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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)

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).
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• 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,
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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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