Non-recursive half-band filter

A non-recursive half-band filter having a filter length N for processing a complex input signal s(2kT) and for converting the complex input signal for into a real output signal s(kT) at double the input sampling frequency, said half-band filter comprising:

• input means for receiving samples of a complex input signal s(2kT) at a sampling frequency fA'=1/2T;
• means for processing and converting the received complex input signal into a real output signal s(kT) at an output sampling frequency fA=2fA' said means having complex coefficients h(l), where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, with alternating purely real and purely imaginary values, said processing and converting means including means for modulating the pulse response of a half-band filter h(l) with exclusively real values and h(l)=h(-l) for all .vertline.l.vertline..ltoreq.(N-1)/2 and h(l)=0 for l=.+-.2, .+-.4, . . . , onto a complex carrier of a frequency of .+-.1/4 of the output sampling frequency fA=2fA' to yield
• h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l .multidot.e.sup.j.phi.0 .multidot.h(l),
• with the null phase .phi.0 of the complex carrier being an integer multiple m of .pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . .);
• and wherein said complex input signal includes a real component and an imaginary component, and said processing and converting means is divided into first and second branches, said first branch comprising a chain (N-1)/2 delay members each having a time delay of 2T, means for weighting said real and imaginary components of the complex input signal and means for forming a plurality of difference signals, the weighting means weights the real component s.sub.r (2kT) with a value equal to a function of h(l) of the pulses response and said forming means feeds the weighted component to the first delay member of the chain of delay members and subtracts the weighted component from the output signal of the last delay member of the chain to produce a difference signal, said difference signal furnishing every second sample of the real filter output signal s(kT);
• said weighting means weights additional momentary values of the real component s.sub.r (2kT) of the filter input signal with a value equal to other functions of h(l) of the pulse response and said forming means adds these additional weighted values to a transversal signal of the chain of delay members at further points;
• and said second branch has a further delay member with a time delay of T.multidot.(N-3)/2 whose input receives the imaginary component s.sub.i (2kT) of the complex input signal which has been weighted with a value equal to a function of h(0) and whose output yields every second time-shifted sample of the real filter output signal s(kT).

A non-recursive half-band filter having a pulse response h(l), where l=-(N-1)/2 to (N-1)/2 and N is an odd filter length, said filter comprising means for receiving a real input signal s(kT) and means for converting the real input signal s(kT) into a complex output signal s(kT), where k is a running index, while maintaining a sampling frequency fA=1T, said converting means modulating the pulse response h(l) onto a complex carrier with a frequency of .+-.1/4 of the sampling frequency fA=1/T, to yield

• h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA'.phi.0) =j.sup..+-.l .multidot.e.sup.j.phi.0 .multidot.h(l),
• with the null phase .phi. 0 of this frequency being an integer multiple m of .pi./2 (.phi.0=m.multidot..pi./2 where m=1, 2, 3, . . .), and wherein said converting means includes a chain of (N-1)/2 delay members having a center delay member, means for forming a plurality of difference signals and means for weighting the formed difference signals, where each sample of the input signal s(kT) is routed into the chain of (N-1)/2 delay members each having a delay time of 2T, and the center delay member is divided into two members each having a delay time of T;
• said difference signal forming means subtracting from the output signal of the last delay member of the chain the input signal of the first delay member of the chain to form a first difference signal, subtracting from the output signal of the penultimate delay member of the chain the input signal of the second delay member of the chain to form a second difference signal, subtracting from the output signal of the third to last delay member of the chain the input signal of the third delay member of the chain to form a third difference signal, and so on until the output signal of each delay member is processed;
• said weighting means includes an adder and weighting each said difference signal by a value equal to a respective function of h(l) of the pulse response and summing the weighted difference signals via the adder to yield either the real or the imaginary component of the filter output signal s(kT);
• said forming means forms a fourth difference signal from the center delay member of the chain where the input signal is delayed by a delay time T.multidot.(N-1)/2 and is weighted with a value equal to the function j(0), which results in the other of the imaginary component and the real component of the filter output signal s(kT).

A non-recursive half-band filter having a filter length N and a pulse response h(l) where l=-(N-1)/2 to (N-1)/2 and the filter length N is odd, said half-band filter comprising:

• means for receiving a complex input signal, and means for converting said complex input signal s(kT), where k is a running index, into a real output signal s(kT), while maintaining a sampling frequency fA=1/T, said converting means modulating the pulse response h(l) with reference to the sampling frequency fA, onto a complex carrier at a frequency of .+-.fA/4 to yield
• h(l)=h(l).multidot.e.sup.j(.+-.2.pi.lfA/4fA+.phi.0) =j.sup..+-.l .multidot.e.sup.j.phi.0 .multidot.h(l)
• with the null phase .phi. 0 of this frequency being an integer multiples m of .pi./2 (.phi.0=m.multidot..pi./2 where m=0, 1, 2, 3, . . .), and wherein said complex input signal has a real component and an imaginary component;
• and said converting means includes a chain of (N-1)/2 delay members each having a time delay of 2T and a center delay member divided into two members, with each divided member having a delay time of T, means for weighting the real and imaginary components with a value equal to a function of h(l), and means for forming a plurality of difference signals;
• and the weighting means weights the imaginary component s.sub.i (kT) with values equal to functions of h(l) of the pulse response and said forming means feeds the weighted component to the first delay member of the chain and subtracts the weighted component from the output signal of the last delay member of the chain to form a difference signal, said difference signal furnishing the real filter output signal s(kT);
• said weighting means weights additional momentary values of the imaginary component s.sub.i (kT) of the filter input signal with a value equal to other functions of h(l) of the pulse response and said forming means adds the additional weighted values to a transversal signal of the chain of delay members at further points;
• and the weighting means further weights the real component s.sub.r (kT) of the complex filter input signal with the function h(0), and additively feeds the weighted signal to the transversal signal of the chain at the center of said center delay member.