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FaaDoOEngineers.com
Time Frequency AnalysisMathematical Preliminaries
S. R. M. Prasanna
Dept of ECE,
IIT Guwahati,
Time Frequency Analysis – p. 1/26
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Development of Relations
Time domain description of signals
Frequency domain description of signals
Frequency domain in terms of time domain
Time domain in terms of frequency domain
Time-frequency description
Time Frequency Analysis – p. 2/26
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Time Description of Signals
Energy Density or instantaneous powerHow much energy a signal has?s(t) is given signal
Energy Density: |s(t)|2 energy per unit time at time t
Fractional Energy: |s(t)|2∆t fractional energy in theinterval ∆t at time tTotal Energy: E =
∫
|s(t)|2dt
Time Frequency Analysis – p. 3/26
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Time Description (contd.)
Characterizing time waveformsx is a random variable and p(x) its density function
Mean: µ =∫
xp(x)dx
Variance: σ2t =
∫
(x− µ)2p(x)dx
Mean (µ) gives gross characterization of densityi.e., where density is concentratedStd. Dev. (σt) indicates how the density isconcentrated around the mean.
t is a random variable and |s(t)|2 its density functionMean or Average Time: < t >=
∫
t|s(t)|2 dtVariance or Duration:T 2 = σ2
t =∫
(t− < t >)2|s(t)|2dt =< t2 > − < t >2
< t2 > is defined similar to < t >
Time Frequency Analysis – p. 4/26
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Time Description (contd.)
Description of Time Waveforms
Duration of SignalIn 2σt most of the density is concentratedT or σt is an indication of the duration of signalIf σt is small then most of the signal is concentratedaround meanSignal is of short durationIf σt is large then most of the signal is spread over along timeSignal is of long duration
A figure of signal w/f to illustrate < t > and T = σt
Time Frequency Analysis – p. 5/26
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Time Description (contd.)
Illustration of < t > and σt computation
s(t) = (α/π)1/4e−(t−t0)2/2
|s(t)|2 = |(α/π)1/4e−(t−t0)2/2|
2= (α/π)1/2e−(t−t0)
2
< t >=√
α/π∫
te−α(t−t0)2
dt = t0
< t2 >=√
α/π∫
t2e−α(t−t0)2
dt = 1/2α + t0
σ2t = < t2 > − < t >
2= 1/2α
Note: Use∫
uv′ = uv −∫
u′v
Time Frequency Analysis – p. 6/26
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Frequency Description of Signals
s(t) is given signal and S(ω) its spectrum
Energy Density Spectrum: |s(ω)|2 energy per unitfrequency at frequency ω
Fractional Energy: |S(ω)|2∆ω fractional energy in theinterval ∆ω at frequency ω
Total Energy: E =∫
|S(ω)|2dω
Parceval’s Theorem: E =∫
|s(t)|2dt =∫
|S(ω)|2dω
Time Frequency Analysis – p. 7/26
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Frequency Description (contd.)
ω is a random variable and |S(ω)|2 its density function infrequency
Mean or Average Frequency: < ω >=∫
ω|S(ω)|2dω
Variance or Bandwidth:B2 = σ2
ω =∫
(ω− < ω >)2|S(ω)|2dω
=< ω2 > − < ω >2
< ω2 > is defined similar to < ω >
Time Frequency Analysis – p. 8/26
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Frequency Description (contd.)
Description of Spectrum of the signal
Bandwidth of SignalIn 2σω most of the density is concentratedB or σω is an indication of the bandwidth of signalIf σω is small then most of the signal is concentratedaround meanSignal is of small bandwidthIf σω is large then most of the signal is spread over alarge bandwidthSignal is of large bandwidth
A figure of signal spectrum to illustrate < ω > andB = σω
Time Frequency Analysis – p. 9/26
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Average and Density Functions
t and ω are R.Vs. Any function of these R.Vs.
The average of any time function, g(t), is< g(t) >=
∫
g(t)|s(t)|2dt
The average of any frequency function, g(ω), is< g(ω) >=
∫
g(ω)|S(ω)|2dω
Energy densitySignal amplitudes present in the total bandwidth ofthe signal.No information about the different frequencycomponents present.
Energy density spectrumFrequency components present in the total durationof the signal.No information about when those frequenciesexisted.
Time Frequency Analysis – p. 10/26
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Time Frequency Analysis
Time Frequency AnalysisMathematical and physical ideas needed tounderstand and describe how the frequencies arechanging in time
Time Frequency Analysis – p. 11/26
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Alternative Relation to Avg. Freq.
Frequency in terms of time
Avg. Freq.: < ω >=∫
ω|S(ω)|2dω =∫
s∗(t)1jddts(t)dt
Avg. Freq. can be computed directly from s(t) withoutcomputing its S(ω)
WKT, S(ω) =∫
t s(t)e−jωtdt and S∗(ω) =
∫
t s∗(t)ejωtdt
|S(ω)|2 = S(ω)S∗(ω)
< ω >=∫
ω|S(ω)|2dω
< ω >=∫
ω ω∫
t′ s(t′)e−jωt
′
dt′∫
t s∗(t)ejωtdtdω
< ω >=∫
ω
∫
t
∫
t′ ωs(t′)s∗(t)ejω(t−t′)dtdt′dω
ddte
jω(t−t′) = ejω(t−t′)jω
Time Frequency Analysis – p. 12/26
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Avg. Freq. Relation (contd.)
< ω >=∫
ω
∫
t
∫
t′ s(t′)s∗(t)1
jddte
jω(t−t′)dtdt′dω
∫
ω ejω(t−t′)dω = δ(t− t′)
∫
t s∗(t)1
jddt
∫
t′ s(t)δ(t− t′)dt′dt
< ω >=∫
s∗(t)1jddt(s(t))dt
Proceeding on the similar lines
< ω2 >=∫
ω2|S(ω)|2dω =∫
s∗(t)(1jddt)
2s(t)dt =∫
| ddts(t)|2dt
< ωn >=∫
ωn|S(ω)|2dω =∫
s∗(t)(1jddt)
ns(t)dt =∫
| ddts(t)|ndt
Time Frequency Analysis – p. 13/26
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Frequency Operator
For frequency function in time domain only
Frequency operator is defined as: W = 1jddt
Repeated use of W denoted by Wn
Wns(t) = (1j )n dn
dtns(t)
Average of a frequency function calculation from timefunction:< g(ω) >=
∫
g(ω)|S(ω)|2dω
Using < ω >=∫
ω|S(ω)|2dω =∫
s∗(t)1jddts(t)dt
< g(ω) >=∫
s∗(t)g(W)s(t)dt
< g(ω) >=∫
s∗(t)g(1jddt)s(t)dt
Take g(ω), replace ω by 1jddt , operate on s(t), multiply by
s∗(t) and integrate.Time Frequency Analysis – p. 14/26
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Time Operator
For time function in frequency domain only
Time operator is defined as: T = −1jddω
Repeated use of T denoted by T n
T nS(ω) = (−1j )n dn
dωnS(ω)
Average of a time function calculation from frequencyfunction:< g(t) >=
∫
g(t)|s(t)|2dt
< g(t) >=∫
S∗(ω)g(T )S(ω)dω
< g(t) >=∫
S∗(ω)g(−1jddω )S(ω)dω
Take g(t), replace t by −1jddω , operate on S(ω), multiply
by S∗(ω) and integrate.
Time Frequency Analysis – p. 15/26
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Complex Time and Frequency Functions
We are aware of Complex Frequency Function:S(ω) = A(ω)ejφ(ω) = Sr + jSi
Can we have similar thing in time domain?
Useful in time frequency descriptions
Complex Time Function: s(t) = A(t)ejψ(t) = sr + jsi
What should be the values of A(t) and ψ(t)
Equivalently values of sr and si
Time Frequency Analysis – p. 16/26
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Mean Freq. using Freq. Operator
Mean Frequency:Ws(t) = WA(t)ejψ(t)
= 1jddtA(t)ejψ(t) = (ψ′(t) − jA
′(t)A(t) )s(t)
Therefore, the mean frequency is< ω >=
∫
ω|S(ω)|2dω =∫
s∗(t)1jddts(t)dt
=∫
(ψ′(t) − jA′(t)A(t) )A2(t)dt
Since < ω > is always real, second term should be zero.
Second term is zero because it is a perfect differentialthat integrates to zero.
< ω >=∫
ψ′(t)|s(t)|2dt =∫
ψ′(t)A2(t)dt
Time Frequency Analysis – p. 17/26
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Concept of Instantaneous Frequency
< ω >=∫
ψ′(t)|s(t)|2dt =∫
ψ′(t)A2(t)dt
Avg. frequency is obtained by integrating ψ′(t) withdensity over all time.
ψ′(t) is instantaneous value of the quantity for which weare calculating the average.
Hence it is termed as instantaneous frequency,ωi(t) = ψ′(t)
Time Frequency Analysis – p. 18/26
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Bandwidth Equation
B2 = σ2ω =
∫
(ω− < ω >)2|S(ω)|2dω
B2 = σ2ω =
∫
s∗(t)(1jddt− < ω >)2s(t)dt
B2 =∫
|(1jddt− < ω >)s(t)|2dt
B2 =∫
|1jA′(t)A(t) + ψ′(t)− < ω > |2A2(t)dt
B2 =∫
(A′(t)A(t) )2A2(t)dt+
∫
(ψ′(t)− < ω >)2A2(t)dt
B.W. is the avg. of two terms, one depending on theamplitude and other depending on phase.
What is the significance of these two terms?
It will be apparent in joint time and frequencydescription
Time Frequency Analysis – p. 19/26
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AM and FM Contributions to B.W.
B2 =∫
(A′(t)A(t) )2A2(t)dt+
∫
(ψ′(t)− < ω >)2A2(t)dt
First term averages an amplitude term over all time
Second term averages a phase dependent term
It is natural to define AM and FM contributions by
B2AM =
∫
A′2(t)dt
B2FM =
∫
(ψ′(t)− < ω >)2A2(t)dt
With B2 = B2AM + B2
FM
Fractional contributions: rFM = BFM
B and rAM = BAM
B
Time Frequency Analysis – p. 20/26
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Duration and Mean Time
Mean Time: < t >= −∫
φ′(ω)|S(ω)|2dω
Duration:T 2 = σ2
t =∫
(B′(ω)
B(ω) )2B2(ω)dω +∫
(φ′(ω)+ < t >)2B2(ω)dω
Group Delay: tg(ω) = −φ′(ω)
Amplitude and frequency variations of the spectrum willcontribute to the duration.
Spectral Amplitude Modulation (SAM)
Spectral Phase Modulation (SPM)
T 2SAM =
∫
B′2(ω)dω
T 2SPM =
∫
(φ′(ω)+ < t >)2B2(ω)dω
With T 2 = T 2SAM + T 2
SPM
Time Frequency Analysis – p. 21/26
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Covariance of a Signal
To find how time and instantaneous frequency arerelated?
< tψ′(t) >=∫
tψ′(t)|s(t)|2dt
Average of time multiplied with the instantaneousfrequency
If time and frequency are not related then< tψ′(t) >=< t >< ψ′(t) >=< t >< ω >
So excess of < tψ′(t) > over < t >< ω > is a goodmeasure of how time is correlated with instantaneousfrequency.
Covtω =< tψ′(t) > − < t >< ω >
Correlation coefficient: r = Covtω
σtσω
Time Frequency Analysis – p. 22/26
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Covariance in terms of Spectrum
In this case let tg represent time and ω representfrequency.
Covtω =< tgω > − < t >< ω >
with < tgω >= − < ωφ′(ω) >= −∫
ωφ′(ω)|S(ω)|2dω
When the two identities will be equal?∫
tψ′(t)|s(t)|2dt = −∫
ωψ′(ω)|S(ω)|2dω
Time Frequency Analysis – p. 23/26
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When Covariance will be zero?
Covariance is an indication of how inst. freq. and timeare related
When inst. freq. does not change, then covarianceshould be zero
Example: s(t) = A(t)ejω0t
where the amplitude mod is arbitrary, but no changeinst. freq.
Covtψ′(t) =∫
tω0|A(t)|2dt = ω0 < t >
But since < ω >= ω0, we have < ω >< t >= ω0 < t >
Therefore covariance and correlation coefficient areequal to zero
Time Frequency Analysis – p. 24/26
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Characteristic Function
F.T. of a density is called the characteristic function
Chr. Fn. for the energy density spectrum isR(τ) =
∫
|S(ω)|2ejτωdω =∫
s∗(t)ejτWs(t)dt
ejτW is the translation operator and thereforeR(τ) =
∫
s∗(t)s(t+ τ)dt
R(τ) compares or correlates the signal at two differenttimes and hence it is termed as autocorrelation function
Inversely we have: |S(ω)|2 = 12π
∫
R(τ)e−jωτdτ
Similarly, Chr. Fn. in freq. domain isR(θ) =
∫
|s(t)|2ejθtdt =∫
S∗(ω)ejθτS(ω)dω =∫
S∗(ω)S(ω − θ)dω
Hence |s(t)|2 = 12π
∫
R(θ)e−jtθdθ
Time Frequency Analysis – p. 25/26
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Non Additivity of Spectral Operators
Conceptual difficulties in TFA are due to basicproperties of signals and spectra
Freq. content is not additive in TF plane
In T or F plane, s = s1 + s2 and S = S1 + S2
However, for energy density|S|2 = |S1 +S2|
2 = |S1|2 + |S2|
2 +2Re{S∗1S2} 6= |S1|
2 + |S2|2
Thus the freq. content is not the sum of the frequencycontent of each signal
When two signal are added, the waveforms may addand interfere in all possible ways to give differentweights to original frequencies
Mathematically, energy density spectrum is theabsolute square of the sum of the spectra, which resultsin nonlinear effects
Time Frequency Analysis – p. 26/26
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