GB2254745A - Square root anti-symmetric filters - Google Patents

Abstract

A filter is produced having a real and an imaginary part, such that for positive frequencies the frequency characteristic F of the filter is F = A + jS. The real part A is a skew-symmetric filter and S is a real symmetric filter symmetric about the cut point. S is related to A by S = 2ROOT [A(1-A)]. Once A has been chosen the filter can be derived. A method of forming the filter involves deriving symmetrical and antisymmetrical windows from the filters frequency characteristic A and S and combining the windows to form an asymmetric window.

Description

SOUARE-ROOT ANTI-SYMMETRIC FILTERS FIELD OF THE INVENTION This invention relates to filters for video signals and in particular to antisymmetric or skew-symmetric filters.

BACKGROUND TO THE INVENTION Anti-symmetric filters, sometimes known as Nyquist filters, are useful because they serve as an analogue bridge between sampled signals. If a train of samples is passed through a low-pass anti-symmetric filter which cuts at half the sampling frequency to form a continuous analogue signal, the original samples can be recovered if the signal is sampled in the correct phase, as shown in Fig. l. This forms the basis of many signal processing applications. Strictly, the term nantisymmetric is a misnomer because the anti symmetry is offset by the value of 1/2 at a definite frequency rather than by zero at zero frequency. This distinction must be pointed out because true antisymmetric functions will arise in the following discussion.Offset antisymmetric functions will, from now on, be referred to as skew-symmetric functions. For the purposes of the following discussion a function A is skew symmetrical if A(f) + A(2fc - f) = I where fc is the cut frequency.

The above spectral property can be explained in terms of the pulse response of the filter which has regularly-spaced zeros at half the reciprocal of the cut frequency. The frequency characteristic can be regarded as the convolution of an ideal rectangular characteristic with an arbitrary symmetrical spectral characteristic, as shown in Fig. 2.

As the edge of the ideal characteristic passes through the arbitrary function, the convolution integral traces out the integral of the function so that that the ideal sharp edge is replaced by this integral.

As the arbitrary function is symmetrical its integral is skew-symmetric.

Since convolution in the frequency domain is simple multiplication in the inverse domain, the (sin s)/x pulse response of the ideal filter with its regular zeros is simply multiplied-by the inverse transform of the arbitrary spectral function which thus acts as a window. Thus, if one starts with a pulse characteristic that is the inverse transform of an ideal filter and windows it symmetrically, a skew-symmetric filter can be guaranteed.

These observations can be generalised to band-pass filters wherein samples of a carrier-based signal can be carried at baseband and recovered with a bandpass filter.

A popular window is the Hanning window which is another name for the raised cosine function. Its transform is of the form (sinc x)/l-x2) which is very like the original function over the region bounded by the first zeros, and very small beyond it, leading to the approximation that it is its own transform. Both are shown in Fig. 3. The integral of the Hanning window is of the form x - sin x, which could be called a raised cosine edge, whilst the integral of its transform is rather more difficult to express but both integrals are skew-symmetrical.

If the filter is digital, and the cut frequency is an integral sub-multiple of the sampling frequency, then the regular zeros appear as zero valued coefficients. This makes skew-symmetric filters attractive from the point of view of saving hardware. Thus, for example, a filter which cuts at 1/4 of the sampling frequency has zero even coefficients except for the centre term. This forms the basis of the CCIR REC 601 chrominance filter There are, however, many other applications in which a square-root skew-symmetic filter, rather than a skew-symmetric filter, is needed.

For example, in sub-Nyquist sampled PAL, the product of the pre- and post- filter is required to be skew-symmetrical. In Clean PAL, four filters are involved in which two products of two filters are required to be skew-symmetrical whilst two cross products are required to be symmetrical. If the overall cost, in terms of coefficients, is to be minimised then this will be achieved if the filters have equal numbers of coefficients, leading to the requirement of the square root. In these two applications the filters are two-dimensional and whilst the condition can be met vertically in the special case of a first-order filter, it can only be met horizontally by assuming an ideal bandpass filter.

SUMMARY OF THE INVENTION The present invention is defined in the independent claims to which reference should be made.

Preferred and advantageous features are set out in the dependent claims.

DESCRIPTION OF DRAWINGS Embodiments of the invention will now be described, by way of example, with reference to the accompanying drawings in which: Figure 1 shows how samples of a digital signal may be recovered from an analogue signal by sampling in the correct phase; Figure 2 shows how the frequency characteristic of a skew-symmetric filter can be regarded as the convolution of an ideal rectangular characteristic with an arbitrary symmetrical spectral characteristic; Figure 3 shows a Hanning window (a) and its transform (b); Figure 4 shows a filter with a cosine edge (a) and its square root (b); Figure 5 shows the convolution of an ideal rectangular characteristic with a pure imaginary antisymmetrical function;; Ficrure 6 shows wave forms at each of the steps (a) to (h) in the design of an asymmetrical window, and the conjugate solution (i); Figure 7 shows symmetric (a), antisymmetric (b) and asymmetric windows for a trangular window; Figure 8 shows symmetric (a) antisymmetric (b) and asymmetric windows for a Gaussian window; Figure 9 shows symmetric (a) antisymmetric (b) and asymmetric windows for an inverse Hanning window; Figure 10 shows the frequency characteristics of symmetrical (a) and asymmetrical (b) 99 term filters; and Figure 11 shows the frequency characteristics of symmetrical (a) and asymmetrical (b) 49 term filters.

One way of designing square-root skew-symmetrical digital filters is by way of the frequency sampling method in which the desired characteristic is specified at a number of equi-spaced points in the frequency domain and the inverse transform is deduced. Such a process results in a filter coefficient pattern which is much greater in extent than the pattern of the filter of which it is the square root. This is because of the sharper characteristic. For example, Fig. 4 shows a filter with a cosine edge and its square root. The cusp in the square root causes many more terms to appear in the transform.This is surprising since if the response of the skew-symmetric digital filter is regarded as a polynomial of degree 2n in the z-transform variable, z, whose coefficients are the 2n + 1 coefficients of the filter, then the square root of the polynomial could be expected to be of degree n with n l 1 terms. This is a paradox, assuming that the root exists.

THE IMPORTANCE OF PHASE The clue to the paradox lies in the assumption that the filter spectral characteristics are real. If, instead, they are allowed to contain imaginary components, then the paradox can be resolved. The condition that is sought is not, now, that the square of the filter should be skew-symmetrical but that the square of its modulus should be real skew-symmetrical, that is, F.Fw = A where denotes the conjugate and A is a real skew-symmetrical function.

The pre- and post-filters are then different, one being F and the other F* whose inverse transform is the reflection of that of F.

The trick is to regard the complex function, F, as composed of separate real and imaginary parts and to repeat the derivation via the convolution argument. Thus the real part proceeds as before with a real, symmetrical arbitrary function, giving a skew-symmetrical edge; the imaginary part, however, involves convolution with a pure imaginary function which must be antisymmetrical if it is practically realisable.

Thus its total integral will be zero so that it will not give a contribution unless near the edge of the ideal filter where its partial integral will give a symmetrical function. This is shown in Fig. 5.

Note that the function must be reflected before integration. Just as the convolution with the real function corresponds to windowing in the inverse domain with a symmetrical function, so the convolution with the imaginary function corresponds to windowing with an antisymmetrical function which is the inverse transform of the imaginary spectral function. Since the resultant frequency characteristic is the sum of the real and imaginary parts, so the symmetrical and antisymmetrical windows add to give a resultant asymmetrical window.

THE METHOD OF DERIVATION The addition of the symmetrical imaginary function to the skew-symmetrical real edge gives a complex asymmetric edge in the frequency domain. It remains to be shown whether this can have the correct properties. Let the filter be expressed, for positive frequencies, as F = A + jS where A is a real skew-symmetric function and S is a real symmetric function, that is, symmetrical about the cut frequency.Then, F.Fw = (A + iS).(A - iS) = A2 + s2 If this is skew-symmetrical about fc then A2(f) + S2(f) + A2(2fc - f) + S2(2fc - f) = 1 But as A is skew-symmetric and S is symmetric about fc A(f) + A(2fc - f) = I and S(f) - S(2fc - f) = 0 so that A2(f) + (1 - A(f)32 = 1 - 2S2(f) i.e.

A2(f) + S2(f) = A(f) i.e. F.Fw is equal to the original skew-symmetric function, A, or S = (A(I Thus, once A is given, S is derived from it. From this, we see immediately, for example, that at the cut frequency where A is 1/2, S is also 1/2, the modulus of F is 1/J2 and the phase of F is 45 degrees.

It would be desirable to give an explicit analytical expression for the antisymmetrical window in terms of the symmetrical window.

Unfortunately, it is not possible to do this but only to set down a series of steps for derivation of the window as follows.

l. Choose a symmetrical window function.

2. Take the transform 3. Integrate to form A, giving the shape of the skew-symmetrical edge(s) 4. Form A(I - At to give S.

5. Differentiate S and reflect it to find the pure imaginary spectral function.

6. Take the inverse transform to give the antisymmetrical window.

7. Combine this with the original symmetrical window to give the final window.

Examples of applying these steps to particular windows are given below.

THE PROBLEMS Although, at first sight, this series of steps appears plausible, there is a problem in that the quantity A(l - A) must be everywhere positive if S is real. Unfortunately, the transform of any finite symmetrical window will be unbounded and probably have negative lobes so that the function A will also have negative lobes. This, in turn, will lead to the function S being undefined in those regions. The question then is, has the result any validity? It could be argued that we have needlessly restricted the solution by assuming that S is real; if S is allowed to be complex then it is pure imaginary where A or 1 - A is negative and so jS is real and adds to A, i.e. it modifies the skew-symmetrical frequency characteristic. At the same time, these regions contribute to the transform of the symmetrical part of the window, thereby modifying it.

This difficulty can be avoided by starting with the transform of the symmetrical window, preventing it from having negative lobes and working back to the symmetrical window itself as well as forwards to the anti symmetrical window.

A second difficulty is that there is no guarantee that the derived antisymmetrical window will have the same bounds as the symmetrical window or be bounded at all. If the symmetrical "windows is unbounded, through starting with a bounded transform, then the result is even more difficult to assess. As a practical window must be finite in extent there will then, in these cases, be a further step of truncation which will modify the frequency characteristics. The proof of the gain of the method is then the effect on the frequency characteristics of this truncation compared with any other method attempting to deliver the same antisymmetrical characteristic, having the same number of terms.

PRACTICAL EXAMPLES The first example will be dealt with in detail as it is one to which a relatively simple mathematically explicit solution can be given.

Functions at each point in the derivation will be shown. In the other examples, the functions are similar and only the final antisymmetrical and asymmetrical windows will be shown.

INVERSE COSINE WINDOW This example starts at step 3 by defining the edge shape of the frequency characteristic to be half a sine wave. The skew-symmetrical filter, over the transition region, takes the characteristic, for positive frequencies, A = 0.5(1 - sin #z). IZI < 1/2 where z = (f - f=)/ft as shown in Fig. 6(d) where fc is the cut frequency and ft is the transition band. Back-tracking over steps l and 2, differentiation to give the transform of the symmetrical window, allowing for positive frequencies and shifting, yields 0.5(W/f-)cos ny, lyl < 1/2 where y = f/ft shown in Fig. 6(c). As can be seen, this is a positive half cycle of a cosine wave, hence the name of the window. The inverse transform, the symmetrical window, is w(x) = (cos #x)/(l - 4x2) where x = ftt shown in Fig. 6(a). Note that this is unbounded and dies away as l/x2 with its first zero at x = 3/2 and successive zeros at 5/2, 7/2 etc.

An alternative expression for the symmetrical window, which will be more helpful, is w(x) = (#/4)[sin(x + 1/2) + sinc(x regarding it as the convolution of the inverse transform of a square pulse of width ft with the inverse transform of a sinusoid of periods 2ft as shown in Fig. 6(b). Note that the lobes of the two sinc functions oppose except over the central region.

Now, going forwards, step 4 yields S - 0.5 /[(1 t sin #z)(l - sin ttz)) = 0.5 cos nz, |z| < 1/2 which has the same form as Fig. 6(c) and differentiation and reflection, to find the transform of the antisymmetrical window, allowing for the frequency shift, yields 0.5(#/ft)sin ny, lyI < 1/2 shown in Fig. 6(e). Remembering that this is pure imaginary, the inverse transform is w(x) = -2x(cos #x)/(l - 4x2) shown in Fig. 6(f) which again can be written as w(x) = (#/4(sinc(x + 1/2) - sinc(x as shown in Fig. 6(g), regarding it as the same convolution but with the sinusoid shifted in phase.Note that the lobes of the two sinc functions now reinforce except over the central region where they cancel exactly at the origin.

It can now be seen immediately that the combination of the symmetrical and anti symmetrical windows cancels one sinc function and reinforces the other. Thus the resultant window is w(x) = (W/2)sinc(x + 1/2) as shown in Fig. 6(h). Note that this has a value of unity at the origin but reaches a peak value of n/2 when x = -1/2, i.e. t = -l(2ft) and dies away as 1/x. The fact that the peak is advanced corresponds to the fact that there is a phase lead at the cut frequency.

Had the conjugate solution been required, S would have been reversed in sign, leading to an anti symmetrical window of w(x) = (tut/4) (sinc(x - 1/2) - sinc(x + l/2)1 When combined with the symmetrical window, this would have been cancelled and reinforced the opposite sinc functions, leading to the resultant window w(x) = (E/2)sinc(x - 1/2) as shown in Fig. 6(i). The peak now has a lag, corresponding to the phase lag at the cut frequency. It will be noted that the anti symmetric and resultant windows die away more slowly than the asymmetric window although all three have zeros atthe same places. This is because of the discontinuities in the transform of the antisymmetric window.

As a particular example, if the filter is low-pass and the cut frequency is chosen to be a quarter of the sampling frequency then the coefficients of the infinitely sharp filter are given by c(i) = (1/2) sinc (i/2) In this set, every even coefficient is zero except c(0) which is 1/2.

If, now, the transition band is chosen to be twice the cut frequency so that x = i/2 then the symmetrical window is given by w(i) = (n/4)[sinc((i + l)/2) + sinc((i - l)/2)j whilst the anti symmetrical window is given by w(i) = (it/4)[sinc((i + l)/2) - sinc((i - In these, every odd value is zero except those at +1 and -1.

Consequently, when these windows multiply the coefficients of the infinitely sharp filter, the product is zero everywhere except at -l, 0 and +1. At these places the resultant is 1/4 1/2 1/4 for the symmetrical window and 1/4 0 -1/4 for the antisymmetrical window. Thus, combining the symmetrical and antisymmetrical contributions, we obtain 1/2 1/2 0 for one filter and 0 1/2 1/2 for the conjugate filter. This is the only known example of a exact solution to the problem and has been known for many years.

TRIANGULAR WINDOW Fig. 7 shows the windows. This is an example of a symmetrical window that is both bounded and has an all-positive transform, thus enabling the derivation to start at step 1. As the transform, being sinc2, is unbounded and dies away only as the inverse square, the function A does not form a fast edge so that it is not a serious contender.

Nevertheless, the anti symmetrical window is bounded to the same limits as the symmetrical window and appears to have discontinuities (Figure 7b). The irregularities are probably due to computation errors caused by the slow decay of the funtions. The asymmetrical window has a similar discontinuous shape (Figure 7c).

GAUSSIAN WINDOW Fig. 8 shows the windows. This is an example of a well-behaved function in that it has no discontinuities of derivatives, its transform is always positive and dies away more quickly than the previous example.

However, both the window and its transform are unbounded, leading to an approximate solution. As can be seen from Fig. 8(b) the antisymmetrical window reaches its peak well within the symmetrical window, at about 81% of the standard deviation and it dies away reasonably quickly. The asymmetrical window reaches its peak at about 60% of the standard deviation.

INVERSE HANNING WINDOW Fig. 9 shows the windows. This is an example which starts at step 2 by assuming a Hanning window for the transform of the window. As remarked above, the inverse transform, shown in Fig. 9(a), is well behaved in that, although unbounded, it has very little energy beyond the first zero, leasing to negligible effects if truncated at this point. As the Hanning window is all-positive, no difficulties arise in defining S.

Although the anti symmetrical window is substantially contained within the first zeros of the symmetrical window, the rate of decay is dissapointingly slow. This, naturally, applies also to the asymmetrical resultant.

ASSESSMENT The final test is whether filters derived by other methods have an inferior performance for the same number of terms or need more terms to achieve the same performance as filters derived by the method described here. To discover this, 99 and 49 term asymmetrical filters were designed, using the inverse cosine window and compared with 99 and 49 term symmetrical filters designed by the frequency sampling method. The cut frequency was chosen to be 3/8 of the sampling frequency and the transition band was chosen to be 1/10 of the cut frequency as these were parameters of an actual application.

Figs. 10 and 11 show the results. As can be seen, the asymmetric filters perform better than their symmetric counterparts as measured by the amplitude of the first overshoot although the improvement is disappointingly small. Care had to be taken when truncating the windows of the asymmetric filters to give equal numbers of terms either side of the peak of the window rather than the central coefficient of the unwindowed sharp cut filter, otherwise the asymmetric filters were inferior.

Alternatively, asymmetric filters can be found which give the same degree of overshoot as the asymmetric filters. These are 115 and 65 terms respectively for the asymmetric 99 and 49 term filters. Again, the improvement is relatively small.

PRACTICAL EXAMPLE Tables 1 and 6 show coefficient values for a practical filter calculated using a Gaussian window as described previously. It is believed at present that the Gaussian window is the most preferred symmetrical window.

Table I is the symmetrical window, the figures in the x column represent points along the x axis in figure 8(a) although they do not correspond to the legends on that axis. The figures in the Y(?) column represent the filter coefficient values.

Table II shows the coefficients of the anti-symmetrical window and corresponds to the window of Figure 8(b) derived by the function S = [A(l-A)].

Table III shows the coefficients of the asymmetrical window which are the sum of the coefficients for table I and II.

The coefficients for Table IV are for an actual high pass filter which cuts sharply at 3/4 of the Nyquist limit of half the sampling frequency fs/2. This is in fact a set of coefficients for an ideal filter and table V shows the coefficients which result from windowing this filter with the asymmetrical window of table C. Thus, table V is therefore the product of tables III and IV which cuts gently as 3/4 fs/2. The cut starts at ss fs/2 and finishes at fs/2 and is 1/ 2 2 at 3/4 (fs/2).

Table VI is a cut down version of the filter of table V whose frequency characteristic is practically indistinguishable from that of table V.

However, many fewer terms are required reducing cost.

The building of an actual transversal filter is straightforward once the coefficients of the filter-have been decided and no further description is necessary.

Picture file SRCPRG: [HDPAL.FILIERS.ASYMM]GAUSSHORH2.WIN;1 z y x Y(?) 0 0 "32 0.00000000 0 0 -31 0.00000000 0 0 -30 0.00000000 ~ 0 0 -29 0.00000000 0 0 -28 0.00000000 0 0 -27 0.00000000 0 0 -26 0.00000006 0 0 -25 0.00000018 0 0 -24 0.00000066 0 0 -23 0.00000215 o 0 -22 0.00000644 0 0 -21 0.0000186 0 0 . 20 0.00005138 0 0 -19 0.00013453 0 0 -18 0.00033545 0 0 -17 0.00079608 0 0 -16 0.00179815 TABLE 1 0 0 -15 0.00386590 0 0 -14 0.00791097 0 0 -13 0.01540846 0 0 -12 0.02856553 0 0 -11 0.05040556 0 0 -10 0.08465797 VAL@ES 0 0 -9 0.13533527 0 0 -8 0.20592427 0 0 -7 0.29823411 0 0 -6 0.41111231 0 0 -5 0.53940749 0 0 -4 0.67363846 0 0 -3 0.80073738 0 0 -2 0.90595520 0 0 -1 0.97561097 0 0 0 1.00000000 O O 1 0.97561097 0 0 2 0.90595520 0 0 3 0.80073738 0 0 a 0.67363846 0 Q 5 0.53940749 0 0 6 0.41111231 0 0 7 0.29823411 0 0 8 0.20592427 0 9 0.13533527 0 0 10 0.08465797 0 0 11 0.05040556 0 0 12 0.02856553 0 13 0.01540846 0 0 14 0.00791097 0 0 15 0.00386590 0 0 16 0.00179815 0 0 17 0.00079608 0 0 18 0.00033545 0 0 19 0.00013453 0 0 20 0.00005138 0 0 21 0.00001866 0 0 22 0.00000644 o 0 23 0.00000215 0 0 24 0.00000066 0 0 25 0.00000018 O 0 26 0.00000006 0 0 27 0.00000000 0 0 .28 0.00000000 0 0 29 0.00000000 0 0 30 0.00000000 0 0 31 0.00000000 TABLE 1 (cont.) 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Claims (18)

1. CLAIMS l. A filter for filtering a video signal, comprising means for filtering the signal according to a first real function (A) skew-symmetrical in the frequency domain, means for filtering the signal according to a second imaginary function (jS) symmetrical about the cut frequency in the frequency domain, and means for summing the signal filtered by the first and second filter functions to form a filter having an asymmetrical frequency characteristic F = A t jS, for positive frequencies, wherein the first and second filter functions are related such that 5 = 4[A(lA)
2. 2. A filter according to Claim 1, wherein the product of the filter function F and its conjugate F* is skew-symmetrical about the filter cut frequency.
3. 3. A filter according to Claim l or 2, comprising means for producing from the skew-symmetrical real filter A a symmetrical window function, and means for producing from the symmetrical imaginary filter an anti-symmetrical window function.
4. 4. A filter according to Claim 3, wherein the symmetrical window is an inverse cosine window.
5. 5. A filter according to Claim 3, wherein the symmetrical window is a triangular window.
6. 6. A filter according to Claim 3, wherein the symmetrical window is a Gaussian window.
7. 7. A filter according to Claim 3, wherein the symmetrical window is an inverse Hanning window.
8. 8. A method of making a filter for a video signal, the filter having a frequency characteristic F being defined by F = A + jS, for positive frequencies, where A is a real skew-symmetrical filter and S is a real filter symmetrical about the cut frequency and S =4(A(l-A)i, comprising the steps of: (a) selecting a symmetrical window function; (b) obtaining from the symmetrical window function the skew-symmetrical filter A; (c) deriving the symmetrical filter S from the skew-symmetrical filter A; (d) Forming an anti-symmetrical window from the derived symmetrical filter; and (e) combining the symmetrical and anti-symmetrical windows to form an asymmetrical window.
9. 9. A method according to Claim 8, wherein the step of obtaining the skew-symmetrical filter A comprises the steps of: (f) taking the transform of the selected symmetrical window function; and (g) integrating the transform to form the skew-symmetrical filter A
10. 10. A method according to Claim 8 or 9, wherein the step of (h) differentiating the symmetrical filter and reflecting it to obtain a pure imaginary spectral function; and (i) taking the inverse transform of the spectral function to obtain the anti-symmetrical window.
11. II. A method of making a.fil-ter for a video signal, the filter having a frequency characteristic F being defined by F = A + jS, for positive frequencies, where A is a real skew-symmetrical filter and S is a real filter symmetrical about the cut frequency and S JEA(I-A)I, comprising the steps of: (a) selecting the skew-symmetrical filter A; (b) forming from the skew-symmetrical filter A a symmetrical window function; (c) forming from the skew-symmetrical filter A the symmetrical filter S; (d) forming from the symmetrical filter S an anti-symmetrical window; and (e) combining the anti-symmetrical and symmetrical windows to form an asymmetrical window for the filter F.
12. 12. A method according to Claim 11, wherein the step of forming the symmetrical window function comprises: (f) differentiating the skew-symmetrical filter functiion to form the transform of the symmetrical window; and (g) taking the inverse transform of the symmetrical window transform to form the symmetrical window.
13. 13. A method according to Claim 11 or 12, wherein the step of forming the anti-symmetrical window comprises: differentiating and reflecting the symmetrical filter to form a pure imaginary spectral function; and (j) taking the inverse transform of the pure imaginary spectral function to form the anti-symmetrical window.
14. 14. A method according to any of Claims 8 to 13, wherein the symmetrical window is the positive half cycle of a cosine wave.
15. 15. A method according to any of Claims 8 to 13, wherein the symmetrical window is a triangular window.
16. 16. A method according to any of Claims 8-13, wherein the symmetrical window is a Gaussian window.
17. 17. A method according to any of Claims 8-13, wherein the symmetrical window is an inverse Hanning window.
18. 18. A filter made according to the method of any of Claims 8 to 17.