Background of the Invention Ghosts are produced in a receiver usually because a signal arrives at the receiver through differ- ent transmission paths. For example, in a system having a single transmitter, the multipath transmission of a signal may occur because of signal reflection. That is, the receiver receives a transmitted signal and one or more reflections of the transmitted signal. As another example, the multipath transmission of a signal may occur in a system having plural transmitters that transmit the same signal to a receiver using the same carrier fre- quency. A network which supports this type of transmis- sion is typically referred to as a single frequency network.

When a signal reaches a receiver through two or more different transmission paths, an interference pat- tern results. In the frequency domain, this interference pattern is manifested by a variable signal amplitude along the frequency axis. An interference pattern which results when the ghost is 100% is shown in Figure 1.

This interference pattern has amplitude nulls or near amplitude nulls at certain frequencies. Therefore, any information contained in the received main signal in the neighborhood of these frequencies is likely lost because the signal to noise ratio near these frequencies is below a usable threshold.

A variety of systems have been devised to deal with the problems caused by ghosts. For example, spread spectrum systems deal very adequately with the problem of a 100% ghost by spreading the transmitted data over a substantial bandwidth. Accordingly, even though a 100% ghost means that some information may be lost in the neighborhood of frequencies corresponding to the ampli- tude nulls, a data element can still be recovered because of the high probability that it was spread over frequen- cies which do not correspond to the amplitude nulls.

Unfortunately, the data rate R associated with spread spectrum systems is typically too low for many applica- tions. (The data rate R is defined as the number of data bits per Hertz of channel bandwidth.) It is also known to use a matched filter in a receiver in order to deal with the problem of a ghost.

In this approach, data is transmitted as a data vector.

The matched filter correlates the received data with reference vectors corresponding to the possible data vectors that can be transmitted. Correlation of the received main signal to the reference vector correspond- ing to the transmitted data vector produces a large peak, and correlation of the received main signal to the other possible reference vectors produces small peaks. Accord- ingly, the transmitted data vector can be easily deter- mined in the receiver. Unfortunately, the data rate R typically associated with the use of matched filters is still too low for many applications.

When high data rates are required, equalizers are often used in a receiver in order to reduce ghosts of a main signal. A classic example of a time domain equal- izer is an FIR filter. An FIR filter convolves its response h (t), shown generally in Figure 2, with a re- ceived signal. The received signal contains the main signal and the ghost of the main signal. The FIR filter produces an output having a large peak representative of the main signal. Ghosts of the main signal have small components in the output of the FIR filter. However, as shown in Figure 2, the values al, a, a3,... of the taps of an FIR filter depend on the value of a and, in order to perfectly cancel a 100% ghost using an FIR filter, the value a of the FIR filter response must approach 1. As the value a approaches 1, the values of the taps of the FIR filter do not asymptotically decrease toward zero. Therefore, the FIR filter becomes infi- nitely long if a 100% ghost is to be eliminated, making the FIR filter impractical to eliminate a 100% ghost.

An example of a frequency domain equalizer 10 is shown in Figure 3. The frequency domain equalizer 10 includes a Fast Fourier Transform (FFT) module 12 which performs a Fast Fourier Transform on the received signal in order to transform the received signal to the fre- quency domain. A multiplier 14 multiplies the frequency domain output of the FFT module 12 by a compensation vector which includes a row of coefficients Ai. An in- verse FFT module 16 performs an inverse FFT on the multi- plication results from the multiplier 14 in order to transform the multiplication results to the time domain.

Figure 4 illustrates an exemplary set of coef- ficients Ai which may be used by the frequency domain equalizer 10. The coefficients Ai are chosen so that, when they and the FFT of the received signal are multi- plied by the multiplier 14, the coefficients Ai cancel the ghost in the received signal leaving only the main signal. It should be noted that the coefficients Ai should have infinite amplitudes at the frequencies where the interference pattern has a zero amplitude. However, the coefficients Ai cannot be made infinite as a practi- cal matter. Accordingly, the coefficients Ai are cut off at these frequencies, which means that information in the received main signal is lost at the cut off frequencies so that the output of the inverse FFT module 16 becomes only an approximation of the transmitted data.

Moreover, it is known to use empty guard inter- vals between the vectors employed in the frequency domain equalizer 10 of Figure 3. The guard intervals are shown in Figure 5 and are provided so that received vectors and ghosts of the received vectors do not overlap because such an overlap could otherwise cause intersymbol inter- ference. Thus, the guard intervals should be at least as long as the expected ghosts. It is also known to use cyclic extensions of the vectors in order to give the received main signal an appearance of periodicity.

Accordingly, a Fast Fourier Transform of the received signal and a Fourier Transform of the received signal appear identical.

The invention disclosed in U. S. Application 09/158,730 filed September 22,1998 is directed to an equalizer which overcomes one or more of the above noted problems. According to this invention, a vector domain equalizer 20 as shown in Figure 6 relies on vectors to distribute the transmitted data in both time and fre- quency so that the vectors are essentially random in the time and frequency domains. Accordingly, in a heavily ghosted channel, all data can be recovered with small noise enhancement, and any enhanced noise that does exist is near white.

The vector domain equalizer 20 includes an inverse vector domain transform 22 and a vector domain transform 24 which are separated by a channel 26. Ac- cordingly, the inverse vector domain transform 22 may be part of a transmitter, and the vector domain transform 24 may be part of a receiver. The inverse vector domain transform 22 performs a matrix multiplication between an input data block and a transform matrix. The input data block may include any number of data elements arranged in a row. These data elements may be bits, symbols, or any other suitable data entities. The transform matrix comprises a plurality of vectors arranged in columns, and each vector of the transform matrix preferably has a length commensurate with the number of data elements of the input data block. Also, the number of vectors of the transform matrix should preferably be commensurate with the number of data elements in the input data block.

Accordingly, if there 256 data elements in the input data block, the transform matrix should preferably have 256 vectors each having 256 elements. The output of the inverse vector domain transform 22 is an output data block having a number of data elements commensurate with the number of data elements of the input data block.

Thus, if there are 256 data elements in the input data block 32, the output data block has 256 data elements.

The vector domain transform 24 performs a matrix multiplication between the received main signal and a plurality of receiver vectors VR. The data trans- mitted through the channel 26 is received, for example, as a row vector. During matrix multiplication, the vector domain transform 24 multiplies each component of the received row vector by a corresponding component in a first column of the receiver vectors VR, and sums the multiplication results to produce a first component ri of a vector ri at the output of the vector domain transform 24. The vector domain transform 24 next multiplies each component of the received row vector by a corresponding component in a second column of the receiver vectors VR, and sums the multiplication results to produce a second component r2 of the output vector ri, and so.

Assuming no channel distortion such as may be caused by channel interference, and assuming that the vector domain transform 24 uses the same vectors as are used by the inverse vector domain transform 22, the matrix multiplication performed by the vector domain transform 24 produces the original input data block.

However, if channel distortion exists, the actual output data block produced by the vector domain transform 24 will not be equal to the original input data block.

Accordingly, a training session is invoked where the vectors of the vector domain transform 24 are adjusted according to channel distortion such that, in the pres- ence of channel distortion, the data of the original input data block is recovered.

The invention of U. S. Application 09/158,730 works quite well. However, the present invention pro- duces similar results but with fewer calculations.

Summary of the Invention In accordance with one aspect of the present invention, an equalizer for processing blocks of data comprises a finite filter and a post-processor. The finite filter has an output, and the finite filter is arranged to substantially eliminate a ghost from a re- ceived signal in order to provide a substantially ghost free signal at the output. The post-processor is ar- ranged to apply a window function to the output of the finite filter. The window function has a duration sub- stantially equal to a duration of a block of data.

In accordance with another aspect of the pres- ent invention, an equalizer comprises a pre-processor, a finite filter, and a post-processor. The pre-processor applies coefficients b to a received main signal and a ghost of the received main signal in order to modulate the received main signal and the ghost. The finite filter applies coefficients a to the modulated received main signal and ghost in order to substantially eliminate the ghost. The post-processor applies coefficients c as a window function to the received main signal in an output of the finite filter in order to remove the modu- lation imposed on the received main signal by the coeffi- cients b.

In accordance with yet another aspect of the present invention, a method of substantially eliminating a ghost of a received main signal containing data blocks comprises the following steps: a) applying coefficients a to the received main signal and the ghost in order to substantially eliminate the ghost, thereby producing a substantially ghost-free signal, wherein the coefficients a have a duration longer than a duration of a data block; and, b) applying coefficients c to the substantially ghost-free signal, wherein the coefficients c form a window function having a duration substantially equal to the duration of a data block.

Brief Description of the Drawings These and other features and advantages of the present invention will become more apparent from a de- tailed consideration of the invention when taken in conjunction with the drawings in which: Figure 1 shows an interference pattern which could result when two signals in the same frequency band are received by a receiver at substantially the same time; Figure 2 illustrates the response of an FIR filter which is commonly used as a time domain equalizer in a receiver in order to eliminate ghosts; Figure 3 illustrates a frequency domain equal- izer which is used in a receiver in order to eliminate ghosts; Figure 4 illustrates an exemplary set of coef- ficients Ai that are used by the frequency domain equal- ize of Figure 3 in order to cancel ghosts; Figure 5 illustrates guard intervals which may be used between transmitted vectors in systems employing equalizers; Figure 6 illustrates an equalizer which in- cludes a vector domain transform pair (i. e., a vector domain transform and an inverse vector domain transform); Figure 7 illustrates a first embodiment of an equalizer in accordance with the present invention; Figure 8 illustrates a second embodiment of an equalizer in accordance with the present invention ; Figure 9 illustrates a first embodiment of the response of the pre-processor of the equalizers shown in Figures 7 and 8; Figure 10 illustrates the response of a convolver of the equalizer shown in Figure 7; Figure 11 illustrates the real part of the response of a multiplier of the equalizer shown in Figure 8; Figure 12 illustrates the imaginary part of the response of a multiplier of the equalizer shown in Figure 8; Figure 13 illustrates a first embodiment of the response of a post-processor of the equalizers shown in Figures 7 and 8; Figure 14 illustrates a second embodiment of the response of the pre-processor of the equalizers shown in Figures 7 and 8; Figure 15 illustrates a second embodiment of the response of a post-processor of the equalizers shown in Figures 7 and 8; Figure 16 is a time domain illustration of a received main signal and its ghost; Figure 17 illustrates the output of the pre- processor response in the time domain; Figure 18 illustrates a third embodiment of an equalizer in accordance with the present invention; Figure 19 illustrates a fourth embodiment of an equalizer in accordance with the present invention; Figure 20 illustrates the real part of the response of a multiplier of the equalizer shown in Figure 19; Figure 21 illustrates the imaginary part of the response of a multiplier of the equalizer shown in Figure 19; and, Figure 22 illustrates a response of a post- processor of the equalizer shown in Figure 19.

Detailed Description An equalizer 100 according to the present invention is shown in Figure 7 and includes a pre-proces- sor 102, a finite filter 104, and a post-processor 106.

The pre-processor 102 of the equalizer 100 multiplies the signal received from the channel by coefficients b. The signal received from the channel is designated in Figure 7 as Data In. The pre-processor 102 is a modulation operation that modulates the received main signal and its ghost so that the ghost is less that the received main signal. Accordingly, the ghost is no longer a 100% ghost.

The finite filter 104, as shown in Figure 7, is a convolver 108. Accordingly, the multiplication results of the pre-processor 102 are convolved in the convolver 108 with coefficients a. The convolution performed by the convolver 108 eliminates the ghost from the multipli- cation results of the pre-processor 102.

The post-processor 106 multiplies the convolu- tion results from the convolver 108 by coefficients c so that the output of the post-processor 106 is the data transmitted into the channel. The data at the output of the post-processor 106 is designated in Figure 7 as Data Out. The post-processor 106 reverses the effects of the modulation imposed by the pre-processor 102 and applies a window function to the output of the convolver 108. This window function has a duration which is substantially equal to the duration of a Data In block.

Because the post-processor 106 applies a window function to the output of the convolver 108 so that a Data Out block temporally matches a corresponding Data In block, the convolver 108 may be implemented, for example, as an FIR filter, such as that described above in connec- tion with Figure 2. That is, because of the window function applied by the post-processor 106, the number of taps of an FIR filter need not be infinite but may be limited to a reasonable number. For example, these taps may have a duration that is twice the duration of a Data In block.

A controller 109 is provided to measure the time interval, d, separating the received main signal and its ghost. As discussed below, the interval d may be used in shaping the coefficients b, a, and c. The con- troller 109 supplies the coefficients b to the pre-pro- cessor 102, supplies the coefficients a to the convolver 108, and supplies the coefficients c to the post-proces- sor 106. The controller 109 also synchronizes the pre- processor 102, the convolver 108, and the post-processor 106 to each block of data moving through the equalizer 100. Each two blocks of data may be separated by a guard interval.

Figure 8 illustrates an equalizer 110 which is equivalent to the equalizer 100 shown in Figure 7 and which includes a pre-processor 112, a finite filter 114, and a post-processor 116. The finite filter 114 includes a Fast Fourier Transform 118, a multiplier 120, and an inverse Fast Fourier Transform 122. Thus, whereas the finite filter 104 operates in the time domain, the finite filter 114 operates substantially in the frequency domain where the multiplier 120 applies complex coefficients A (described below) to the frequency domain output of the Fast Fourier Transform 122.

Accordingly, the pre-processor 112 of the equalizer 110 multiplies the signal received from the channel by the coefficients b. Again, the pre-processor 112 is in effect a modulation operation that modulates the received main signal and its ghost so that the ghost is unequal to the received main signal. Accordingly, the ghost is no longer a 100% ghost. The multiplication results of the pre-processor 112 are transformed to the frequency domain by the Fast Fourier Transform 122, the multiplier 120 multiplies the frequency domain multipli- cation results from the Fast Fourier Transform 118 by the complex coefficients A in order to eliminate the ghost from the multiplication results of the pre-processor 112, and the inverse Fast Fourier Transform 122 transforms the ghost-free, frequency domain, modulated received main signal to the time domain. The post-processor 116 multi- plies the output from the finite filter 114 by the coef- ficients c in order to reverse the effects of the modula- tion imposed by the pre-processor 102 and to apply a window function to the output of the inverse Fast Fourier Transform 122, as described above.

A controller 124 measures the interval d, supplies the coefficients b, A, and c to the pre-proces- sor 112, the multiplier 120, and the post-processor 116, respectively, and synchronizes the pre-processor 112, the finite filter 114, and the post-processor 116 to each block of data moving through the equalizer 110.

The coefficients b applied by the pre-proces- sors 102 and 112 may be discrete steps as shown by way of example in Figure 9. Each of these steps has a width along the time axis equal to the interval d, which is the time interval separating the received main signal and its ghost. Also, the ratio of the amplitude of any one step to the amplitude of the next previous step is a,. where a is a constant and is preferably less than one. In the example shown in Figure 9, a is 0.8. Moreover, the coefficients b are applied as a block to each Data In block and, therefore, the difference between to at the beginning of the block of coefficients b and tbld at the end of the block of coefficients b is commensurate with the length in time of a Data In block plus d, where d, as discussed above, is the time interval separating the received main signal and its ghost. For example, if each Data In block has a duration of 256 samples times and d has a duration of 32 sample times, then the difference between to and tb is 288 sample times, as shown in Figure 9. In addition, there should be an appropriate guard interval on each side of the block of coefficients b.

It is noted that the coefficients b modulate both the received main signal and its ghost, so that the amplitude of the ghost after application of the coeffi- cients b is preferably less than the amplitude of the received main signal. Thus, if a received main signal 130 and its ghost 132 are shown in Figure 16 as impulses separated by the interval d along the time axis, the signal 130 and its ghost 132 may have the appearance shown in Figure 17 after application of the coefficients b. It is also noted that the coefficients b perform a window function in the sense that any energy received in the intervals outside of the Data In block plus the interval d is eliminated.

The coefficients a applied by the finite filter 104 are shown in Figure 10 by way of example. As can be seen from Figure 10, the coefficients a are applied as in the case of an FIR filter. Each adjacent pair of these coefficients are separated by the interval d. Also, the ratio of the magnitude of any one coefficient to the magnitude of the next previous coefficient is the con- stant a. Because a is less than one, the magnitudes of the coefficients a asymptotically decrease toward zero.

The coefficients a preferably occupy a space in time that is twice as long as a Data In block. For example, if a Data In block has a duration of 256 sample times, then the coefficients a preferably have a duration of 512 sample times. As a result of the application of the coefficients a by the finite filter 104, the ghost in the output from the pre-processor 102 is eliminated.

The coefficients A applied by the multiplier 120 are shown in Figures 11 and 12 by way of example.

Because the output of the Fast Fourier Transform 122 is complex, the coefficients A must also be complex. Ac- cordingly, the coefficients A have a real part shown in Figure 11 and an imaginary part shown in Figure 12. As can be seen from Figures 11 and 12, the coefficients A are based upon the delay d and the ratio a. Again, the duration of each of the real and imaginary parts of the coefficients A is preferably twice as long as the dura- tion of a Data In block. As a result of the application of the coefficients A by the multiplier 120, the ghost in the output from the pre-processor 112 is eliminated.

The coefficients c applied by the post-proces- sors 106 and 116 may be discrete steps shown by way of example in Figure 13. Each of these steps has the width d along the time axis. Also, a, which is the ratio of the amplitude of any one step to the amplitude of the next succeeding step in the case of the coefficients c, is preferably less than one. In the example shown in Figure 13, a is 0.8. Moreover, the coefficients c are applied as a block to the output of the finite filter 104 and the output of the inverse Fast Fourier Transform 120 and, therefore, the difference between to at the begin- ning of the block of coefficients c and tc at the end of the block of coefficients c is commensurate with the length in time of a Data In block. The difference be- tween to and tc is not required to include d which, as discussed above, is the length of time separating the received main signal and its ghost, because the ghost has already been eliminated. For example, if a Data In block has a duration of 256 samples times, then the difference between to and tc is also 256 sample times. In addition, there should be an appropriate guard interval on each side of the block of coefficients c. The coefficients c reverse the modulation imposed on the received main signal by application of the coefficients b. The coeffi- cients c also provide a window function so that a Data Out block at the output of the finite filters 104 and 114 has a duration which substantially matches the duration of a Data In block. Accordingly, the number of impulses in the response of the finite filters 104 and 114 need not be infinite in order to eliminate a 100% ghost, but may instead be a practicable number.

The coefficients b and c as described above in relation to Figures 9 and 13 generally require a priori knowledge of d. The coefficients b and c described below in relation to Figures 14 and 15 require no a priori knowledge of d. The curve for the coefficients b as shown by way of example in Figure 14 is such that the ratio of the amplitude of the curve at any point xl along the time axis to the amplitude of the curve at a point x2 is the constant a, where xl and x2 are separated by d, where d may have any value, and where x2 occurs earlier along the time axis than xl. The constant a is prefera- bly less than one. In the example shown in Figure 14, a is 0.8. Moreover, as before, the coefficients b are applied as a block to a Data In block and, therefore, the difference between to at the beginning of the curve and tbd at the end of the curve is commensurate with the length in time of a Data In block plus d where d, as discussed above, is the length of time separating the received main signal and its ghost. In addition, there should be an appropriate guard interval on each side of the block of coefficients b.

The curve for the coefficients b as shown in Figure 14 is given by the following equation: where x is a point along the time axis between to and tb+d, a is as described above, ko is a constant such that b has a desired value at the point to, and k, is related to d.

The curve for the coefficients c as shown by way of example in Figure 15 is such that the ratio of the amplitude of the curve at any point xl along the time axis to the amplitude of the curve at a point x2 is a, where xl and x2 are separated by d, where d may be any value, and where x2 occurs later along the time axis than xl. As shown in Figure 15, a is 0.8. As before, the coefficients c are applied as a block to the output of the finite filter 104 and the inverse Fast Fourier Trans- form 120 and, therefore, the difference between to at the beginning of the block of coefficients c and tc at the end of the block of coefficients c is commensurate with the duration of a Data In block. The difference between to and tc is not required to include d because the ghost has already been eliminated. In addition, there should be an appropriate guard interval on each side of the block of coefficients c. The coefficients c reverse the modulation imposed on the signal by application of the coefficients b. Also, as discussed above, the coeffi- cients c provide a window function so that a Data Out block at the output of the finite filters 104 and 114 has a duration that substantially matches the duration of a corresponding Data In block.

The curve for the coefficients c as shown in Figure 15 is given by the following equation: x c = koak' (2) where x is a point along the time axis between to and tc, a is as described above, ko is a constant such that c has a desired value at the point to, and kl is related to d.

It is noted that the number of calculations performed by the transforms shown in Figure 6 increases in accordance with n2 as n increases, where n is the number of data elements in a data block. It is further noted that the number of calculations performed by a convolver, such as the convolver 108 of Figure 7, also increases in accordance with n as n increases. However, the number of calculations performed by the finite filter 114 of Figure 8 increases in accordance with nlogn as n increases. Thus, the calculations performed by the equalizer 110 are considerably fewer than the calcula- tions performed by the transforms of Figure 6.

Two embodiments of an equalizer in accordance with the present invention have been discussed above in relation to Figures 7 and 8. However, other embodiments of an equalizer in accordance with the present invention are possible. For example, as shown in Figure 18, a pre- processor 150 multiplies the received main signal and its ghost by the coefficients b. The output of the pre- processor 150 is transformed to the frequency domain by a Fast Fourier Transform 152, and a multiplier 154 multi- plies the frequency domain output of the Fast Fourier Transform 152 by the complex coefficients A such as those shown in Figures 11 and 12. A convolver 156 convolves the output of the multiplier 154 with the coefficients C in order to recover the data that was transmitted through the channel. The coefficients C in this case must be complex. Also, an inverse Fast Fourier Transform, which is the complement of the Fast Fourier Transform 152, is located in the transmitter and transforms the signals therein to the time domain for transmission through the channel.

Figure 19 illustrates an equalizer 160 which includes a finite filter 162 and a post-processor 164.

The finite filter 162 includes a Fast Fourier Transform 166, a multiplier 168, and an inverse Fast Fourier Trans- form 170. The signal received from the channel is trans- formed to the frequency domain by the Fast Fourier Trans- form 166, the multiplier 168 multiplies the frequency domain signal from the Fast Fourier Transform 166 by complex coefficients A in order to eliminate the ghost from the received signal, and the inverse Fast Fourier Transform 170 transforms the ghost-free, frequency domain signal to the time domain. The post-processor 172 multi- plies the output from the finite filter 162 by the coef- ficients c in order to apply a window function to the output of the inverse Fast Fourier Transform 170 so that each Data Out block at the output of the finite filter 162 has a duration that substantially matches the dura- tion of its corresponding Data In block being processed by the equalizer 160.

A controller 172 measures the interval d in order to determine the coefficients A, supplies the coefficients A and c to the multiplier 168 and the post- processor 164, respectively, and synchronizes the finite filter 162 and the post-processor 164 to each block of data moving through the equalizer 160.

The coefficients A applied by the multiplier 168 are shown in Figures 20 and 21 by way of example.

Because the output of the Fast Fourier Transform 166 is complex, the coefficients A must also be complex. Ac- cordingly, the coefficients A have a real part shown in Figure 20 and an imaginary part shown in Figure 21. As can be seen from Figures 20 and 21, the coefficients A are based upon the interval d and the ratio a. Again, each of the real and imaginary parts of the coefficients A preferably have a duration that is twice as long as the duration of a Data In block. As a result of the applica- tion of the coefficients A by the multiplier 168, the ghost in the output from the pre-processor 102 is elimi- nated.

The coefficients c applied by the post-proces- sor 164 are shown by way of example in Figure 22. Be- cause there is no pre-processor in the equalizer 160 that modulates both the received main signal and the ghost, the coefficients c are not required to undo the effects of any modulation. Therefore, the coefficients may have a constant non-zero value within the window from to and tc. The coefficients c shown in Figure 22 are applied as a block to the output of the finite filter 162 and, therefore, the difference between to at the beginning of the coefficients c and tc at the end of the coefficients c is commensurate with the duration of each Data In block. As before, if a Data In block has a duration of 256 samples times, then the difference between to and tc is also 256 sample times. In addition, there should be an appropriate guard interval on each side of the block of coefficients c. The coefficients c provide a window function that limits each Data Out block at the output of the finite filter 162 to a duration that substantially matches the duration of its corresponding Data In block.

Accordingly, the number of impulses in the response of the finite filter 162 need not be infinite in order to eliminate a 100% ghost, but may instead be a practicable number.

Certain modifications and alternatives of the present invention have been discussed above. Other modifications and alternatives will occur to those practicing in the art of the present invention. For example, because the present invention operates most satisfactorily in the presence of ghosts and other linear distortions, the term ghost as used herein in connection with the present invention includes ghosts and/or other linear distortions.

Moreover, the coefficients b have been shown above as non-complex coefficients. However, the coeffi- cients b may be complex, such as where the received main signal is a QAM signal.

Accordingly, the description of the present invention is to be construed as illustrative only and is for the purpose of teaching those skilled in the art the best mode of carrying out the invention. The details may be varied substantially without departing from the spirit of the invention, and the exclusive use of all modifica- tions which are within the scope of the appended claims is reserved.