FaaDoOEngineers.com

Time Frequency AnalysisMathematical Preliminaries

S. R. M. Prasanna

Dept of ECE,

IIT Guwahati,

prasanna@iitg.ernet.in

Time Frequency Analysis – p. 1/26

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com

FaaDoOEngineers.com

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

ww

w.jntuw

orld.com