Simple and Accurate Algorithms

74 Pages·2015·1.66 MB·English

Simple and Accurate Algorithms

for Sinusoidal Frequency Estimation

Hing Cheung So

http://www.ee.cityu.edu.hk/~hcso

Department of Electronic Engineering
City University of Hong Kong

H. C. So Page 1
Outline

 Introduction

 Common 1D Signal Models

 Key Ideas in Algorithm Development

 Proposed Algorithms

 Common 2D Signal Models

 Key Ideas in Algorithm Development

 Proposed Algorithms

 List of References

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Introduction

What is sinusoidal frequency estimation? [1]-[3]

Determine the frequency of a sinusoidal signal

 Consider a sinusoid s(t) = Acos(ωt + θ), the frequency is ω in
radian or ω /(2π) in Hz

 The problem of sinusoidal frequency estimation is to
estimate ω given a noisy version of s(t) and the major
difficulty is that the frequency is nonlinear in the signal

 Once the frequency is known, the amplitude and phase
parameters are easily obtained as they can be
transformed as linear parameters

Similar terminologies include spectral analysis, spectral line
estimation, harmonic retrieval
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Application Areas

 Wireless communications

e.g., frequency shift keying (FSK) signal demodulation:

s(t) = cos(ω t) or s(t) = cos(ω t)?
1 2

 Audio and speech signal processing

 e.g., speech and music analysis using harmonic model:

M
x(t) = a(t) ∑ c cos(mω t + φ )
m 0 m
m=1

where is the fundamental frequency or pitch
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 Source localization

 Position of a target can be obtained via direction-of-
arrival (DOA) estimation from signals received at an
antenna array

 DOA estimation model can be converted to the problem
of frequency estimation
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 Biomedical engineering

e.g., nuclear magnetic resonance (NMR) or magnetic
resonance spectroscopy (MRS) signal analysis

M
jφ (−λ + jω )t
y(t) = ∑ A e m e m m + w(t)
m
m=1

 Power electronics

e.g., reliable frequency measurement in a power system
is important for effective power control, load restoration
and generator protection, and smart grid [4]

 Instrumentation and measurement

e.g., IEEE Standard for Digitalizing Waveform Recorder
(IEEE Std. 1057-1994) [5]
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Common 1D Signal Models

 Complex tone model:

M
jφ (−λ + jω )n
x = ∑ A e m e m m + q , n = 0,1,, N − 1
n m n
m=1

where {A }, {φ }, {λ } and {ω } are constants while q
m m m m n
is a zero-mean white noise

j(ωn+φ)
Simplest case: x = Ae + q
n n

Using nonlinear least squares (NLS), optimum frequency
estimation is achieved from:

N −1 ~ 2
~ ~
(Aˆ,ωˆ ,φˆ) = arg min ∑ x − Ae j(ωn+φ)
~ ~ n
~
A,ω,φ n=0

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 Real tone model:

M
x = ∑ A cos(ω n + φ ) + q , n = 0,1,, N − 1
n m m m n
m=1

Simplest case: x = Acos(ωn + φ) + q
n n

Using NLS, optimum frequency estimation is achieved
from:

(Aˆ,ωˆ ,φˆ) = arg min N∑−1 (x − A~ cos(ω~n + ~φ))2
~ ~ n
~
A,ω,φ n=0

As the cost functions are multi-modal, global solution is
not guaranteed

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Key Ideas in Algorithm Development

 Linear prediction (LP) property of sinusoids

M
 M (damped) complex sinusoid: s = − ∑ a s
n i n−i
i=1

where {a } are LP parameters characterized by
i
frequencies

j(ωn+φ)
e.g., for s = Ae :
n
jω jω
s = e ⋅ s , a = −e
n n−1 1

2M
 M (damped) real sinusoid: s = − ∑ a s with a = a
n i n−i i 2M −i
i=1
and a = 1
2M

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e.g., for s = Acos(ωn + φ)
n

s = 2 cos(ω) ⋅ s − s , a = −2cos(ω), a = 1
n n−1 n−2 1 2

Two advantages of LP:

 Nonlinear frequency parameters are transformed into
linear {a } which simplifies the estimation process
i

 Amplitude and phase parameters do not appear in the
LP signal model which means that less parameters are
needed for estimation

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