Abstract:
A non-recursive half-band filter having a filter length N and complex coefficients for receiving either a real input signal s(kT) or a complex input signal s(2kT) and for processing and converting the received input signals into either a complex output signal s(2kT) or a real output signal s(kT) wherein the complex coefficients operate at a function of h(l) where l=-(N-1)/2 to (N-1)/2 to (N-1)/2 and the filter length N is odd. As a result it is possible to convert a real input signal into a complex output signal, by modulating its pulse response to a complex carrier of the frequency equal to 1/4 or 3/4 of the sampling frequency, where the null phase of this frequency is an integer multiple of π/2. It is also possible to convert a complex input signal into a real output signal, by modulating its pulse frequency to the complex carrier of a frequency signal to the input sampling frequency or half thereof, where the null phase of this frequency is an integer multiple of π/2.

Description:
This application is a continuation of application Ser. No. 07/408,493, filed Aug. 18, 1989, now abandoned. 
    
    
     BACKGROUND OF THE INVENTION 
     The invention relates to a non-recursive half-band filter. Such filters have become known from the paper by Bellanger et al, entitled, &#34;Interpolation, Extrapolation, and Reduction of Computation Speed in Digital Filters,&#34; published in IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-22, No. 4, August, 1974, pages 231-235. 
     The known half-band filters process real input signals into real output signals. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a non-recursive half-band filter that makes it possible to convert a real input signal into a complex output signal, or vice versa, in an inexpensive manner. 
     The above object is achieved according to a first aspect of the invention by a non-recursive half-band filter with complex coefficients for processing a real input signal s(kT) by having the sampling frequency fA=1/T and for converting this real input signal s(kT) into a complex output signal s(2kT), wherein the filter complex coefficients h(l), where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, have alternating purely real and purely imaginary values, and therefore no complex values in the fullest sense, and wherein the pulse response of a half-band filter h(l) having exclusively real values and the characteristics h(l)=h(-l) for all |l|≦(N-1)/2 and h(l)=0 for 1=±2, ±4, . . . , is modulated onto the complex carrier of a frequency of ±1/4 of the input sampling frequency fA=1/T to yield 
     
         h(l)=h(l)·e.sup.j(±2πlfA/4fA+φ0) =j.sup.±l ·e.sup.jφ0 ·h(l) 
    
     and the null phase φ0 of this complex carrier is an integer multiple m of π/2 (φ0=m·π2 where m=0, 1, 2, 3, . . . ). 
     The above object is achieved according to another aspect of the invention by a non-recursive half-band filter with complex coefficients for processing a complex input signal s(2kT) and for doubling the sampling frequency fA&#39;=1/2T to fA=2fA&#39; and for converting this complex input signal s(2kT) into a real output signal s(kT), and wherein the filter complex coefficients h(l), where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, alternatingly have purely real and purely imaginary values, and therefore no complex values in the fullest sense, and the pulse response of a half-band filter h(l) having exclusively real values and the characteristics h(l)=h(l) for all |l|≦(N-1)/2 and h(l)=0 for l=±2, ±4, . . . , is modulated onto the complex carrier of a frequency of ±1/4 of the output sampling frequency fA=1/T to yield 
     
         h(l)=h(l)·e.sup.j(±πlfA/4fA)+φ0) =j.sup.±l ·e.sup.jφ0 ·h(l) 
    
     and the null phase φ0 of this complex carrier is an integer multiple m of π/2 (φ0=m·π/2 where m=0, 1, 2, 3, . . . ). 
     The above object is achieved according to still a further object of the invention by a non-recursive half-band filter, which converts a real input signal s(kT) into a complex output signal s(kT) where k is a running index, while maintaining the sampling frequency fA=1/T, in that the filter pulse response h(l), where l=-(N-1)/2 to (N-1)/2 and N is an odd filter length, is modulated onto a complex carrier at a frequency of ±1/4 of the sampling frequency fA=1/T, to yield 
     
         h(l)=h(l)·e.sup.j(±2πlfA/4fA+φ0) =j.sup.±l ·e.sup.jφ0 ·h(l) 
    
     and the null phase φ0 of this frequency is an integer multiple m of π/2 (φ0=m·π/2 where m=0, 1, 2, 3, . . . ). 
     The above object is achieved according to still a further aspect of the invention by a non-recursive half-band filter, which converts a complex input signal s(kT) where k is a running index into a real output signal s(kT), while maintaining the sampling frequency fA=1/T, in that the filter pulse response h(l) with reference to the sampling frequency fA, where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, is modulated onto the complex carrier at a frequency of ±fA/4 to yield 
     
         h(l)=h(l)·e.sup.j(±2πlfA/4fA+φ0) =j.sup.±l ·e.sup.jφ0 ·h(l) 
    
     and the null phase 0 of this frequency is an integer multiple m of π/2 (φ0=m·π/2 where m=0, 1, 2, 3, . . . ). 
     The novel non-recursive half-band filter according to each of the first two aspects of the invention permits the conversion of real digital input signals into complex digital output signals with a simultaneous reduction of the sampling frequency by a factor of two, and the conversion of complex digital input signal into real digital output signals with a simultaneous increase in the sampling frequency by a factor of 2. The novel non-recursive half-band filter according to the latter two aspects of the invention permits the conversion of real digital input signals into complex digital output signals while maintaining the sampling frequency, and the conversion of complex digital input signals into real digital output signals, likewise while maintaining the sampling frequency. 
     These relatively inexpensive half-band filters are thus suitable as digital pre-filters or post-filters for digital systems employed to process complex signals and as digital partial filters in an arrangement of anti-aliasing filters for band limitation while complying with the sampling theorem. The advantage of the half-band filter lies in its linear phase and simultaneously its low cost. In each case, the smallest possible sampling frequency required on the basis of the sampling theorem can be employed. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     There now follows a description with reference to the drawing figures in which: 
     FIG. 1 is a block circuit diagram for the digital filter according to the invention. 
     FIGS. 2a to 2c depict several amplitude responses of half-band filters plotted over frequency. 
     FIGS. 3 and 4 show particularly favorable circuit variations of the half-band filter. 
     FIG. 5 is a block circuit diagram for a half-band filter used to process a complex input signal into a real output signal, according to the present invention. 
     FIG. 6 shows, in schematic, circuit details of the filter shown in FIG. 5, with this circuit having been developed by transposition from that of FIG. 3, i.e., by reversing all directions indicated by arrows and replacing a branching switch for an adder and vice versa, and by replacing a demultiplexer with a multiplexer. 
     FIG. 7 shows, in schematic, another circuit arrangement of the filter shown in FIG. 5, this circuit arrangement being developed from that of FIG. 4 in a manner similar to that of FIG. 8. 
     FIG. 8 is another embodiment of a block circuit diagram for the digital filter according to the invention. 
     FIGS. 9a to 9c show several amplitude responses of half-band filters plotted over frequency. 
     FIGS. 10 and 11 show particularly favorable circuit variations of the half-band filter. 
     FIG. 12 is a block circuit diagram of a transposed, inversely operated half-band filter for processing a complex input signal into a real output signal. 
     FIGS. 13 and 14 respectively show, in schematic, circuit details of the filter according to FIG. 12, with the circuits being respectively developed by transposition from FIGS. 10 and 11, i.e. by reversal of all directions indicated by arrows and the replacement of a branching switch with an adder and vice versa. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     In FIG. 1, the real input signal s(kT) is applied by halving the sampling rate to a digital half-band filter DF which generates therefrom the complex output signal s(2kT). 
     The amplitude frequency response of a prototype half-band filter is shown in FIG. 2a; the pass band of this filter extends from -fA/4+Δf to +fA/4-Δf, and its stop band also has a width of fA/2-2Δf. It is a further characteristic of this half-band filter that the transition from the stop band to the pass band is steady and takes place over a width of 2Δf. This transition range is symmetrical to fA/4. A further characteristic of the half-band filter is that its ripple in the pass band and in the stop band is identical, namely δ1=δ2=δ. In such a filter, there results a pulse response h(l) where 1=0 to N-1 and the filter length N is odd, and it follows that every second value is identical to zero, with the exception of the central main value (see FIG. 2, page 233, in the abovecited, paper by Bellanger et al). 
     FIG. 2b shows the frequency response |H|. It can be seen that this frequency response has been shifted to the right by the frequency fA/4 relative to the frequency response of the prototype half-band filter. In addition, FIG. 2b shows that the spectrum |S| of a real input signal s(kT) sampled at the sampling frequency fA has been inserted; because of sampling with fA, this input signal spectrum is periodically repeated in frequency ranges [m·fA, (m+1/4)·fA] in the normal position and in frequency ranges [(m+1/4)·fA, (m+1)·fA] in the inverted position where m=. . . , -1, 0, +1, . . . The input signal s(kT), applied to the half-band filter according to the invention without any change in the sampling rate would thus suppress the inverted position between fA/2 and fA and of course all its repetitions and would simultaneously generate a complex signal s(kT). Halving the sampling rate now results in the desired spectra, with the normal position being repeated in each instance in a pattern of fA/2=fA&#39;, where fA&#39; is the new sampling rate (see FIG. 2c). 
     At this point, it should be noted that a complex signal in the inverse position is obtained at the output of the half-band filter if the frequency response of the prototype half-band filter according to FIG. 2a is shifted by -fA/4 or, equivalently, by +3fA/4. 
     FIG. 3 now shows a detailed embodiment of a half-band filter according to the invention. 
     First, however, it should be noted, with reference to FIGS. 2a-c that the halving of the sampling rate is carried out only after filtering. This sequence for the procedure according to FIGS. 2a-c should be formally adhered to. However, according to the invention, the half-band filter can be divided into two branches, each of which is supplied from the start with every second sample of the input signal. However, this means nothing other than that the halving of the sampling rate can take place directly at the filter input, as shown schematically in the block circuit diagram of FIG. 1. 
     Accordingly, the detailed circuit embodiments of FIGS. 3 and 4 include an input-side demultiplexer switch Sw which supplies the input signal s(kT) to the upper branch and then to the lower branch, in each case at the rhythm of the sampling rate fA&#39;=fA/2. 
     Both FIG. 3 and FIG. 4 show, as an example, a realization for a filter length of N=11. Accordingly, the lower branch incorporates a delay member 4T with a time delay of (N-3)·T/2=4T, while the upper branch includes a chain of five delay members 2T with a time delay of 2T. 
     The circuit arrangement of FIG. 3 can be employed for two variations namely for a modulation phase angle φ0=0 and φ0=π corresponding to m=0 and m=2. The output signal of the delay member of the lower branch is weighted (multiplied) with h(0)=1/4 and thus yields the real component S r  (2kT) of the output signal. For m=2 weighting occurs with -1/4. The further processing of the upper branch now takes place in such a way that (N+1)/4=3 difference signals are formed: 
     the first difference signal equals the output signal of the first delay member minus the input signal of the last delay member; 
     the second difference signal equals the output signal of the second delay member minus the input signal of the penultimate delay member; and 
     the third difference signal equals the output signal of the third delay member minus the input signal of the third last, i.e. the middle, delay member. 
     Next, these difference signals are weighted (multiplied) and summed by adder A and thereby yield the imaginary component of output signal s(2kT). The weighting is effected according to the following tables. 
     Examples for N=11 and h(-l)=h(l), where l=0, 1, . . . , 5, corresponding to the prototype half-band filter according to the frequency response curve of FIG. 2a: 
     
                       TABLE 1______________________________________m = 0 (for m = 2 in each case with the opposite sign for thecomplex coefficients h = Re(h) + jJm(h))l     -5      -3      -1    0    1     3     5______________________________________Re(h) 0       0       0     h(0) 0     0     0Im(h) -h(5)   h(3)    -h(1) 0    h(1)  -h(3) h(5)______________________________________ 
    
     
                       TABLE 2______________________________________m = 1 (for m = 3 in each case with the opposite sign for thecomplex coefficients)l     -5      -3      -1    0    1     3     5______________________________________Re(h) h(5)    -h(3)   h(1)  0    -h(1) h(3)  -h(5)Im(h) 0       0       0     h(0) 0     0     0______________________________________ 
    
     The realization according to FIG. 4 takes place in the same manner as that in FIG. 3; the sole difference is in the other null phase value φ0=m·π/2 where m=1 and 3, the only consequence of which is a different weighting and an exchange of filter branch outputs. 
     FIG. 5 shows the block circuit diagram for the reversed use of the half-band filter of FIG. 1, namely for the generation of a real output signal from a complex input signal. To this end, there must be a transposition of the circuits presented above, which results in a reversal of the directions of all arrows and the replacement of a branching switch BS w  for adder A and vice versa, as well as the replacement of a demultiplexer with a multiplexer. In a corresponding manner, the circuit embodiment of FIG. 6 is derived from FIG. 3 and the circuit of FIG. 7 is derived from FIG. 4. Thus, both FIGS. 6 and 7 show, as an example, a realization for a filter length N=11 where m=0 or 2 in FIG. 6, and m=1 or 3 in FIG. 7. 
     In FIG. 8, the real input signal s(kT) is fed to digital half-band filter DF which generates therefrom the complex output signal s(kT). 
     FIG. 9a shows the amplitude frequency response of a prototype half-band filter; its pass band extends from -fA/4 +Δf to +fA/4-Δf (half value) and its stop band also has a width of fA/2 -2Δf. It is a further characteristic of the half-band filter that the transition from the stop band to the pass band is steady and takes place over a width of 2Δf. This transition region is symmetrical to fA/4. A further characteristic of the half-band filter is that its ripple is the same in the pass band as in the stop band, namely δ1=δ2 =δ. In such a filter, there results a pulse response h(l) where l=0 to N-1 and the filter length N is odd with the result that every second value is equal to zero, except for the central main value (see in this connection also FIG. 2 at page 233 of the above-cited paper by Bellanger et al). 
     FIG. 9b shows the frequency response |H|. It can be seen that this frequency response is shifted to the right by the frequency fA/4 relative to the frequency response of the prototype half-band filter. FIG. 9b additionally shows the spectrum |S| of a real input signal s(kT) sampled at sampling frequency fA. Due to the sampling at fA, this signal is periodically repeated in frequency ranges [m·fA, (m+1/2)·fA] in the normal position and in frequency ranges [(m+1/2)·fA, (m+1)·fA] in the inverse position where m=. . . , -1, 0 +1, . . . Thus the inverse position of real input signal s(kT), applied to the half-band filter according to the invention without a change in sampling rate, and of course all of its repetitions are suppressed between fA/2 and fA and at the same time a complex signal s(kT) is generated, (see FIG. 9c). 
     At this point it should be mentioned that a complex signal in the inverse position is obtained at the output of the half-band filter if the frequency response of the prototype half-band filter of FIG. 9a is shifted by -fA/4 or, the equivalent, by +3fA/4. 
     FIG. 10 now shows a detailed embodiment of a half-band filter according to the invention. 
     FIG. 10 as well as FIG. 14 shows exemplary realizations for a filter length N=11 including a chain of six delay members (2T, T), four of which having a delay time of 2T and two, which are disposed symmetrically between the other four delay members, a delay time of T. 
     The circuit of FIG. 10 can be employed for two realizations of the invention, namely for a modulation phase angle φ0=0 and φ0=π, corresponding to m =0 and m=2. The output signal of the delay members of the left half of the chain is weighted (multiplied) with h(0)=1/2 and thus provides the real component s r  (kT) of the output signal. For m=2, weighting occurs with -1/2. The further processing in the delay chain now takes place in such a way that (N+1)/4=3 difference signals are formed: 
     the first difference signal is equal to the input 
     signal of the first delay member minus the output signal 
     of the last delay member; 
     the second difference signal is equal to the input signal of the second delay member minus the output signal of the penultimate delay member; and 
     the third difference signal is equal to the input signal of the third delay member minus the output signal of the third last, i.e. the middle, delay member on the right. 
     Next, these difference signals are weighted (multiplied) and summed by an adder A and thereby yield the imaginary component of output signal s(kT) The weighting is effected according to the following tables. 
     Examples for N=11 and h(-l)=h(l), where l=0, 1, . . . 5, corresponding to the prototype half-band filter according to the frequency response curve of FIG. 2a: 
     
                       TABLE 11______________________________________m = 0 (for m = 2 in each case with the opposite sign for thecomplex coefficients h = Re(h) + jJm(h))l     -5      -3      -1    0    1     3     5______________________________________Re(h) 0       0       0     h(0) 0     0     0Jm(h) -h(5)   h(3)    -h(1) 0    h(1)  -h(3) h(5)______________________________________ 
    
     
                       TABLE 12______________________________________m = 1 (for m = 3 in each case with the opposite sign for thecomplex coefficients)l     -5      -3      -1    0    1     3     5______________________________________Re(h) h(5)    -h(3)   h(1)  0    -h(1) h(3)  -h(5)Jm(h) 0       0       0     h(0) 0     0     0______________________________________ 
    
     The realization according to FIG. 11 takes place in the same manner as that in FIG. 10; the sole difference is in the other null phase value φ0=m·π/2 where m=1 and 3, the only consequence of which is a different weighting. 
     FIG. 12 shows the block circuit diagram for the reversed use of the half-band filter of FIG. 8, namely for the generation of a real output signal from a complex input signal. For this purpose, there must be a transposition of the circuits presented above, which results in a reversal of the directions of all arrows and the replacement of a branching switch BS w  adder A and vice versa. In a corresponding manner, the circuit embodiment of FIG. 13 is derived from FIG. 10 and the circuit of FIG. 14 is derived from FIG. 11. Accordingly, both FIGS. 13 and 14 show a realization for a filter length N=11 where m=0 or 2 in FIG. 13, and m=1 or 3 in FIG. 14.