Optimized Sparse Transversal Filter

FIG. 1 illustrates conventional adaptive transversal filter;

FIG. 2 shows the error surface for the filter of FIG. 1;

FIG. 3A illustrates a novel ghost cancellation circuit;

FIG. 3B illustrates the ghost cancellation circuit of FIG. 3A with additional control structure;

FIGS. 4A-4B illustrate the circuit of a novel adaptive sparse transversal filter used in the ghost cancellation circuits of FIGS. 3A and 3B;

FIG. 5 shows an example of the filter of FIG. 4 having two filter taps;

FIG. 6 is a matrix structure described in the specification;

FIG. 7 is a transformed error surface

FIGS. 8 and 9 are the results of simulation of the ghost cancellation circuits; and

FIG. 10 illustrates a particular technique of ghost cancellation circuit adaptation.

In order to facilitate description of the invention, a simple geometrical relationship based on Wiener filter theory is derived. Algorithms for clustering (i.e. locally grouping free multipliers at needed taps) based on the formulation are presented.

1. Adaptive Filtering

In a finite impulse response (FIR) adaptive filters as shown in FIG. 1 the multiplier values or tap weights and signal are represented by R.sup.m vectors W.sup.T ={W.sub.o . . . W.sub.m-1 } and X.sup.T ={X(n) . . . X(n-m+1)}, respectively. The filter output may then be written as y(n)=X.sup.T W=W.sup.T X. To tune the filter output to create a good replication of the reference signal d(n) from input X(n), the error .uparw..epsilon.(n).uparw..sup.K =d(n)-y(n).uparw..sup.K must be minimized. For ease of calculation it is desirable to set K=2 obtaining

Expectation operator E is then applied to average out any noise as shown below.

where e is the mean square error (MSE).

Minimization can then take place by writing the gradient of e with respect to the tap weights obtaining .gradient., and then setting this quantity to `0` thus solving for the optimal weight vector W* as shown in FIG. 2 below. W* is known as the Wiener solution with E[XX.sup.T ]=R being the input autocorrelation matrix.

The condition .gradient.=0 yields the optimal solution W* given by eqn. (4).

FIG. 2 shows an example of a two weight error surface. The quadratic shape of the error surface is determined by R and the center point is W*. Below we describe how this surface is cut when multipliers are deleted from the FIR filter.

From eqn. (4), we can see the difficulty in calculating W*. Long term averaging and complex matrix operations are needed. It is simpler to start with an initial guess of W* and then follow the gradient down the error surface to reach the actual optimum. Because .gradient. in eqn. (3) is also difficult to calculate, the least mean square (LMS) gradient estimate may be used. Instead of using an actual gradient, the expectation is dropped to obtain

The gradient estimate .gradient.(n) in final form is obtained after solving for .differential..epsilon.(n)/.differential.W from .epsilon.(n)=d(n)-X.sup.T W.

If .gradient.(n) is time averaged at a particular W on the error surface, .gradient..fwdarw..gradient. (closeness increasing with averaging time, weights are iterated to reach W* by following the gradient as shown below. ##EQU1##

Since the constant .mu. determines the rate of descent, the time average of .gradient. and thus precision is sacrificed when using large .mu.. There is a compromise between speed and accuracy of adaptation.

A thorough treatment of adaptive filtering can be found in the text: Widrow & Stearns, "Adaptive Signal Processing", Prentice-Hall (1985).

2. Ghost Cancellation

The image portion of a video signal such as occurs in television broadcasting, is not known to the receiver before reception, thus making it undesirable as a reference signal [d(n)]for adaptive filtering. In the system according to the invention a standardized reference signal is inserted in the vertical interval test signal (VITS) during vertical retrace. A circuit in the receiver generates an identical reference signal simultaneously with the VITS transmitted reference (which is corrupted by an unknown linear ghosting function h(n) before it is received). A signal x(n)=d(n)*h(n) is applied to an adaptive FIR filter input, the top weights may then be adapted to minimize the error e. The FIR filter then approximates inverse ghosting function h.sup.-1 (n); when the picture portion of the signal is passed through the FIR significant ghost reduction takes place. A FIR filter cannot provide complete cancellation, however, because ghosting function h(n) is a finite convolution, and thus has an all zero transfer function. The inverse all pole filter must have infinite impulses response (IIR) for complete cancellation.

The error surface of an IIR adaptive filter is not quadratic and may also have local minima. Additionally, the LMS gradient estimate has recursive terms, making it more complex than the FIR counterpart. Below, we show that it is possible to adapt in FIR mode and retain the complete cancellation provided by an IIR structure; in effect, the best attributes of FIR and IIR filtering can be combined.

FIG. 3A illustrates an adaptive ghost cancellation circuit according to the invention. This circuit operates in a convergence mode or adaptation in which adaptation takes place, and in a ghost cancellation mode. It requires only a single FIR adaptive filter.

The signal transmission path is represented as comprising a direct path 1 as well as an indirect path 2 having a transfer function H(s), the former representing direct reception and the latter reception due only to an echo. The combined signals of these two path components is applied to the ghost canceller circuit at input 10. It should be kept in mind that elements 1, 2 and the indicated summation are not physical elements of the system, but representations of the signal transmission path. The physical invention starts at input 10.

If a reference signal R(s) is transmitted as part of the video signal, its component X(s) of the received signal is given by

and is applied to the subtracter circuit 11. The ghost cancellation circuit is illustrated in the cancellation mode in which the output &(s) of the subtracter circuit 11 is applied to an adaptive filter 15, and the filter output is fed back to the subtracter circuit 11. Thus, the ghost cancellation circuit output is given by

If H'(s,t) H(s) after convergence time T, then

As can be seen, if H'(s,T.sub.c).apprxeq.H(s), complete ghost cancellation is approached. This adjustment of H'(s,T.sub.c) occurs in the convergence or adaptation mode, where H'(s,T.sub.c) performs system identification of echo channel 2 having transfer function H(s).

In the convergence mode the switches 16, 17 and 18 are set to the positions opposite those shown in the drawing. The reference signal generator 20 develops a reference signal identical to the predetermined reference signal R(s) in the received video signal, and this reference signal is applied through the respective switches 17 and 18 to the adaptive filter 15 and to the subtracter circuit 11. The circuit through switch 18 corresponds to the direct transmission path 1 and the circuit through switch 17 corresponds to the echo signal path. Consequently, the sum of the inputs to the subtracter circuit 11 should be X(s) and -X(s) so that the subtracter circuit output Y(s) effectively be null.

Any non-zero value for Y(s) can be thought of as an error signal representative of the difference between the indirect path transfer function H(s) and the filter transfer function H'(s, t.sub.c). This error signal is applied through switch 16 to the block 21 marked error correction. This block represents a circuit for determining the magnitude of the error signal and for determining the optimum filter tap weights to reduce the error according to some criterion such as the least mean square error. It could be physically realized by a microprocessor based microcomputer for calculating the tap weights according to the equations derived below. The important point is that in the convergence mode the FIR filter 15 is tuned so that its transfer function is made to replicate the indirect signal path.

FIG. 3B illustrates the ghost cancellation circuit according to the invention shown in FIG. 3A but with control for switching between operating modes.

Control circuit 25 is responsive to the received video signal for determining when the vertical retrace interval occurs. Control circuit 25 applies a control signal to the switches 16, 17 and 18 to switch them to the settings for the convergence mode during the vertical retrace interval so that the transmitted reference signal R(s) can be used for adaptation of the filter 15. During the reception of the image part of the video signal the switches 16, 17 and 18 are set for the ghost cancellation mode. Switching between convergence and ghost cancellation modes occurs every time a reference VITS appears, creating the effect of having an IIR adaptive filter continuously available to process the picture.

The indirect or echo channel discrete transfer function, h(n), is sparse because most of the response values over time are zero or insignificant. In the convergence mode, a FIR filter is tuned to replicate the echo transfer function; therefore, many multipliers along the delay taps of the transversal filter are of `0` value, meaning they effectively are unused. Consequently, a FIR filter with a reduced number of moveable multipliers can be employed; multipliers are placed to replicate the clustering properties of the echo transfer function.

FIGS. 4A-4B show the adaptive cluster transversal filter. Considering adaptation, our problem is to distribute the multipliers only to the tap locations where they are needed and then adjust them to the correct value. This new filter structure is equivalent to a normal M tap FIR filter having M-N of the multipliers set to zero; this kind of tap weight pattern would be expected when identifying a non-smeared or smeared echo sequence that could be approximated by N active components. Suppose that the N multipliers are distributed in a known pattern that is different from the correct pattern of the active taps required to replicate the echo. We can find out how the optimum associated with this known tap weight pattern is related to that of the global optimum of the filter having multipliers at all M taps. In fact, if we only know the optima of a sufficient number of different patterns containing N multipliers, we can combine this information to reconstruct what would be the global M tap optimum. The zero and non-zero taps can then be identified, leading to the correct multiplier distribution. We call this filter an adaptive sparse transversal filter.

The situation is elaborated in the two tap example of FIG. 5. The single multiplier is first connected to the left delay tap and is optimized to setting W.sub.1 '. When connected to the right delay tap, the optimum is found to be W.sub.2 '. After combining these coordinates to form W.sub.c * it is seen that this is a point different from global optimum W*. Below, we derive a linear mapping A.sub.c that can relate W.sub.c * to W*. In the formulation, there are N multipliers to be distributed among M taps of the transversal filter.

To start, Eqs. (3) and (4) are combined so that the gradient is written with respect to W*, where R=E[XX.sup.T ], as follows.

Since the search for minima is being constrained to certain components of W being held to `0`we define switching matrix S=diag{s.sub.K }, s.sub.k =1,0. sK=1 corresponds to taps having multiplier, where s.sub.K =0 is the converse, and the following holds

Multiplying W by S thus nulls the appropriate elements. We additionally wish to pre-null corresponding elements of vector .gradient. so as to ignore those components in the optimization procedure. Since some of the elements of W are constrained, the remaining free elements can assume a different optimal solution than the original solution obtained in (3) and (4). To distinguish this new solution, the constrained weight vector will be called W.sub.s. The constrained optimum, W.sub.s *, is obtained by setting S.gradient.=0, as displayed below. Setting=0 and solving for W.sub.s yields the constrained optima, as shown below.

Eq. (12) is written in space, where (11) is in R.sup.M ; N<M. The subspace operations of (12) are necessary as a pre-requisite for finding the relationship of W.sub.c * and W in R.sup.M. MXM matrix [SRS] is singular since its rank.ltoreq.N, since not all S.sub.K =1. Consequently, we cannot invert [SRS] directly. We instead need to work with a sub-matrix derived from [SRS] that performs mappings on the non-nulled sub-space of W-space. This NXN sub-matrix can be constructed by permuting all the null columns of [SRS] to the right and null rows to the bottom, leaving the sub-matrix nested in the upper left corner of the full MXM matrix. We note that to preserve correspondence, elements of W.sub.s * are permuted in the same fashion as the column of [SRS], and the rows of [SR] are permuted to match the rows of [SRS]. A matrix Z will be constructed to perform the permutations. Premultiplying by Z permutes a matrix's rows while post-mulitplying by Z.sup.T permutes the columns. The equation below shows how Z is formed from S=diag {sK}, where K=1, . . . , M. ##EQU2##

In (13), Z starts from an all null MXM matrix where `1` elements are added only at the indices indicated. While column index `j` is scanned as `K` goes from 1 to M row index `i` is incremented by 1 starting from 0 each time s.sub.K =1; when S.sub.k =0, the row index is also incremented by 1 but starts from

Thus, every row and column has a single `1` in it with the rest of the elements being `0`. An example is shown in FIG. 6 to aid understanding.

The submatrix of ZSRSZ.sup.T can be constructed to be non-singular since it is a function of R and S, both under our control, and not dependent on echo h(n). Z has the following properties:

Applying the described permutation to (12), yields

Considering the non-nulled elements of vectors to be active, we have defined the transformation W*.fwdarw.[Z SR]W* which is a mapping R.sup.M .fwdarw.R.sup.N. Similarly, the bottom M-N elements of ZW.sub.s * are nulled by the zero elements in nested matrix [Z SRS Z.sup.T ], making both sides of (15) elements of R.sup.N. We therefore write (15) as

where the apostrophe indicates the bottom nulled elements of the vectors have been truncated, making them N dimensional; additionally, R' is an NXN invertible matrix corresponding to [Z SRS Z.sup.T ]. With this, (ZW.sub.s *); can be solved, however, it is desirable to find the solution and preserve the identity of the nulled vector elements for later operations. To solve (15) as described, operation -1.quadrature. is defined to perform inversion only on the nested sub-matrix within [Z SRS Z.sup.T ] and place this new matrix back in the same position as the original sub-matrix. Therefore,

To further simplify we note that [Z SRS Z.sup.T ].sup.-1 =Z SQS Z.sup.T for some matrix Q having corresponding elements to inverse sub-matrix of R', also S=S.sup.n. Applying this to (16) and using (14) yields

Finally using (17) and (14) we obtain

In (18) the nulled elements of W.sub.s * are not only preserved, but also permuted to match corresponding elements of W*. The active elements of W.sub.s * relate in order, to the s.sub.K =1 elements in S=diag {s.sub.K }. To summarize, (18) takes R, nulls out appropriate rows and coluns, permutes remaining rows and columns to form an N.times.N sub-matrix, inverts the sub-matrix, and de-permutes the inverted submatrix so that elements occupy the original non-nulled locations creating a `sub-space inverse of R'. This `subspace inverse of` R is post multiplied by R yielding the R.sup.M .fwdarw.R.sup.N mapping W*.fwdarw.W.sub.s * having preserved nulled elements.

Suppose that W.sub.si * correspond to switching matrices S.sub.i, i=1, . . . , T with

Then there are subspaces E.sup.Ni R.sup.M, with W.sub.si *.epsilon.E.sup.Ni and .andgate..sub.i E.sup.Ni =.phi. and UiE.sup.Ni =R.sup.M. Accordingly we can build W.sub.c in R.sup.M from the E.sup.Ni subspaces as shown; using (18), ##EQU3## (19a) and (19b) are R.sup.M .fwdarw.R.sup.M, and A.sub.c is invertible.

It is clear how subspace optima, W.sub.i ', in FIG. 5 are constructed. Since (18) shows that only one optimum exists for a given matrix S (or subspace), the optimum must be found on the contour tangent to the axis constraining the sub-space.

Now that mapping As has been found, it is easy to see how a cluster can be formed. Referring to the two tap example of FIG. 7, we see that W.sub.c * is constructed in the same manner as in FIG. 5. Both components of W.sub.c * are non-zero. If we were to use (19) and then find A.sub.c.sup.-1, the global optimum W*, can be calculated. Notice that W* has a single non-zero component, hence, only one multiplier is needed is this simple cancellation filter.

3. Examples

Combing

A switching pattern that meets the conditions needed to obtain a valid W.sub.c * usable in (19) is the comb. We will describe a 3 to 1 comb, but this approach is not restricted to a 3 to 1 ratio. Assuming that the transversal filter in FIG. 4 has M taps, let us say that we find (for example) the worst case echo requires that only N multipliers are needed in this structure at any given time. Let M=3N. To form, this comb, distribute the N weights evenly over the 3N taps, leaving a gap of 2 taps between multipliers. The comb can be advanced. More specifically:

(i) Distribute N weights evenly over the first of every third 3N taps; LMS converge and store weight values forming W.sub.s1 *

(ii) Distribute N weights evenly over the second of every third 3N taps; LMS converge and store weight values forming W.sub.s2 *

(iii) Distribute N weights evenly over the third of every third 3N taps; LMS converge and store weight values forming W.sub.s3 *

(iv) Combine stored weight values forming W.sub.c *

In the sequence described, the comb advances position only after each W.sub.si has converged to its final value. A useful variation would be to advance the comb after each LMS cycle of (7) thus updating W.sub.c * every 3 cycles. Intermediate values of W.sub.c * provided with this "cyclic comb" may be desirable in giving continuous subjective improvements to the picture as final convergence is reached.

1. Clustering with R=.LAMBDA.

a) Find W.sub.c * either by advancing or cyclic comb.

b) Select N non-zero values of weights; assign the N multipliers to related locations and set to corresponding weight values. (Some components of W.sub.c * would only be zero under ideal conditions with perfect convergence. The correct objective could be achieved by selecting the N maximum magnitude weight values).

2. Clustering with R.noteq..LAMBDA.

This algorithm uses steps a) and b) from above. However, the following intermediate step must be added between a) and b).

a') calculate W*=A.sub.c.sup.-1 Wc. W* is used in step b).

Since A.sub.c.sup.-1 is a function of R and {S.sub.1, . . . , ST}, it may be precalculated and stored. It is best to find a reference signal and switching pattern (i.e. a different comb ratio) to lead to a very simple A.sub.c.sup.-1 structure, as this will facilitate multiplication later.

In simulations, a sin x/x reference signal R(s) was chosen for its smooth spectral characteristics (any frequency null would make tuning at that point indeterminate). To match video bandwidth, the sin x/x rolloff was set at 5 MHz. Sampling was done at 3x bandwidth with a 3 to 1 comb switching pattern. A.sub.c was calculated using (19) and found to be practically indistinguishable from (l/r.sub.ii)R. To show the utility of the mapping, a single echo that can be described as having W*={0,0, . . . , 1, . . . , 0,0} was processed with a 3 to 1 Advancing Comb (with LMS as described in this section under `Combing`) to obtain W.sub.c *. W* was also mapped through A.sub.c to find W.sub.c * according to (19b). FIG. 8 shows the results of the Advancing Comb while FIG. 9 that of the mapping.

Another embodiment of the invention will be discussed with reference to FIG. 10. A single echo is mapped through matrix A.sub.c, which is effectively Toplitz (i.e., A.sub.c ={a.sub.ij }={a.sub.i-j }), and has a very dominant main diagonal. As in FIG. 9, the impulse of W* is mapped to an impulse with side-lobes in W.sub.c *. The peak position of W.sub.c * is the same as the W* pulse. Because of the described structure of A.sub.c, we can say that any component of W* maps to the same spot in W.sub.c * with similar side-lobes; superposition can therefore be applied for multiple echoes. In this case the maximum magnitude components of W.sub.c * a high probability of matching the non-zero elements of W*. If it is known a priori that the most complex echo will have .ltoreq.N components, then it is highly likely that these will be the N maximum components of W.sub.c *. This relationship would not work if the side-lobe of an echo sufficiently lowered the peak amplitude of another echo so as to not be counted in the set of N selected W.sub.c * elements; the more dominant the main diagonal of A.sub.c, the less likely this event.

3. Course-Fine Clustering Projecting Forward through A.sub.c for a 3x sampled sin x/x; 3 to 1 comb.

a) Find W.sub.c * either by advancing or cyclic comb

b) Select N maximum magnitude elements of W.sub.c *; assign the N multipliers to related locations and set to corresponding weight values (considered as initial condition)

c) With the weights set as in b), preform final LMS convergence. This fine tunes the weights to match W*.

Selection of the maximum magnitude elements in step b) can be by a predetermined threshold, or by setting a threshold value so that the number of elements of W.sub.c * having values exceeding the threshold equals N.