Performance analysis of AM-FM estimators - ePrints@IISc

PERFORMANCE ANALYSIS OF AM-FM ESTIMATORS Jitendru

Kumur

Gupta.

S.

Chandra Sekhar and 7:

!i

Sreenivns'

Dept.

of

Electrical Commlinication Engg., Indian Institute of Science, Bangalore - 560 012, INDIA.

'e-mail:tvsree@ece.iisc.emet.in

ABSTRACT

We address the problem of decomposing a handpass signal into ainplilude arid frequency modulated components (AM and FM respectivelyj. Several estimators have been pro- posed in literature. each of which works under specific as- sumptions. In this paper, we p e r f o m a comparative study of some popular techniques hy studying their performance with changes in the modulation parameters andthe tradeoffs involved. We also study their performance in the presence of handpass white Gaussian noise.

1. INTRODUCTION

Most practical signals, such as speech, audio. biomedial sigonls are nonstationary in nature and have time-varying spectral characteristics. The time-varying spectral charac- teristic is usually, a direct consequencehf the signal gen- eration process. The spectral variability with time can be decnmpnsed into ampIitudeP,variability and frequency vari- ability. Most sources convey information throu& modula- tions of amplitude (amplitude modulation or AM) and fre- quency (frequency modulation o r FM) of a steady camer which serves a s a means of transporting the information contained in the modulations. The basic and simplest signal processing model of such a source is therefoi-e an AM-FM combination.

Given a bandpass signal, it is passihle to decompose it into an AM-FM combination. C)n the other hand, given an AM-FM combination, we can synlhesize a handpass signal.

Several such AM-FM Combinations can be used to repre- sent complcx signals such as speech and audio. Decompos- ing a bandpass signal into its A M and FM parts has been addressed by many researchers and a number of techniques have been puhlished in the literature. Each technique works under specific assumptions. Also, the performance of these techniques in the presence of handpass noise has not been addressed.

The present work is motivated by the following ques- tions:

I . How do the techniques perform for large camer fre- quencies and large frequency deviations'?

2. How does noise affect thc performmce of these mcth- ods?

2. SIGNAL M O D E L

The signal model is given as:

z ( t ) = A(t)cos(Znf,t

+

2 r k f ~ L ( T ) ~ T ) ( I ) where A ( t ) is the amplitude modulation. The phase mod- ulation component, represented by $ ( t ) is given by $ ( t ) = 2a fct+2akf

f,

rn(r)dT. The 'instwtaneous frequency'(lF), fi(t) is defined as

f ; ( t )

=

&%

^{= j c}

+

k f r n ( t ) . which shows modulation of frequency by the signal m ( t ) . In terms of the angular frequency, w i ( t ) = 27ifi(t). The methmis that will bc discussed use the discrete-time version of the above signal(fbr implementation purposes), which, with nor- malized sampling period of unity is given as

m

A[n] is the discrete-time AM and th6 IF is given as ].rif =

& ( $ [ n ) - $ ( n - l ) ) = f , + k ~ r n [ n ] . The angular frequency is given by w,[n] = Zaf,[n].

3. METHODS FOR AM-FM DECOMPOSITION Of the several methods that exist, we,have chosen the fol- lowing popular ones for a comparative study:

I . Auditory Motivated AM-FM Decomposition [ I ] , 2. Teager Energy based algorithms 121,

3. Positive time-Frequency distrihution approach 13.61.

4. Hilbert transform approach 141.

I n llic suhscclioiis that follow, u'e briefly review tliese metli- od5.

3.1. Anditory Motivated AM.lPM decompositioa l h i s inetliod purely rclies on only the ampliludc envclopc, i.c., llic magnitude of the oulput of a linear lilter (chal.de.

1eriPcd hy inipulsc rcspoiisc. h[n]), b y exploiting tlic prop- city of filter transcluction, i.e., llic linear l i l l c r oulpul ciin hc ohtaincd approximately by sweeping

w[n]

through the lil- ler's tl-ansfcr function [I]. The amplilude envelope of the instantaneous output i s given by tlic approximalion y[n] cs O.SA[n].N(u[n]). Only the magnitudc of tlic iiltcr rcspoiisc

H ( w )

i s u s e d

( H ( w )

i s a zero-phase film). To solve for llie AM and PM w c need atleast two filter outputs. A pair o f handpass lilters arc used. 71Vo kinds o f filters arc commonly uscd

-

Picccwisc-linear filters and Gaossian fillers.

For two Piecewise-linear tilters,

H l ( w )

=

a l w +

^{01 and}

I12(w) = azw

+

^bz. If the input i s A[n]cos(w[n]). llic out- puts can he approximated as yl[n] cs

alA[n]w[n] + h A [ n ]

mid y2[71] g

a ~ A [ n ] w [ n ]

i- bnA[n]. These equations can be solved ror

A[$

antlw[n]

Thc assumptions ;u'c lliiit l l i e AM i s slnwly varyiiig

arc shown i n Fig. I. We prefcr lliesc ¹⁰ tlinsc prnposcd in [ 1

I

hccausc these can Iiandlc liirgcr swings i n frcrlucncy. I'm smaller swings in frcqucncy. hutlr yield idciilical IosuIis.

3.2. Tcager Energy Based Algnrittims

Oi'rccent inlcrcst liab been the use nf'l'e;igcr cnwgy for AM- FM dccompnsitioii 121. The cncrgy opcl-alol- (dcnotcd h y

Q[:c[n]) on a signal :K[n] i s iicliircd iis Q { : c [ n ] ) ) ^x z2[n] - :c[?L-

I]z[n. +

^{I ] . Thcrc arc} two popular algorithms iiamcly, DESA-I and DESA-2 (DESA st;mdr for Discrctc Encrgy Scpel-ation AIgorithm). The cxprcssions fnr AM and FM using llic DESA-I algorilhm iirc given h y

W ; [ l l ] M 0.6COS-*(l - t 2 [ 7 1 ] ) (6)

whcrc

44

^{= z[n}

+

^{l ] - z[n - I ] zind}

cn[n]

⁼

3v

^LY'{:r

4]

DESA-2 algol-ilhm can eslimalc IF sucli tlliit [ I

< w;[n] <

f.

linplcrnentatioii delails iirc discussed i n 121. 1 3 d i lhcsc algorithms i-equirc tliat llic AM Ihc very slowly varying and that frequency dcvialion of tlic signel iicross h e carrier frc- qucncy hc "cry s m x l l .

Fig. 1. Picccwisc-lincar iuditory Iillcrs used for AM-PM demodulation

(bairdwidtli iniicli less tlian the carricr Ircqucncy. f c ) and Lliat the FM deviation i s limited lo the region where holh the bandpass filters havc posilivc nr negative slope (a slrict rcyuiremcnt fnr piccewisc lincer filters). 1f there i s apriori knowledgc of (lie center frequency, frequency devialion and handwidtli o f AM, then appropriate fillers can h e designed 1 0 pcrlbrm AM-PM decomposition Othcrwisc, we need 10

use a hank

d

overlapping hendpass fillers (which rcprcsenl

3.3. Positive Time-frequency Distrilmtion Approach

111 h i s approach, ii pusitivc limc-frcqucncy distriholiniiiPI'FD), . r ( t , w ) i s used l o estimate llic I'M i'ollowcd hy AM csliina- lion using colrcrenl time-varying demi~dulalioii 131. Cdreii- Posch positive tinrc-frequcncy distrihulion 161 i s dcsigiictl such h a t the tiiiic and frequency inarginals, iiistiinlancnus Vrequency properly arc salislicd. Fnr an AM-FM conihina- Lion signal, computing Llic avcl-age frcqucncy nf tlic FTPD [ S ] at each timc-instant givcs the FM i.c.,

rum

w ~ ( t + , ) c ~

I"=

/ ' ( t , U ) l L

liic audilory lilterlxink lront-end) lo liltcr thc input signal.

The oiilpiils of lillers with the dominanl cncrgy (implying coincidencc of thc liltcr spcc1l.a w i l h input signal spectrum)

w;(L) = ^L-- ( 7 )

can he used lo pcrform AM-FM dccomposilion

I n this work. we assumed apriori informalion about ,fc

and iwquoncy swing. 71Vo filters used i n (lie simulations

Using lhis, thc itisl~inlaneous phase i s given as

$ ( t )

= J:" ^~ w i ( ~ ) h i s computed. l'lic in-phase and quadlniure cnmponcnls 01

TENCON 2003

/956 the AM are obtained as:

A d t ) =

~ ' m z ( ~ ) ~ o ~ ( g ( t ) ) ~ , ~ ( t , 7 ) d 7

m (8)

+ m

A , ( t ) =

1,

z ( 7 ) s i n ( g ( t ) ) h i p ( t , 7 ) d 7 (9) where h l , ( t , 7 ) is the time-varying impulse response of a low-pass filter with varying cutoff frequencywi(t) and pass- band gain equal to two. The AM is obtained by combining these results :IS A ( t ) =

,/-

t ) + . A Z ( t ) . The F'TFD has to be obtained iteratively staning with the spectrogram. Spec- trogramcomputation depends upon the choice of a window.

As a result, the window length plays a crucial role in lim- iting the performance of the algorithm. The method yields satisfactory results as long as the assumption that the signal is nearly stationary within the window holds.

3.4. Hilbert Transform Approach

This is perhaps, the oldest of techniques for AM-FM sep- aration [4]. Given a signal,

z ( t ) .

we compute its Hilbert transform %(t). The analytic signal is defined as c ( t ) = z ( t )

+ @(t)

⁼ a(t)eJ@(t), where

a ( t )

is the AM and g ( t ) is the phase modulation component. The FM can he ob- tained as f i ( t ) =

&%.

The conceptual paradox associ- ated with this method is that to obtain time-varying charac- teristics of the signal, which are local in nature, we use the Hilbert transform which is computed on a global basis.

For implementation purposes, we use the discrete-time Hilbert transform.

4. PERFORMANCE ANALYSIS

The focus qf the present study is to understand, by simu- lations, the limitations of these techniques for AM-FM de- composition for different carrier frequencies, frequency de- viations and presence'of noise.

4.1. Effect of Carrier Frequency and Frequency Dena- tion

For the noise-free case, extensive simulations were carried out hut only results for the sinusoidal AM, and sinusoidal FM Combination are discussed due to lack of space.

1024 samples of an AM-FM combination were gener- ated. 1.5 cycles of a sinusoid was used as the AM with 25% modulation depth(pa). The FM consisted of 3 cycles of a sinusoid with a frequency deviation (Af). In rela- tion to the carrier frequency fc, this is specified by the ra- tio pf =

y .

The results of two experiments are reported here - performance as a function'of frequency deviation for ifixed carrier frequency (Experiment 1) and performance as

Fig. 2. Performance analysis as a function of p f for f c = O.1Ht (top row) and fc = 0.2Hz (bottom row). [a],[c]AM- MAE(dB) and [bl,[dlFM-MAE(dB).

a function of carrier frequency for a fixed frequency devia- tion (Experiment 2). Expt.1 was performed with f,=O.IHz, p f = 0.2,0.4,0.6. and fe=0.2Hz, p f = 0.2,0.4,0.6. The error measure used was the Mean Absolute Error (MAE) and defined as

where 0 is either the AM or FM and idenotes its estimate, giving respectively the MAE in AM and FM denoted by AM-MAE and FM-MAE. These are plotted in decihel (dB) scale in Figs. 2 and 3.

From Fig. 2, we conclude that

I . All methods exhibit an increase in error with modu- lation depth for a fixed carrier frequency.

2. The maximum increase in error with modulation depth for a fixed fc is exhibited by DESA algorithms.

3. Consistently, DESA-I outperforms DESA-2, 4. Amongst auditory filtering based AM-FM separation.

Piecewise-linear filters consistently outperform Gaus- sian filters in AM estimation. The reverse is true for FM estimation.

5 . For the PTFD approach. with changes in carrier fre- quency and modulation depths, the FM estimation does not suffer and showsody marginal degradation. How- ever, AM estimation degrades.

6 . For low carrier frequencies and large modulation depths.

the Hilben transform approach is the one that has

95% of the signal energy is concentrated. The bandpass fil- tered noise w a suitably scaled to achieve a desired SNR.

We consider two different AM-FM combinations I '

:-

z .m I . Sinusoidal AM, A[n] = 1

+

0 . 2 5 c o s ( 2 ? i 9 ) , sinu- soidal FM, f i [ n ] = 0.18+0.045sin(Z?i~),O

5

n _<

2. Exponential AM, A[n] = e'"(O.')*, linear FM, f;[nJ =

,a1

0 2 N - 1 .

a

'~

~ $ 0

0.1

+ Y , o 5 n 5

N

-

1.

The parameters were chosen such that in the absence of noise, all techniques perform nearly identically. For each

z s

combination, SNR was varied from 4dB to 32dB in steps of

~z-2

4dB. For each SNR, 40 Monte-Carlo realizations were used to obtain the average MAE measure. Finite data can cause

:;a

f = % ~ , ( ( 1

Fig. 3. Performance Analysis ils a function of ^fc for pf =

%%(top row) and pf, = 50% (bottom row). [a].[clAM- MAE(dB).and [b],[dlFM-MAE(dB).

least error in both AM and FM. It is surprising to notc that the Hjlhen transform approach that computes lo- cal information on a global basis has the least error unlike specifically designed tools to extract local in- formation such as the other methods discussed here.

From Fig. 3, we conclude that

I . The Hilben transform approach offers the best per- formance in AM estimation for small-to-large carri- ers with small-to-large modulation depths. The FM periormance suffers for high carriers with high mod- ulation depths.

2. Piecewise:linear.filters outpertom Gaussian filters in FM estimation' even for high carriers with large or small modulation depths. The reverse i s tlue for AM estimation. The same conclusion was drawn from Fig. 2

. . .

3

3. The PTFD based AM estimation has high errors for smalltlarge modulation depths for large carriers. FM estimation however does not face similar problems.

4. DESA-I. consistently, outperforms DESA-2, 4.2. Effect ofNoise

We study the effect of bandpass white Gaussian noise. Since AM-FM signals are essentially bandpass in nature, we con- sider the effects of only in-band noise. Sample realizations from a whitc Gaussian noise process were generated and bandpass filtered to give handpass w,hite noise. The band- pass region was selected as the frequency zone in which

large errors at the ends of the window. Hence, they were ig- nored in computing the error measure. Otherwise, theerror measure will be severely biased. The results are tabulated in Tables. Land Z(values rounded to the nearest integer). One common trend exhibited by all ofthem is the decrease in er- ror with increase in SNR. This is natural and expected. The following conclusions are in order:

I . A t very low SNRs, the PTFD'approach offers the best performance for all combinations considered and hence is more robust.

2. DESA-I performs better than DESA-2 even in the presence of noise.

3. Almost all techniques show a liiicar decrease in error with increase in SNR. For every one-dB rise ih SNR, the error reduces by roughlyhne-dB but for Hilbert transiomi where the error initially follows this rule but saturates for very high SNRs. We call this, the 'one-dB rrrlc'.

4. At high SNRs, owing to their simplicity, the DESA algorithms may he preferred. At low SNRs. the PTFD approach is preferred though computationally more

complex. . .

It must also he noted that if some apriori informatiun is available about the smoothness of the AM andlor FM, then the estimates given by any of these methods can be further improved by filtering(linear or nonlinear depending on the nature of estimation errors). For example. for IF with dis- continuities(not reported here). Hilbert transform approaches yield estimates with spiky errors which can be reduced us-

ing nonlinear filters. _{. .}

5.' CONCLUSIONS

In this paper, we studied the performance of different AM- FM decomposition methods with and without noise. In the

TENCON 2003 /958

Tahle 1. Performance analysis for sinusoidal AM and sinusoidal FM, Mean Absolute Error(dB) in AM and FM as a func- tion of SNR(dB). (T1:Piecewise-linear auditory filters, T2Gaussian auditory filters, T3:DESA-I,T4DESA-2.T5:PTFD.

T6:Hilbert transform)

Table 2. Performance analysis for exponential AM andlinear FM, Mean Absolute Error(dB) in AM and FM as a function of SNR(dB). (T1:Piecewise-linear auditory filters, TZGaussian auditory filters, T3:DESA- l,T4DESA-2,T5:PTFD, T6Hilben transform)

absence of noise, the effect of parameters such as carrier frequency and frequency deviation was studied. The effect of bandpass white Gaussian noise on the performance of the AM-FM estimation techniques was also studied. The observations and conclusions were given at the end of each experiment.

6. REFERENCES

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