation function envelopes to estimate time delays in energy propagation problems. The principles behind such applications are now developed for both nondispersive and dispersive propagations.

    13.2.1 Correlation and Envelope Definitions

    Let x(t) and y(i) represent zero mean value stationary random data with autocorrela-tion functions R^ix) and Ryy(x) and a cross-correlation function R^x). Let the associated two-sided autospectral density functions be S^if) and 5 y y ( / ) and the associated two-sided cross-spectral density function be S^if). As defined in Section 5.2.1, such associated functions are Fourier transform pairs. For each stationary random record x(t) or y{t), let x(t) = 3^\x{t)\ and y(t) = 3V\y{t)\ be their Hilbert transforms. Because Hilbert transforms are linear operations, it follows that x(t) and y(t) will also be stationary random data. Various correlation and spectral density functions for jc(i) and y(r) can be defined the same as for x(t) and y(i) to yield R^(), Ryy(x), Rxy{x) and Sjaif), Syy(f), S^tf), where the associated functions are Fourier

  • HILBERT TRANSFORMS FOR CORRELATION FUNCTIONS 485

    txansform pairs. To be specific, the definitions are

    *() = E[x(t)x(t + x)}

    R99(r)=E\y(t)y(t + x)] (13.65)

    Rxy(x)=E[x(t)y(t + x)\

    Then their Fourier transforms give

    S s ( t ) = &[R&(z)]

    Syy{f) = ^[Ryyir)} (13.66)

    Except for a scale factor (which is unimportant for theoretical derivations, since it will appear on both sides of equations and therefore cancel out), one can also use the definitions

    Six(f)=E\r(f)X(f)\=E[\X(f)\2]

    Syy(f)=E[Y*(f)Y(f)]=E[\Y(f)\2} (13.67)

    Sxy(f)=ElX*(f)Y(f)}

    Mixing functions and their Hilbert transforms leads to /?^(), Rxy(x) and the associated S^f/), S^if) defined by

    Rxy(x)=E[x(t)y(t + x)}

    Rxy(x)=E[x(t)y(t + x)} 1 " '

    Sxy(f)=F[Rxy{z)]

    Special cases are R^x), () , 5 ^ (/), and S^if). Except for a scale factor, as noted previously, one can also define Sxy(f) and Sayif) directly by

    Sxy(f)=E[X'(f)Y(f)]

    Sty(f)=EiX*(f)Y(f)}

    Envelope functions for R^x), Ryy(x), and Rxy(x) are defined in terms of Hilbert transforms Rxx(x), Ryy(x), and R^x) as follows. The envelope function for Rxx(x) is

    4 W = ( 4 W + 4 ) ] ' / 2 (13-71)

    The envelope function for Ryy(x) is

    A W ( T ) = [ ^ ( T ) + ^ ( T ) ]1 / 2 (13.72)

    The envelope function for /?^() is

    Axy{x) = [Rxy{x)+R2

    xy{x)]1'2 (13.73)

    Various relationships and properties for all of the above quantities will now be stated.

  • THE HILBERT TRANSFORM

    13.2.2 Hilbert Transform Relations

    Table 13.2 gives a number of useful relationships for autocorrelation (auto-spectral density) and cross-correlation (cross-spectral density) functions. Proofs of these results represent straightforward exercises.

    Some results in Table 13.2 are worthy of attention. Specifically,

    () = X[Rxx(x)\ = -Mr) (13.74)

    In words, the Hilbert transform of ^ ) is the cross-correlation function between x(t) and its Hilbert transform x(t). Also, note that , () is an odd function of r because Rxx(x)lS an even function of . This gives

    Rxx(0) = Ra(0) = 0 (13.75)

    Thus, a zero crossing of R ^ T ) occurs at = 0, correspondi...

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