The frequency estim ation of a single tone corrupted by additive w hite G aussian noise has received significant atten tio n over th e last decades due to its wide applicability in signal processing. In this chapter, a com putationally fast and statistically im proved hybrid single tone estim ator is proposed, which outperform s other recently proposed approaches. Nu merical sim ulations indicate th a t, in contrast to many other techniques, th e performance of the hybrid estim ato r is essentially independent of the underlying frequency component. Furtherm ore, it has also been exam ined for how power dam ping and frequency spread affect the perform ance of th e estim ator.
2.1 Introduction
Frequency estim ation is a topic widely occurring in signal processing and can be roughly classified into two m ain p aram eter estim ation problems:
• Single tone estim ation: where th e signal is a single, constant-frequency sinusoid, cor rupted by some noise.
• M ulti-carrier frequency estim ation: where there are several carriers of harmonically
S e c tio n 2 .1 . I n tr o d u c tio n 12
related or un related frequencies present as in the m ultiple-carrier based estim ators to be discussed in C hapter 3.
In this chapter, discussion will be confined to single tone estim ation. The problem of es tim ating th e frequency of a sinusoid in noise has received much atten tion in the literature as the problem arises in m any areas of applied signal processing, such as biomedicine, com m unications and ra d a r [Edd93, W ai02, Jen96a]. One often encounters a need to find a low com putational com plexity estim ate of the frequency com ponent of d a ta which are assumed to consist of a single com plex sinusoid corrupted by additive w hite G aussian noise, and the topic has, as a result, a ttra c te d significant interest over th e last decades (see, e.g., [LRP73,RB74, Tre85, Kay89, LM89, LW92, Cla92, CKQ94, H95, KNC96, FJ99, QuiOO, Fow02, Mac04, Kle05], and the references therein). T he problem can be briefly sta te d as follows; consider the d ata sequence
y(t) = Pel^ t+es> -I- n(t), (2.1.1)
where (3 E R, u and 6 E [—7r, n) denote the determ inistic b u t unknow n am plitude, fre quency, and initial phase, respectively, of a complex sinusoid. F u rther, n(t) is circular zero
mean complex G aussian w hite noise w ith variance g\ . T hen, given th e sequence y(t), for
t = 0 , . . . , iV — 1, th e problem is simply to estim ate accurately u w ith the lowest possible
com putational complexity. In [RB74], Rife and Boorstyn derived th e m axim um likelihood
(ML) estim ator of lj and proposed a statistically efficient approxim ate ML approach in
volving both a com bined coarse and fine search using the fast Fourier transform (FFT) algorithm. However, zeropadding is often required to obtain sufficient resolution, requiring
0 ( N ' \ o g 2 N ') operations, where N ' is the size of the desired frequency grid, w ith typically N ' N . Furtherm ore, an iterative linear prediction (ILP) approach requiring G ( N log2 N )
operations was suggested in [BW02] showing similar perform ance to th a t of the ML estim a tor. A variety of phase-based m ethods requiring only O ( N ) operations have been developed. In [Tre85], for exam ple, T re tte r proposed a phase-based approach simplifying the problem
S e c tio n 2 .1 . in t r o d u c t io n 13
to a linear regression on th e phase. The m ethod is based on a phase unw rapping algorithm ,
requiring a very high signal-to-noise ratio (SNR) (S N R 1), here defined as
R2
S N R = t L , (2.1.2)
to work well. Later, K ay proposed a modified version of T re tte r’s algorithm avoiding the use of the phase unw rapping algorithm [Kay89]. The m ethod, here term ed K ay’s weighted phase average (KW PA) estim ator, was claimed to be unbiased and its variance attain s the Cramer- Rao lower bound (CRLB) for sufficiently high SNR (a higher threshold th a n the ML method). Later, analysis showed th a t th e KW PA m ethod is in general biased, and th e SNR for which the CRLB is achieved depends on th e underlying frequency [CKQ94,LW92,Qui00]. For much of the frequency range, th e K W PA m ethod can not handle the circular n atu re (rotation) of frequency correctly. As a result, th e focus of recent contributions has m ainly been aimed at reducing the SNR threshold [KNC96], th e frequency dependency of th e threshold [Cla92], or both [FJ99,Mac04].
In this chapter, we propose a hybrid m ethod combining th e ideas in [KNC96, FJ99, Mac04] to show perform ance close to th a t of ML or ILP, b u t only requiring O ( N ) operations. The hybrid estim ate is based on an initial coarse estim ate of th e unknow n frequency using th e unweighted linear prediction (UW LP) m ethod [LRP73, Kay89]; this estim ate is used to remove the frequency dependence of the SNR threshold. T his SNR threshold is then further reduced via a com bination of using an averaging filter, as suggested in [KNC96], and an outlier removal scheme as proposed in [Mac04]. Finally, a refined frequency estim ate is formed along th e lines proposed in [KNC96, FJ99]. In Section 2.2, several basic estim ators are discussed, including th e ML estim ator in Section 2.2.1, th e U W LP in Section 2.2.2, the KW PA in Section 2.2.3; then the proposed hybrid estim ator is given in Section 2.2.4. Section 2.4 contains numerical exam ples, w ith a conclusion in Section 2.5.
S e c tio n 2 .2 . S in g le T o n e E s tim a to r s 14
2.2 Single Tone Estim ators
In this section, an overview of recent single frequency estim ators is presented as well as the proposed hybrid estim ator. In this section, all th e results have been obtained using 103 M onte Carlo sim ulations.
2.2.1 M aximum Likelihood Estim ator
If the additive noise, n(t), in (2.1.1) is a zero mean w hite G aussian process, th e ML estim ator of the frequency u> in (2.1.1) is the maximizer of the likelihood function (also called the joint probability density function) of the sequence {?/(£)}, given as
& M L E = arg m a x /
(y;£),
(2-2-1)U)
where ujmle is th e ML estim ate, w ith the likelihood function defined as
f te ’® = J ^
n exp(-^2Y ,
(»(*) - /?ei<"‘+<’)) 2) . (2.2.2)where y = [2/(0), • - - , y ( N — 1)]T, o 2n = the noise variance and £ = [/?,<*;, 0]r , w ith [-]T
denoting the transpose operation. Maximizing (2.2.1) is equivalent to minimizing
N - 1
u m l e = argin in (y(t) - j3el{uJt+e)) 2 , (2.2.3)
t=o
which can be seen to be th e nonlinear least squares (NLS) estim ation problem . It can readily be shown th a t th e NLS estim ate of the frequency of a single sine wave buried in white noise is given by th e peak of the periodogram of th e d a ta sequence [SM05], i.e.,
Um l e = arg m ax P (uj), (2.2.4)
S e c tio n 2 .2 . S in g le T o n e E s tim a to r s 15
where |x| denotes th e m odulus of the scalar x.
Rife &; B oorstyn [RB74] proposed a numerical m ethod similar to the ML estim ator, which involves a coarse and a fine search. The coarse estim ate is obtained by choosing the frequency having th e greatest m agnitude in the periodogram , as discussed above. A finer estim ate is obtained using a m ethod such as the secant m ethod. Generally, zeropadding is required to o btain sufficient resolution when using th e F F T , requiring 0 ( N ' log2 iV7) opera
tions, where N ' is th e size of the desired frequency grid, w ith typically N ' N . Thus, the
estim ator is not com putationally efficient. However, it is statistically efficient with the low est SNR threshold am ong the variously proposed estim ators, exhibiting estim ation variance identical to the corresponding CRLB given by [RB74]
6
CRLB& = jV(jV2 _ 1) g j Vfl {r a d /s a m p le )2, (2.2.6)
where u denotes the estim ated frequency and the C R L B & is th e lowest variance which a statistically unbiased estim ato r may exhibit for a given SNR and N . Over the last decades, the ML estim ato r’s high com putation cost has led to a search for altern ative m ethods th a t approach its statistical performance, bu t with less com putation.
2.2.2 Unw eighted Linear Predictor
As suggested in [Tre85], the d a ta model in (2.1.1) can be w ritten as
j/(t) = [1 + v(t)]/3eil-ult+e\ (2.2.7)
where
v(t) = (2.2 .8 )
is a complex w hite sequence. Let vr (t) and Vi(t) denote th e real and th e im aginary parts of
v(t), respectively. Then, for high SNR,
S e c t io n 2 .2 . S in g le T o n e E s t im a t o r s 16 -15 2.-25 -35 -45 -55 -65 -1 0 SNR (dB)
F ig u r e 2 .1 . T h e M SE of th e U W LP e stim ato r as a function of th e SNR, N = 24.
allowing th e approxim ation
y(t)s r (2.2.10)
where
(j){t) = u t + 0 + Vi(t). (2.2.11)
Thus, th e additive noise has been converted into an equivalent phase noise Vi(t) w ith variance
[Kay89] ^
v a r M t ) ) = & = _ L _ . (2.2 .12)
M ost of th e recent phase-based approaches exploit th is approxim ation, allowing th e phase to be approxim ately estim ated from th e difference of th e adjacent phase values suggested by
S e c tio n 2 .2 . S in g le T o n e E s tim a to r s IT
Kay [Kay89], i.e.,
A <f>(t) = arg [y*(t)y(t + 1)] « u + Vi(t + 1) - u*(t), (2.2.13)
where (•)* denotes th e complex conjugate, suggesting the so-called unweighted linear predic tor (U W LP) [LRP73,Kay89]
ujc — arg
t = 0
The UW LP m ethod is also term ed th e autocorrelation estim ator in ultrasound blood velocity estim ation [KNK085], as discussed in th e later chapters. It is straightforw ard to show th a t the UW LP estim ator is unbiased, b u t statistically inefficient w ith variance [Kay89, CKQ94]
v a r (* ' ) = ( N - i y S N R ( 2 2 1 5 ) giving the ratio
variujr) N
= — . (2.2.16)
C R L Bqj 6 V j
Figure 2.1 shows th e m ean square error (MSE) of th e UW LP estim ator as a function of SNR, w ith different frequencies denoted by different plots therein. As is clear from th e figure, the U W LP m ethod is statistically inefficient for all exam ined frequencies across the whole SNR range, unable to reach the CRLB. However, given th e fact th a t th e U W LP is very simple w ith very low com putational cost, it can be adequate as a coarse estim ator, as will be used in our proposed hybrid estim ator discussed later.
(2.2.14)
2.2.3 Kay’s W eighted P h ase A verager
A nother basic co m pu tatio nally efficient estim ator discussed here is K ay’s weighted phase
average (KW PA) m ethod [Kay89]. Recall A i n (2.2.13), th e K W PA m ethod is expressed
as
N - 2
& K W P A ~
E
w(t)A4>(t), (2.2.17)S e c tio n 2 .2 . S in g le T o n e E s tim a to r s 18
where C j k w p a denotes th e KW PA estim ate, and the parabolic window
<“ is»
The approxim ation in (2.2.13) will hold for a very high SNR (S N R 1) following the
approxim ation in (2.2.9) and (2.2.10). For this condition C j k w p a can be shown to be an
unbiased estim ator following th e derivation below
N - 2 E {ljKw p a} = ^>2 w ( t ) E { A ( f ) ( t ) } t=o N - 2 t=o N - 2
~ ^ 2 w (i )E{uj + Vi(t + 1) - Vi(t)}
=o N - 2 t = 0 N - 2 U t = 0 — UJ (2.2.19)
where the last equality follows from
N - 2
J 2 w (t ) = 1- (2.2.20)
t = 0
Similarly, as in (2.2.19), th e K W PA estim ator, for high SNR, can be shown to have the following variance [Kay89]
var(u,KIVPA) = n (n 2 _61 )s n r (2.2.21) which is identical to th e CRLB in (2.2.6). As discussed earlier, th e KW PA m ethod is in general biased and (2.2.21) will only hold tru e for high SNR and very low frequency range [CKQ94, LW92, QuiOO].
S e c tio n 2 .2 . S in g le T o n e E s t im a t o r s 19
It has been shown th a t one of th e drawbacks of the KW PA m ethod is the unavoidable
w rapping errors w hen th e phase uj approaches —7r /+7T, making the SNR threshold dependent
on underlying frequency [FJ99]. This is because the cum ulative errors from noise component
Vi(t) will som etim es m ake th e estim ate lower th a n u and sometimes higher than ui. If
A (f){t) exceeds —7r/ + 7r, th en it gets w rapped over the —7r / + n boundary and thus aliasing occurs resulting in higher variance as shown in Figure 2.2. A nother drawback of the KWPA m ethod is th a t its perform ance highly depends on SNR [Kay89]. W hen th e SNR drops below a certain threshold as shown in Figure 2.2, the perform ance of th e KW PA m ethod falls off rapidly, exhibiting threshold behavior a t a higher SNR which is also confirmed in Figure 2.2. Also, unlike the ML m ethod, as th e d a ta length N increases, th e SNR threshold of the KWPA m ethod slowly increases [LM89], which is illustrated in Figure 2.3. As Kay pointed
out [Kay89], th e coefficients of th e parabolic window are responsible for ujkwpa attaining
the CRLB. If we let w(t) = (2.2.22) then (2.2.17) becomes ^ N - 2 u u w p a = t=0 = W w - 1) - * ( 0)] = [v,(N - 1) - ^ ( 0)] (2.2.23)
which is term ed th e unw eighted phase average (UWPA) estim ator and is unbiased, b ut again becomes inefficient statistically as shown in Figure 2.4. This is because th e UWPA estim ator in (2.2.23) discards useful inform ation by allowing common inform ation in adjacent term s of the sum to cancel. T his is th e direct result of ignoring th e colouring in the noise term
S e c t io n 2 .2 . S in g le T o n e E s t im a t o r s 20 -15 -35 -45 -55 -65 -1 0 SNR (dB)
F ig u r e 2 .2 . T h e M SE of th e exam ined K W PA estim ato r as a function of th e SNR, N = 24.
2 .2 .4 T h e Proposed Hybrid Estim ator
Given th e lim itatio n s of th e K W PA estim ato r discussed above, various im proved and ex tended m eth od s have been proposed for reducing th e SN R thresho ld and th e frequency dependency of th e thresho ld (see, e.g., [KNC96, Cla92, FJ99, BW 02, Mac04], and th e refer ences therein). In th is section, th e proposed hybrid m ethod com bines th e ideas in [KNC96, FJ99,M ac04] to o b ta in th e perform ance close to th a t of ML or ILP, b u t only requiring 0 ( N ) operations.
As suggested in [Cla92,BW 02,M ac04], th e U W LP estim ate is used to form a dow nshifted signal, yd(t), to remove th e frequency dependence of th e SNR threshold, i.e.,
S e c t io n 2 .2 . S in g le T o n e E s t im a t o r s 21 N = 12 N = 16 - - N = 20 - t - N = 24 •••• CRLB -1 5 2 .- 2 5 -35 -45 -55 -65 -1 0 -5 0 5 10 15 20 25 30 SNR (dB)
F ig u r e 2 .3 . T h e M SE of th e exam ined K W PA estim ato r as a fu n ctio n of th e SNR, u —
0.75?r.
In [KNC96], Kim et a l proposed using a sim ple K -ta p m oving average filter to sm ooth irregularities and rand om variations prior to th e frequency estim atio n as a way to reduce th e SNR threshold. Such an averaging can be show n to lower th e SN R threshold up to 10 log10 K dB. However, as such an averaging will severely re stric t th e allowed frequency range down to ( —T r /K ,ir /K \, th e m ethod in [KNC96] is lim ited to signals w ith frequencies near zero. T his is because th e finite im pulse response (F IR ) averaging filter is essentially a low pass filter. Herein, it is noted th a t th e frequency co n ten t of th e dow nshifted signal,
S e c t io n 2 .2 . S in g le T o n e E s t im a t o r s 2 2 -15 tj -2 5 -35 -45 -55 (0 = 0 co = 0.25ji — (D = 0.57i —t— CO = 0.75k CRLB -6 5 >- -10 SNR (dB)
F ig u r e 2 .4 . T h e M SE of th e exam ined U W PA estim a to r as a fun ctio n of th e SNR, N = 24.
signal as
1 K ~ l
» M - 5f 5 X * + *). (2.2.25)
fc=0
Sim ilar to (2.2.13), th e adjacent phase difference of (2.2.25) can be form ed as
A </)f (t) = arg [y}(t)yf (t + 1)] = u f + u c(t), (2.2.26)
where uc(t) is given by (2.A .9) for a general K (see A p pend ix 2.A for fu rth er details). It is w orth noting th a t th e noise process u c(t) will now be coloured due to th e average filtering.
As shown in [LM89], th e SNR threshold behavior of th e phase-based frequency esti m ators is affected by cum ulative ±27r phase errors resulting from th e effect of th e additive noise. T his effect can be countered by in tro du cin g an ou tlier d etectio n scheme. Recently, an
S e c tio n 2 .2 . S in g le T o n e E s tim a to r s 23
/
I Begin j
X
Step 1: Coarse estimate using 'UWLP' 1
N - 2
\co
= arg'Step 2: Frequency downshift
^ - r E / t O X Z + l )
1 r=0
r ... [3;r f ( 0 = X 0 e
Step 3: Improve SNR using FIR ^
-ICQJ
£=0 Step 4: Form the phase difference
| A <pf (t) = arg [ y f (t ) y f (/ + !)]
Step 5: Outlier removal
I
|A M 0 =