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ADSP-25-TF-TFDs-EC623

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 Features from Time-Frequency Concepts
• Time-frequency    signal analysis and processing concerns the
 analysis and processing of signals with time varying frequency
 content.
• Such   signals are best represented by a Time-Frequency Distri-
 bution (TFD).
• TFD    is intended to show how the energy of the signal is dis-
 tributed over the two-dimensional time-frequency space.
• Processing   of signal may then exploit the features produced by
 the concentration of signal energy in two dimensions(time and
 frequency) instead of only one (time or frequency).
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Mono-component and Multi-component Signals:
•A   mono-component signal is described in the (t, f ) domain by one
  single ridge corresponding to an elongated region of energy con-
  centration.
• Furthermore,   interpreting the crest of the ridge as a graph of
  instantaneous frequency (IF) versus time, we require the IF of a
  mono-component signal to be a single valued function of time.
•A   multi-component signal may be described as the sum of two or
  more mono-component signals.
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Instantaneous Frequency (IF) and Mean Frequency:
• The    Instantaneous frequency of a mono-component signal is
                                            1 ′
                                fi (t) =      φ (t)
                                           2π
        where φ(t) is the instantaneous phase of the signal s(t).
•A   multi-component signal will have a separate IF for each com-
  ponent.
• The    IF is a detailed description of the frequency characteristics of
  a signal.
• This   contrasts with the definition of mean frequency defined next.
• The    mean frequency of a signal is
                                     ∞
                                    0 f.   | S(f ) |2 df
                             fo =     ∞
                                     0 |   S(f ) |2 df
        where S(f ) is the Fourier Transform of the signal s(t).
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• f0   is the first moment of | S(f ) |2 with respect to frequency: i.e.,
  we define the mean frequency as if | S(f ) |2 were the probability
  density function of the frequency.

Instantaneous Frequency and Time Delay:
• Instantaneous      frequency of a signal indicates the dominant fre-
  quency of the signal at a given time.
• Let   us now seek the dual of the IF, indicating the dominant time
  when a given frequency occurs.
• If z(t)   is an analytic signal with the Fourier Transform

                               Z(f ) = Aδ(f − fi)

        where A is in general complex, the dominant frequency is fi.
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• Taking    the inverse Fourier transform of Z(f ) gives
                                     z(t) = Aej2πfi t

• The   instantaneous phase of z(t), denoted by φ(t), is
                            φ(t) = arg(z(t)) = 2πfit + arg(A)

        so that fi = 2π φ′(t)
                      1


• Now     let us repeat the argument with the time and frequency
 variables interchanged.
• The   signal
                                    z(t) = aδ(t − τd)

        where a is in general complex, is an impulse at τd.
• The   Fourier transform of the signal is
                                    Z(f ) = ae−j2πf τd
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       and its phase, denoted by θ(f ), is

                        θ(f ) = arg(Z(f )) = −2πf τd + arg(a)

       so that τd = − 2π θ′(f ). τd is termed as time delay.
                       1


• If z(t)   is an analytic signal with the Fourier transform Z(f ), then
 the time delay (TD) of Z(f ), denoted by τd(f ), is
                                                  1 ′
                                    τd(f ) = −      θ (f )
                                                 2π
        where θ(f ) = arg(Z(f )).
• Notice     that the definitions of τd and fi are similar except that time
 and frequency are interchanged and the time delay has an extra
 minus sign, hence time delay is termed as the dual of instanta-
 neous frequency.
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Mean Instantaneous Frequency and Group Delay:
• Let z(t)   be a band pass analytic signal with center frequency fc.
• Let   its Fourier transform be

                                Z(f ) = M (f − fc )ejθ(f )

        where the magnitude M (f − fc), and phase θ(f ) are real.
• If   the signal has linear phase in the support of Z(f ), i.e., if θ(f ) is
  a linear function of f whenever Z(f ) is non-zero, we can let

                            θ(f ) = −2πτpfc − 2πτg [f − fc]

       where τ p and τg are real coefficients with dimensions of time.

                         Z(f ) = M (f − fc )e−j(2πτpfc+2πτg [f −fc])

                           = e−j2πτpfc .M (f − fc ).e−j2πτg [f −fc]
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• Taking    the Inverse Fourier transform of Z(f ), we find

                             z(t) = m(t − τg ).e2πfc(t−τp )

        where m(t) is the Inverse Fourier transform of M (f ).
• Now     because M (f ) is real, m(t) is Hermitian, [i.e., m(−t) = m∗(t)], so
 that | m(t) | is even.
• Hence τg    is the time about which the envelope function is sym-
 metrical, for this reason,τg is called the group delay.
• The    phase of the oscillatory factor is −2πfcτp, where τp is called the
 phase delay.
• If   an analytic signal has the Fourier transform

                                Z(f ) =| Z(f ) | ejθ(f )
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     then the Group Delay(GD) of the signal is
                                               1 ′
                                τg (f ) = −      θ (f )
                                              2π
        and the phase delay of the signal is
                                             1
                               τp(f ) = −       θ(f )
                                            2πf

• The   relation for time delay (τd) and group delay are same. However
 the physical interpretations are different.
• The   time delay applies to an impulse, where as the group delay
 applies to the envelope of a narrow band signal.
• Now    consider the dual of the above.
• Let z(t)   be a time limited signal centered on t = tc, and let

                               z(t) = a(t − tc)ejφ(t)
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       where a(t) and φ(t) are real. If the signal has constant in-
 stantaneous frequency in the support of z(t), i.e., if φ(t) is a linear
 function of t wherever z(t) is non zero, we can let,

                            φ(t) = 2πf0tc + 2πfm [t − tc]

      where f0 and fm are real coefficients with dimensions of fre-
 quency.
                          z(t) = a(t − tc)ej(2πf0tc +2πfm[t−tc ])

                            = ej2πf0tc .a(t − tc).ej2πfm[t−tc ]

• Taking   the Fourier transform of z(t) we find

                           Z(f ) = A(f − fm ).e−j2π[f −f0 ]tc

     where A(f ) is the Fourier transform of a(t).
• Now a(t)   is real, A(f ) is Hermitian, so that | A(f ) | is even.
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• Hence fm    is the frequency about which the amplitude spectrum
 is symmetrical; for this reason, fm is called mean instantaneous
 frequency.
• For   the signal, z(t) =| z(t) | ejφ(t), the mean instantaneous frequency
 is
                                            1 ′
                                fm (t) =      φ (t)
                                           2π
• Thus   the mean instantaneous frequency is same as the instanta-
 neous frequency defined earlier, but the physical interpretations
 are different.
• The   instantaneous frequency has been derived for a tone, where
 as the mean instantaneous frequency applies to the spectrum of
 a short duration signal.
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  Heuristic Formulation of Time-Frequency
                Distributions

 • Given    an analytic signal z(t) obtained from a real signal s(t), we
  seek to construct a time-frequency distribution ρz (t, f ) to represent
  precisely the energy, temporal and spectral characteristics of the
  signal.

The Time-Frequency Distribution:

 • For   a monocomponent signal, it is reasonable to expect that the
  TFD should take the form of a knife-edge ridge whose crest is a
  graph of the IF law in the (t, f ) plane.
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 • Mathematically,     the knife-edge is idealized as a delta function with
  respect to frequency.

Knifes-Edge IF Indication:
 • Noting    that ρz is a function of frequency f and represents a kind
  of spectrum, we may reasonably require ρz to be Fourier transform
  of some function related to the signal.
 • Let   us call this function as Signal Kernel and give it the symbol Kz .
 • So    the signal kernel can be written as Kz (t, τ ), and the TFD is

                               ρz (t, f ) = F {Kz (t, τ )}
                                          τ →f

 • To   find a suitable form for Kz (t, τ ), for simplicity, let us first consider
  the case of unit amplitude mono component FM signal.

                                     z(t) = ejφ(t)
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• Whose   instantaneous frequency is
                                                φ′ (t)
                                       fi (t) =
                                                 2π

• We   would like the TFD of z(t) at any given time to be a unit
 delta function at the instantaneous frequency, so the ”instanta-
 neous frequency” reduces to the ordinary Fourier transform in the
 constant frequency case.
• That   is, we want
                                 ρz (t, f ) = δ(f − fi (t))

• Substituting   into the definition ρz (t, f ) we have

                             δ(f − fi (t)) = F {Kz (t, τ )}
                                              τ →f
                                                                      ′
                  Kz (t, τ ) = F −1{δ(f − fi (t))} = ej2πfi(t)τ = ejφ (t)τ
                              τ ←f
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 • Because φ′(t)   is not directly available, we write
                                          φ(t + τ /2) − φ(t − τ /2)
                             φ′ (t) = lim
                                     τ →0             τ
  and use the approximation
                                     1
                             φ′ (t) ≈ [φ(t + τ /2) − φ(t − τ /2)]
                                     τ
  which is called the central finite difference (CFD) approximation.
  Substituting φ′(t) into Kz (t, τ ) gives the signal kernel.
                               Kz (t, τ ) = ejφ(t+τ /2).e−jφ((t−τ /2))

                              Kz (t, τ ) = z(t + τ /2).z ∗ (t − τ /2)

The Wigner Distribution:

 • Substituting Kz (t, τ )   in ρz (t, f ) we get
                         ρz (t, f ) = F {z(t + τ /2).z ∗(t − τ /2)}
                                       τ →f
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                                     ∞
                      ρz (t, f ) =        z(t + τ /2).z ∗(t − τ /2)e−j2πf τ dτ
                                     −∞

• The    above equation is given the symbol Wz (t, f ) and is called the
 Wigner Distribution (WD).
                               1
• The    approximation φ′(t) ≈ τ [φ(t + τ /2) − φ(t − τ /2)] is exact if φ is a linear
 function of t, i.e. if φ′(t) is constant.
• It   is also exact if φ(t) is quadrature, i.e. if φ(t) is linear.
• Thus    the WD gives an unbiased estimate of the IF for a complex
 linear FM signal.
• The    constant frequency real signal
                                          s(t) = cos(2πfct)

 leads to the kernel
                               Ks(t, τ ) = s(t + τ /2)s(t − τ /2)
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                               = cos(2πfc(t + τ /2)cos(2πfc(t − τ /2)
                                   1                 1
                                 = cos(2πfc(2t)) + cos(2πfcτ )
                                   2                 2
Note:cos(A + B) + cos(A − B) = 2cos(A)cos(B)
Let A = t + τ /2 & B = t − τ /2
      cos(2t) + cos(τ )

 • Taking   Fourier transforms with respect to τ gives the WD
                            1           1           1
                 Ws(t, f ) = δ(f − fc) + δ(f + fc) + cos(2πfc(2t))δ(f )
                            4           4           2
 • The    terms in δ(f ∓ fc) are naturally expected and arise because s(t)
   may be expressed as a sum of complex sinusoids at frequencies
   ±fc.

 • The    term in δ(f ) is an artifact arising because the non linearity
   of the WD causes interaction between the positive and negative
   frequency terms.
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• In   a similar way, we can find the Ks(t, τ ) for a non-windowed linear
 FM signal
                                                            α
                                      s(t) = Acos(2π(f0t + 2t2 ))

                                   Ks(t, τ ) = s(t + τ /2)s(t − τ /2)

                        s(t + τ /2) = Acos(2π(f0(t + τ /2) + α (t + τ /2)2))
                                                             2


                        s(t − τ /2) = Acos(2π(f0(t − τ /2) + α (t − τ /2)2))
                                                             2


                                   A = f0(t + τ /2) + α (t + τ /2)2)
                                                      2


                                   B = f0(t − τ /2) + α (t − τ /2)2)
                                                      2


                            2cos(A)cos(B) = cos(A + B) + cos(A − B)

        cos(A + B) = cos(2π(f0(t + τ /2) + α (t + τ /2)2 + f0(t − τ /2) + α (t − τ /2)2))
                                           2                              2

                                           2                                         2
       = cos(2π((f0t + f0 τ + α t2 + α τ4 + 2. α .t. τ ) + (f0t − f0 τ + α t2 + α τ4 − 2. α .t. τ )))
                          2   2      2         2     2               2   2      2         2     2
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                                                                2
                                  = cos(2π(2f0t + αt2 + α τ4 ))

     cos(A − B) = cos(2π(f0(t + τ /2) + α (t + τ /2)2 − f0(t − τ /2) + α (t − τ /2)2))
                                        2                              2

                                        2                                         2
    = cos(2π((f0t + f0 τ + α t2 + α τ4 + 2. α .t. τ ) − (f0t − f0 τ + α t2 + α τ4 − 2. α .t. τ )))
                       2   2      2         2     2               2   2      2         2     2


                                      = cos(2π(f0τ + αtτ ))

                                      = cos(2π((f0 + αt)τ ))

                                        = cos(2π(fi(t)τ ))

 when fi(t) = f0 + αt
                                                                                      2
              Ks(t, τ ) = 1 A2 cos(2π(fi(t)τ )) + 1 A2 cos(2π(2f0t + αt2 + α τ4 ))
                          2                       2


• Taking   Fourier transform with respect to τ gives the WD

                                    Ws(t, f ) = F {Ks(t, τ )}
                                                 τ →f
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                                                                                  2
         = 1 A2δ(f − fi (t)) + 2 A2δ(f + fi (t)) + 1 A2 F {cos(2π(2f0t + αt2 + α τ4 ))}
           4
                               1
                                                   2
                                                     τ →f

 • The   terms in δ(f ∓ fi(t)) are naturally expected, while the last term
  in the signal kernel gives rise to a continuum of artifacts in the
  WD.
 • These        artifacts, which greatly diminish the usefulness of the WD
  for real signals, are removed by modifying the WD with the ana-
  lytic signal, which is termed as Wigner-Ville Distribution.

The Wigner-Ville Distribution (WVD):

 • The   Wigner-ville Distribution (WVD) of a signal s(t), denoted by
  Wz (t, f ),   is defined as the WD of its analytic associate, i.e.,

                           Wz (t, f ) = F {z(t + τ /2).z ∗(t − τ /2)}
                                       τ →f

  where z(t) is the analytic associate of s(t).
                                                                              www.jntuworld.com
• The     name ‘Wigner-Ville distribution”, as opposed to ‘Wigner dis-
 tribution”, emphasizes the use of the analytic signal.
• Wigner     proposed WD in the context of quantum mechanics.
• Ville   ported WD in a signal processing context.
• Noting     that a signal can have a time-dependent frequency con-
 tent, Ville sought an instantaneous spectrum having the attributes
 of an energy density and satisfying the so-called marginal condi-
 tions.
• The     integral of the TFD Wz (t, f ) with respect to frequency is the
 instantaneous power |z(t)|2.
• The     integral of the TFD Wz (t, f ) with respect to time is the energy
 spectrum |Z(f )|2.
• We    know that,
                                                                              www.jntuworld.com
                            Wz (t, f ) = F {Kz (t, τ )}
                                       τ →f

• The   signal kernel Kz (t, τ ) is also called the instantaneous auto cor-
 relation function (IAF) of z(t).

				
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posted:8/22/2011
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