I

318 IEEE TRANSACTIONS ON BROADCASTING, VOL. 39, NUMBER 3, SEPTEMBER 1993

AUTOMATIC RECOGNITION OF INTERMODULATION BEAT PRODUCTS IN CABLE TELEVISION PICTVRES

Pingnan Shi and Rabab K. Ward Department of Electrical Engineering

University of British Columbia Vancouver, B.C., Canada V6T 124

Abstract

Recent& the feasibility of automatically detecting the presence of intermodulation beat products in cable TV pictures has been es- tablished. It has been found that the nvo-dimensional Fourier trans- form of a picture impaired with intermodulation beat products con- tains distinguishable pulses. Each product results in four pulses of special characteristics. In this paper we firrther analyze and study the inninsic properties of these pulses. Based on these properties we have designed a method which can detect these pulses, even if the picture is also impaired by snow noise, as long as the intermod- dation impairment is visible to the human eyes.

Introduction

Intermodulation beat products (referred to as intermodulation beats or beats hereafter) are caused by the nonlinearity of over- driven amplifiers in the cable distribution network [l]. If the fRquency of a beat is precisely equal to that of the victim channel’s video c ~ e r , the beat is not noticeable on the TV screen. But if the frequency is somewhere else in the video bandwidth, parallel diagonal bars would appear (see Figure 1 for an example). If it is near the color subcarrier of the victim channel, the color beat impairment would result. Since the presence of intermodulation beats indicates problems in the cable network as well as causes annoyance to the viewers, the automatic detection of the presence of these beats is of great relevance to cable operators.

The causes and effects of intermodulation beat products have been extensively studied in the past two decades [2, 3, 4, 5 , 6, 71. However, the automatic detection of this type of impairments has become a subject of study only recently [8, 91. Here, we continue the work in [9] and propose a detection scheme which has much higher detection rate and is more robust than the ad-hoc detection scheme suggested in that paper.

It has been shown that the two-dimensional Fourier transform of an intermodulation beat has distinctive features [9]. It is com- prised of four pulses grouped into two pairs which are symmetrical around the position of the DC term. The two pulses in either pair are separated by exactly 4 indices if an N x N point discrete Fourier transform (DFT) is used (see Figure 2 for an example). Based on these features, an ad-hoc detection method, which is based on thresholding the amplitude spectrum, is suggested [9]. This method declares the presence of a beat if the four pulses with the afm-mentioned relationship are found.

Figure 1 Picture “House”

Figure 2 The Amplitude Spectrum of “House”

However, further study shows that it is difficult to find the optimum threshold value since it changes with the pictures to be examined, as well as with the frequencies and amplitudes of the intermodulation beats. Furthermore, two of the pulses may have zero amplitude. In this paper we present a detection method

0018-9316/93$03.00 0 1993 IEEE

wlrich8voidr therediflicultier by no( *mplicrdethrerhdding. Instead, It utilizes the intrinsic properties of an intermodulation beat. Experiments show that this method is able to detect an intermodulation beat as long as the impairment it causes is visible to the human eyes.

Snow noise is another common impairment of concern. It introduces many pulses in the spectrum in a random fashion, and cane of these pulses are likely to be misclassified as pulses caused by intermodulation. We have tested our method with snow noise impaired pictutes and the experiments show that it does not give such misclassification. With the amplitude thresholding method [9], however, such misclassification is unavoidable.

In the following sections, we will describe the digitization process of a TV picture and derive the DFT of a podon of the digitized picture of an intermodulation beat. Then the relevant pper t ies of a pulse resulting from a beat are shown. Our detection acheme is then described and some experimental results are given, followed by conclusions. In the appendix the derivation of two parameters used in our detection scheme is detailed.

The Digital Representation of a N Picture

The effect of an intermodulation beat on a TV signal transmit- ted in the cable can be modeled as a sinusoid being superimposed on the signal. The resulting one-dimensional temporal signal is then mapped on the TV screen, forming a two-dimensional spatial signal (noise picture). The equations characterizing this mapping and the Fourier transform of the noise picture are provided in [9]. It is shown there that the two-dimensional Fourier transform of a beat has four pulses with distinct relationship.

In [9], the continuous two-dimensional Fourier transform of the sampled version of the noise picture is derived and studied. This is a common approach which serves well the purpose of revealing the spectrum propemes of the intermodulation beats. However, since in our detection scheme we deal with discrete Fourier transform of parts of digitized pictures, we shall first derive the discrete Fourier transform of an arbitrary portion of the digitized noise picture.

In the remaining of this section we shall derive the discrete vetsion of the noise picture. Then in the next section we will derive tbe DFT of an arbitrary portion of the digitized noise picture.

The mster scanning of a TV picture is illustrated in Figure 3. where X ud Y arc. the width and height of the screen respectively. Tbe -pinta cbe odd numbered lines b t , and after one field (265.5 lines in NTSC system) paints the even numbered lines.

X

-1 0 1 2 3 4

Y rm 5

0 1 2 3 4

YI c--c--, 5 mT I

L L M ,

Figuw 3 Interlaced Scanning

319 The intennoduktion beat in the time domain is described as

f(t) = cos (wtt + 4). (1)

After being scanned on the TV screen, it becomes a two- dimensional spacial function f(x,y). Note that in the following derivation 4 is set to zero since its value doesn't affect the result.

The time variable t and the spacial coordinates (z,y) has the following relationship. When (2, y) lies on the odd lines,

ut=- 2k - l ( X + H ) $2, (2) 2 where v is the scanning speed, Ay is the distance between two adjacent horizontal lines, and H is the distance corresponding to horizontal flyback time. When (z,y) lies on the even lines,

L - 2 ut = - 2 + l ) ( X + H ) + v + 2 (AY ( X + H ) + 5, (3)

where Mt is the number of lines per frame (525 lines for NTSC system) and V is the distance corresponding to vertical flyback time. The resulting two-dimensional spacial function, therefore, is

To find the discrete form of the TV picture, f ( n , m ) , let us denote the number of samples taken for each line as Nt and the sampling period as r . Then we have

(5)

and x = nvr. (6)

Let m denotes the mth horizontal line, i.e., & = m. The digitized picture can now be expressed as

X = N ~ v T ,

(7)

for 1 5 n 5 Nt and 1 5 m 5 Mt. This equation can be rewritten

(8)

m ( ~ t ( + ( N t ~ + f ) + n ~ ) ) m odd { m ( w t ( " + , " - ' ( ~ t s + $ ) + : + n r ) ) meven' f (n, 4 =

as cos (wn(n + am - a ) ) cos (wn(n + am + b ) )

m odd m even'

where w, = wtr,

N t + $ 2 '

a = -

(9)

(11) V

and b = (Mt - l ) a + -.

U T

Both a and b can be determined once Nt is given.

Spectrum of a Portion of the Digitized Picture ~~ ~

Consider a portiQn of a digitized noise picture, whose upper left corner is at (nl, ml) of the digitized picture and whose size is N x M (see Figure 3). This subpicture can be expressed as

(12) cm(w,(n + a m - a + nl + am]))

n + a m + b + nl + aml))

for 0 5 n 5 N - 1 and 0 5 m 5 M - 1. For the convenience of DFI; both M and N me assumed even numbers.

m + ml odd m + ml even' fp(n7m) = { (

320 To obtain Fp(l, k), the two-dimensional discrete Fourier trans-

form of fp (n , m ) , we note that f p ( n , m) can be written as the sum of two complex exponential functions:

1 (13) s(n,m) +s ' (n ,m) 2 fp (n ,m) =

where g(n , m) = e:wn(n+am+d) (14)

g'(n, m ) = , -zw,(n+nm+d) and

(15) The parameter d varies with respect to the variable m, i.e.,

(16) dl = n1 + u(m1 - 1) dz = n1 + am1 + b

m + ml odd m + m l wen

d = { The DFT of g(n,m) is

N - 1 M - 1 G(I , E ) = e : w n ( n + a m + d ) e - j ( v + y )

n=O m=O N - 1 M - 1

- - e:wnne--r?nl e t w , ( a m + d ) e - t s m k

n=O m=O

where

m+ml odd + e:wn(am+dz)e-:+zt

M / 2 - 1 m+ml even

t - - elwn(2Q?7L'-Qml + a + d l ) -Z$(?m'-m 1 + 1 ) L

m'=O M / 2 - 1

+ e:wn(2a"-aml+d2) -t$(?m'-")l

"=O

= (e:(U"(a+dl)-%k) + e z w n d 2 )

M/2-1 e : ( ~ m l k - w n a m l ) ez"(w,?n-ga)

"=O = (e:(wn(a+di)-zk) + e:wnd2)

where 1.J is the floor operator. Substituting (18) into (17), we get

Similarly, the DFT of g'(n,m) is

Now we study the shape of the discrete amplitude spectrum lFp(Z,k)l. We shall show that it is comprised of four pulses of distinguishing characteristics.

The amplitude of G(1,k) is

x ]cos (?(a + dl - d 2 ) - Zk) 1 M

= 2 x C(Z) x D ( k ) x E ( k ) (23)

Thus IG(1, k)l is a separable function where C(1) vanes only in the horizontal direction, and D ( k ) and E ( k ) vary only in the vertical direction.

Usually e is not an integer. Let

where h is an integer and 6 is a real number in the interval / c l 5 0.5. Then the denominator of C(Z),

has a global minimum at 1 h (mod N ) . Thus, the term C(1) has a global maximum at Z (mod N ) . A typical plot of C(I) is shown in Figure 4. For the special case E = 0, the nominator of C(l) ,

[* + $1

is equal to zero, but the denominator is equal to zero only when 1 E h (mod N ) . The limiting value of C(I = h (mod N ) ) can be shown to be N . That is, in this case C ( / ) is an impulse.

To analyze the term D ( k ) , we let

where g is a integer and [ is a real number in the interval I f 1 5 0.5. Then the denominator of D ( k ) ,

2?r (sin ($k - awn) 1 = /sin ;i?(k - g - 01, (28)

has a global minimum at k g (mod A I ) , and another one at k + g (mod M ) . Thus, the term D ( k ) has two pulses, one centered at IC L* + $1 (mod .AI) and the other at k 9 + [* + $1 (mod M ) .

t

Figure 4

As k varies from 0 to M - 1, E ( k ) varies from (cos(p)( to around lcos (9 - a)l, where 'p = ? ( a + dl - d 2 ) . That is, E ( k ) varies almost half a cycle over the whole spectrum domain. In the neighborhood of any of the two pulses of D( k ) , the variation in E ( k ) is relatively very small, thus the effect of E ( k ) can be considered as a multiplying constant on one pulse and a different multiplying constant on the other pulse. When E ( k ) happens to be zero, one of the two pulses will disappear. However, the other pulse at a vertical distance of away will get it$ maximum value since E ( k + 9) = lcos (q - sk - f ) 1 = 1. Thir effect has been taken into consideration in our detection scheme.

In general, JG(I,k)l has two pulses with the same hor- izontal coordinate but separated by 9 indices in the ver- tical direction. One pulse is at ([&J.], [&,,I ,,) and the other is at ( [ ~ L J . ] ~ , where [ . I ] , ,

1. + $1 (mod Y ) . Similarly, lG'(l ,A)l also has two pulses, one at ( [ N - gun],, [nil - $3.+] ,r) and the other at ( [ N - +n] N , [+ - %aun]M).

Therefore, the amplitude spectrum of an intermodulation beat has four pulses which are at the above mentioned locations. That is, there are four pulses grouped into two pairs which are symmetrical around the origin of the spectrum domain, and the two pulses in either pair are separated by exactly indices in the vertical direction. We will refer to these pulses as intermodulation pulses in the following sections to distinguish them from pulses caused by other impairments such as snow noise.

The two-dimensional DFT of an unimpaired picture is known to concentrate in the low frequency region of the picture [lo]. Two of the four intermodulation pulses, however, always occur in the higher half of the vertical frequency region of the spectrum. Therefore, theoretically intermodulation impairment can always be detected except for the rare case when the frequency of the beat is too close to that of the victim channel's video camer and the two pulses in the higher half of the vertical frequency region are too small due to the effect of the term E ( k ) .

The Detection Scheme

It is clear now that the essence of detecting intermodulation impairments is the detection of the intermodulation pulses (referred to as pulses in this section). Here, a method for detecting these pulses is developed first before proposing our intermodulation detection scheme.

321 For our detection scheme, the Fourier spectrum is shifted so

that the DC term which is located at the origin ( l=k=O) now appears in the middle of the spectrum domain (see Figure 2) . We then divide the spectrum domain into two halves: right and left, or four quadrants: left upper, left lower, right upper, and right lower. For an intermodulation beat, its four pulses are located in the spectrum domain one at each quadrant.

Pulse Detection

To detect a pulse, a sliding window is used over the two- dimensional spectrum domain and the function within the window is checked to see if it is a pulse. Since the spectrum of a real function is symmetrical around the origin and pictures are real functions, the window actually only need to move over half of the spectrum domain. In our case, the window moves over the right half.

The discrete amplitude spectrum of a one-dimensional sinu- soidal function consists of two pulses. Each of the vertical and horizontal slices of an intermodulation pulse has the same shape as that of such pulses. To show that, consider the intermodulation pulse which is located at ([gun] N , [ E a u , , ] AI). Its amplitude is

(29) 1 2

JFp(I,k)l rz -/G(2. k ) I .

From (22), we get

Note the omission of the term [cos (?(a + d l + d l ) - Gk)l, which is due to the fact that the change in this term in the neigh- borhood of the pulse is relatively very small.

To find the values of the pulse along its horizontal slice, we fix IC. We get

Note that Function (31) is the same as one of the pulses in the amplitude spectrum of a one-dimensional sinusoidal function. This is also true for the vertical slice of the pulse. This property is utilized in our pulse checking method.

To simplify the detection process, we slice the two-dimensional function within the window along both horizontal and vemcal di- rections with the cross point at the centre of the window. We then check if each of the sliced one-dimensional functions is a one- dimensional pulse. If both of them are, we conclude that the corre- sponding two-dimensional function is also a pulse. Although it is not always true that a two-dimensional function is a pulse when its both vertical and horizontal slices are one-dimensional pulses, the possible misclassification is overweighed by the efficiency of the simplification. Moreover, in our detection scheme described later, measures are taken to minimize such possible misclassification.

In order to determine whether or not a function is a pulse, some matched-filtering techniques may be used. However, since the shape of the pulse is not fixed, matched filtering would be com-

322 plicated and require much higher computing power. To simplify the pulse checking process, we utilize the two characteristics of a pulse, namely, narrowness and symmetry.

An intermodulation pulse is narrow and almost symmetrical around its maximum point. If we can measure the narrowness and the symmetry of a function, then we can use these measures to see if the function is a pulse. But first, precise definitions of narrowness and symmetry are needed in order to measure them.

Let us examine the following one-dimensional function of a mangle, tri(x), which is shown in Figure 5. It is straightforward

Figure 5 The Triangle Function

to define the narrowness 6, and the symmetry 6, of this triangular function as follows:

and

(33) tr i (z)dx

a

To measure the narrowness and the symmetry of an arbitrary positive function f(z) within a interval [W~, I l ’~ ] whose centre point is at x = W, we first find the equivalent triangular function in the sense that the following equations are satisfied

b = W, (34)

and W2 j tr i (z)dz = / .f(.)dz. (37)

After the parameters a, b, and c are determined from the above equations, the narrowness and the symmetry of the equivalent triangular function are calculated from (32) and (33) respectively. They are taken as the narrowness and the symmetry of the function

By combining the above equations ((32) to (37)), the narrow- ness and the symmetry of the function f ( .) can be obtained directly from the following equations:

w2

b W

f ( X I -

and

6, = (39)

Now, since we have the measurement of narrowness and symmetry of a function, we are able to determine if a function is a pulse by checking if the following conditions are satisfied:

Both E, and sn are determined theoretically from the spectrum of a one-dimensional sinusoidal function, and their derivations are given in the appendix.

Intermodulation Detection Scheme

As mentioned earlier, we have simplified the pulse checking process by assumingthat a two-dimensional function is a pulse if its two sliced one-dimensional functions are one-dimensional pulses. This simplification, however, introduces possible misclassification.

To minimize such misclassification, we divide. the digitized TV picture to many subpictures. If there is intermodulation impairment in the picture, then it should be present in all of the subpictures. Furthermore, the intermodulation pulses should be located at the same places and have similar shapes in the spectra of all the subpictures, while the contributions of the original (unimpaired) contents of the subpictures vary from one subpicture to another. Therefore, if there is a pair of pulses separated by T indices which occur at the same locations in the right half planes of the spectra of all subpictures, we know that they are intermodulation pulses.

Still, it is not guaranteed that every intermodulation pulse in every subpicture can be detected. For example, the subpicture may have some high frequency content which obscures the pulses. Furthermore, one of the pulses in the pair may be too small to detect or may even have zero magnitude.

Taking these into consideration, we introduce a term called occurrence rute. The occurrence rate (OR) of a pulse when m subpictures are used is defined as

#of occurences O R =

2n1 ‘

where # of occurrences means the number of times a pulse occurs in all the subpictures. Here, the meaning of “occur” is different from what is commonly used.

To count the number of occurrences, we check the possible pulses in the right half plane of the spectrum of each subpicture and record their locations. In the spectrum of each subpicture, if a pulse at location (n,m) where m < + is found, we consider that a pulse “occurs” once at that location; if m 2 $, we consider that a pulse “occurs” once at location (n, m - F). Thus, if a pair of pulses, one at (n, m) and the other at (n, m + T) where m < y, are found, we consider that a pulse “occurs” twice at location (n, m). We call this pulse as the representative pulse-which is always located in the right upper quadrant of the spectrum domain. Therefore, if m subpictures are taken, then the number of occurrences of the representative pulse of a beat should range from m to 2m, that is, the occurrence rate of this pulse is between 50% to 100%.

Our detection scheme is as follows:

I-

Step 1.

Step 2.

Step 3.

Step 4.

Since the

Divide the picture into a set of subpictures and cal- culate their spectra; Find all the candidate pulses using the pulse check- ing method, Calculate the occurrence rate of each representative pulse; When the occurrence rate exceeds a pre-set thresh- old, intermodulation impairment is declared.

occurrence rate of the representative pulse of a beat

323 snow noise at 26dB (signal-to-noise ratio); “Shopping” is impaired with intermodulation beats at 40dB; “Boats” is an unimpaired picture; “Snow26jnt27” is “snow26db” superimposed with an intermodulation beat of 27dB.

ranges from 50% to loo%, we set the threshold at 50%. From our experiments, we have found that this threshold value gives the maximum success.

There are many ways of dividing a picture to subpictures. The way we use is shown in Figure 6. The rationale behind this scheme is to get the subpictures spreaded over the picture as much as possible.

Figure 7 Picture “Snow26db”

sub1 sub2 1

sub3 SUM

sub5

sub6 sub7

SUM sub9

Figure 6 A Scheme to Divide a Picture to Subpictures

Experimental Results

We have tested our scheme on a set of TV pictures which consists of the following four types: 1) unimpaired pictures, 2) pictures impaired by intermodulation beat(s), 3) pictures impaired by snow noise, and 4) pictures impaired by both snow noise and intermodulation beat(s). All of the pictures have the same size of 480 x 512 pixels.

In the experiment, some of the intermodulation impaired pic- tures were obtained by letting the TV signal go through a non-linear amplifier before feeding it to a VCR whose output is then sent to a digitizer. The rest of the intermodulation impaired pictures were digitized from video tapes which document TV impairments. The snow noise impaired pictures were also digitized from those tapes. The fourth type of pictures were obtained by superimpos- ing snow noise impaired pictures with simulated intermodulation impairments.

Our detection scheme has been successful on all of the test pictures, i.e., it has successfully detected the presence of the inter- modulation impairment in both type 2 and type 4 pictures, while not misclassifying the other two types of pictures.

Four pictures, representing the four types of test pictures, are shown in Figures 7, 8, 9, and 10. “Snow26db” is impaired with

Figure 8 Picture “Shopping”

I

324

Figure 10 Picture “Snow26-int27”

The amplitude spectra of the above mentioned pictures are shown in Figures 11, 12, 13, and 14. Every plot is actually the spectrum of one of the nine subpictures of the corresponding picture. The size of each subpicture is 64 x 64. The DC terms in all the spectra are reduced in value in order to show the impairments more clearly.

Figure 11 Spectrum of “Snow26db”

(a,2) (41.5) 1 Figure 12 Spectrum of “Shopping”

Figure 13 Spectrum of “Boats”

Figure .14 Spectrum of “Snow26jnt27”

For the picture shown in Figure 8, two beats were introduced by using a non-linear amplifier. The temporal frequency of one of the beats was observed at 1.5Mhz by using a spectrum analyzer. The frequency of the other beat could not be determined because its amplitude was not large enough. Looking at Figure 12, it becomes clear that there are actually two beats present. For this picture, in the right half plane of its spectrum, all of the two pulses of the first beat, whose temporal frequency is 1.5Mhz and whose representative pulse is thus at (43,2) , were detected in four of the nine subpictures. And in each of the other five subpictures, only one pulse was detected. For the other beat, whose representative pulse is at (41,5), the pulse at (41.5) was detected in every subpicture but the other pulse was not detected. Since the possibility that some pulses may be swamped in the picture content or too small to be detected has been taken into consideration, our detection algorithm has successfully detected the presence of both beats.

For the picture shown in Figure 10, a beat was superimposed on the snow noise impaired picture. Theoretically, the two pulses corresponding to this beat should occur at the locations (35,1) and (35,33). The pulse at (354 does appear in Figure 14, but the other pulse does not appear as it is swamped by the contents of the picture and snow noise. As a matter of fact, in eight of the nine subpictures, only the pulse at (35,1) was detected; only in one of the nine subpictures all two pulses were detected. Moreover, various other pulses (due to snow noise) also present in Figure 14. Despite these, our detection algorithm has successfully detected the presence of this beat.

Table 1 shows the results of using our detection method on the four example pictures. Our detection method found that picture “shopping” has two intermodulation beats whose representative pulses are. at location (432) and (413) respectively as indicated by *’S. The picture “snow26-int27” is found to have a beat whose representative pulse is at (35,I). For other pictures, the occurrence rates of the pulses are too small (below 50%) and, rightly, are not considered as intermodulation pulses.

Table 1 Experiment Results of Some Typical Pictures

(38.7)

(41.7)

snow26db I shopping

2 11% (49.24) 4 22%

2 11% (57,7) 3 17%

In a secondary experiment, we have tested our detection scheme on a set of four pictures. To each picture a beat was super- imposed whose amplitude and frequency were varied to generate a set of impaired pictures. The impaired pictures were shown to observers who was to identify whether or not the pictures had been impaired. The result of the observers’ identification coincided with our detection method, i.e., our method has been successful in de- tecting the presence of a beat as long as the impairment is visible to the observers.

Conclusions

For the purpose of automatically detecting the intermodulation impairment in cable TV pictures, we have derived the digital form of a picture formed by an intermodulation beat and the discrete Fourier transform of an arbitrary portion of the picture. We have found that the DFT of any portion contains four pulses. These pa!ses are narrow and almost symmetrical. Two of the pulses may sometimes be too weak to detect or even have zero magnitude.

We have presented a robust detection method which is able to detect the intermodulation impairment as long as it is visible. Also it can distinguish the intermodulation pulses from other types of pulses. In our experiments, our method has been successful in detecting the presence of intermodulation beats, even in the case when the pictures are also corrupted by snow noise.

Acknowledgment

This work was funded by the Canadian Cable Labs Fund of the Rogers Cable Television Company, grant number 5-55252.

325

Appendix: Theoretical Derivation of tS And en

Let’s consider the function

f ( n ) = cos ( w n ) ,

w = - + at, where

27rm N

where m is an integer in the range of (0, N - I], and

-- 2 ; 5 t 5 y . 2 N

The N-point discrete Fourier transform of this function is

(43)

(for a detailed derivation see [ 1 13). The amplitude spectrum of .f( n ) has two pulses, one at 1% + 31 and the other at :V - 1% + $1.

At the neighborhood of one of the pulses, say the one located at 1% + 41, the amplitude spectrum is

where K is a positive constant. Substituting (42) into (45), we get

From this equation, it can be deduced that

IF(Z = m)l = m?x{IF(l)l}. (47)

From (46), it can be seen that when t = 0, IF(/ = m)l = T<X and IF(Z # m)l = 0, i.e., F(1) is an impulse; when ( # 0, IF(r)l is a pulse which spreads out around 1 = m; when 5 = &e, the pulse is exactly symmetric around m f 4.

Using a window of size five centered at 1 = m, the values of IF(!)[ within the window [m - 2, m + 21 are respectively equal to

A A Ai-. A . (48)

where A = Klsin NE/. Since 1[1 5 &, and N is large, the above values can be approximated as

lsin (-9 - C) I ’ lsin (- - () I ’ lsin (()I ’ lsin I + - E ) I ’ l c in (4 - c) I

(49) A A A A A - _ _ _ _ - -

% + [ ’ $ + [ ’ I 5 l ’ K - r ‘ S - r . Using (49) and the definitions of & and 6,,, we get (the

derivation is straightforward but lengthy, and is thus not shown here)

6, 5 0.282, (50)

and 2 5 6, 5 5.2 .

Therefore, we have found the limits on the symmetry and narrow- ness of a pulse:

cs = 0.282, (52)

and E, = 5.2. (53)

326

Note that the given definitions of the two parameters Ss and 6, are for continuous functions. To obtain these two parameters for the discrete function shown in (49), Trapezoidal approximation [12] is used to evaluate the integrals in the definitions.

References

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[3] R. G. Meyer, M. J. Shensa, and R. Eschenbach, “Cross mod- ulation and intermodulation in amplifiers at high frequencies,” IEEE J . Solid-state Circuits, vol. SC-7, pp. 1632, Feb. 1972.

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[8] Q. Zhang and R. K. Ward, “Automatic monitoring of cable TV pictures,” in ICASSP, pp. In 549-552, Mar. 1992.

[9] R. K. Ward and Q. Zhang, “Automatic identification of impairments caused by intermodulation distortion in cable television pictures,” IEEE Trans. on Broadcasting, vol. 38, pp. 60-68, Mar. 1992.

[IoIC. D. J. V. Rensburg, G. D. Jager, and A. L. Curle, “The measurement of signal to noise ratio of a television broadcast picture,” IEEE Transactions on Broadcasting, vol. 37, no. 2, pp. 35-43, June 1991.

[ 1 1]M. Cartwright, Fourier Methods for Mathematicians, Scientists, and Engineers. Ellis Horwood, 1990.

[12]P. Henrici, Elements of Numerical Analysis. John Wiley & Sons, 1967.

Pingnan Shi was born in Luzhou, China on January 11, 1963. He received the B.A.Sc. degree from Chongqing University, China in 1982, the M.A.Sc. and Ph.D. degrees from the University of British Columbia, Vancouver, Canada in 1987 and 1991 respectively, all in Electrical Engineering.

His early research was in control theory and robotics. From 1987 to 1991, he worked

on neural network implementation of image processing algorithms. Since 1991, he has been working at UBC, first as a post-doctoral fellow and then as a research engineer, on cable television quality monitoring and measurement; and cable television noise cancelling.

Rabab Kreidieh Ward was born in Beirut, Lebanon. She received the B. Eng. from the University of Cairo, Egypt in 1966, and the Masters and Ph.D. from the University of California, Berkeley, in 1969 and 1972, respectively, in electrical engineering.

Until 1979, she was a senior lecturer in the Electrical Engineering Department at the University of Zimbabwe. Since

1979, she has been with the Electrical Engineering Department at the University of British Columbia, Vancouver, Canada, where she is currently a Professor and a member of the Center for Integrated Computer Systems Research. Her early research was in cyber- netics, detection and system theory. Since 1984, she has devoted all her research efforts to signal and image processing, including detection, recognition, encoding, restoration, enhancement, and ap- plications to cytometry, microscopy, mammography, stellar images and cable television monitoring, measurement and noise cancelling.