The Uncertainty Principle in Signal Analysis

• Was first proposed by Gabor in 1946.

• It is a fundamental statement regarding Fourier transform pairs.

• We have uncertainty principle in quantum mechanics which infers

that we cannot place an object at a given position with required

velocity.

• However it should be noted that we can choose the position and

velocity of the object independently at will but not both simulta-

neously.

• Gabor observed this and used the same to explain the similar

phenomenon occurring in signal analysis.

ww

w.jntuw

orld.com

• From the Fourier analysis it is known that a narrow waveform

yields a wide spectrum and a wide waveform yields a narrow spec-

trum.

• Thus both time waveform and frequency spectrum cannot be

made arbitrarily small simultaneously and this fact is explained

as uncertainty principle.

• There is a misconception that it is not possible to measure t-f en-

ergy density of a given waveform as a consequence of uncertainty

relation.

• However, uncertainty principle of waveform is not concerned with

the measurement of t-f energy density distributions.

• Instead, it states that if the effective bandwidth of a signal is W

then the effective duration cannot be less than about 1/W and

ww

w.jntuw

orld.com

conversely.

• In summary, if the density in time is |s(t)|2 and the density in fre-

quency is |S(jw)|2, since s(t) and S(jw) are related through Fourier

relation, there is a relation between the densities also.

• The relation is such that if one density is narrow, then the other

is broad. That is all the uncertainty principle.

• To emphasize further, it is not that both time and frequency

cannot be made narrow, but the densities of time and frequency

cannot both be made narrow.

• It should be clear that the uncertainty principle never applies to

a single variable.

• It is always a statement about two variables.

• Further, it does not apply to any two variables, but only two

ww

w.jntuw

orld.com

variables whose associated operations do not go together.

• That is I cannot narrow the width simultaneously in time and

frequency domains.

Some Results for Proving Uncertainty Principle

1

j

d

dts(t) =

1

j

d

dt(A(t)ejφ(t))

=1

j(A(t)

d

dtejφ(t) + ejφ(t) d

dtA(t))

=1

j(A(t)jφ

′(t)ejφ(t) + ejφ(t)A

′(t))

= (A(t)ejφ(t)(φ′(t) − j

A′(t)

A(t))

ww

w.jntuw

orld.com

1

j

d

dts(t) = s(t)(φ

′(t) − j

A′(t)

A(t))

1

j

d

dts(t) =

1

j(A(t)

d

dtejφ(t) + ejφ(t) d

dtA(t))

< ω >= µω =

∫

ω

ω.|S(jω)|2dω =

∫

t

s∗(t).(1

j

d

dt)s(t)dt

< ωn >=

∫

ω

ωn.|S(jω)|2dω =

∫

t

s∗(t).(1

j

d

dt)ns(t)dt

Proof :

< ω >= µω =

∫

ω

ω.|S(jω)|2dω

S(jω) =

∫

t

s(t).e−jωtdt

ww

w.jntuw

orld.com

S∗(jω) =

∫

t

s∗(t).ejωtdt

|S(jω)|2 = S(jω).S∗(jω)

< ω >=

∫

ω

ω(

∫

t

s∗(t).ejωtdt).(

∫

t

s(t′).e−jωt′dt′)dω

< ω >=

∫

ω

∫

t

∫

t′

ω.s∗(t)ejω(t−t′).s(t′)dt′dtdω

∂

∂tejω(t−t′) = ejω(t−t′).jω

< ω >=

∫

ω

∫

t

∫

t′

s∗(t)

ww

w.jntuw

orld.com

(1

j

∂

∂tejω(t−t′)).s(t′).dt′dtdω

∫

ω

ejω(t−t′)dω = δ(t − t′)

< ω >=1

j

∫

t

∫

t′

s∗(t)∂

∂tδ(t − t′)s(t′).dt′dt

< ω >=1

j

∫

t

s∗(t)∂

∂t(

∫

t′

s(t′)δ(t − t′)dt′)dt

< ω >=1

j

∫

t

s∗(t)d

dts(t)dt

< ω >=

∫

t

s∗(t)(1

j

d

dts(t))dt

ww

w.jntuw

orld.com

< ω2 >=

∫

ω

ω2.|S(jω)|2dω

=

∫

ω

ω2(

∫

t

s∗(t).ejωtdt).

∫

t

s(t′).e−jωt′dt′)dω

< ω2 >=

∫

ω

∫

t

∫

t′

ω2.s∗(t)ejω(t−t′).s(t′)dt′dtdω

∂

∂tejω(t−t′) = ejω(t−t′).jω

∂2

∂2t=

∂

∂t(∂

∂tejω(t−t′))

=∂

∂t(jωejω(t−t′))

ww

w.jntuw

orld.com

= jω(jωejω(t−t′))∂2

∂t2

= j2ω2ejω(t−t′) ∂n

∂tn

= jnωnejω(t−t′) < ω2 >=

∫

ω

∫

t

∫

t′

s∗(t)(1

j2

∂2

∂t2ejω(t−t′)).s(t′).dt′dtdω

< ω2 >=

∫

t

∫

t′

s∗(t)(1

j2

∂2

∂t2

∫

ω

ejω(t−t′)dω).s(t′).dt′dt

< ω2 >=

∫

t

∫

t′

s∗(t)(1

j2

∂2

∂t2δ(t − t′)).s(t′).dt′dt

< ω2 >=

∫

t

s∗(t)(1

j2

∂2

∂2t

∫

t′

s(t′).δ(t − t′)).dt′dt

ww

w.jntuw

orld.com

< ω2 >=

∫

t

s∗(t)(1

j2

d2

dt2(s(t))dt

< ωn >=

∫

t

s∗(t)(1

jn

dn

dtn(s(t))dt

Further,

< ω2 >=

∫

ω

ω2.|S(jω)|2dω

⇒ < ω2 >=

∫

t

s∗(t)((1

j

d

dt)2(s(t))dt

⇒ < ω2 >= −

∫

t

s∗(t)(d

dt)2(s(t))dt

s(t) =1

2π

∫

ω

S(jω)ejωtdω

ww

w.jntuw

orld.com

d

dts(t) =

1

2π

∫

ω

S(jω)jωejωtdω

d

dts(t) =

j

2π

∫

ω

ωS(jω)ejωtdω

d2

dt2s(t) =

d

dts(t)

d

dts(t)s(t)

d2

dt2s(t) =

j2

2π

∫

w

w2.S(jw).ejwtdw

d

dts(t) =

j

2π

∫

w

w.S(jw).ejwtdw

|d

dts(t) |=|

j

2π

∫

w

w.S(jw).ejwtdw |

ww

w.jntuw

orld.com

|d

dts(t) |2=|

j

2π

∫

w

w.S(jw).ejwtdw |2

|d

dts(t) |2= (

j

2π

∫

w

w.S(jw).ejwtdw |)(−j

2π

∫

w

w.S(jw).e−jwtdw)

|d

dts(t) |2= (

j

2π

∫

w

S(jw).ejwtdw |)(−j

2π

∫

w

w2.S(jw).e−jwtdw)

|d

dts(t) |2= −1.s∗(t).

d2

dt2s(t)

< w2 >=

∫

w

w2. | S(jw) |2 dw =

∫

t

s∗(t)(1

j

d

dt)2s(t)dt

< w2 >= −

∫

t

s∗(t).d2

dt2.s(t)dt

ww

w.jntuw

orld.com

< w2 >=

∫

t

|d

dts(t) |2 dt

< w2 >=

∫

t

| s′(t) |2 dt

Uncertainty Principle:

TeBe ≥ 1/2

Proof:

Be = σ2w =

∫

w

w2|S(jw)|2dw =

∫

t

|s′(t)|2dt

T 2e = σ2

t =

∫

t

t2|s(t)|2dt

and therefore,

T 2e B2

e = σ2t σ

2w =

∫

t

|ts(t)|2dt

∫

t

|s′(t)|2dt

ww

w.jntuw

orld.com

• From the Schwartz inequality,∫

|f(x)|2dx

∫

|g(x)|2dx ≥ |f∗(x)g(x)dx|2

• Taking f = ts and g = s′ gives

σ2t σ

2w ≥

∫

t

|ts∗(t)s′(t)dt|2

• Let, s(t) = Aejφ(t) ⇒ s∗(t) = Ae−jφ(t)

s′(t) = d/dts(t) = d/dt(Aejφ(t))

= A.ejφ(t).jφ′ + ejφ(t)d/dtA

= jφ′A.ejφ(t) + A′.ejφ(t)

ts∗(t).s′(t) = tA.e−jφ(t)(jφ′A.ejφ(t) + A′ejφ(t))

= tj.φ′A2 + tA.A′

• This further simplifies to

ts∗(t).s′(t) =1

2

d

dtA2 −

1

2A2 + jtφ′(t)

ww

w.jntuw

orld.com

• The first term is a perfect differential and integrates to zero.

• The second term gives 1/2 since we assumed the signal is normal-

ized.

• The third term∫

tφ′(t)dt = Covtw, covariance of the signal.

• Therefore,

σ2t σ

2w ≥

∫

|ts∗(t).s′(t)dt|2

= | − 1/2 + jCovtw|2

= (1/2)2 + Cov2tw

= 1/4(1 + 4Cov2tw)

σtσw ≥ 1/2(√

1 + 4Cov2tw)

ww

w.jntuw

orld.com

• At the extreme the signal can be stationary under which Covtw = 0

⇒ σtσw ≥ 1/2

i.e. TeBe ≥ 1/2

ww

w.jntuw

orld.com

Short Term Fourier Transform (STFT)

• STFT is the most widely used method for studying non-stationary

signals.

Basic Idea of STFT:

• Suppose we listen to a piece of music that lasts an hour where

in the beginning there are violins and at the end drums. If

we Fourier analyze the whole hour, the energy spectrum will

show peaks at the frequencies corresponding to the violins and

drums. That will tell us that there were violins and drums, but

will not give us any indication of where the violins and drums

were played.

• How do we handle this?

ww

w.jntuw

orld.com

– Breakup the hour into five minute segments and Fourier an-

alyze each interval.

– Upon estimating the spectrums of each segment we will see

in which five minute intervals the violins and drums occurred.

• If we want to localize even better, we break up the hour into

one minute segments or even smaller time intervals and Fourier

analyze each segment.

• This is the basic idea of the STFT

– Breakup the signal into small time segments and Fourier

analyze each time segment to ascertain the frequencies that

existed in that segment.

• The totality of such spectra indicates how the spectrum is

varying in time.

ww

w.jntuw

orld.com

• Can this process be continued to achieve finer and finer time

localization? Can we make the time intervals as short as we

want?

• No, because after a certain narrowing the answer we get for

the spectrum become meaningless and show no relation to the

spectrum of the original signal.

• The reason is that we have taken a perfectly good signal and

broken it up into short duration signals.

• But short duration signals have inherently large bandwidths,

and the spectra of such short duration signals have very little

to do with the properties of the original signal.

• This should be attributed not to any fundamental limitation,

but rather to a limitation of the technique which makes short

ww

w.jntuw

orld.com

duration signals for the purpose of estimating the spectrum.

• Thus sometimes the STFT technique works well and sometimes

it does not.

• It is not the uncertainty principle as applied to the signal that

is the limiting factor.

• It is the uncertainty principle as applied to the small time in-

tervals that we have created for the purpose of analysis.

• The distinction between the two should be kept in mind and

should not be confused.

• In STFT the properties of the signal are scrambled with the

properties of the window function. Unscrambling is required

for proper interpretation and estimation of original signal.

• Despite of all these factors, the STFT is ideal in many respects.

ww

w.jntuw

orld.com

• It is well defined, based on reasonable physical principles, and

for many signals and situations gives an excellent time-frequency

structure consistent with our intuition.

• However, for certain situations it may not be the best method

available in the sense that it does not always give us the clearest

possible picture of what is going on.

• Hence other methods have been developed.

The STFT:

• Given s(t) which represents a signal having time varying spectra.

• The windowed/short duration signal is obtained as

st(τ ) = s(τ )h(t − τ )

ww

w.jntuw

orld.com

where, h(t) is the window function placed around t.

• Since windowed signal st(τ ) emphasizes the signal around the

time t , the Fourier transform will reject the distribution of

frequency around that time:

St(w) =

∫

st(τ )e−jwτdτ

St(w) =

∫

s(τ )h(t − τ )e−jwτdτ

• St(w) is termed as STFT.

• The energy density spectrum at time t is therefore

Psp(t, w) =| St(w) |2 = |

∫

s(τ )h(t − τ )e−jwτdτ |2

ww

w.jntuw

orld.com

• Such a plot is more commonly termed as spectrogram.

• For each different time t, we get a different spectrum and the

totality of these spectra is the time-frequency distribution (Psp).

Uncertainty Principle for the STFT:

• A short duration signal obtained by windowing is given by

st(τ ) = s(τ )h(t − τ )

• The normalized short duration signal at time t is given by

ηt(τ ) =s(τ )h(t − τ )

√

∫

| s(τ )h(t − τ ) |2 dτ

• DR. is square root of total energy in windowed signal.

ww

w.jntuw

orld.com

• This normalization ensures that

∫

| ηt(τ ) |2 dτ = 1

i.e. total normalized energy will be unity for any t.

• The STFT of ηt(τ ) is given by

Ft(jw) =

∫

ηt(τ ).e−jwτdτ.

• We can define all the relevant quantities such as mean time,

duration, and bandwidth in the standard way, but they will be

time dependent.

Mean Time:

ww

w.jntuw

orld.com

< τ >t=

∫

τ | ηt(τ ) |2 dτ

< τ >t=

∫

τ | s(τ )h(τ − t) |2 dτ∫

| s(τ )h(τ − t) |2 dτ

Duration:

T 2t =

∫

(τ− < τ >t)2. | ηt(τ ) |2 dτ

T 2t =

∫

(τ− < τ >t)2. | s(τ )h(τ − t) |2 dτ

∫

| s(τ )h(τ − t) |2 dτ

Mean Frequency:

< w >t=

∫

w | Ft(jw) |2 dw

Bandwidth:

ww

w.jntuw

orld.com

B2t =

∫

(w− < w >t)2 | Ft(jw) |2 dw

T 2t =

∫

(τ− < τ >t)2 | ηt(τ ) |2 dτ

• Let < τ >t= 0 then,

T 2t =

∫

τ 2 | ηt(τ ) | dτ

• Similarly, < w >t= 0 then,

B2t =

∫

w2 | Ft(jw) | dw

B2t =

∫

| η′t(τ ) |2 dτ

ww

w.jntuw

orld.com

T 2t B2

t =

∫

| τηt(τ ) | dτ.

∫

| η′t(τ ) |2 dτ

∫

| f(x) |2 dx

∫

| g(x) |2 dx ≥ |

∫

f∗(x)g(x)dx |2

• Let, f = τηt & g = η′t

T 2t B2

t ≥ |

∫

τη∗t (τ )η′t(τ )dτ |2

• Substituting and simplifying we get

T 2t B2

t ≥1

2

• This is the uncertainty principle for the STFT.

• It is a function of time, the signal, and the window.

ww

w.jntuw

orld.com

• It should not be compared with the uncertainty principle ap-

plied to the signal.

• It is important to understand this uncertainty principle, because

it places limits on the technique of the STFT procedure.

• However, it places no constraints on the original signal.

• It is true that if we modify the signal by the technique of STFT,

we limit our abilities in terms of resolution.

• Hence the search for new time-frequency analysis tools.

ww

w.jntuw

orld.com