ANew DFT-Based Frequency Estimator forSingle-Tone Complex Sinusoidal Signals

Luoyang Fang", Dongliang Duant and Liuqing Yang"* Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA

t Department of Electrical and Computer Engineering, University of Wyoming, Laramie, WY 82071, USAEmails:Luoyang.Fang@colostate.edu.duandongliang@gmail.com. lqyang@engr.colostate.edu

Abstract-Frequency estimation for single-tone complex sinu­soidal signals under additive white Gaussian noise is a classicaland fundamental problem in many applications, such as com­munications, radar, sonar and power systems. In this paper, wepropose a new algorithm by interpolating discrete Fourier trans­form (DFT) samples. Different from other existing interpolationmethods for frequency estimation, our algorithm is based ona much simpler expression and has mathematically tractablebias expression in closed form, which can potentially assistfutu~ bias correction. Simulations confirm that our proposeda1gonthm outperforms all existing alternatives in the literaturewith comparable complexity.

I. INTRODUCTION

Frequency estimation for sinusoidal signals is widely usedin many applications, such as communications, radar, sonarand power systems. In the literature, many techniques for thefrequency estimation of complex sinusoidal signals have beenproposed. Among them, there are a good number of very lowcomplexity methods based on the discrete Fourier transform(DFT) samples [1-9].

It is well known that the Fourier transform of a perfect,infinitely-long, continuous-time, single-tone signal will bean impulse located exactly at the frequency of the signal.However, in practical applications, the signal always hasfinite duration and is often discretely sampled. Therefore,the frequency spectrum, obtained by Discrete Time FourierTransform (OTFT), will no longer be a perfect impulse butwill take a sine shape. Moreover, with OFT, the signals arealso discrete in frequency. This will limit the resolution offrequency estimations. In other words, there is no guaranteethat the true frequency is exactly sampled. Zero padding inthe time domain before DFT is a technique that can introducemore samples to the frequency domain without altering thespectrum shape, which in tum leads to finer sampling intervalsin the frequency domain.

In the literature, the DFT-based frequency estimation isusually carried out in two steps. In the first step, a coarseestimate of the frequency is made by locating the DFT sampleclosest to the true frequency. In the second step, assuming thatthe coarse estimate in the first step is correct, the frequencydeviation between the location obtained from the first stepand the true frequency is further estimated to obtain a finerfrequency estimate. Many algorithms employing the two-step

This work is in part supported by Department of Energy under DOE SGIL533877.

978-1-4673-3/12/$31.00 ©2013 IEEE

strategy have been proposed in the literature. They all share thesame coarse estimate in their first step by selecting the OFTsample with the maximum magnitude, and their differenceslie in the fractional frequency deviation estimate in the secondstep [3-5, 7, 9]. Moreover, the algorithms in [3-5, 7, 9] allutilize the OFT sample with the maximum magnitude andits nearest neighbors to calculate the frequency deviation byinterpolation. As we will show later, all the existing estimatorsdescribed above are biased due to the approximations in theiralgorithm development.

In this paper, following the same two-step strategy, wepropose a new frequency estimation algorithm. We adoptthe same coarse estimate in the first step, and use a moresystematic approach for the second step. To better estimate thefrequency deviation, the spectrum shape information, such aslocation of DFT samples and their magnitude value, are betterutilized. In specific, zero padding is used in our algorithmto guarantee that the samples of interest are located withinthe main lobe of the sine spectrum. In addition, a specificzero-padding number to observation window length ratio ischosen such that the fractional frequency deviation estimateis expressed nicely and simply. Although approximation isalso used in our estimator resulting in bias, quite unlike otherexisting algorithms, a closed-form expression for the biasterm can be readily computed for our algorithm. This canbe potentially utilized for further bias correction. In addition,analysis of the noise-dependent bias is provided and usedto explain an interesting behavior of the performance ofour estimator. Simulations show that our proposed algorithmoutperforms all existing OFT-based alternatives.

This paper is organized as follows. We present the signalmodel and problem formulation in Section Il, The Cramer-Raolower bound and existing estimators are reviewed in SectionIII. The development of our proposed algorithm together withits bias analysis is given in Section IV. Simulations arepresented in Section V to show the performance comparisons.Concluding remarks will be presented in Section VI.

II. SIGNAL MODEL AND PROBLEM FORMULATION

A continuous-time single-tone complex sinusoidal signal isgiven as follows:

x(t) == Aej (21r j t+ ¢ ) + w(t) , (1)

where f is the signal frequency to be estimated, A and ¢are the unknown magnitude and initial phase respectively,

and w(t) is the additive complex white Gaussian noise withvariance 0- 2 • In practice, the measurement device can onlyobtain a sampled and finite-windowed version of the originalsignal. In this paper, the rectangular window is adopted as in[3-5, 7, 9]. Denote the sampling frequency as Is and windowsize as N. Then the discrete-time signal is as follows:

x[n] = Aej (27ft n +cP) +w[n] . (2)

where w[n]sare additive white Gaussian noise with per samplevariance 0-2 • Accordingly, the M -point (M ~ N) DFT of x[n]is

As an interpolation method, the algorithm development will assume absenceof noise. However, the noise performance will be analyzed, simulated andcompared in the sequel.

where km is the DFT index of the sample with the maximumamplitude, 8 E [-0.5, 0.5] captures its fractional deviationfrom the true frequency.

Accordingly, incorporating the noise in our original model,X[k] can be rewritten as

X[k + i]=Ae1"'e1"NMl (Ii-i) sin (~(8 - i)) + W[k +i].m sin (~(8 _ i)) m

(5)With this signal model, and assuming that km can be

correctly located at reasonable level of noise, we can estimatethe fractional frequency deviation 8 to obtain the frequencyestimate.

If M > N, (M - N) zeros need to be padded to the originalobservations before taking the DFT. The W[k]s are Gaussiannoise samples in frequency domain, which will be white ifM = N and colored if M > N.

To illustrate our frequency estimation strategy, we plot theDFT and DTFT of a noise-free single-tone signal in Fig. 1. Asshown in Fig. 1, the continuous spectrum of a noise-free finite­length single-tone sinusoidal signal obtained by DTFT is a sinefunction with the peak at its frequency I and main-lobe width2£8 .The DFT samples in Eq. (2) are samples of the continuousspectrum as shown in Fig. 1 with sampling interval fl.! = -tA.Therefore, X [k] corresponds to the value of the spectrum atfDFTF = kfl./· Clearly, as long as I is not an integer multipleof tJ..f, DFT will not provide the value at the true frequency,where the peak of the continuous spectrum is located. TheDFT sample with the maximum amplitude will be the closestto the true frequency as shown in Fig. 1. Therefore, the truefrequency can be expressed as

III. CRAMER-RAO LOWER BOUND AND EXISTINGESTIMATORS

A. Cramer-Rao Lower BoundThe Cramer-Rao lower bound (CRLB) indicates the best

performance that an unbiased estimator can achieve. TheCRLB can be obtained by the diagonal elements of the inverseof the Fisher information matrix J, whose elements are givenby J i j = -E{HoiHo:j } , where HO:i = a~i logp(x; a).Here, a is the vector of parameters to be estimated, x is thevector of available observations, and p(x; a) is the probabilitydistribution function of x parametrized by a. For our signalmodel, a = [f, A, ¢]T. The CRLB was investigated in [1] and[5], and accordingly,

A 6f2var{t} ~ CRLB = 47r2"(N (';v2 _ 1) , (6)

where N is the number of samples, and 'Y = ~ is thesignal-to-noise ratio (SNR). As Eq. (6) demonstrates, largerobservation window size N and higher SNR 'Y will bothimprove the accuracy of frequency estimator. Although allestimators mentioned in this paper are biased, we still usethe CRLB as a reference for the root mean squared errorperformance.

B. Existing EstimatorsThere are many DFT-based frequency estimators in the

literature [3-5, 7, 9]. They all first take the N-point DFT of anN -point signal and obtain the DFT sample with the maximummagnitude. And then, assuming they obtain the correct km inthis step, they utilize its neighbors to estimate the fractionalfrequency deviation 8 by different interpolation methods. in[3], two candidate estimates 81 and 82 are obtained by therelationships between the maximum and its left neighbor andbetween the maximum and its right neighbor respectively, and

Fig. 1. Magnitude of DTFf (dash line) and M -point DFf samples of anoise-free single-tone signal x(t) = Aej ( 27r jt+cP) with sampling frequencyIs and observation window size N. In this figure, index of the DFf samplewith the maximum magnitude km = 4.

(4)I = (km + 8)tJ..I ,

(9)

(10)

(11)

TABLE IEXISTING ESTIMATORS

0:1 = Real(X[k'm - 1]1 X[k'mD

0:2 = Real(X[k'm + 1]/X[k'mD

Quinn94[3] 61 = 0:1/(1 - 0:1), 62 = 0:2/(1 - 0:2)

if 61 > 0 and 62 > 0,6= 62

else,6 = 61

0:1 = Real(X[km - I]/X[km D0:2 = Real(X [k'm + 1]/X [k'mD

61 = 0:1/(1 - O:l)i 62 = 0:2/(1 - 0:2)

d1 = 6~ ' d2 = 6~

Quinn97[4] 1 2 ,/6 ( di+ I-VI)kl = - log(3d1 + 6dl + 1) - - log v'J4 24 dl + 1 + i1 2 ,/6 ( d2 + 1 - VI )

k2 = - log(3d2 + 6d2 + 1) - - log -If4 24 d2 + 1 + i

6 = (61 + 62) + kl + k22

0:0 = IX[kmll~

0:1 = X[k'm - I]X· [k'm]

Macleod98 [5]0:2 = X[k'm + I]X· [k'm]

d=Real (0:1 - 0:2)

Real (20:0 + 0:1 + 0:2)

6 = (.Jl+8i2 - 1)/(4d)

Jacobsen07 [7] A { X [k'm - 1] - X [k'm + 1] }6 = Real

2Xrk'l'J'l.1 - Xrk'l'J'l. - 11 - Xrk'l'J'l. + 11

Candanll[9] g = ta.n,,/N Rea.l { X[k", - 1] - X[k", + 11 11rIN 2X[k'm] - X[k'm - 1] - X[km. + 1] Jx * denotes the conjugate of X, Real denotes the real part

one of them is selected according to their signs. In [4], theestimator in [3] was improved by taking interpolation of thesetwo estimators. The estimator proposed in [5] utilized both themagnitude and phase information of three OFT samples. In[7], an estimator based on the simple relationship of the OFTsample with the maximum magnitude and its two neighborswas proposed. However, it was only based on ad-hoc empiricalobservation rather than strict mathematical derivation. In [9],not only the mathematical derivation of the estimator in [7]was presented, but also a bias correction was proposed toreduce its error floor at high SNR. The detailed algorithmimplementations are shown in Table I. All the estimatorsdescribed above are biased due to the approximations in theiralgorithm development. Among the existing estimators, [9] hasthe best noiseless bias performance.

obtain the two neighbors of the maximum:

NIsin (.!I. (6 - 1))X[k + 1] == Ae j (7r 2N (6-1)+¢) 2 (7)m sin (2~ (6 - 1)) ,

X[km

-1] = Aej (1r r;j\/ (<l+l l+tPl sin (~(8+ 1)) . (8)sin (2~ (6 + 1))

In practice, we will have N ~ 2, and recall that the fractionaldeviation 6 E [-0.5,0.5]. As a result, 0 < 2~ (1 ± 6) < ~.

The frequency spacing between neighboring OFT samplesis l::t..1 == ~ == 1L and the width of main lobe of sine functionafter DTFf is 1(. Accordingly, the number of samples in themain lobe is If x liB == 4 as shown in Fig. 1. Therefore, in

2N x B

the noiseless case, IX [km ± 1]I will be guaranteed in the mainlobe of the sine spectrum and never be zero-valued.

By taking the ratio of the magnitude of the two neighboringOFT samples of the maximum, from Eqs. (7) and (8), we canobtain a simple expression as follows:

I

X[km + 1] I == sin(2N(1 + 6))X[km - 1] sin(2~(1- 6)) ·

As shown above, the ratio in Eq. (9) only contains the infor­mation of 6 which indicates the true frequency location. Usingtrigonometric identities, Eq. (9) can be derived as follows:

I

X[km + 1] I sin(2N) cos (2N 6) + cos(2N) sin (2N 6)

X[km - 1] sin(2~) cos (2~6) - COS(2~) sin (2~6)

tan(2N) + tan (2N6)tan(2~) - tan (2~6) ·

where cos (2~6) i= 0 and cos (2~) i= 0 since 0 < 2~ < ~and 0 ::; 2~6 < ~. Then, tan (2~6) can be obtained by Eq.(10) as follows:

IX[k'm+l] 1- 1

( 1r) ( 1r) X[km-l]tan 2N 6 == tan 2N I X[km+l] 1 + 1 .

X[km.- 1]

Since 2~ 6 is very small, the approximation tan (2~6) ~ 2~ 6can be applied to obtain the estimate of 6 in a closed-formexpression as follows:

B. Bias Analysis

One advantage of our algorithm is that we have a muchsimpler relationship for the interpolation method and we canobtain a closed-form expression for the noiseless bias, whichcan be utilized for bias correction for further performanceimprovement. In this section, we will derive the noiseless biasexpression and will also analyze the bias behavior in the pres­ence of noise to understand the SNR-dependent performance.

IV. PROPOSED ALGORITHM AND ITS BIAS ANALYSIS

A. Proposed Algorithm

Following the same two-step strategy and the same first-stepcoarse estimate, we propose a new algorithm to improve theaccuracy of the fractional frequency deviation 6 estimation inthe second step. For simplicity, we will describe the algorithmdevelopment in the noiseless case.

In our proposed algorithm, to obtain an alternative relation­ship between the OFT samples enabling better interpolation,we pad zeros with the same length as the data window beforethe OFT. In other words, we take a 2N-point OFT for an N­point signal. From Eq. (5), let M == 2N and i == ±1, we can

8== tan(2N) IX[km + 1]1-IX[km -1]1 .2~ IX[km + 1]1 + IX[km -1]1

Accordingly, by Eq, (4), the frequency estimate is

A A IsI == (km + 6)2N .

(12)

(13)

The bias of the fractional frequency deviation 6 is definedas:

bs = E{8} - 6 . (14)

Then accordingly, by Eqs. (4) and (14), the bias of thefrequency estimate will be

(15)

Since the relationship between the gap bias blj and the fre­quency bias bf is linear, it is sufficient to analyze the fractionalfrequency deviation bias, blj.

As a function of 6, tan (~6) can be expanded in Taylorseries as:

tan (~8) = ~ 8 + ~ (~ ) 3 83 + 125(~) 5 85 + h. o. t.

where h.o.t. stands for higher order terms.Therefore, in the noise-free case, by Eqs. (12) and (14),

b8 = 8- 8 ~ ~ (~f 83· (16)

In the presence of noise, the SNR-dependent bias is definedas

(17)

A property of blj(,) can be obtained as follows:

Theorem 1 There is at least one SNR value "( such thatblj("() = o.

Fig. 2. Noiseless bias comparison with observation window length N = 8and N = 32.

shown in Appendix A and B, from Eqs. (19) and (20), wehave

(21)

Therefore, from Eq. (18), lim"Y---+o blj('Y) = -6.On the other hand, when the SNR "{ -t 00, it will

approximate the noiseless case. Then accordingly, the biasblj(+(0) shares the same sign of 6 itself as suggested by Eq.(16).

In summary, we have:

where

IX[km + 1]1

, if 'Y -t 0, if'Y -t +00

where sgn( .) denotes the sign function. As demonstratedabove, the signs of bias in the high SNR and low SNR casesare opposite. Hence, there must be at least one SNR value "(such that blj('Y) = O. •

This is also verified by simulations as shown in Table II.

v. PERFORMANCE COMPARISONS

In this section, we compare the existing frequency estima­tion algorithms described in Table I in the literature with ourproposed algorithm in terms of the noiseless bias and the rootmean square error (RMSE) in the presence of noise.

A. Noiseless Bias Performance

It has already been shown that the algorithm proposed in[9] has the best noiseless bias performance among the existingalgorithms. Therefore, in this subsection, we will only comparethe bias between our proposed algorithms and the algorithmin [9].

Under the noiseless circumstance, the bias performance withobservation length N = 8 and N = 32 is shown in Fig. 2.It can be observed that the bias is a periodic function of thesignal frequency, f. This is because the fractional frequencydeviation 6 is a periodic function of the frequency f with aperiod of ilf, and the bias is a function of 6 as shown in Eq.(16). In addition, it can be observed that within a period, the

IX[km -1]1 ~

IX[km + 1]1 + IX[km -1]1 'IX[km -1]1

where the subscripts r and i denote the real and imaginaryparts, respectively. Similarly, we have

Proof: From Eqs. (12) and (17), we can obtain

b8(-Y) tan(7r/M)E {IX[km + 1]1-IX[km -1]1} _ 6'TrIM IX[km + 1]1 + IX[km -1]1

ta:~~M) (E {Y+ 1 } - E {Y- 1 } ) - 8 , (18)

To analyze the property of blj(,), the representation ofIX[km + 1]1 in Gaussian noise is presented in Appendix A.When SNR "{ -t 0, taking the dominant term of Eq. (22), wehave

N (Wr[km - 1]2 + Wilkm - 1]2). (20)2"{

Since Wr[km+ 1],Wi[km+ 1],Wr[km -1], and Wi[km -1]are uncorrelated identically standard Gaussian distributed as

TABLE IIBIAS btS OF PROPOSED ESTIMATOR IN THE PRESENCE OF NOISE

SNR(dB) 10 15 20 25 30 35 40 45 Noise free8 = -0.5, I = 43.65 0.0748 0.0406 0.0216 0.0113 0.0056 0.0024 0.0006 -0.0003 -0.0016

8 = -0.4, I = 45 0.1307 0.0222 0.0001 -0.0004 -0.0009 -0.0009 -0.0008 -0.0008 -0.00088 = -0.3, I = 46.25 0.0052 0.0011 0.0001 -0.0003 -0.0004 -0.0003 -0.0003 -0.0003 -0.00038 = -0.2, I = 47.5 0.0160 0.0019 0.0002 -0.0001 -0.0000 -0.0001 -0.0001 -0.0001 -0.0001

In this simulation, Is = 200Hz and N = 8.

bias, bo, is a monotonically increasing function of 181 as statedin Eq. (4).

Comparing these two algorithms, first we find that they havevery similar bias performance within some range. However,the period of our proposed algorithm is one half of Candan'sin [9] for the same window length, because of the zero paddingused in our method. This results in a significant performancegain in some range of frequencies, such as f E [25,43.75], asshown in Fig. 2.B. RMSE Performance

In this part, the RMSE defined as E[(f _j)2] is com-pared among the proposed method an exiting ones listed inTable L

As shown in Fig. 3, for f == 38.75 Hz, our proposedestimator has a significant RMSE performance gain comparedwith existing alternatives. This comes from the noiseless biasgain as shown in Fig. 2. For the range of frequency whereour algorithm shares similar bias performance with [9], ourproposed algorithm still outperforms the others at low-to­medium SNR as shown in Fig. 4. In fact, there is almost a2 dB gain for our algorithm compared to Candan's in [9] at Fig. 3. RMSE comparison with N = 8, I = 38.75 Hz and Is = 200 Hz.

low-to-medium SNR with the gain G defined as:RMSEc

G == 20 loglO RMSE 'p

where the subscripts c and p correspond to Candan's algo­rithm [9] and our proposed one, respectively. Our proposedalgorithm and Candan's algorithm share the same error floorat high SNR because they have same bias as indicated inFig. 2. It is also worth noticing that although all the DFT­based algorithms including ours are heavily dependent on thecorrectness of the first coarse step which can be bad at lowSNR, we see from the low SNR performance in Figs. 3 and4 that our algorithm is the least affected by the wrongness inthe first step.

In Fig. 5, a very interesting phenomenon is that when SNRis from 40 to 60 dB, there is a dip in the RMSE curve for ouralgorithm, with a value even smaller than the error floor. Thisphenomenon results from the behavior of the SNR-dependentbias stated in Theorem 1 and shown in Table II. When 8 == 0.5,the SNR value at which bias crosses zero is in the range from40 to 60 dB as Table II shows. At this range, the bias alreadydominates the RMSE performance compared with the noisevariance. Therefore, a dip in this SNR range can be observedin this special case. However, this is generally not true forother frequency f, since they have an earlier zero-crossing Fig. 4. RMSE comparison with N = 8, I = 45 Hz and Is = 200 Hz.

SNR point for the bias where the noise still dominates theRMSE performance.

2 ~ 1 NIX[km + 1]1 = AX+l 1 + - - x W + -2--(Wr[km + 1]2 + Wi[k m + 1]2) ,X+l 2')' X+ 1 2')'

where W = cos (N2~)1r(8 -1) +4» Wr[km + 1] +jsin (N2~)7r(8-1) +4» Wi[km + 1].

(22)

ApPENDIX BNOISE CORRELATION ANALYSIS

The autocorrelation of noise, W [k], obtained by the M­point DFT of the original N -point signal can be obtained asthe follows:

Rw(l) = E{W[k]W* [k + l]}N-l

= L{e-j~knei~(k+l)n}E{w2[n]}

n=O

Fig. 5. Special case when f = 56.25 Hz and 8 = 0.5 for N = 8 andIs = 200 Hz.

VI. CONCLUSIONSIn this paper, we proposed a new DFT-based frequency

estimation algorithm by estimating the fractional frequencydeviation between the location of the DFT sample with themaximum amplitude and the true frequency. Zero padding isused in our algorithm to obtain a better relationship betweenthe fractional frequency deviation and the DFT sample ampli­tudes. We also derive a closed-form expression for the biasof our algorithm. Simulations showed that our proposed algo­rithm has a better performance than all existing alternatives,in the sense of both noiseless bias and RMSE performance.

ApPENDIX AREPRESENTATION OF IX[km + 1]1 IN GAUSSIAN NOISE

From Eq. (3),

IX[km + 1]1 = IAei(~Nll1"(8-1)H)X+l + W[km + 1]1 '

where X+1 = s~~(~(tl~l?) and W[km+ 1] is the complexGaussian noise. No~a1izing the noise term, we have

IX[km + 1]1 = AX+1JXn+1] + Xt[+I] ,where

( N - l )~Xr[+I]=cos 2N7r (8 -1)+¢ + 2X2 Wr[km + l ],+1·"1

( N - l )~Xi[+I]=sin 2N7r (8 - 1)+¢ + 2X2 Wi [km+ l ]'+1·"Y

with Wr[km + 1] rv N (0, 1) and Wi[km + 1] rv N (0, 1).Hence, we obtain the expression of IX[km + 1]1 in Gaussianrandom variable in Eq. (22). Similar expression for IX[km -1] Ican be derived.

With M = 2N in our algorithm,

{

NU2, ifl=O

Rw(l) = ~j *, if 1= ±1l-e N

0, if l # 0 and l # ±1

The distance between W[km + 1] and W[k m - 1] is 2 andRw(2) = O. Therefore, W[k m + 1] and W[k m - 1] areuncorrelated.

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