The Uncertainty Principle in Signal Processing
(A Short Note)
Sandeep Palakkal
Indian Institute of Technology Madras, India,
email: sandeep.dion@gmail.com
Remark 1 The uncertainty principle (TUP) is one of the fundamental concepts in signal processing. There is actually nothing uncertain in the uncertainty principle, but this name is widely used because of its similarity with the Heisenberg's uncertainty principle in physics. Before formally state this principle, a few definitions are in order.
Definition 2 (Localization time of a signal) Let x(t)x(t) be an L2(R)L2(R) signal, where L2(R)L2(R) is the set(Hilbert space) of all square-integrable signals and RR stands for the set of real numbers. Now, the point t¯t¯ in time around with the signal is localized may be defined as its first moment with respect to its energy spread in time. That is,
t
¯
=
∫
t
|
x
(
t
)
|
2
d
t
/
∥
x
∥
2
,
t
¯
=
∫
t
|
x
(
t
)
|
2
d
t
/
∥
x
∥
2
,
(1)
where ∥x∥2∥x∥2 is the energy of the signal. Note that the limits of integration are -∞-∞ and ∞∞ in all the formula in this document.
Definition 3 (Localization frequency of a signal) Similarly, we can define the frequency ω¯ω¯ around with the Fourier transform of x(t)x(t) is localized. Denoting the Fourier transform of x(t)x(t) by x^(ω)x^(ω), we have
ω
¯
=
1
2
π
∫
ω
|
x
^
(
ω
)
|
2
d
ω
/
∥
x
∥
2
ω
¯
=
1
2
π
∫
ω
|
x
^
(
ω
)
|
2
d
ω
/
∥
x
∥
2
(2)Definition 4 (Time-duration and bandwidth of a signal) Now, the time-duration and bandwidth of a signal may be defined as the second central moments of energy distribution of the signal in time domain and the Fourier domain as given below
σ
t
2
=
∫
(
t
-
t
¯
)
2
|
x
(
t
)
|
2
d
t
/
∥
x
∥
2
σ
t
2
=
∫
(
t
-
t
¯
)
2
|
x
(
t
)
|
2
d
t
/
∥
x
∥
2
(3)
σ
ω
2
=
1
2
π
∫
(
ω
-
ω
¯
)
2
|
x
^
(
ω
)
|
2
d
ω
/
∥
x
∥
2
σ
ω
2
=
1
2
π
∫
(
ω
-
ω
¯
)
2
|
x
^
(
ω
)
|
2
d
ω
/
∥
x
∥
2
(4)Theorem 5 (The uncertainty principle) The product of time-duration and bandwidth of a signal are bounded from below by 1212. That is,
σ
t
σ
ω
≥
1
2
σ
t
σ
ω
≥
1
2
(5)For convenience, let us assume that t¯t¯ and ω¯ω¯ are zeros, which, if they are originally not, can be obtained by time-shifting and modulation. Note that these two operations do not change the time-duration as well as the bandwidth of the signal, only which we are interested in. Therefore, the new assumptions do not limit the generality of the proof. Using Cauchy-Schwartz inequality,
∫
t
x
(
t
)
x
'
(
t
)
d
t
2
≤
∫
t
2
x
(
t
)
2
d
t
∫
|
x
'
(
t
)
|
2
d
t
.
∫
t
x
(
t
)
x
'
(
t
)
d
t
2
≤
∫
t
2
x
(
t
)
2
d
t
∫
|
x
'
(
t
)
|
2
d
t
.
(6)
On the right-hand side (RHS), the first tem is σt2∥x∥2σt2∥x∥2, whereas the second term can be written as follows using the Parseval's theorem
1
2
π
∫
|
ω
|
2
|
x
^
(
ω
)
|
2
d
ω
=
σ
ω
2
∥
x
∥
2
.
1
2
π
∫
|
ω
|
2
|
x
^
(
ω
)
|
2
d
ω
=
σ
ω
2
∥
x
∥
2
.
(7)
Therefore, the RHS of Equation 6 is equivalent to the product of time-duration and bandwidth of the signal. From the left-had side (LHS)
1
2
∫
t
∂
x
2
(
t
)
∂
t
d
t
=
1
2
t
x
2
(
t
)
-
∫
x
2
(
t
)
d
t
-
∞
∞
.
1
2
∫
t
∂
x
2
(
t
)
∂
t
d
t
=
1
2
t
x
2
(
t
)
-
∫
x
2
(
t
)
d
t
-
∞
∞
.
(8)
Since x(t)∈L2(Rx(t)∈L2(R, the limit value of tx2(t)tx2(t) at infinity is zero (To see this, recall that since x(t)x(t) is square integrable, its decay is of O(1/t)O(1/t)); this implies that tx(t)=O(1/t)tx(t)=O(1/t), which, in the limit as tt tends to infinity, is zero.). Therefore, the RHS of Equation 8 reduces to -∥x∥2/2-∥x∥2/2. Substituting in Equation 6, we have
σ
t
2
σ
ω
2
≥
1
4
σ
t
2
σ
ω
2
≥
1
4
(9)Remark 6 There are variations in expressing TUP, which we now describe. For convenience, let us assume that ∥x∥2=1∥x∥2=1 and, again, ω¯=0ω¯=0 and t¯=0t¯=0.
Remark 7 (Variation 1) Suppose we define the bandwidth as
D
ω
2
=
∫
ω
2
|
x
^
(
ω
)
|
2
d
ω
.
D
ω
2
=
∫
ω
2
|
x
^
(
ω
)
|
2
d
ω
.
(10)
Then from Equation 4, Dω2=2πσω2Dω2=2πσω2. Substituting in Equation 9,
D
ω
2
σ
t
2
≥
π
2
.
D
ω
2
σ
t
2
≥
π
2
.
(11)Remark 8 (Variation 2) Defining the bandwidth as
Δ
f
2
=
∫
f
2
|
x
^
(
f
)
|
2
d
f
,
where
f
=
ω
/
(
2
π
)
,
Δ
f
2
=
∫
f
2
|
x
^
(
f
)
|
2
d
f
,
where
f
=
ω
/
(
2
π
)
,
(12)
we have Δf=σω2/(4π2)Δf=σω2/(4π2). Therefore, TUP becomes
Δ
f
2
σ
t
2
≥
1
16
π
2
.
Δ
f
2
σ
t
2
≥
1
16
π
2
.
(13)Remark 9 The variations in the way TUP is expressed may cause confusion. However, it should be noted that all of them are the same. Probably, assuming that the signal is localized around zero in both time and frequency axes, and is normalized to have unit energy, the best way to express TUP may be
∫
-
∞
∞
t
2
|
x
(
t
)
|
2
d
t
1
2
π
∫
-
∞
∞
ω
2
|
x
^
(
ω
)
|
2
d
ω
≥
1
4
∫
-
∞
∞
t
2
|
x
(
t
)
|
2
d
t
1
2
π
∫
-
∞
∞
ω
2
|
x
^
(
ω
)
|
2
d
ω
≥
1
4
(14)
By substituting ω=2πω=2π in the above equation, other variations can be easily derived.
Remark 10 In Heisenberg's uncertainty principle, the place of the energy density |x(t)|2/∥x∥2|x(t)|2/∥x∥2, or equivalently |x^(ω)|2/∥x∥2|x^(ω)|2/∥x∥2, is taken by the probability density. Therefore, instead of the time-bandwidth product, we have the product of two standard deviations, which, according to the theorem, are bounded from below by a constant. This reflects the involvement of some uncertainty. However, in signal processing, there is no uncertainty involved, and the name uncertainty theorem is a misnomer.
Remark 11 TUP is expressed as an inequality. Is this inequality tight? That is, does the time-bandwidth product ever achieve the lower-bound? The answer is well-known: yes, for Gaussian signal. That is,
x
(
t
)
=
exp
(
-
α
t
2
)
,
α
>
0
.
x
(
t
)
=
exp
(
-
α
t
2
)
,
α
>
0
.
(15)Recall that Equation 6 is the Cauchy-Schwartz inequality. It becomes an equality when the two functions are linearly related. That is,
x
'
(
t
)
=
c
t
x
(
t
)
,
c
=
arbitrary
constant
.
x
'
(
t
)
=
c
t
x
(
t
)
,
c
=
arbitrary
constant
.
(16)
This condition is only satisfied by the Gaussian function. From Equation 15,
x
'
(
t
)
=
-
2
α
t
exp
(
-
α
t
2
)
=
c
t
x
(
t
)
,
c
=
constant
.
x
'
(
t
)
=
-
2
α
t
exp
(
-
α
t
2
)
=
c
t
x
(
t
)
,
c
=
constant
.
(17)
In passing we note that the function exp(αt2)exp(αt2) also satisfies the condition given in Equation 16 but does not belong to L2(R)L2(R).
Remark 12 Having understood TUP, a very important problem is this: find band-limited functions which are maximally concentrated around the origin in time. Mathematically, find x(t)x(t), say of unit norm, such that it is band-limited to ω0ω0 (x^(ω)=0,|ω|>ω0)x^(ω)=0,|ω|>ω0) and for a given T∈(0,∞)T∈(0,∞)
α
=
∫
-
T
T
|
x
(
t
)
|
2
d
t
α
=
∫
-
T
T
|
x
(
t
)
|
2
d
t
(18)
is maximized. It turns out that [4], [2] the solution is the function having the largest eigen value satisfying
∫
-
T
T
x
(
τ
)
sin
(
ω
0
(
t
-
τ
)
)
π
(
t
-
τ
)
d
τ
=
λ
f
(
t
)
∫
-
T
T
x
(
τ
)
sin
(
ω
0
(
t
-
τ
)
)
π
(
t
-
τ
)
d
τ
=
λ
f
(
t
)
(19)
The eigen functions satisfying the above equation are called the prolate spheroidal wave functions.
-
Gabor, D. (1946). Theory of Communication. J. Inst. Elec. Engg., 93, 429–457.
-
Landau, H. J. and Pollak, H. O. (1961, Jan.). Prolate spheroidal wave functions, Fourier analysis and uncertainty – II. Bell Syst. Tech. J., 40, 65–84.
-
Mallat, Stéphan. (1999). A Wavelet Tour of Signal Processing. (2). San Diego, California, USA: Academic Press.
-
Slepian, D. and Pollak, H. O. (1961, Jan.). Prolate spheroidal wave functions, Fourier analysis and uncertainty – I. Bell Syst. Tech. J., 40, 43–63.
-
Vetterli, Martin and Kovačević, Jelena. (1995). Wavelets and Subband Coding. Englewood Cliffs, New Jersey: Prentice Hall PTR.
