Estimating Periodic Signals
Debasis Kundu
Department of Mathematics & Statistics
Indian Institute of Technology Kanpur
Most of this talk has been taken from the book ”Statistical Signal Processing”,
by D. Kundu and S. Nandi.
August 26, 2012
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Some Definitions:
What is a Signal?
A signal is a function that conveys information
about the behavior or attributes of some
phenomenon (Wikipedia)
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Some Definitions:
Different Examples:
1
Daily Gold price.
2
Monthly expenditure in a family
3
ECG signal of a human being.
4
Satellite images.
5
Textures
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Some Definitions:
What is Signal Processing?
Signal Processing may broadly be considered to
involve the recovery of information from physical
observations. The received signal is usually
disturbed by external or internal noises. Due to
random nature of the signal, statistical techniques
play important roles in analyzing the signals.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Examples: ECG Signal
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Examples: Rumford Data
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Example: Vowel Sound
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Example: Variable Star Brightness Signal
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Example: Airlines Passenger Data
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Some More Definition
What is a Periodic Signal?
A signal (function) which repeats after a fixed
period of time.
f (t) = f (t 0 );
where t 0 = t mod T
Example:
y (t) = A cos(ωt) + B sin(ωt).
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Fourier Transform
A smooth mean zero periodic function can be written as
y (t) =
∞
X
Ak cos(kωt) + Bk sin(kωt),
k=1
and it is known as the Fourier expansion of y (t).
Most of the times y (t) is corrupted with noise, hence we use
y (t) =
∞
X
Ak cos(kωt) + Bk sin(kωt) + X (t)
k=1
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Sinusoidal Signal
Since it is impossible to estimate infinite number of parameters,
the following model has been used
y (t) =
p
X
Ak cos(ωk t) + Bk sin(ωk t) + X (t);
k=1
where p < ∞. Often the problem boils down to estimate Ak ’s,
Bk ’s, ωk ’s and p based on a sample of size n, namely
y (1), . . . , y (n).
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Periodogram Estimators
The most used and popular estimation procedure is the
periodogram estimators. The periodogram at a particular
frequency is defined as
2
n
1 X
y (t)e −iωt I (ω) = n
t=1
or equivalently
1
I (ω) =
n
n
X
!2
y (t) cos(ωt)
t=1
Debasis Kundu
1
+
n
n
X
!2
y (t) sin(ωt)
t=1
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Periodogram Estimator
Consider the following sinusoidal signal: Sinusoidal Example 1:
y (t) = 3.0(cos(0.2πt)+sin(0.2πt))+3.0(cos(0.5πt)+sin(0.5πt))+X (t)
Here X (t)’s are i.i.d. N(0,0.5)
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Examples: Sinusoidal Signal
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Periodogram Estimator
Consider the following sinusoidal signal: Sinusoidal Example 2:
y (t) = 3.0(cos(0.2πt)+sin(0.2πt))+0.25(cos(0.5πt)+sin(0.5πt))+X (t)
Here X (t)’s are i.i.d. N(0,2.0)
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Examples: Sinusoidal Signal
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Least Squares Estimators
The model can be seen as a non-linear regression model:
y (t) = ft (θ, p) + X (t)
where
ft (θ, p) =
p
X
Ak cos(ωk t) + Bk sin(ωk t)
k=1
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Least Squares Estimators
Assuming p is known, the most natural estimators will be the least
squares estimators and they can be obtained as follows:
n
X
t=1
"
y (t) −
p
X
#!2
Ak cos(ωk t) + Bk sin(ωk t)
k=1
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Theoretical and Numerical Issues
1
It does not satisfy the standard sufficient condition of
Jennrich or Wu.
2
The least squares may not be consistent.
3
The asymptotic distribution of the least squares estimators are
√
not n consistent.
4
Numerically it is a challenging problem.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Separable Regression Technique
The model can be written as follows
Y = A(θ)β + e
where

cos(ω1 )

..
A(θ) = 
.
sin(ω1 )
..
.
...
..
.
cos(ωp )
..
.

sin(ωp )

..

.
cos(nω1 ) sin(nω1 ) . . . cos(nωp ) sin(nωp )
β T = (A1 , B1 , . . . , Ap , Bp ), e T = (X (1), . . . , X (n)).
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Separable Regression Technique
The least squares estimators can be obtained by minimizing
Q(θ, β) = (Y − A(θ)β)T (Y − A(θ)β)
with respect to the unknown parameters.
Note that if θ is known then
βbT (θ) = (A(θ)T A(θ))−1 AT (θ)Y .
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Separable Regression Technique
The least squares estimators of θ can be obtained by minimizing
T b
b
Q(θ, β(θ))
= Y − A(θ)βbT (θ)
Y − A(θ)β(θ)
with respect to θ.
It is equivalent in saying minimize
Q(θ) = Y T (I − PA )Y ,
where PA is the projection matrix as PA = A(AT A)−1 AT
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Approximate Least Squares Estimators
Note that minimizing
Q(θ) = Y T (I − PA )Y ,
is equivalent to maximizing
R(θ) = Y T PA Y .
Approximate n1 (AT A) = I . Therefore,
1
1
Q̃(θ) = Y T AAT Y =
n
n
n
X
t=1
Debasis Kundu
!2
y (t) cos(ωt)
1
+
n
Estimating Periodic Signals
n
X
t=1
!2
y (t) sin(ωt)
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Complex Exponential
The sum of sinusoidal model has a very close resemblance with the
corresponding model
y (t) =
p
X
Ak e iωk t + X (t)
k=1
Here y (t)’s are complex valued, Ak and Bk are complex valued,
0 < ωk < 2π. The problem remains the same, estimate the
unknown parameters based on y (t)’s.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Prony’s Equation
Prony in 1795 observed the following interesting facts: If
µ(t) =
p
X
Ak e βk t ;
t = 1, . . . , n,
k=1
here Ak ’s and βk ’s are real and βk ’s are distinct, then there exists
g0 , . . . , gp such that

   
µ(1)
. . . µ(p + 1)
g0
0
 µ(2)
 g1  0
.
.
.
µ(p
+
2)

   

  ..  =  .. 
..
..
..

  .  .
.
.
.
gp
µ(n − p) . . .
µ(n)
0
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Prony’s Equation
The constants g0 , . . . , gp do not depend on Ak ’s, they depend only
on βk ’s.
βk ’s can be obtained from gk ’s as follows: Consider the following
polynomial equation
g0 + g1 x + . . . + gp x p = 0,
then e β1 , . . . , e βp are the roots of the above polynomial equations.
Once βk ’s are obtained, Ak ’s are obtained using simple linear
regression method.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Prony’s Equations
Similar results are true in case of complex exponential also, i.e. if
µ(t) =
p
X
Ak e βk t ;
t = 1, . . . , n,
k=1
here Ak ’s and βk ’s are complex.
Similarly, if
µ(t) =
p
X
Ak cos(ωk t) + Bk sin(ωk t);
t = 1, . . . , n,
k=1
here Ak ’s and Bk ’s are real, and 0 < ωk < 2π.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Prony’s Equation
It is immediate that if there is no error then Ak ’s and βk can be
recovered from µ(t)’s without any problem.. Now suppose
y (t) =
p
X
Ak e βk t + e(t);
t = 1, . . . , n,
k=1
here Ak ’s and βk ’s are real and βk ’s are distinct, and e(t)’s are
small mean zero error. Then it is expected

   
y (1)
. . . y (p + 1)
g0
0
 y (2)
 g1  0
.
.
.
y
(p
+
2)

   

  ..  ≈  .. 
..
..
..

  .  .
.
.
.
y (n − p) . . .
y (n)
gp
0
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Prony’s Equation
Therefore, if


. . . y (p + 1)

. . . y (p + 2)


A=

..
..


.
.
y (n − p) . . .
y (n)
y (1)
y (2)
..
.
 
g0
 g1 
 
g =.
 .. 
gp
then we want to solve
Ag = 0 ⇔ AT Ag = 0 ⇒
g is an eigen vector corresponds to 0 eigenvalue of AT A.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Numerical Issues
1
It is a highly non-linear problem. The least squares surface
has several local minima.
2
Most of the time the standard Newton-Raphson algorithm
may not converge.
3
Even if they converge, often it converges to the local
minimum rather than the global minimum.
4
If p is large, it becomes a higher dimensional optimization
problem, extremely accurate initial guesses are required for
any iterative procedure to work well.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Sequential Estimation Procedures
It is based on the facts that the components are orthogonal and it
works like this
First minimize
n
X
(y (t) − A cos(ωt) − B sin(ωt))2
t=1
with respect to A, B and ω.
Take out their effect from y (t), i.e. consider
b cos(b
b sin(b
ỹ (t) = y (t) − A
ω t) − B
ω t)
Repeat the procedure p times.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Advantage
It reduces the computational burden significantly. For example if p
= 25, instead of solving a 25 dimensional optimization problem, we
need to solve 25 one dimensional optimization problems. It does
not have any problem about initial guess or convergence.
It produces the same accuracy as the least squares estimators.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Super Efficient Estimators
When p = 1, the Newton-Raphson algorithm will be of the
following form:
Q 0 (ω)
ω (j+1) = ω (j) − 00
Q (ω)
After few pages of calculations it has been suggested
ω (j+1) = ω (j) −
1 Q 0 (ω)
4 Q 00 (ω)
It not only converges, it produces estimators which are better than
the least squares estimators.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Main Asymptotic Results
1
2
Least squares estimators are consistent under mild
assumptions on the errors.
Least squares estimators have the convergence rate n−3/2 .
3
Sequential estimators have the same convergence rate as the
least squares estimators.
4
Asymptotic variances of the super efficient estimators are
smaller than the least squares estimators.
5
Prony’s estimators are not consistent.
6
Periodogram estimators are consistent, but it has the
convergence rate n−1/2 .
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Estimation of p
1
Consider the number of peaks of the periodogram function.
2
It can be very misleading.
3
In the least squares procedure, consider residual sums of
squares.
4
It can be very misleading too.
5
Information theoretic criterion.
6
Cross validation technique.
7
Likelihood ratio approach.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Information Theoretic Criterion
AIC (k) = n ln Rk + 2(3k)
BIC (k) = n ln Rk +
1
ln n(3k)
2
EDC (k) = n ln Rk + Cn k.
Here Cn satisfies certain conditions namely
Cn
−→ ∞
n
Cn
−→ ∞.
ln ln n
Choose that model for which AIC (k), BIC (k) or EDC (k) is
minimum
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Information Theoretic Criterion
Which Cn to choose?
Resampling technique can be used to compute PCS for each Cn
and choose that Cn for which the PCS is maximum.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Outline
1
Introduction
2
3
Basic Estimation Procedures
Least Squares and Approximate Least Squares
Estimators
4
Prony’s Estimators
5
Two Other Important Estimators
6
Asymptotic Properties
7
Estimation of p
8
Some Related Models
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Compartment Model
Consider the following real valued model:
y (t) =
p
X
Ak e βk t + e(t);
t = 1, . . . , n
k=1
Here Ak ’s and βk ’s are real numbers. The number of components
p may be known or unknown. The problem is to estimate Ak ’s and
βk ’s based on y (t)’s.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Fundamental Frequency Model
Consider the following model:
y (t) =
p
X
[Ak cos(kλt) + Bk sin(kλt)] + e(t)
k=1
Here λ is the fundamental frequency, and it has p harmonics. The
problem remains the same.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Chirp Signal Model
Consider the following model:
y (t) =
p
X
Ak cos(λk t + βk t 2 ) + Bk sin(λk t + βk t 2 ) + e(t)
k=1
The problem is to estimate the frequency and frequency rates.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Partially Sum of Sinusoidal Model
Consider the following model:
y (t) = a + bt +
p
X
[Ak cos(ωk t) + Bk sin(ωk t)] + e(t)
k=1
The problem is to estimate the unknown parameters.
Debasis Kundu
Estimating Periodic Signals
Introduction
Basic Estimation Procedures
Least Squares and Approximate Least Squares Estimators
Prony’s Estimators
Two Other Important Estimators
Asymptotic Properties
Estimation of p
Some Related Models
Thank You
Debasis Kundu
Estimating Periodic Signals