The instantaneous frequency of a sinewave squelched bandlimited signal

LTRASONIC IMAGING 8, 285-295

(1986)

THE INSTANTANEOUS FREQUENCY OF A SINEWAVE SQUELCHED BANDLIMITED Jaleh

Komaili',

P. V. Sankars,

1 Rockwell

Leonard

International, Newport Beach,

2 Department University

A. Ferrariz,

SIGNAL

and S. Leeman

4311 Jamboree CA 92658

Road

of Electrical Engineering of California, Irvine Irvine, CA 92717

3 Department of Medical Engineering and Physics King's College School of Medicine and Dentistry Denmark Hill, London SE5 BRX, U.K.

We study the instantaneous frequency of a signal which is formed by the summation of a sinewave and a bandlimited signal. We refer to the composite signal as a sinewave squelched or simply a squelched signal. We study the behavior of the squelched signal at its real zeros and extrema. In particular when the sinewave frequency is equal to the largest frequency in the signal's spectrum and its amplitude is greater than the signal's maximum amplitude, the analytic signal corresponding to this signal (hereafter resultant abbreviated as RZ signal) has only real zeros which contain all of the bandlimited signal's information. FM systems often use zero crossing techniques in the demodulation process. For a process with a normal amplitude distribution, we derive a relationship between the zero crossing density of an RZ converted signal and that of the original signal. of these analyses is in ultrasonic FM imaging. We One application give an explanation for the behavior of the FM imaging system as a function of amplitude and frequency of the added cosine. 0 1986 Academic Press.

Inc.

Key

words:

I.

FM demodulation; ultrasound; zero

instantaneous crossing density.

frequency;

squelch;

INTRODUCTION A bandlimited

signal,

x(t),

may be written

in amplitude/

phase

form

as x(t) d(t)

where

= a(t)cos(d(t)) = w,t+a(t).

(1.1)

(1.2)

we defines the center frequency (rad) and o(t) is the phase deviation. Occasionally we may substitute f, (Hz) for ws. The Hilbert transform of a signal is denoted by the symbol (^) and its derivative by (I). We can show that h

x(t)

= a(t)sind(t).

(1.3) 0161-7346/86 285

All

Copyright 0 1986 rights of reproduction

$3.00

by Academic Press, Inc. in any form reserved.

KOMAILI

The envelope

and the

instantaneous

a(t)

frequency

that when a(t) however, undefined state, i.e., lim d'(t) a(t)->0

a(t)->0

are

given

by (1.4) (1.5)

approaches

zero,

4' (t)

approaches

= $

a

(1.6)

h

lim

of the signal

= [x2(t)+i2(t)]l'2 h' = (x x - i&a*(t).

d'(t) We note, recurrent

ET AL.

h

d/dt

( x x - xx')

d/dt

(a'(t)

3 0

(1.7)

U

)

To avoid this situtation, we can add a cosine wave of frequency w and We borrow the term squelch frzm the amplitude A to the signal. communications literature to describe the noise suppression associated the addition of the cosine to the signal prior to demodulation. with The resulting signal is given by y(t)

with

Hilbert

= x(t) + s(t) = a(t)cos$(t)

(1.8) + Acosost

,

(1.9)

transform h

y(t)

(1.10)

= a(t)sin4(t)+Asinwst

Computing the envelope and instantaneous (1.4) and (1.5), we obtain e(t)

frequency

of

= [y*(t)+j*(t)]l'*

y(t)

via

Eqs.

(1.11)

= [a2(t)+A2+2aAcos(4(t)-ost)]

l/2 (1.12)

a2~f+A2ws+a'Asin(&wst)+aA(ws+$')cos(~-~st) s'(t)

=

a 2 + A2+ 2aAcos(4-wst) (1.13)

Equation (1.13) indicates that the output of the ideal FM demodulator is no longer undefined when the envelope of the signal approaches zero. We also know that the envelope of a signal contains information about its One method of converting the complex zeros of a complex zeros [1,2]. signal to real zeros is by the addition of a cosine to the signal as in Eq. (1.9). The amplitude and frequency of the added cosine must satisfy the following two conditions: 1)

2)

A 2 sup(a(t)l ws t max(d(t)).

286

INSTANTANEOUS FREQUENCY OF SQUELCHED SIGNAL

Therefore, obtain a signal, zeros converted analytic form number of zeros

depending on the values of A and o in Eq. (1.9), we y(t), which is a combination of ?he envelope ( complex to real zeros) and phase ( zero crossings) of the of the input signal, x(t). If w - max(d(t)), then the of the analytic signal of y(t) isSequal to that of s(t).

These concepts have been used in ultrasonic imaging to obtain images which contain a combination of the envelope and phase information. As w and A are varied, however, we have noticed different effects on the out&t of the FM demodulator. The ultrasonic imaging system described in [3] utilizes such a demodulation process. In this case, x(t) corresponds to the backscattered ultrasound signal. Hence, the imaging system produces an output which is a combination of the rf signal's instantaneous frequency and envelope. Recently this system was used for diagnostic tissue characterization [3,4]. In the presence of hepatic fibrosis, Aufrichtig et al. [3] report that conventional AM images of liver have less diagnostic value than the corresponding FM demodulated images. In a similar study Seggie et al. report on the resemblance of the images obtained by a real zero conversion WC) imaging system ( another implementation of the FM demodulation system) to thresholded AM images [5]. This observation prompted us to conduct a theoretical investigation of the behavior of the output of an FM demodulator when a cosine of known amplitude and frequency, squelch, is added to the signal. Some FM systems use zero crossing techniques in the demodulation process [2,6]. In a statistical sense, the zero crossing density of a normal signal with Gaussian spectrum is equivalent to its RMS frequency In section III we give the relationship between the zero [7,8,91. crossing density, ZCD, of x(t) and the ZCD of y(t). We also show that the ZCD of y(t) is a function of the ZCD of x(t) and the variance of The variance of x(t) is a function of the signal envelope. x(t). Hence, the FM demodulator output, O'(t), contains amplitude and phase The frequency demodulated images, in some areas, information of x(t). show a strong resemblance to conventional envelope derived images.

II.

EFFECT OF SQUELCH SIGNAL AMPLITUDE AND FREQUENCY ON THE OUTPUT OF AN FM DEMODULATOR

In this section we consider the behavior demodulator whose input is the squelched defined points, e.g., the zeros and extrema The results are summarized in table I. e'(t),

In the for

Case a In this

of the output of an FM signal, y(t), at some wellof x(t), s(t) I and y(t).

remainder of this section we examine several cases of wS and A: x(t)

case,

<< s(t)

for

one or both i) ii)

all

t (negligible

of the following

a(t) small cos d(t) z 0

->

the

demodulator

signal),

s(t)zO.

two conditions

qS(t) = (2k+l)?r/2

exists: ;

k=O,l,.

In other words, the envelope of x(t) is very small or the phase spans n/2. Then, y(t) is approximately equal to s(t), i.e., y(t)

-

x(t)

+ s(t)

;

s(t).

287

output,

of

x(t)

KOMAILI

Table

y(t)

-

0

y(t)

22

1.

at

Output

ET AL.

of

FM demodulator,

an exfrem”m

212

s(t)

22

H

= 0

s(t)

an extremum

2, * a $ + A us+

12

2

at

a’A

k’ a* x(t)

=

0

+

hs

A ~,-a

$

a+A

+ (-1) 2

(a

a

+

A

us-

[A

2

2 )

us-

2a

a 0

a+‘ALd

]

2 a+A

2

x?

1 4

if

y(r)

at

an extremum 2

I

Aw+aA

A---2 a+A

2

of Aw x(c)

6 a(t)

$‘-O

l

A2U o’-

2 a+A

0

+A

2

+=o

2

+

25

a+A

a+A

The output

of an FM demodulator s'(t)

=

under

these

conditions

is

simply

w S

Case b If the follows

A << a(t)

(negligible

amplitude of s(t) the signal, i.e., y(t)

The output

= x(t)

of FM demodulator s'(t)

is

squelch), very

+ s(t) is

small,

s(t)#O. then

the

composite

signal

= x(t). given

by

= 4'(t).

When s(t) is added to x(t) in order to define the instantaneous frequency of the signal as a(t) -> 0, A is small, and in conventional envelope detected ultrasound imaging systems a constant gray level is assigned to w . This is equivalent to assigning a lower bound to the FM demodulator gutput. Case c

fs << f, + u; where

20 is

the bandwidth

of x(t).

It is known that the number of zeros of the sum of two analytic signals on and inside a closed curve is equal to the number of zeros of the function with larger modulus [1,2,10-121. Therefore, if A is larger than the maximum of a(t), then the frequency of s(t) determines the number of zeros of y(t). It is also known that bandlimited and bounded signals can be characterized by their zeros [l] and that the signal's Fourier series terms can be calculated from the signal's zero locations

288

also

INSTANTANEOUS FREQUENCY OF SQUELCHED SIGNAL

apart from a multiplicative constant [14]. In other wards, and A > a(t) for smaller than the maximum signal's frequency, then y(t) has as many zeros as s(t) and fewer zeros than x(t). Case d

if ws is every t,

fs = 0 + f,

In this

case,

y(t)

has as many zeros

as x(t)

if

A > max(a(t)). Notice that bandwidth. real zeros. Case e

adding s(t) to the signal, does not x(t), We refer to this signal as an RZ signal because

change the it has only

fs >> 0 + f,

In this case, if A is larger than max(a(t)), then y(t) is an RZ signal, however, the number or zeros of y(t) is considerably larger than that of the original signal. The output of an FM demodulator with input y(t) approaches wS' i.e., s'(t)

-> us.

In the last two cases a fixed gray level is assigned to ws and sets a threshold in the case of FM imaging. In cases (a) and (b), we showed that when x(t) is small ( a(t) --> 0 or the phase spans s/2), the output of the FM demodulator is approximately w . In this case, the output is mapped to a constant gray level assigned ?o w . When x(t) is significant with respect to s(t), the output is approxima?.ely d'(t). We now derive expressions for the behavior of x(t), s(t), and y(t) at their zeros and extrema. We begin with the zeros and extrema of the original signal, the behavior of the output of an x(t) I and examine ideal FM demodulator when a zero or extremum of s(t) or y(t) occurs simultaneously with zeros of x(t). Case 1

At a real x(t)

zero

= a(t)

of x(t) cosd(t)

= 0. Assuming

the

envelope,

a(t)

#

0,

we

obtain cosj(t) Case

= 0 =>

At a real

la

Combining

zero

= 0 => a(t)

Eqs.

(2.1)

and a real

cosd(t)

and (2.2)

zero

k = O,l,....

of y(t).

(2.1) (2.2)

we find

(2.3) k

in Eq.

for

= -A coswst.

= 0

sinwst==(-1) Substitution

= (2k+l)*/2

of x(t)

y(t)

coswst

4(t)

(1.13)

for

k=O,l,...

(2.4)

yields

a2+'+A2ws+a*(ws+d')

,g'=

a2+ A2+2aA Aws+ad =

’

(2.5)

a+A

289

KOMAILI

To

obtain

Eq.

(2.5),

we

use

the

ET AL.

following

trigonometric

sin(d-wst)=sind

cosost-cosd

sinwst

(2.6)

cos(+wst)=cosd

coswst+sind

sinwst.

(2.7)

Equation (2.5) implies of the FM demodulator, instantaneous frequency

that at a real zero B'(t) is a function of x(t).

Case

of

In

lb

At

this

identities

a real

case,

zero

x(t)

substitution

and

in

an

Eq.

A2wi-a2$

of

x(t) both

extremum

(1.13)

and v(t). the the envelope

of

output and the

y(t).

yields 2

2 , 0 =

of

] 1/*

+a'(-l)k[A2wz-a2d' (a2+A2)ws-2a2+6'

(2.8) In

deriving

Eq.

y'(t)=0

(2.8), =>

sinwst

we have

a4'(-l)k=

the

relationships

-Awssinwst k

-ad'(-1)

=

used

(2.9)

AWS

2 cos.ast

-

[l

l/2

*;y

-

1

(2.10)

S

Case

lc

At

a real

zero

of

x(t)

and

a real

zero

of

s(t).

We have s(t)=A

We obtain

8'

coswst=O

by

=>

wst=k?r

substituting

;

the

k=O,l,...

above

(2.11)

equations

in

Eq.

(1.13),

i.e.,

a*d'+A*ws+aA(ws+d') o'= a2+A2+2aA ad'+Aws a+A (2.12) Case In

Id this

At case

a real sin

zero wst

of

x(t)

= 0 and

and substitution

2' 2 a 4 +A ws+a'A tl'= a2+A2

an

extremum yields

of

s(t).

INSTANTANEOUS FREQUENCY OF SQUELCHED SIGNAL

If

t is

also

an extremum

of y(t),

*'=

Case 2

If

At an extremum

then

d'=O

and

A2ws+a'A 2 a+A

of x(t)

(2.13)

2

and a(t),

x'(t) = 0 => a'(t) cosfj(t)-a(t) 4'(t) sind(t) this is also an extremum of the envelope, then a'(t) = 0.

Assuming

a(t)

f 0, and substituting a(t)d'(t) sin4 b'(t)

1)

2)

sin#(t) = 0 => d(t) = 0.

Eq.

(2.15)

= 0.

(2.14) (2.15)

into

(2.14),

we obtain

= 0 => = kr

for

k=O,l,...

If we assume b'(t) is an analytic function has to assume an extremum when 4'(t) function bounded by:

within one period, = 0. But d(t) is

then d(t) a periodic

-?r5l#5x. Hence,

d(t)

= k?r

Therefore, We continue

for

Case 2a At extrema

of x(t),

(2.2) k cos+=(-1)

holds

&(t)

a(t)

and a real

zero

= kn.

of y(t)

and

; k=O,l,...

(2.2)

implies

coswst-

-a/A

(-l)k.

in Eq.

(1.13)

Substitution

then

= a(t)cosd(t)+Acoswst.

case Eq.

Equation

(2.16)

when x(t) and a(t) are both at their extrema. the analysis for the squelched signal, y(t)

In this

k-0,1,....

(2.17)

(2.18) yields

a2d'+A2ws+aA(d'+ws)[-(-l)2ka/A] 8' -

=W s .

a2+A2+2aA[-(-l)2ka/A] Therefore. and a(t).

when a real zero the instantaneous

Case 2b At extrema

of x(t),

of v(t) coincides freauencv of y(t)

a(t)

and a local

(2.19)

with an extremum is ws .

extremum

of

x(t)

of y(t).

Then, y'(t)

= a'(t)

cosd

- a+'

sin4

291

- Aws sinost

= 0.

(2.20)

KOMAILI

Substitution

ET AL.

yields:

y'(t)

= -Aw s sinwst

o’=

= 0 => sinwst

= 0 => WSt = kn

(2.21)

2/+2ws+aA(ws+~‘) a2+A2+2aA aq5 +Aus

=

a+A (2.22)

Then,

,#I'=0

=>

Eauation (2.21) is eauivalent to s'(t)=O: also attains its extremum at t. Case 2c At extrema

of x(t),

a(t)

it

and a zero

imulies

of the

that

the

sinewave

s(t).

We have s(t)=A

coswst=O

=> coswst=O

=> wst=(2k+l)r/2

for

k=O,l,... (2.23)

After

some manipulation,

we obtain

a2qS'+A20 S

Ba2+A2

(2.24) Then,

=>

4' = 0

Case 2d At extrema In this Under

Then

#'=

a(t)

s'(t)

conditions *'=

22 a+A

of x(t),

case we have these

A2 ws

o'=

= -A ws sinwst

equation ad'+

and an extremum

(1.13)

= 0

of s(t) => wst = kn.

becomes

Aw S

(2.25)

a+A 0 => o'=

Aws 7x--.

Table I shows the relationships which exist between the output of and the envelope and frequency the FM demodulator, when driven by y(t), x(t). We find that the output of the components of the original signal, FM demodulator is a function of both the envelope and frequency of x(t) unless A >> max{a(t)) in which case the output is ws. Zero

[2,61.

of x(t)

crossing density is In the next section and the ZCD of y(t).

often used as a method of we examine the relationship

292

FM demodulation between the ZCD

INSTANTANEOUS FREQUENCY OF SQUELCHED SIGNAL

III.

RELATIONSHIP

BETWEEN ZCD OF x(t)

x(t),

The expression for is given by [13]:

the zero

AND ZCD OF y(t)

crossing

density

of a

normal

process,

(3.1) If

the

spectrum

of the (x:)2=

normal section, 3.1

signal

4(fz+

is

Gaussian

shaped,

we can show

[7,8,9]:

02) = 4 fZms.

Then, in a statistical sense, the ZCD of a normal process with a spectrum is equivalent to its RMS frequency. In the next we derive similar expressions for the squelched signal, y(t).

ZERO CROSSING DENSITY OF SQUELCHED SIGNAL

The zero crossing density of a signal is given by process also has a Gaussian shaped spectrum, relationship between ZCD and RMS frequency. If x(t)

has a normal

distribution,

y(t) is also normal, autocorrelation

= x(t)

because function

= RX(~)+(A

+ Acoswst s(t), by

is

AL/2

> Rx(O)

of Ry(r)

deterministic.

The

COSW~T.

(3.3)

function of y(t) is equal Hence, the autocorrelation function of x(t) plus a periodic term due to s(t). We notice

If the provides a

then

the added signal, of y(t) is given

2 /2)

Eq. (3.1). Eq.(3.2)

that

Ry(7)

has the

then

Ry(r)

is

same form

as y(t). Taking

an RZ function.

to

autocorrelation

Therefore,

if

the derivative

we obtain (3.4)

Taking

the

derivative

of R'y(7) 1L

we obtain

w

R;(T)=R;(~).A~+ Substituting for the zero

($j20

R" (0) and R (0) crgssing dengity

=

-1 2 II

(3.5)

cosws7

R;(O)-(A*,'2) Rx(0)+A2/2

in Eq. (3.1), of y(t) w;

we obtain

1.

293

an expression

(3.6)

KOMAILI

We can

relate

the

($2

crossing

- -

Eq.

(3.7)

=>

into

Eq.

of

R;(O)

=

(3.6)

yields

A2

(.$12+ (9j2II

density

1 R;(O) 712 Rx(O)

=

Substituting

zero

ET AL.

y(t)

to

x(t)

using

Eq.

(3.1).

-n2(X;)*Rx(0).

(3.7)

w2

2n2Rx(0)

'

(3.8)

= A2 *Rx(O)

'+ Finally, (X;)2+ (x$2

2fzA2/Rx(0)

=

of

For a Gaussian by f rms of x(t)

afZms

(Gj2= 0

IV

(3.9)

2 1+

*

2RxG’)

shaped substituting

spectrum Eq.

we (3

can find the ZCD of y(t) 2) in Eq. (3.9) we obtain

in

terms

+4A2fi/Rx(0) (3.10) A2/Rx(0)

+ 2

CONCLUSIONS

The behavior signal input the instantaneous has been used To provide we simplified defined points. simplifications. function of original signal, conclusion, crossing density x(t).

of has in

the output of an FM demodulator with a squelched been studied analytically. The output of this system, frequency of the squelched signal given by Eq. (1.13), ultrasonic imaging and tissue characterization.

a better understanding of the operation of this FM system, the general output equation, Eq. (1.13), at some wellTable I provides the collective results of these We note that the output of the FM demodulator is a both the amplitude and the instantaneous frequency of the x(t). In a statistical sense, we arrived at the same Eq. (3.10), by obtaining the relationship between the zero of the squelched signal, y(t), and the original signal,

REFERENCES [l]

[2]

Voelcker, II, Proc.

H.B.,

Towards

IEEE 54,

Requicha, A.A.G., 328 (1980).

353, Zeros

a unified

735-755 of

theory (1966).

entire

functions,

294

of

modulation,

Proc.

parts

IEEE

68,

I

and

308-

INSTANTANEOUS FREQUENCY OF SQUELCHED SIGNAL

[3]

Ferrari, L.A., Fridenberg, Aufrichtig, D., Lottenberg, S., R.M., G., Cole-Beuglet, C., and Leeman, S., Frequency demodulated Kanel, ultrasound imaging -an evaluation in the liver, Radiology 160 5964 (1983)

[4]

Hoefs, J., Aufrichtig, D., Lottenberg, S., Kanel, G., Dormer, B., Ferrari, L.A., Leeman, S., and Friedenberg, R., A noninvasive evaluation of hepatic fibrosis using frequency demodulation of ultrasound signals, Digestive Diseases and Sciences, to appear (1986).

[5]

Seggie, image

D.A., Doherty, processing for Proceedings, vol. 575, F.G., Reading,

G.M., Leeman, S., and Deighton, H.V., instantaneous frequency mapping, 114-120, (1985).

[6]

Stremler, Wesley,

Introduction Massachusetts,

[7]

Rice, S.O., Mathematical J. 24, 46-156 (1945).

[8]

Flax, S.W., Pelt, N.J., Glover, Spectral characterization M., ultrasound, Ultrasonic Imaging

[9]

Ferrari, L.A., of attenuation,

Jones,

analysis

J.P.,

Churchill, R.V., Brown, J.W., and applications, 3rd ed., New York, NY 1960).

[ll]

Bar-David, functions,

An implicit

Control

Haavik, S.J., (University

[13]

Kay, S.M., and Sudhaker, R., analyzer, IEEE Trans Acoustics, 34, 96-104, (1986). Papoulis, Processes, 1984.)

noise,

V.M.,

66-72

Bell

(Addison

Syst.

Tech.

theorem 24,

In viva

for

36-44,

Complex variables Book Company, Inc., bounded

A

zero

Msc.

crossing-based and

Signal

A., Probability, Random Variables, 2nd ed., (McGraw-Hill Book Company,

295

bandlimited

(1974).

The conversion of the zeros of noise, of Rochester, Rochester, NY, 1966). Speech,

measurement

(1986).

and Verhey, R.F., 300, (McGraw-Hill

[12]

[14]

24,

sampling and

systems,

SPIE

G.H., Gutmann, F.D., and McLachlan, and attenuation measurements in 5 95-116 (1983). vol.

[lo]

Information

of random

and Gonzalez,

Ultrasonics,

I.,

to communication 1977).

Digital

spectrum

Processing

and Inc.,

Thesis,

ASSP

Stochastic New York, NY,