Improved Frequency Estimation by Interpolated DFT Method

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Procedia

Engineering

www.elsevier.com/locate/procedia

2012 International Workshop on Information and Electronics Engineering (IWIEE)

Improved Frequency Estimation by Interpolated DFT Method

Chao Gong

*

, Daoxing Guo, Bangning Zhang and Aijun Liu

Institute of Communications Engineering, PLA University of Science and Technology,, Nanjing, Jiangsu 210007, China

Abstract

An improved method for estimating the frequency of a single complex sinusoid in complex additive white Gaussian noise is proposed. The method uses a modified version of the iterative interpolation strategies of Aboutanios and Mulgrew which are asymptotically unbiased and normally distributed with a variance that is very close to the Cramér–Rao bound (CRB). The A&M algorithms require the calculation of four additional discrete Fourier transform (DFT) coefficients with two iterations, whereas the new algorithm only needs two additional coefficients by exploiting the standard DFT coefficients at the first iteration. It is shown by simulation results that the new algorithm maintains the same performance as the A&M estimators.

Keywords：Frequency Estimation; DFT; Interpolated Algorithm

1. Introduction

Frequency estimation of a complex sinusoid in complex additive white Gaussian noise is relevant to a wide range of areas such as radar, sonar, communications, and biomedicine to name a few [1]. Many methods focusing on this task and presented in scientific literature can be classified either in time-domain methods [2-3] or in frequency-domain methods [1, 4-6]. Time-domain methods are computationally simple, but usually achieve the CRB only at high signal to noise ratios (SNR). The frequency-domain methods are efficient even at moderate/low SNR, but the computational load is a bottleneck for real-time implementation. In reference [1], Aboutanios and Mulgrew (A&M) proposed two iterative interpolation estimators based on fast Fourier transform (FFT), both of which were shown to achieve identical

*

Corresponding author.

E-mail address: [email protected]

Open access under CC BY-NC-ND license.

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asymptotic performances. In this letter, we modify the A&M algorithm for the purpose of reducing computational load and delay.

2. System Model

Consider the estimation of the frequency of a complex exponential, which is given by

s k_{( )}=Aej[2πkf f/ s+θ]+w k_{( ),} k=_0,1,_LN−₁ (1)

where the signal amplitude , frequency , phase , and sampling frequency are deterministic but unknown constant and is the number of samples. The value of is assumed to be in the interval , while noise is assumed to be a zero-mean complex white Gaussian process having variance and SNR is given by . Rife and Boorstyn [4] showed that the maximum-likelihood (ML) estimator of the frequency is given by the argument of the periodogram maximizer, i.e.,

{

}

2 1 2 ML 0

ˆ _{arg max} _{( )} _{arg max} N _{( )} j k

k f Y λ − s k e− π λ = ⎧ ⎫ ⎪ ⎪ = = _⎨ _⎬ ⎪ ⎪ ⎩

∑

⎭ (2)

The numerical maximization of (2) is not a computationally simple task and may suffer from convergence and resolution problems [5]. Therefore, it is common to estimate the frequency of a sinusoid by a two-step process comprising a coarse estimator followed by a fine search [1, 4-6].

3. Frequency Estimation

The coarse estimation stage is usually implemented using the maximum bin search (MBS) as coarse approximations of the periodogram maximize [6]. This consists of calculating the -point FFT of the sampled signal and then locating the index of the bin with the highest magnitude. The coarse search returns the index of the bin with the largest magnitude.

Interpolated DFT method is used for fine frequency estimation. A bin within the spectrum mainlobe is chosen as reference bin for interpolation, which can be expressed as

1 2 ˆ/ 0 0 ( ) N j kg N k X −s k e− π = =

∑

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The relationship between and the frequency of the signal is given by

ˆ s g f f N δ + = (4)

where is a residual in the interval if is really within the spectrum mainlob. The purpose of the fine frequency estimator is to obtain an estimate of . We set, without loss of generality, and . Ignoring the noise terms, we can express the Fourier coefficient with an offset from the reference bin as

2 ( ) 1 ˆ 2 ( )/ 2 ( )/ 0 1 ( ) 1 j p N j k g p N j p j p N k e X s k e e e π δ π θ π δ − − − + − = − = = −

∑

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For and , can be further given by

1 2 ₀ , 1 2 j e j X_p Ne X p j p p πδ _δ _δ θ πδ δ δ ⎛ ₋ _⎞⎛ _⎞ ⎜ ⎟ ≈ − _⎜ _⎟≈ = ± ⎜ ⎟_⎝ − _⎠ − ⎝ ⎠ (6) From (5), we can get the estimation of as

0 ˆ p _, ₁ p pX p X X δ = = ± − (7)

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In order to obtain a real-valued estimation of , we modify (6) as follow 0 0 2 0 0 0 0 0 Re( ) ˆ _{, = 1} Re( ) p p p p pX X p X X p X X X X X X X δ ∗ ∗ ∗ ∗ ∗ = ≈ ± − − (8)

where the sign “ ” represent conjugate. is chosen according to magnitude of . With the estimation formula of , the whole frequency estimation process can be summarized as follow,

(i) Course search:

Let S=FFT( ) and ( )s Y n = S n( ) , 2 n=0LN−1

Find ˆ arg max

{

( )

}

n m= Y n . (ii) Iteration 1: If Y m(ˆ− ≥1) Y m(ˆ+ then 1) g m pˆ= ˆ, = −1,X₀ =S m X( ),ˆ _p =S m(ˆ− ,1) else g m pˆ= ˆ, =1,X₀ =S m X( ),ˆ _p =S m(ˆ+ .1) Calculateδ by (8). ˆ₁ (iii) Iteration 2: Let 1 1 ˆ ˆ 2 ( )/ 0 ( ) , 0.5 N j k m q N q k X − s k e− π + +δ q = =

∑

= ± . If X−_0.52≥ X_0.52 then g mˆ= + −ˆ δˆ1 0.5,p=1, , X0 =X−0.5 Xp =X0.5, else g mˆ= + +ˆ δˆ1 0.5,p= −1, , X0 =X0.5 Xp =X−0.5. Calculate δ by (8). ˆ2 Finally ˆ ˆ ˆ1 ˆ2 0.5 s m p f f N δ δ + + − = .

It’s obvious that the proposed algorithm only requires the calculation of two additional DFT coefficients, whereas the A&M algorithms need four such coefficients with two iterations. The time delay expended for the calculation of coefficients at the first iteration is also avoided. Besides, real division which is more suitable for hardware realization is used in the new interpolation formula instead of complex division in the A&M algorithm 1, and the calculation of absolute value in the A&M algorithm 2 is unnecessary.

4. Simulation Results

As a basis for comparison, the CRB [4] is given by

2 2 2 2 6 (2 ) ( 1) s f f N N σ π ρ =

− . The performances of the

proposed algorithm and the A&M algorithm 1 are compared in figure 1 with an average of 10000 times at an SNR of 0 dB. A&M algorithm 1 is chosen as it performs better than algorithm 2 at iteration 1. The

number of samples used in simulation is N =1024.

At iteration 1, the performance of proposed algorithm is inferior to A&M algorithm especially in the center of the interval as shown in figure 1. However the performance of both algorithms is very close to the CRB over the entire interval at iteration 2. Fig. 2 presents the simulation results of the noise performance of both algorithms as a function of the SNR, averaged over 10000 simulations. Both algorithms exhibit almost identical performances that are on the CRB curve. The threshold effect that is characteristic of the ML estimator, and results from the coarse estimation stage, is visible. In computer

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-0.50 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6

Offset From Bin Center,δ

R at io of E st im at or E rr or V ar ianc e to C R B Proposed:Iteration 1 Proposed:Iteration 2 A&M Alg1:Iteration 1 A&M Alg1:Iteration 2

Fig. 1 Ratio of the variance of the proposed algorithm and the A&M algorithm 1 to the CRB as a function of δ

-20 -15 -10 -5 0 5 10 10-4 10-3 10-2 10-1 100 101

Signal to Noise Ratio,dB

S tan tar d D ev ia tion o f F re que nc y E st im at es ,H z CRB:N=1024 Proposed:N=1024 A&M:N=1024 CRB:N=256 Proposed:N=256 A&M:N=256 CRB:N=64 Proposed:N=64 A&M:N=64

Fig. 2 Standard deviation of the frequency estimation error of the proposed algorithm and the A&M algorithm as function of SNR with N =1024, N =256, and N =64

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simulation, the

δ

ˆ₁ and

δ

ˆ₂ are limited to the interval [-1,1] in order to avoid unreasonable big value when the SNR is under the threshold.

5. Conclution

We have proposed an improved estimator for single frequency estimation by interpolated DFT method. The estimator consists of a coarse search followed by a new fine search algorithm. Compared with the A&M algorithm, it requires lower computational load and shorter delay while maintain the same performance as proved by simulation.

References

[1 ] Aboutanios, E., and Mulgrew, B.: ‘Iterative frequency estimation by interpolation on fourier coefficients’, IEEE Transactions on Signal Processing, 2005, 53, (4), pp. 1237–1242,.

[2 ] Fu, H., and Kam, P.Y.: ‘Improved weighted phase average for frequency estimation of single sinusoid’, Electron. Lett., 2008, 44, (3)

[3] Awoseyila, A.B., Kasparis, C. and Evans, B.G. : ‘Improved single frequency estimation with wide acquisition range’, Electron. Lett., 2008, 44, (3)

[4] Rife, D. C., and Boorstyn, R. R.: ‘Single tone parameter estimation from discrete-time observations’, IEEE Trans. Inf. Theory, 1974, vol. IT-20, (5), pp. 591–598

[5] Quinn, B. G. : ‘Estimating frequency by interpolation using Fourier coefficients’, IEEE Trans. Signal Process., 1994, 42, (5), pp. 1264–1268

[6] Aboutanios, E. : ‘A modified dichotomous search frequency estimator’, IEEE Signal Process. Lett., 2004, 11, (2), pp. 186– 188