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.

1. Single-Tone Parameter Estimation from
discrete-Time Observations
Ankara University
Faculty of Engineering
EEE department

2. CONTENTS
1. INTRODUCTION
2. BOUNDS
3. MAXIMUM-LIKELIHOOD ESTIMATION
4. ALGORITHM
5. THRESHOLD EFFECT AND SIMULATIONS
6. SUMMARY

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