In this topic, three main aspects of the problem will be considered:
1-The lower bounds of the Cramer-Rao(CR) estimation error will be examined.
2- Maximum Likelihood (ML) estimators for signal parameters will be developed and analyzed.
3- The practical estimation algorithms and simulation results will be discussed.
➢ The frequency estimation algorithm has a threshold effect which will also be discussed.
3. 1. INTRODUCTION
In this topic, three main aspects of the problem will be considered:
1-The lower bounds of the Cramer-Rao(CR) estimation error will be examined.
2- Maximum Likelihood (ML) estimators for signal parameters will be developed and analyzed.
3- The practical estimation algorithms and simulation results will be discussed.
➢ The frequency estimation algorithm has a threshold effect which will also be discussed.
In general the signal has the form σ𝒊
𝒌
𝒃𝒊 𝐞𝐱𝐩[𝒋 𝒘𝒊𝒕 + 𝜽𝒊 ]
The real part of the signal s(t) = 𝑏0cos (𝑤0𝑡 +𝜃0), and s(t) = 𝑏0sin (𝑤0𝑡 +𝜃0).
The computer input will be two sample vectors:
𝑋 = [𝑋0, 𝑋1, …., 𝑋𝑁−1and 𝑌= [𝑌𝑂, 𝑌1, …., 𝑌𝑁−1]
where :
ෝ
𝑤(t) is the Hilbert transform of the noise w(t).
The problem is estimating the parameters of single-frequency tones from a finite number of noisy discrete-time
observations.
4. where, if 𝑤, 𝑏, and 𝜃 are all unknown,
If Z = 𝑿 + 𝒋𝒀 , then the joint probability density function (pdf) of the elements of the sample vector Z
when the unknown parameter vector is a is given by
1. INTRODUCTION
5. 2. BOUNDS
The unbiased CR bounds are the diagonal elements of
the inverse of the Fisher information matrix 𝐽, from
𝑓(𝑍; 𝜃) the elements of 𝐽 are :
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
The most general case is of all elements of ∝ unknown.
The matrix J, from (6), is then
where
6. 3. MAXIMUM-LIKELIHOOD ESTIMATION
A. General
where
and Re [∙] means the real part of [∙].
(15)
(16)
(18)
(17)
Since σ 𝑋𝑛
2
and σ 𝑌𝑛
2
are constants , we can drop them from 𝐿0
and maximize 𝐿:
After substitution the 𝜇𝑛and 𝑣𝑛 into (16)
7. 3. MAXIMUM-LIKELIHOOD ESTIMATION
B. All Parameters Unknown
where arg [∙] means the argument or phase of [∙] , taken mod 2𝜋 for
convenience, then
Let ෝ
𝑤 be the value of 𝑤 that maximizes 𝐴(𝑤) .
Finally, the value of b that maximizes (20) is
(21)
(20)
(19)
which gives
(22)
The numbers
𝑏 and ෝ
𝑤 are the ML estimates of 𝑤0 and 𝑏0.
Relationship to Discrete Fourier Transform
From (23) and the definition of 𝐴 𝜔 ,
(24)
(25)
The dots along the curve on Fig. 1 are the { 𝐴𝑘 } points
This relationship indicates that the coarse
(approximate) estimates of ෝ
𝑤𝑀𝐿, and
𝑏𝑀𝐿
The ML estimate of
𝜃 is
(23)
8. 3. MAXIMUM-LIKELIHOOD ESTIMATION
C. Summary of Algorithms
For the above-mentioned, the results can be summarized.
If 𝒘𝟎 is unknown then ෝ
𝒘𝑴𝑳 maximizes
if the phase is known,
if the phase is unknown,
If 𝑏0, is unknown then
𝑏𝑀𝐿 is equal to
if the phase is known but the frequency is unknown
Finally, መ
𝜃𝑴𝑳 is equal to
9. 3. MAXIMUM-LIKELIHOOD ESTIMATION
D. Properties of ෝ
𝑤
The ML estimates of 𝑤0 , have the following properties.
1) The pdf of ෝ
𝑤 is symmetrical about 𝑤0, mod 𝑤𝑠.
2) var (ෝ
𝑤) is proportional to 𝑤𝑠
2
and independent of 𝜃0
Noise Model:
Let {𝑉
𝑛} be a set of independent Rayleigh random variables
with parameter 1.
(26)
(27)
and
Let {∅𝑛} be a set of independent random variables uniformly
distributed over (- 𝜋, 𝜋).
(25)
using the noise model.
(29)
(28)
where (30)
and
(31)
let
𝛽 be the value of 𝛽 in the range (-𝜋, 𝜋) that maximizes 𝐴 𝑤 .
(32)
➢ the bias of ෝ
𝑤 has the following values
Proof: Recall that
10. 4. ALGORITHM
The coarse search
where
Thus
(33) (37)
(36)
(34)
(35)
When 𝑀 is 2𝑁, 4 𝑁, or 8 𝑁.
The Fine Search:
The fine search algorithm determines the value of 𝑤 closest to 𝑤𝑙
that maximizes 𝐴 𝑤 .
The secant method is used to compute successive estimates
of frequency ෝ
𝑤 .
The global maximum of 𝐴 𝑤 and the ML estimates.
When there is no noise it can be shown that
For coarse search, we calculate 𝐴 𝑤 at the set of
frequencies {𝑤𝑘} defined by
The output of the coarse search is the value of 𝑤𝑘,called 𝑤𝑙
that corresponds to the largest member of the set {𝐴 𝑤𝑘 }.
11. 5. THRESHOLD EFFECT AND SIMULATIONS
A. General
where
(38)
➢ Consider the estimation of the frequency of a single
complex tone. Assume the phase is unknown.
Suppose the tone frequency
is 𝑤0 = 𝑤𝑠/2, with M = N.
Thus, 𝐴 Τ
𝑁 2 should be the largest. The coarse search should
give 𝑙 = N/2. If 𝑙 ≠ N/2 an outlier has occurred.
The mse was approximated when 𝑙 = N/2 by CR bound
to an unbiased estimator, which is defined from (17),
(17)
The mse when an outlier occurs as
(39)
12. 5. THRESHOLD EFFECT AND SIMULATIONS
(43)
Where 𝑞 is the probability of an outlier. Let the total mse be 𝑤𝑒.
Then 𝑤𝑒
2
The rms error is
The total mse is the weighted sum of the two contributions,
(40)
(42)
(41)
B. Probability of an Outlier
The probability of an outlier 𝑞 is represented in the equation (42).
13. 5. THRESHOLD EFFECT AND SIMULATIONS
C. Approximate RMS Frequency Error
➢ The previous formula from equation (42) for 𝑤𝑟𝑚𝑠 was used
to calculate the rms error for several values of N as shown in
Figure 4.
➢ The simulations described in the previous included level
estimates according to 6) (Section 3-C), if frequency and
phase are unknown. In every case, the rms level errors were
almost equal to the CR bounds.
➢ Threshold effects were not observed.
➢ The next question is, what about different signal
frequencies? When the simulation was applied with M = 64,
N = 16, and 𝑓0 = 2120 Hz, using - 10, - 5, 0, and 5-dB SNR.
14. 5. THRESHOLD EFFECT AND SIMULATIONS
D. Effect of 𝒕𝟎
var ෝ
𝒘 = (10)
if phase is known if phase is unknown
var ෝ
𝒘 = (14)
The correct choice for 𝑡0 = − (N − 1 /2]T (43).
15. 6. SUMMARY
This has been an introductory study of the problem of estimating the frequency and level of a sinusoidal
(complex sinusoidal) signal from a finite number of noisy observations of the signal.
➢ Equations describing the CR lower bounds of the variance of estimation errors
were derived.
➢ ML estimators were derived and we showed their relationship to DFT.
➢ An algorithm suitable for implementation on a digital computer
is presented.
➢ The algorithm almost always yields ML estimates.
➢ an expression for the algorithm's threshold behavior was derived.
➢ The results of this paper support the current use of the DFT for tone
parameter estimation