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

Ant -symmetric filters, sometimes known as Nyσuist 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. 1. This forms the basis of many signal processing applications. Strictly, the term "antisymmetric" is a misnomer because the antisymmetry 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(2f«= - f) = 1 where f _c 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 (sine x)/l-x ² } 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;

Figure 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 specif ed 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 + 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.F-

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 imagine ^{* •} Darts and to repeat the derivation via the convolution argvur_ . Thus the real part proceeds as before with a real, symmetr;cal 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.F* = (A + jS).(A - jS) = A ² + S ²

If this is skew-symmetrical about f _c then

A ² (f) + S ² (f) + A ² (2f - f) + S ² (2f _c - f) = 1

But as A is skew-symmetric and S is symmetric about f_

A(f ) + A(2f _β - f) = 1 and

S(f) - S(2f _c - f) = 0 so that

A ² (f) + [1 - A(f)] ² = 1 - 2S ² (f) i.e. k-(f) + S ² (f) = A(f)

i.e. F.F* is equal to the original skew-symmetric function, A, or

S = _[A(1 - A}]

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//2 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.

1. 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(l - A)] 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 antisymmetrical 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 "window" 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 itz), jzj < 1/2 where z = (f - f_)/f_

as shown in Fig. 6(d) where f_ is the cut frequency and f_ is the transition band. Back-tracking over steps 1 and 2, differentiation to give the transform of the symmetrical window, allowing for positive frequencies and shifting, yields

0.5(π/f_)cos ity, jy| < 1/2 where

Y = f/f _*

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 πx)/(l - 4x ² ) where x = f_t

shown in Fig. 6(a). Note that this is unbounded and dies away as 1/x ² 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) = (ιt/4)[sin(x + 1/2) + sinc(x - 1/2)1

regarding it as the convolution of the inverse transform of a square pulse of width f_ with the inverse transform of a sinusoid of "period" 2f_ as shown in Fig. 6(b). Note that the lobes of the two sine functions oppose except over the central region.

Now, going forwards, step 4 yields

S = 0.5 /[(l + sin _z)(l - sin πz}] = 0.5 cos πz, jz( < 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(τι/fr)sin πy, jyj < 1/2

shown in Fig. 6(e). Remembering that this is pure imaginary, the inverse transform is

w(x) = -2x(cos _x)/(l - 4x ² )

shown in Fig. 6(f) which again can be written as

w(x) = (τx/4[sinc(x + 1/2) - sinc(x - 1/2)]

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 sine 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 antisymmetrical windows cancels one sine function and reinforces the other. Thus the resultant window is

w(x) = (ιt/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 π/2 when x = -1/2, i.e. t = -l(2f-c) 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 antisymmetrical window of

w(x) = (π/4) [sinc(x - 1/2) - sinc(x + 1/2)]

When combined with the symmetrical window, this would have been cancelled and reinforced the opposite sine functions, leading to the resultant window

(x) = (τt/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 antisymmetric and resultant windows die away more slowly than the asymmetric window although all three have zeros at the 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) sine (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) = (_/4)[si c((i + l)/2) + sinc((i - 13/2} ]

whilst the antisymmetrical window is given by

w(i) = (τt/4)[sinc((i + l)/2) - sinc((i - 1J/2)]

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 -1, 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 sine ² , 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 antisymmetrical 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 antisymmetrical 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 tr- same performance as filters derived by the method describe here. To discover this, 99 and 49 term asymmetrical filters were design .d, 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 Ny uist 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 fs/2 and finishes at fs/2 and

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.

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