DE10309262B4 - Method for estimating the frequency and / or the phase of a digital signal sequence - Google Patents

Abstract

Method for estimating the frequency (Δf) and / or the phase (Φ) of a digital input signal sequence (x _ν ) with the following procedural steps:
- Carrying out a coarse frequency estimate (22) using a discrete Fourier transform and
- subsequent implementation of a fine frequency estimation (23) by means of interpolation,
_{in the case of the coarse} frequency estimate (22) for generating a coarse estimate (Δf ^ coarse) of the frequency, a discrete Fourier transformation (25) of the input signal sequence (x _ν ) for generating a _{Fourier sequence (X μ} ), an amount formation (26) of the Fourier sequence (X _μ ) and a determination of the maximum of the amount (| X _μ |) or a function (| X _μ | ² ) of the amount of the Fourier sequence, where the Fourier sequence (X _μ ), the amount (| X _μ | ) or a function (| X _μ | ² ) of the amount of the Fourier sequence according to the index (μ _max ), at which the maximum of the amount (| X _μ |) or the function (| X _μ | ² ) of the amount of the Fourier sequence ( X _μ ) is present, is shifted, and then an inverse discrete Fourier transformation (29) is carried out to generate a back-transformed signal sequence (y _{ν ) and the back-transformed signal sequence (y ν} ) with the rough estimate (Δf ^ _coarse ) for generation. .

Description

the The invention relates to a method for estimating the frequency and / or the phase of a digital signal sequence.

procedure for detecting the frequency and / or the phase of a digital signal are used in digital signal processing, for example for mobile radio technology needed. There is therefore a need for a robust process that can be used universally and that can be used with little numerical effort.

From the DE 101 38 963 A1 discloses a method for estimating parameters of a CDMA signal. Among other things, the frequency offset and the phase offset of a digital signal with respect to a reference or synchronization signal are estimated there. This method is specially tailored to the orthogonal code channels of a CDMA signal and can therefore not be used universally.

Of the The invention is based on the object of a method for estimating the Frequency and / or the phase of a digital input signal sequence to create, which requires little numerical effort, even with a low signal-to-noise ratio works, converges quickly and is universally applicable.

the The object is achieved by the features of claim 1.

Of the The invention is based on the concept, initially under a large frequency estimate Use a discrete Fourier transform to perform and afterward make a fine frequency estimate by means of interpolation. The rough estimate of the Frequency by searching for the maximum in the frequency domain enables a rough determination the frequency in the context of the frequency resolution of the Fourier transform. The following interpolation method relies on this information the frequency rough estimate. Although only linear interpolation is preferred, the interpolation method converges quickly.

the Subclaims contain advantageous developments of the invention.

in the A special interpolation formula is used within the scope of the present application derived, which is advantageously used in the interpolation. The calculation of the fine estimate the frequency is preferably repeated iteratively, in which next Iteration stage a compensated signal sequence is used.

A Estimate for the Phase can be carried out by evaluating all sampled values of the input signal sequence arithmetic averaging can be calculated.

claim 9 relates to a digital storage medium, claim 10 relates to a Computer program product and claims 11 and 12 relate to one Computer program for implementation of the method according to the invention.

the The invention is described in more detail below with reference to the drawing. In the drawing show:

1 a block diagram of a digital transmission system on which the invention is based;

2 the transmission model in the equivalent low-pass range;

3 a system-theoretical model in discrete-time baseband representation;

4th a schematic representation of a matrix;

5 a block diagram to explain the method according to the invention and

6th a block diagram of an advantageous development of the method according to the invention.

Below will initially for better understanding According to the invention, a model of a digital transmission system on which the invention is based explained.

α

Allgemeiner Summationsindex

a

Komplexe zu sendende Symbolfolge

aν

ν-tes Symbol der Folge a

aν,I

Imaginärteil des ν-ten Symbols der Folge a

aν,R

Realteil des ν-ten Symbols der Folge a

arg{∙}

Argumentbildung

β

Allgemeiner Summationsindex

δ(t)

Diracimpuls (Delta-Funktional)

δνμ

Kronecker-Operator

ε

Normierter Zeitversatz

E{∙}

Erwartungswert

f(∙)

Allgemeine Funktion

Δf

Zu schätzender Frequenzversatz zwischen Sender und Empfänger

Δferr

Verbleibender Schätzfehler am Ausgang der Feinschätzstufe des uML

Δferr,limit

Definierte Schranke für den maximal zulässigen Schätzfehler am Ausgang der Feinschätzstufe

Δfrest

Durch die Feinschätzung des uML-Schätzers zu schätzender Frequenzversatz, der nach erfolgter Kompensation mit dem Grobschätzwert Δf ^_coarse verbleibt

Δf{it}rest

Vom Feinschätzalgorithmus in der it-ten Iteration zu schätzende Frequenz

Δfrest,conv

Maximales Δf_rest, für welches der Feinschätzalgorithmus noch konvergiert

Δfrest,max

Maximaler Frequenzversatz Δf_rest am Ausgang der Grobschätzstufe

Δf~

Probierwert für Δf

Δf^

Schätzwert für Δf

Δf^coarse

Frequenzschätzwert der Grobschätzung des universellen ML-Schätzers

Δf^fine

Frequenzschätzwert der Feinschätzung des universellen ML-Schätzers

Δf^{it}fine

Bei der it-ten Iteration des Feinschätzalgorithmus ermittelter Schätzwert

Δf^fine_total

Summe aller bis zur aktuellen Iteration des Feinschätzalgorithmus ermittelten Schätzwerte Δf ^{it}fine

Δf^int

Schätzwert für Δf_rest am Ausgang der Grobschätzstufe beim Verfahren zur nachträglichen Erhöhung der Frequenzauflösung des uML-Schätzers

f+½·Q,

Versuchswerte für die Schätzung von Δf_rest beim

f-½·Q

Verfahren zur nachträglichen Erhöhung der Frequenzauflösung des uML-Schätzers

fQ

Quantisierung der FFT im Frequenzbereich (Frequenzauflösung)

fS

Symbolrate (= 1/T_S)

fT Sender

Trägerfrequenz des Modulators im Sender

fT Empfänger

Trägerfrequenz des Modulators im Empfänger

F(∙)

Stammfunktion der allgemeinen Funktion f(∙)

FnN

Verteilungsfunktion der normierten Normalverteilung

FFT{∙}

Fouriertransformation

Γ(ν·Ta/Tbeob)

Zeitdiskrete Gewichtsfunktion, die durch Faltung zweier reeller Rechteck-folgen der Länge N_Sym mit einer konstanten Amplitude von 1 entsteht

g

Zu schätzende Verstärkung

g~

Probierwert für g

ĝ

Schätzwert für g

he(t)

Impulsantwort des Empfangsfilters

hges(t)

Impulsantwort der Gesamtübertragungsstrecke

hS(t)

Impulsantwort des Sendefilters

it

Aktuelle Iterationszahl des Feinschätzalgorithmus des uML-Schätzers

IFFT{∙}

Inverse Fouriertransformation

Im{∙}

Imaginärteilbildung

κ

Allgemeiner Summationsindex

k

Zählvariable

K1, K2

Eingeführte Substitutionen für spezielle Zahlenreihen

L(∙)

Log-Likelihood-Funktion

μ

Index für bestimmte Frequenz innerhalb des diskreten Frequenzspektrums

μmax

Index der diskreten Frequenz, bei der das Maximum von Y_μ vorliegt

Maximumbildung der Funktion f(x) bezüglich des Parameters x

Mα,β

Matrixelement in Zeile α, Spalte β

n

Komplexe Rauschfolge (AWGN)

nν

ν-tes Symbol der Rauschfolge n

nν,I

Imaginärteil des ν-ten Symbols der Rauschfolge n

nν,R

Realteil des ν-ten Symbols der Rauschfolge n

nof_it

Gesamtzahl der vom Feinschätzalgorithmus durchzuführenden Iterationen

N

Menge der natürlichen Zahlen

½·N0

Zweiseitiges Leistungsdichtespektrum des Rauschsignals

NFFT

FFT-Länge bei der Frequenz-Grobschätzung des universellen ML-Schätzers

NSym

Länge der gesendeten Symbolfolge (= Schätzlänge)

ν

Index für ein bestimmtes Symbol innerhalb der Symbolanzahl N_Sym

Δω

Zu schätzender Frequenzversatz als Kreisfrequenz

Δῶ

Probierwert für Δω

Δω^

Schätzwert für Δω

Δϖ

Zu schätzender Frequenzversatz als Kreisfrequenz, normiert auf T_a

Δϖ~

Probierwert für Δϖ

Δϖ^

Schätzwert für Δϖ

Phase des ν-ten Gliedes der Folge y_comp

ϕ

Zu schätzender Phasenversatz zwischen Sender und Empfänger

ϕ~

Probierwert für ϕ

ϕ^

Schätzwert für ϕ

ϕ^err,max

Maximaler Phasenschätzfehler

P(∙)

Auftrittswahrscheinlichkeit

P(a ≤ x ≤ b)

Wahrscheinlichkeit, daß der Parameter x zwischen den Grenzen a und b liegt

r

Komplexe Empfangsfolge am Ausgang des zeitdiskreten Ersatzkanals

rν

ν-tes Symbol der Empfangsfolge r

Re{∙}

Realteilbildung

s

Komplexe Sendefolge (Symbolfolge a moduliert mit Δf, ϕ und g)

sν

ν-tes Symbol der Sendefolge s

σ2CR,f

Untere Grenze der Varianz von Δ⨍ ^ (≙ Cramér-Rao-Grenze)

σCR,ϖ

Untere Grenze der Standardabweichung von Δϖ ^ (≙ Cramér-Rao-Grenze)

σ2CR,ϖ

Untere Grenze der Varianz von Δϖ ^ (≙ Cramér-Rao-Grenze)

σCR,ϕ

Untere Grenze der Standardabweichung von ϕ ^ (≙ Cramér-Rao-Grenze)

σ2CR,ϕ

Untere Grenze der Varianz von ϕ ^ (≙ Cramér-Rao-Grenze)

σϖ

Standardabweichung des Frequenzschätzfehlers

σϕ

Standardabweichung des Phasenschätzfehlers

σϕ,krit

Standardabweichung des Phasenschätzfehlers für SNR_krit

SNR

Linearer Signal-Rausch-Abstand

SNRkrit

Kritischer Signal-Rausch-Abstand, bei dem die Frequenz- bzw. Phasenschätzung des uML-Schätzers gerade noch effizient arbeitet

Ta

Abtastzeit

Tbeob

Beobachtungsdauer

TS

Symboldauer

window(ν)

Fensterfunktion zur Beschneidung einer Folge auf eine festgelegte Länge

x

Modulationsbereinigte komplexe Empfangsfolge

xν

ν-tes Symbol der modulationsbereinigten Empfangsfolge x

Xμ

μ-tes Folgenglied der Fouriertransformierten der Folge x

y

Komplexe Folgen, entspricht der inversen FFT des betragsquadrierten Spektrums der Folge x

yν

ν-tes Symbol der Folge y

ycomp

Mit Δf ^_coarse kompensierte Folge am Ausgang der Grobschätzstufe des uML

yν,comp

ν-tes Symbol der Folge y_comp

y{it}ν,com

ν-tes Symbol einer frequenzkompensierten Folge innerhalb der it-ten Iteration des Feinschätzalgorithmus des universellen ML-Schätzers

Yμ

μ-tes Folgenglied der Fouriertransformierten der Folge y

z*

Konjugiert komplexer Wert der Variable z

Z

Komplexer Gesamtzeiger einer Summe

In the context of the present application, the following symbols are used:

α

General summation index

a

Complex symbol sequence to be sent

aν

ν-th symbol of the sequence a

aν, I

Imaginary part of the ν-th symbol of the sequence a

aν, R

Real part of the ν th symbol of the sequence a

arg {∙}

Argument formation

β

General summation index

δ (t)

Dirac pulse (delta functional)

δνμ

Kronecker operator

ε

Normalized time offset

E {∙}

Expected value

f (∙)

General function

Δf

Frequency offset to be estimated between transmitter and receiver

Δferr

Remaining estimation error at the output of the fine estimation stage of the uML

Δferr, limit

Defined limit for the maximum allowable estimation error at the output of the fine estimation stage

Δfrest

Frequency offset to be estimated by the fine estimation of the uML estimator, which remains _{after the compensation with the rough estimated value Δf ^ coarse}

Δf {it} rest

Frequency to be estimated by the fine estimation algorithm in the it-th iteration

Δfrest, conv

Maximum Δf _rest for which the fine estimation algorithm is still converging

Δfrest, max

Maximum frequency offset Δf _rest at the output of the rough estimation stage

Δf ~

Trial value for Δf

Δf ^

Estimated value for Δf

Δf ^ coarse

Frequency estimate of the rough estimate of the universal ML estimator

Δf ^ fine

Frequency estimate of the fine estimate of the universal ML estimator

Δf ^ {it} fine

Estimated value determined in the it-th iteration of the fine estimation algorithm

Δf ^ fine_total

Sum of all estimated values determined up to the current iteration of the fine estimation algorithm Δf ^ {it} fine

Δf ^ int

Estimated value for Δf _rest at the output of the rough estimation stage in the method for subsequently increasing the frequency resolution of the uML estimator

f + ½ · Q,

Test values for the estimation of Δf _rest at

f-½ · Q

Procedure for subsequently increasing the frequency resolution of the uML estimator

fQ

Quantization of the FFT in the frequency domain (frequency resolution)

fS

Symbol rate (= 1 / T _S )

fT transmitter

Carrier frequency of the modulator in the transmitter

fT recipient

Carrier frequency of the modulator in the receiver

F (∙)

Antiderivative of the general function f (∙)

FnN

Distribution function of the normalized normal distribution

FFT {∙}

Fourier transform

Γ (ν Ta / Tbeob)

Discrete-time weighting function that is created by convolution of two real rectangular sequences of length N _Sym with a constant amplitude of 1

G

Reinforcement to be estimated

g ~

Trial value for g

G

Estimate for g

he (t)

Impulse response of the receiving filter

hges (t)

Impulse response of the entire transmission path

hS (t)

Impulse response of the transmission filter

it

Current iteration number of the fine estimation algorithm of the uML estimator

IFFT {∙}

Inverse Fourier transform

In the {∙}

Imaginary part formation

κ

General summation index

k

Counting variable

K1, K2

Introduced substitutions for special series of numbers

L (∙)

Log likelihood function

μ

Index for a specific frequency within the discrete frequency spectrum

μmax

Index of the discrete frequency at which the maximum of Y _μ is present

Maximum formation of the function f (x) with respect to the parameter x

Mα, β

Matrix element in row α, column β

n

Complex Noise Sequence (AWGN)

nν

ν-th symbol of the noise sequence n

nν, I

Imaginary part of the ν th symbol of the noise sequence n

nν, R

Real part of the ν th symbol of the noise sequence n

nof_it

Total number of iterations to be performed by the fine estimation algorithm

N

Set of natural numbers

½ · N0

Two-sided power density spectrum of the noise signal

NFFT

FFT length in the rough frequency estimation of the universal ML estimator

NSym

Length of the transmitted symbol sequence (= estimated length)

ν

Index for a specific symbol within the number of symbols N _Sym

Δω

Frequency offset to be estimated as circular frequency

Δῶ

Trial value for Δω

Δω ^

Estimated value for Δω

Δϖ

Frequency offset to be estimated as angular frequency, normalized to T _a

Δϖ ~

Trial value for Δϖ

Δϖ ^

Estimated value for Δϖ

Phase of the ν-th member of the sequence y _comp

ϕ

Phase offset to be estimated between transmitter and receiver

ϕ ~

Trial value for ϕ

ϕ ^

Estimated value for ϕ

ϕ ^ err, max

Maximum phase estimation error

P (∙)

Probability of occurrence

P (a ≤ x ≤ b)

Probability that the parameter x lies between the limits a and b

r

Complex reception sequence at the output of the discrete-time backup channel

rν

ν-th symbol of the reception sequence r

Re {∙}

Real part formation

s

Complex transmission sequence (symbol sequence a modulated with Δf, ϕ and g)

sν

ν-th symbol of the transmission sequence s

σ2CR, f

Lower limit of the variance of Δ⨍ ^ (≙ Cramér-Rao limit)

σCR, ϖ

Lower limit of the standard deviation of Δϖ ^ (≙ Cramér-Rao limit)

σ2CR, ϖ

Lower limit of the variance of Δϖ ^ (≙ Cramér-Rao limit)

σCR, ϕ

Lower limit of the standard deviation of ϕ ^ (≙ Cramér-Rao limit)

σ2CR, ϕ

Lower limit of the variance of ϕ ^ (≙ Cramér-Rao limit)

σϖ

Standard deviation of the frequency estimation error

σϕ

Standard deviation of the phase estimation error

σϕ, crit

Standard deviation of the phase _{estimation error for SNR crit}

SNR

Linear signal-to-noise ratio

SNRkrit

Critical signal-to-noise ratio at which the frequency or phase estimation of the uML estimator still works efficiently

Ta

Sampling time

Tbeob

Observation period

TS

Symbol duration

window (ν)

Window function for cutting a sequence to a specified length

x

Modulation-adjusted complex reception sequence

xν

ν-th symbol of the modulation-adjusted receive sequence x

Xμ

μ-th term of the Fourier transform of the sequence x

y

Complex sequences, corresponds to the inverse FFT of the spectrum squared by the magnitude of the sequence x

yν

ν th symbol of the sequence y

ycomp

Sequence compensated with Δf ^ _coarse at the output of the coarse estimation stage of the uML

yν, comp

ν th symbol of the sequence y _comp

y {it} ν, com

ν-th symbol of a frequency-compensated sequence within the it-th iteration of the fine estimation algorithm of the universal ML estimator

Yμ

μ-th term of the Fourier transform of the sequence y

z *

Conjugate the complex value of the variable e.g.

Z

Complex total pointer of a sum

AWGN

Additive White Gaussian Noise

const

konstant

CRB

Cramér-Rao-Grenze

DA

Data Aided

dB

Dezibel

FFT

Fast Fourier Transformation

ISI

Intersymbolinterferenz

LDS

Leistungsdichtespektrum

LLF

Log-Likelihood-Funktion

ML

Maximum Likelihood

NDA

Non Data Aided

QRM

Quadratur-Amplituden-Modulation

SNR

Signal-Rausch-Abstand (signal to noise ratio)

uML

Universeller Maximum-Likelihood-Schätzer für Frequenz und Phase

The following abbreviations are used in the context of the present application:

AWGN

Additive White Gaussian Noise

const

constant

CRB

Cramer-Rao border

THERE

Data aided

dB

decibel

FFT

Fast Fourier Transformation

ISI

Intersymbol interference

LDS

Power density spectrum

LLF

Log likelihood function

ML

Maximum likelihood

NDA

Non data aided

QRM

Quadrature amplitude modulation

SNR

Signal-to-noise ratio

uML

Universal maximum likelihood estimator for frequency and phase

The necessity of frequency and phase estimation for carrier synchronization arises from the structure of digital transmission systems. In 1 the basic elements of such a transmission system are shown.

After through a send filter 2 a pulse shaping of the digital source signal to be transmitted from the digital source 1 is carried out in the modulator 3 the modulation of the signal with the carrier frequency f _{T transmitter} . When transmitting over the channel 4th the signal, which is now in the bandpass range, is falsified by the superimposition of interference. In the context of this application, a so-called AWGN channel is assumed, ie the addition of a noise signal with a Gaussian-shaped amplitude and a power density spectrum (LDS) of ½ · N ₀ constant for all frequencies is assumed as interference. The noisy signal arriving at the receiver is transmitted by the demodulator 5 transformed back into baseband with the frequency f _{T receiver.} With the help of the reception filter below 6th the interference power is then reduced. This and a successful scanning in the scanning unit 7th or synchronization in the synchronization unit 8th a decoding of the actually sent signal in the digital sink 9 enables.

As part of the synchronization, in addition to the clock synchronization, a frequency and phase estimation is necessary, since on the one hand the frequency of the local oscillators of the transmitter and receiver changes slightly Δf = f T transmitter - f T recipient (1) differs. On the other hand, there is also a phase shift ϕ between the transmitter and receiver. This frequency or phase offset must be determined by estimation and compensated for in the received signal so that it is in the digital sink 9 the right symbol points can be decided.

Based on the block diagram described according to 1 a transmission model for the equivalent low-pass range can be found 2 create. The time-discrete sequence of complex _{transmission symbols with the symbol duration T S is passed} through the transmission filter 10 formed into the transmission signal with the impulse response h _{s (t).} The frequency and phase offset between transmitter and receiver is shown in the form of a rotary pointer by the multiplier 11 modulated, with an additional gain g being taken into account. After white noise in the channel through the adder 12th was additively superimposed, the filtering is carried out in the receiver by the reception filter 13th with the impulse response h _e (t). The filtered received signal is finally _{sampled at intervals of T a} , it being possible for a time offset ε · T _S to occur with respect to the ideal sampling times.

For a successful frequency and phase estimation, several prerequisites are necessary. On the one hand, the receiver must have a corresponding clock synchronization, ie it will ε =! 0 (2) provided.

On the other hand, the first Nyquist condition must be fulfilled so that an intersymbol interference-free (ISI-free) sampling can take place. For this purpose, the transmission and reception filters must be designed in such a way that the impulse response H total (t) = h s (t) · h e (t) (3) of the total transmission distance fulfills the Nyquist condition:

This is the case when used as a transmission filter 10 and receive filters 13th For example, so-called root-raised cosine filters can be used. In addition, the frequency offset Δf between the transmitter and receiver must be so small that the transmission filter 10 and the receive filter 13th cover sufficiently in the frequency range. The following requirement must therefore be met:

If the first Nyquist condition is met, this can be done in 2 the transfer model shown in 3 discrete-time baseband model illustrated can be simplified. This model represents the starting point for the considerations and simulations in the context of this application.

The symbols a _{ν of} the sequence a to be sent are first used by the multiplier 14th multiplied by a rotation pointer, which models the frequency and phase offset between transmitter and receiver. After amplification with the real factor g by the multiplier 15th the resulting transmission sequence s in the discrete-time substitute channel is replaced by the by the adder 16 added complex noise sequence n disturbed, so that ultimately the symbols r _{ν of} the received sequence r result.

Of the Maximum likelihood (ML) estimator for frequency presented in this application and phase at large Observation lengths is based on a universally applicable frequency and phase estimation algorithm, according to the maximum likelihood principle is working. A universal maximum likelihood (hereinafter referred to as "uML") estimator for frequency and phase is presented below.

The uML algorithm is an exact implementation of the maximum likelihood principle, ie it is the optimal estimator for an AWGN channel. No approximations are used, so that no non-linear phenomena such as cycle slipping of the phase can occur. As a result, the ideal ML estimate is determined even with very low signal-to-noise ratios, ie the estimation accuracy is only determined by the Cramér-Rao limit. The algorithm also works for symbolic alphabets with non-constant amounts | a _ν | optimal.

That Procedure is for the so-called data-aided case (DA) derived, i.e. to carry out the frequency and phase estimation a data sequence is transmitted, which the recipient is known. It should be mentioned, however, that by Application of a non-linearity to remove the modulation the uML algorithm is also used for the non-data-aided case (NDA) can be.

The transfer model follows the estimation problem 3 based on:

vorgegeben und wird im AWGN-Kanal durch eine additive, weiße Rauschfolge n gestört.The undisturbed transmission sequence s is through

specified and is disturbed in the AWGN channel by an additive, white noise sequence n.

The result is the reception sequence r with: r ν = s ν + n ν (7)

• – Frequenzversatz Δf
• – Phasenversatz ϕ
• – Verstärkung g (reell)

On the receiver side, the following parameters of the transmission sequence s are to be estimated from the reception sequence r according to the maximum likelihood principle:

• - Frequency offset Δf
• - phase shift ϕ
• - gain g (real)

The maximum likelihood of the above parameters is estimated by minimizing the log likelihood function. The following equation (8) is the log-likelihood function applicable to the present transfer model or estimation problem:

With the angular frequency Δω = 2π · Δf (9) and by multiplying it out, one obtains for the log likelihood function:

As the date-aided case is present, the transmitted symbol sequence a is known to the recipient. The dependency of the reception sequence r on the symbol sequence a can therefore be removed. For this purpose, the modulation-adjusted sequence x is introduced with x ν = r ν · A ν * (11)

This results in the log-likelihood function:

Below it is now shown that the present multidimensional estimation problem in independent, one-dimensional estimation problems can be divided, i.e. that frequency, Phase and reinforcement can be separated from each other.

From (12) it can be seen that frequency and phase only appear in the last term. The present log-likelihood function is therefore minimized with respect to Δf and ϕ by maximizing the last term. Since the gain g and the prefactor have no influence on the position of the maximum, they can be omitted. This gives the equivalent log-likelihood function to be maximized for the frequency and phase estimation:

The real part formation contained therein may be interchanged with the summation:

This function is at its maximum when the phase ϕ ~ rotates the complex total pointer Z on the real part axis. The following therefore applies to the estimated phase:

It should therefore be noted that the log-likelihood function L ₁ from (14) is maximized in that the frequency Δῶ corresponds to the absolute value of the total vector | z → | maximized and at the same time z → is rotated through the optimal phase on the real part axis. This results in the following log-likelihood function, which is only dependent on the frequency and which in turn must be maximized:

Man now recognizes a similarity between the existing log-likelihood function and the Fast Fourier Transformation FFT. With the help of the FFT, the LLF can be used for certain discrete frequencies are calculated.

mit NFFT = 2k und k ∊ N (18) The definition of the FFT is

with N FFT = 2 k and k ∊ N (18)

The quantization of the FFT in the frequency domain, i.e. the frequency resolution, is:

At the Compare the summation from (16) with the FFT definition one that an equivalence exists if Δω ~ on one of the discrete frequencies of the FFT falls.

erfüllt ist, entspricht die zu maximierende LLF dem Betrag von FFT{x_ν}:

So if the context

is fulfilled, the LLF to be maximized corresponds to the amount of FFT {x _ν }:

auftreten, wenn die zu schätzende Frequenz Δf nicht genau einer der diskreten Frequenzen des Frequenzrasters der FFT entspricht.This means that the frequency estimation can be carried out in two stages. First, a coarse frequency estimate Δf _coarse is determined by looking for the maximum amount of the Fourier transform of x _ν. However, there can be an estimation error

occur when the frequency to be estimated Δf does not exactly correspond to one of the discrete frequencies of the frequency grid of the FFT.

If the FFT length N _{FFT is} correspondingly large, however, the remaining estimation error Δf _rest becomes so small that it can be determined in a subsequent fine estimation stage by linear approximation. However, the problem arises that a linearization of the absolute value function from (16) is not possible.

von der nachfolgend gezeigt wird, daß sie linearisierbar ist. Mit dem Zusammenhang |z|2 = z·z* (24)für komplexe Größen erhält man für die vorliegende LLF:

The following aid leads to the solution: Since the LLF from (16) is real because of the formation of the absolute value and positive for all Δω ~, the position of the maximum of the LLF is not changed by squaring. The result is the equivalent LLF

which is shown below to be linearizable. With the context | z | 2 = z * z * (24) for complex sizes one obtains for the present LLF:

Now the individual sum elements of (24) are entered into a schematic matrix, which is shown in 4th is shown.

• – Die Haupt- und Nebendiagonalen besitzen einen jeweils identischen Drehzeiger e^-j...
• – Die Matrix ist hermetisch, d.h. die Matrixelemente M haben paarweise konjugiert komplexe Werte: Mα,β = M*β,α
• – Die Hauptdiagonale ist reell.

The following symmetry properties can be seen:

• - The main and secondary diagonals each have an identical rotary pointer e ^{-j ...}
• - The matrix is hermetic, i.e. the matrix elements M have complex values conjugated in pairs: M. α, β = M * β, α
• - The main diagonal is real.

These symmetry properties can be used by introducing the following alternative summation indices: ν = α - β and γ 25)

With the help of the matrix, this means that you no longer add up over the rows and columns of the matrix, but add up the matrix elements of each diagonal and then add the results of the individual diagonals. The following relationship is also used for the addition of the pair-wise conjugate complex values: z + z * = 2 · Re {z} (26)

The result is the following representation, equivalent to (24 '):

Since the first term, which describes the main diagonal of the matrix, does not depend on the frequency, it can be omitted together with the prefactor of the second term without shifting the maximum of the LLF. This gives a relatively simple, closed representation:

In this LLF there is a (cyclical) convolution of the form y ν = x ν X * -ν (29) before. It can be seen that the sequence y can be calculated with the help of the Fourier transform: Y μ = FFT {y ν } = | X μ | 2 = | FFT {x ν } | 2 (30)

• – Konjugiert komplexe Symmetrie der Folgenglieder: y-ν = y*ν (31)
• – Die Folge y ist nur im Bereich –(NSym – 1) ≤ ν ≤ NSym – 1 (32)ungleich Null

In addition, y has the following properties:

• - Conjugates complex symmetry of the terms of the sequence: y -ν = y * ν (31)
• - The sequence y is only in the area - (N Sym - 1) ≤ ν ≤ N Sym - 1 (32) not equal to zero

With the defined sequence y, the LLF is simplified to:

With the help of this form of LLF, a fine estimate can be carried out. To understand the rough estimate, however, a further transformation of the LLF is helpful. First, (26) can be applied, which leads to the following representation of the LLF:

Now one can use the symmetry property of y from (31):

Since the rear term y ₀ and the prefactor have no influence on the position of the maximum of the LLF, they can be omitted. This gives the equivalent LLF:

It can be seen that this is the Fourier transform of the sequence y _ν at the frequency Δf ~, since the summation limits can be shifted due to the cyclic convolution of the FFT. Thus the relationship applies:

In addition, it can be seen that the following applies to the minimum length of this FFT: N FFT ≥ 2 * N Sym - 1 (38)

A smaller N _FFT would result in aliasing in the time domain.

However, it should already be pointed out here that the accuracy of a rough estimate with this FFT length is not always sufficient for a reliable convergence of the iterative algorithm of the fine estimation stage. Either a correspondingly larger FFT length is therefore necessary or, in accordance with a preferred development of the method according to the invention, the frequency resolution must be increased _{accordingly after a rough frequency estimate with N FFT} = 2 * N Sym. The method for subsequent refinement of the frequency resolution of the coarse estimate is described below.

• – Grobschätzung von Δf ^ mittels FFT
• – Iterative Feinschätzung mittels linearer Interpolation

In summary, one can see from the previous derivations that a two-stage frequency estimation with the following steps is useful:

• - Rough estimate of Δf ^ using FFT
• - Iterative fine estimation using linear interpolation

• a) Berechne die modulationsbereinigte Folge x nach (11)
• b) Erhöhe durch Zero-Padding (Einfügen von Nullen) am Ende der Folge die Länge von x auf die notwendige FFT-Länge N_FFT
• c) Berechne die Fouriertransformierte X_μ = FFT{x_ν}
• d) Suche nach (21) das Maximum von |X_μ|. Die Betragsbildung ist allerdings numerisch aufwendig. Daher ist es sinnvoller, stattdessen nach (30) das Betragsquadrat von X_μ zu bilden: Yμ = |Xμ|2 Die nun für Y_μ durchgeführte Maximumsuche liefert als Ergebnis den Index der diskreten Frequenz, bei der das Maximum vorliegt: μmax = arg{max(Yμ)} (39)Daraus ergibt sich mit (19) der Grobschätzwert der Frequenz:
  
• e) Berechne die Folge y mit Hilfe der inversen FFT: yν = IFFT{Yμ+μmax} (41)
• f) Kompensiere die Folge y mit dem ermittelten Grobschätzwert der Frequenz:
  

The rough estimate is carried out in the following steps:

• a) Compute the modulation-adjusted sequence x according to (11)
• _{b) Increase the length of x to the necessary FFT length N FFT} by zero padding (insertion of zeros) at the end of the sequence
• c) Calculate the Fourier transform X _μ = FFT {x _ν }
• d) Search for (21) the maximum of | X _μ |. The formation of the amount is, however, numerically complex. Therefore it makes more sense to use (30) instead to form _{the square of the absolute values of X μ:} Y μ = | X μ | 2 The maximum search now carried out for Y _μ provides the index of the discrete frequency at which the maximum is present as a result: μ Max = arg {max (Y μ )} (39) This gives the rough estimate of the frequency with (19):
  
• e) Calculate the sequence y using the inverse FFT: y ν = IFFT {Y μ + μmax } (41)
• f) Compensate the sequence y with the determined rough estimate of the frequency:
  

It can be seen from (40) that the roughly frequency-compensated sequence y _{comp can} alternatively also be calculated with the aid of the modulation theorem:

After the compensation of the sequence y with the coarse frequency estimate Δf _{coarse , the frequency offset Δf fine} remains due to the finite discrete frequency resolution of the FFT, which is to be estimated by the fine estimation stage. The overall estimate is therefore made up of Δf ^ = Δf ^ coarse + Δf ^ fine (44) together.

ermittelt werden, indem man die Ableitung nach der Frequenz bildet und gleich Null setzt:

Here, Δf ^ _fine can be calculated by maximizing the LLF which is equivalent to (33)

can be determined by taking the derivative according to the frequency and setting it to zero:

_{Since the Δω fine} to be estimated is sufficiently small by a corresponding choice of N _{FFT of} the coarse estimation stage, linearization may be used. If one applies the linearization of the exponential which is permissible for small x function e x ≈ 1 + x (47) on the present equation, one obtains:

This ultimately results in the required calculation rule for the fine frequency estimate:

Due to the linearization error of the fine estimation stage, the total estimation value corresponds Δf ^ = Δf ^ coarse + Δf ^ fine not exactly the frequency to be estimated Δf. The iterative application of the fine estimation algorithm creates an improvement here. This means that the linearization error can be reduced as required.

• a) Berechne den Feinschätzwert Δf ^{it}fine der aktuellen Iteration nach der iterativ anwendbaren Berechnungsvorschrift aus (49)
  
  Für die erste Frequenzschätzung wird dabei die am Ausgang der Grobschätzstufe vorliegende, nur mit Δf_coarse kompensierte Folge y_comp verwendet, d.h.
  
• b) Addiere alle bisherigen Feinschätzwerte auf zu Δf ^_{fine_total}:
  
• c) Bilde die für die nächste Feinschätzung benötigte Eingangsfolge durch Kompensation von y_comp mit Δf ^_{fine_total}:
  
• d) Ist die gewünschte Iterationszahl nof_it noch nicht erreicht, so wird die nächste Iteration entsprechend Schritt a) bis c) durchgeführt.
• e) Sobald nof_it Iterationen durchgeführt wurden, steht der Gesamt-Feinschätzwert
  
  als Ausgangsgröße der iterativen Feinschätzstufe zur Verfügung.

The fine estimation of the frequency in nof_it iterations takes place in the following steps:

• a) Calculate the fine estimate Δf ^ {it} fine the current iteration according to the iteratively applicable calculation rule from (49)
  
  The sequence y _comp present at the output of the coarse estimation stage and only _{compensated with Δf coarse} is used for the first frequency estimation, ie
  
• b) Add up all previous fine estimate values to Δf ^ _{fine_total} :
  
• c) Include the input sequence required for the next fine estimate by compensating y _comp Δf ^ _{fine_total} :
  
• d) If the desired number of iterations nof_it has not yet been reached, the next iteration is carried out in accordance with steps a) to c).
• e) As soon as nof_it iterations have been carried out, the total fine estimate is available
  
  available as the output variable of the iterative fine estimation stage.

The total estimated frequency value Δf ^ sought for the frequency Δf to be estimated is calculated from the results of the rough and fine estimation:

After the frequency has been estimated, the phase ϕ can be estimated according to (15) using the frequency estimated value Δf ^:

the estimate of reinforcement g estimator becomes of completeness for the sake of it listed here.

To estimate the gain, the log-likelihood function (12) must be maximized. For this purpose, the derivative of the LLF according to g ~ is set equal to zero:

From this, with the previously determined frequency and phase estimates, the estimate for the gain is calculated as:

The sequence of frequency and phase estimation with the universal maximum likelihood estimator described in detail above is in 5 shown schematically.

This in 5 The illustrated block diagram of the frequency and phase estimator according to the invention 1 is divided into a rough frequency estimator 22nd , a frequency estimator 23 and a phase estimator 24 on. The rough frequency estimator 22nd the digital input signal sequence x _{ν is} supplied. In a block 25th a discrete Fourier transformation takes place, preferably a fast Fourier transformation FFT. The length of the Fourier transformation must be at least twice as long as the length of the input signal sequence x _ν , ie N _FFT ≥ 2 · N _Sym - 1. The Fourier transformation results in the Fourier sequence X in the frequency space _μ . In the block 26th an amount or an amount square is formed. In a block 27 the maximum of the magnitude of the Fourier sequence or, since this is numerically simpler and equivalent, the maximum of the magnitude square of the Fourier sequence is sought.

The index μ _{max of} the discrete value Y _{μ, max of} the Fourier sequence, in which the maximum of the amount or the amount square is present, is applied to the block 28 passed on, in which a conversion into the rough estimate Δf ^ _{coarse of} the frequency is carried out according to formula (40). In the block 29 an inverse Fourier transformation takes place, but not the original square Fourier sequence Y _μ , but the Fourier sequence Y _μ + μ _max shifted by the index μ _max corresponding to the maximum. After compensation with formula (42), the back-transformed signal sequence y _{ν, comp is produced} .

By the connoisseur 23 a fine estimate of the frequency takes place. The preferred case of the iterative approach is shown. In a block 30th the iteration counter it is first set to 1 and the entire fine estimate Δf ^ _{fine_total} is set to zero. In a block 31 the fine estimate of the frequency is calculated according to formula (50). For the next iteration step the iteration counter is it in the block 32 incremented. In block 33 the compensation factor required in formula (53) is made available. The compensation according to formula (53) is finally made by the multiplier 34 .

After calculating the fine estimate iteration contribution of this iteration stage according to formula (50) in block 31 be in the block 35 the fine estimated-Iterationsbeiträge of each iteration to fine estimated total value .DELTA.f ^ added _{fine_total.} If the iteration counter it has reached the number of specified iteration stages nof_it, the total fine estimate value is displayed using the switch, which is only shown symbolically 37 output and in the adder 38 is added to the coarse estimate to produce an overall estimate of the frequency.

The overall estimate of the frequency and the input signal sequence are used by the phase estimator 24 supplied, which calculates the estimated value of the phase according to formula (55).

The estimated value of the phase can be combined with the input signal sequence to the block 39 which calculates the estimated value for the gain according to formula (57).

In the rough frequency estimation of the universal maximum likelihood estimator according to the invention, the estimated frequency value is determined with the aid of a Fast Fourier transformation. According to (38) it should be noted that a minimum FFT length of N FFT ≥ 2 * N Sym - 1 must be observed in order to avoid aliasing effects in the time domain. However, this FFT length may not be sufficient. Due to an excessively large maximum estimation error Δf _{rest, max} at the output of the coarse estimation stage, reliable convergence of the subsequent iterative fine estimation algorithm is not always guaranteed with this FFT length. The additional computational effort and memory requirement due to a corresponding increase in the FFT length in the rough estimation stage would, however, be considerable.

The following method provides a remedy as an advantageous further development of the method according to the invention. Here, a rough estimate is initially included, taking into account (18) N FFT = 2 * N Sym (58) carried out. With little additional effort, by maximizing the corresponding LLF at certain frequencies, the same frequency resolution and thus also the same estimation accuracy as with a rough estimation with N _FFT = 4 * N _{Sym is} achieved. This ensures both the convergence of the fine estimation algorithm and compliance with defined accuracy limits.

According to (19), the FFT of the rough estimation stage with the FFT length has a frequency resolution of

The aim is now to subsequently double this resolution in the area around the found coarse estimate Δf ^ _coarse and thereby the maximum remaining frequency offset Δf _{rest, max} at the output of the coarse estimate 22nd to cut in half.

As with the fine estimation, the starting point is the log likelihood function according to (45), in which the compensation with the rough estimate Δf ^ _coarse is already taken into account:

Maximizing this LLF gives an estimate of the remaining frequency offset Δf rest = Δf - Δf ^ coarse (61) at the output of the rough estimation stage.

berechnet und mit dem Funktionswert von L₄ bei f = 0 vergleicht. Aus den drei Frequenzen wird jene als Schätzfrequenz Δf ^_int ausgewählt, bei der die LLF den größten Wert aufweist und die somit der zu schätzenden Frequenz Δf_rest am nächsten kommt. Der nach der Kompensation der Folge y_comp mit dem Schätzwert Δf ^_int noch verbleibende Frequenzversatz wird anschließend von dem vorzugsweise iterativ arbeitenden Feinschätzer 23 ermittelt.The mentioned doubling of the frequency resolution compared to the resolution f _{Q of} the coarse estimator is now achieved by using the function values of the LLF L ₄ at the frequencies

calculated and compared with the function value of L ₄ at f = 0. From the three frequencies that one is selected as the estimated frequency Δf ^ _int at which the LLF has the greatest value and which thus comes closest _{to the frequency to be estimated Δf rest.} The frequency offset still remaining after the compensation of the sequence y _comp with the estimated value Δf ^ _int is then determined by the fine estimator, which preferably works iteratively 23 determined.

In other words, the above method examines whether one of the frequencies Δf ^ _coarse ± 0.5 · f _O corresponds more precisely than Δf ^ _{coarse to} the frequency Δf to be estimated by the universal maximum likelihood estimator. The same resolution and estimation accuracy are thus achieved as with a rough estimation with N _FFT = 4 · N _Sym .

Mathematically, the determination of the estimated value Δf ^ _int for the subsequent increase in the frequency resolution can be represented as follows:

For a more efficient implementation, e.g. on a digital signal processor, the following can be simplified by swapping the real part formation and summation:

The entire process flow is shown schematically in 6th summarized.

In 6th is first of all the rough frequency estimator 22nd , the frequency estimator 23 and the phase estimator 24 as well as the gain estimator 39 in the same way as in 5 shown. The relevant elements are provided with the same reference numerals, so that a repetitive description is not necessary.

In contrast to 5 is between the rough frequency estimator 22nd and the fine frequency estimator 23 to increase the frequency resolution of the block 40 inserted. In the block 40 will not just be in the block 41 according to formula (45) the log likelihood function L _{4 is} calculated at the point of the rough estimate, but in the blocks 42 and 43 the log-likelihood function L ₄ is calculated according to formula (60) in conjunction with formula (62) or (63) and half a step size of the quantization f _{Q of} the FFT in addition to the rough estimate. In a block 44 the maximum of these three alternatives of L _{4 is} determined and the frequency contribution Δf ^ _{int is} determined from this by the arg function, which is in the adder 45 is added to the rough estimate of the frequency.

The resulting phase compensation values are in the block 46 available in a table and the signal sequence y _{ν, comp} is in the multiplier 47 multiplied by this compensation value.

Claims (12)

Method for estimating the frequency (Δf) and / or the phase (Φ) of a digital input signal sequence (x _ν ) with the following procedural steps: - Carrying out a rough frequency estimation ( 22nd ) using a discrete Fourier transform and - then performing a fine frequency estimation ( 23 ) by means of interpolation, with the rough frequency estimation ( 22nd ) For generating a coarse estimate (.DELTA.f ^ _coarse) frequency of a discrete Fourier transform ( 25th ) the input signal sequence (x _ν ) to generate _{a Fourier sequence (X μ} ), an amount ( 26th ) of the Fourier sequence (X _μ ) and a determination of the maximum of the amount (| X _μ |) or a function (| X _μ | ² ) of the amount of the Fourier sequence, where the Fourier sequence (X _μ ), the amount (| X _μ |) or a function (| X _μ | ² ) of the amount of the Fourier sequence corresponding to the index (μ _max ) at which the maximum of the amount (| X _μ |) or the function (| X _μ | ² ) of the amount of the Fourier sequence (X _μ) is present, is displaced, and then to generate an inversely-transformed signal sequence (y _ν), an inverse discrete Fourier transform ( 29 ) is carried out and the back-transformed signal sequence (y _ν ) is compensated with the rough estimate (Δf ^ _coarse ) to generate a compensated, back-transformed signal sequence (y _{ν, comp} ), and the compensation according to

takes place, where y _ν : the back-transformed signal sequence, y _{ν, comp} : the compensated, back-transformed signal sequence, Δf ^ _coarse : the rough estimate of the frequency, T _a : the sampling period of the digital input signal sequence (x _ν ) and ν: the symbol index of the mean reverse transformed signal sequence. Method according to Claim 1, characterized in that the coarse estimate (Δf ^ _coarse ) of the frequency from the maximum of the amount (| X _μ |) or a function (| X _μ | ² ) of the amount of the Fourier sequence according to

is calculated, where f _Q : the quantization of the discrete Fourier transform in the frequency domain, N _FFT : the length of the _{Fourier sequence (X μ} ) and μ _max : the index of the discrete value of the Fourier sequence at which the maximum of the amount (| X _μ |) or the function (| X _μ | ² ) of the amount of the Fourier sequence (X _μ ) is present. Method according to Claim 1 or 2, characterized in that in order to increase the frequency resolution according to the inverse discrete Fourier transformation ( 29 ) the calculation of a log-likelihood function (L ₄ ) on the deviation zero (Δf ~ _int = 0) from the coarse estimate (Δf ^ _coarse ) of the frequency as well as on a deviation (Δf ~ _int = ± 1 / (4 · N _sym · T _a)) from the coarse estimated value (.DELTA.f ^ _coarse), which corresponds to the half of the quantization (f _Q) of the discrete Fourier transform, and in that to the coarse estimated value (.DELTA.f ^ _coarse) the frequency of a supplementary value (.DELTA.f ~ _int ), which corresponds to the deviation (Δf ~ _int ) from the rough estimate (Δf ^ _coarse ) of the frequency at which the log-likelihood function (L ₄ ) reaches its maximum (max {L ₄ (Δf ~ _int )} ) Has. Method according to one of Claims 1 to 3, characterized in that a fine estimated value Δf ^ _fine according to the frequency

is calculated, where N _{Sym means} the length of the input signal sequence (x _ν ). Method according to Claim 4, characterized in that the calculation of the fine estimated value (Δf ^ _fine ) of the frequency is repeated iteratively. Method according to Claim 5, characterized in that, in successive iteration stages, fine estimate iteration contributions according to

are generated, where it means the iteration index of the it-th iteration stage that the fine estimate iteration contributions of the iteration stages already carried out according to

are added up and that the back-transformed, compensated signal sequence y _{ν, comp} additionally according to

to generate a compensated iteration signal sequence y {it} / ν, comp is compensated, and the next iteration stage is carried out with the compensated iteration signal sequence. Method according to one of Claims 4 to 6, characterized in that the fine estimate (Δf ^ _fine ) of the frequency and the coarse estimate (Δf ^ _coarse ) of the frequency are added to generate an overall estimate (Δf ^) of the frequency. Method according to Claim 7, characterized in that an estimated value of the phase is derived from the overall estimated value of the frequency

is calculated, where x _ν : means the input signal sequence. Digital storage medium with electronically readable Control signals that are so with a programmable computer or digital signal processor can work together that the method according to a of claims 1 to 8 executed will. Computer program product with on a machine-readable carrier stored program code means to carry out all steps according to a of claims Perform 1 to 8 to be able to if the program is on a computer or a digital signal processor executed will. Computer program with program code means to all Steps according to a of claims Perform 1 to 8 to be able to if the program is on a computer or a digital signal processor executed will. Computer program with program code means to all Steps according to a of claims Perform 1 to 8 to be able to if the program is saved on a machine-readable data carrier is.