CA1281382C - Non-recursive half-band filter - Google Patents
Non-recursive half-band filterInfo
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- CA1281382C CA1281382C CA000555352A CA555352A CA1281382C CA 1281382 C CA1281382 C CA 1281382C CA 000555352 A CA000555352 A CA 000555352A CA 555352 A CA555352 A CA 555352A CA 1281382 C CA1281382 C CA 1281382C
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Abstract
Abstract The present invention relates to a non-recursive half-band filter and permits the conversion of a real input signal into a complex output signal, in that its impulse response is modulated onto the complex carrier of a frequency of a 1/4 of the sampling frequency, with the zero phase of this frequency amounting to integer multiples of .pi./2, and with the sampling rate being halved (figure 3), and also permits the conversion of a complex input signal into a real output signal, in that its pulse response is modulated onto the complex carrier of a frequency of the half input sampling rate, the zero phase of this frequency being integer multiples of .pi./2 and the sampling fequency being doubled.
Description
2 ~ 3~ 27371-174 The present invention relatès to a non-recursive half-band fil-ter. Such fil-ters have become known from the paper titled "Interpolation, Extrapolationl and Reduction of Computation Speed in Digital Filters," by Bellanger et al., in IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-22, No. 4, August, 1974, pp. 231 et. seq.
The known half-band filters process real input signals into real output signals.
It was the task 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 a less costly manner.
According to a broad aspect of the invention there is pro-vided a non-recursive half-band filter with complex coefficients for processing a real value input signal s(kT) by halving the sampling frequency fA = l/T and for converting this real value input signal s(kT) into a complex value output signal s(2kT), wherein its complex coefficients _(l) with 1 = -(N-1)/2 -to (N-1)/2 and the unstraight filter length N, have alternating purely real and purely imaginary values, therefore no complex values in the fullest sense; wherein the impulse response of a half-band filter h(l) with exclusively real values, with the characteristics h(l) = h(-l) for all ¦11 < (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 = l/T to give h(l) = h(l) . ei (_ lfA/4fA) ~ 0) = j_l . ei~ . h(l); and wherein the null phase ~0 of this complex carrier is an integer multiple m of ~/2 ( 0 = m. ~/2 with m = 0,1,2,3,...).
The known half-band filters process real input signals into real output signals.
It was the task 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 a less costly manner.
According to a broad aspect of the invention there is pro-vided a non-recursive half-band filter with complex coefficients for processing a real value input signal s(kT) by halving the sampling frequency fA = l/T and for converting this real value input signal s(kT) into a complex value output signal s(2kT), wherein its complex coefficients _(l) with 1 = -(N-1)/2 -to (N-1)/2 and the unstraight filter length N, have alternating purely real and purely imaginary values, therefore no complex values in the fullest sense; wherein the impulse response of a half-band filter h(l) with exclusively real values, with the characteristics h(l) = h(-l) for all ¦11 < (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 = l/T to give h(l) = h(l) . ei (_ lfA/4fA) ~ 0) = j_l . ei~ . h(l); and wherein the null phase ~0 of this complex carrier is an integer multiple m of ~/2 ( 0 = m. ~/2 with m = 0,1,2,3,...).
3 ~X~3~3~ 27371~174 According to another broad aspect of the invention there is provided a non-recursive half-band filter with complex coefficients for processing a complex value input signal s(2kT) and for doubl-ing the sampling frequency fA' = 1/2T to fA =2fA' and to convert this complex value input signal s(2kT) into a real value output signal s(kT), wherein its complex coefficients h(l) with 1 = -(N-1)/2 to (N-1)/2 and the unstraight filter length N have alter-nating purely real and purely imaginary values, therefore no com-plex values in the fullest sense; wherein the impulse response of a half-band filter h(l) with exclusively real values, with the characteristics h(l) = h(-l) for all~ (N-1)/2 and h(l) = 0 for 1 = +2, +4, ... , is modulated onto the complex carrier of a fre-quency of + 1/4 of the input sampling frequency fA = l/T to give h(l) = h(l) ej(+ 2~1fA/4fA~ + ~0) j+l e ~ h(l); and wherein the null phase ~ 0 of this complex carrier is an integer multiple m of ~/2 (~0 = m .~/2 with m = 0, 1, 2, 3, ...).
The non-recursive half-band filter according to the present 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, or the conversion of complex digital input signals into real digital output signals with a simultaneous increase in the scanning frequency by a fac- /
tor of 2. This relatively inexpensive half-band filter is thus suitable as a digital pre-filter or post-filter for digital sys-tems that are used to process complex signals and as a digital partial filter in a system of antialiasing filters for band lim-iting whilst complying with the sampling theorem. The advantage of the half-band filter lies in its linear phase and simultaneously 3a ~ 3~ 27371-174 low cost, whereby in each instance the slnallest possible sampling frequency required on the basis of the sampling theorem can be used.
The following description is based on the figures appendad hereto.
Figure 1 is the block circuit diagram for the digital filter according to the present invention.
In figures 2a to 2c, several amplitude responses of half-band filters are plotted over frequency.
Figures 3 and 4 show particularly favourable circuit varia-tions of the half-band filter.
Figure 5 is the block circuit diagra~ of a half-band filte~
used to process a complex input signal into a real output signal.
Figure 6 is a detailed circuit of the filter shown in figure 5, this circuit having been developed from that in figure 3 by transposition, i.e., by reversing all the directions indicated by the arrows and r~placing a branching switch by an adder and vice versa, and by replacing of a demultiplexer by a multiplexer.
The circuit shown in figure 7 resulted from that in figure 4 in a like manner.
4 1~ 3~'~
In figure 1, the real input signal s(kT) is applied to the half-band filter DF by halving the sampling rate, which generates the complex output signal s(2kT).
The amplitude frequency response of a prototype half-band filter is shown in figure 2a; the pass band of this extends from -fA/4 + Q f to -~fA/4 - d f, and its stop band is also fA/2 - 2 ~ f wide. It is also characteristic of the half-band filter that the transition from the stop-band to the pass-band is constant and takes place at a width of 2 ~ f. This transmission band is symmetrical about fA/4. A further characteristic of the half-band filter is that its ripple in the transmission and the stop band is squal, i.e., ~1 = v2 = ~. In such a filter there is a impulse response h(l) with 1 = o to N-l for the unstraight filter length N, and it follows that each second value is identically zero, with the exception of the central main value (see figure 2, p. 233 in the paper by Bellanger et al that is cited above). Figure 2b shows the frequency range ¦H¦. It can be seen that this frequency range has been shifted to the right to a point around the frequency fA/4 relative to the frequency range of the prototype half~band filter. In addition, in figure 2b the spectrum ¦S¦ of a real input signal s(kT) sampled at the sampling frequency fA has been inserted; because of the sampling with fA this repeats periodically in the frequency range [m.fA, (m+l/2).fA] in the normal position and in the frequency range [(m+l/2)0fA, (m~l).fA)] in the inverted position with m = ~ 1, O, +1... 0 The input signal s(kT), applied to the 5 ~ z~3~3~3~
llf-band filter according to the present invention without any change in the sampling rate, would thus suppress the inverted position between fA/2 and fA, and naturally with all repetitions, and also generate a complex signal s~kT). Halving the sampling rate now results in the dasired spectra, with the normal position being repeated in each instance in the raster of fA/2 = fA', thus in the new sampling rate; see figure 2c.
At this point, it should be noted that a complex signal is obtained at the output of the half-band filter if the frequency range of the prototype half-band filter according to figure 2a is shifted by -fA/4 or, aquivalently, by +3fA/~.
Figure 3 now shows a detailed embodiment of a half-band filter according to the present invention. First, however, with reference to figure 2, it should be stated that the halving of the sampling rate is only carried out after filtering. This sequence for the process as in figure 2 should to be adhered to strictly; according to the invention, however, the half-band filter can be divided into two halves, each of which is supplied from the start with every second sample of the input signal.
This means nothing else, however, than that the halving of the sampling rate can take place directly at the filter in pu~, as is shown in the block schematic diagram in figure 1.
Accordingly, the detailed circuits in figures 3 and 4 incorporate an input-side demultiplexer switch which supplies, on the one hand, the upper branch, and on the other, the lower ~2B13~3~
~anch with the input signal s(kT), in each instance in time at the sampling rate fA' = fA/2.
Both figure 3 and figure 4 show, as an example, a realisation for a filter length of N = 11. Accordingly, the lower branch incorporates a delay element of the time delay (N-3).T/2 = 4T, whereas the upper branch incorporates a chain of five delay elements of time delay 2T. Fi~ure 3 shows two realisations, namely, for a modulation phase angle ~ O = O and ~o = ~ for m = O and m = 2, respectively.
The output signal of the delay element of the lower ~ranch is weighted (multiplied) at h(5) = 1/2 (multiplied) and thus yields the imaginary portion si(2kT) of the output signal. At m = 2, this is weighted at -1/2. The further processing of the upper branch now takes place such that (N+~)/4, i.e., three differential signals are formed:
The first differential signal equals the input signal of the first, minus the ouput signal of the last delay element;
The second differential signal equals the lnput signal of the second, minus the output signal of the second last delay element;
The third differential signal equals the input signal of the third, minus the output signal of the third last, which is to say, the middle delay element.
Next, these differential signals are weighted ~multiplied), and summed and thereby yield the real component of the output 7 ~ 313~3~
c;gnal s(2kT). The weighting is effected according to the following tables.
Examples for N=ll and ht-l) = h(l) for 1 = o, 1, ...5, according to the prototype half-band filter according to the frequency range in figure 2a:
Table 1:
m=O (m=2 with in each instance a changed sign of the complex coefficients h = Ra(h)+jJm(h)) Re(h) O O O h(O) 0 0 0 Jm(h) -h(5) h(3) -h(l) h(l) -h(3) h~5) Table 2:
m=1 (m=3 with in each instance a changed sign of the complex coefficients) Re(h) h(5)-h(3) h(l) -h(l) h(3) -h(5) ... . . ....
Jm(h) O h(O) O O O
The realisation as in figure 4 takes place in the same manner as in the one shown in figure 3; the sole difference is in the other null phase value ~0 = m .Tr/~ with m = 1 and 3, the ~ly consequences of which are a diffe~rent welghting and an exchange of the filter branch outputs.
Figure 5 shows the block schematic diagram for the reversed use of the half-band filter as in figure 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 circuit shown above, which results in a reversal of the directions of all the arrows and the replacement of a branch by an adder, and vice versa, as well as the replacement of a demultiplexer by a multiplexer. In a similar manner, the example circuit in figure 6 is derived from figure 3, and the circuit in figure 7 is derived from that in figure 4.
The non-recursive half-band filter according to the present 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, or the conversion of complex digital input signals into real digital output signals with a simultaneous increase in the scanning frequency by a fac- /
tor of 2. This relatively inexpensive half-band filter is thus suitable as a digital pre-filter or post-filter for digital sys-tems that are used to process complex signals and as a digital partial filter in a system of antialiasing filters for band lim-iting whilst complying with the sampling theorem. The advantage of the half-band filter lies in its linear phase and simultaneously 3a ~ 3~ 27371-174 low cost, whereby in each instance the slnallest possible sampling frequency required on the basis of the sampling theorem can be used.
The following description is based on the figures appendad hereto.
Figure 1 is the block circuit diagram for the digital filter according to the present invention.
In figures 2a to 2c, several amplitude responses of half-band filters are plotted over frequency.
Figures 3 and 4 show particularly favourable circuit varia-tions of the half-band filter.
Figure 5 is the block circuit diagra~ of a half-band filte~
used to process a complex input signal into a real output signal.
Figure 6 is a detailed circuit of the filter shown in figure 5, this circuit having been developed from that in figure 3 by transposition, i.e., by reversing all the directions indicated by the arrows and r~placing a branching switch by an adder and vice versa, and by replacing of a demultiplexer by a multiplexer.
The circuit shown in figure 7 resulted from that in figure 4 in a like manner.
4 1~ 3~'~
In figure 1, the real input signal s(kT) is applied to the half-band filter DF by halving the sampling rate, which generates the complex output signal s(2kT).
The amplitude frequency response of a prototype half-band filter is shown in figure 2a; the pass band of this extends from -fA/4 + Q f to -~fA/4 - d f, and its stop band is also fA/2 - 2 ~ f wide. It is also characteristic of the half-band filter that the transition from the stop-band to the pass-band is constant and takes place at a width of 2 ~ f. This transmission band is symmetrical about fA/4. A further characteristic of the half-band filter is that its ripple in the transmission and the stop band is squal, i.e., ~1 = v2 = ~. In such a filter there is a impulse response h(l) with 1 = o to N-l for the unstraight filter length N, and it follows that each second value is identically zero, with the exception of the central main value (see figure 2, p. 233 in the paper by Bellanger et al that is cited above). Figure 2b shows the frequency range ¦H¦. It can be seen that this frequency range has been shifted to the right to a point around the frequency fA/4 relative to the frequency range of the prototype half~band filter. In addition, in figure 2b the spectrum ¦S¦ of a real input signal s(kT) sampled at the sampling frequency fA has been inserted; because of the sampling with fA this repeats periodically in the frequency range [m.fA, (m+l/2).fA] in the normal position and in the frequency range [(m+l/2)0fA, (m~l).fA)] in the inverted position with m = ~ 1, O, +1... 0 The input signal s(kT), applied to the 5 ~ z~3~3~3~
llf-band filter according to the present invention without any change in the sampling rate, would thus suppress the inverted position between fA/2 and fA, and naturally with all repetitions, and also generate a complex signal s~kT). Halving the sampling rate now results in the dasired spectra, with the normal position being repeated in each instance in the raster of fA/2 = fA', thus in the new sampling rate; see figure 2c.
At this point, it should be noted that a complex signal is obtained at the output of the half-band filter if the frequency range of the prototype half-band filter according to figure 2a is shifted by -fA/4 or, aquivalently, by +3fA/~.
Figure 3 now shows a detailed embodiment of a half-band filter according to the present invention. First, however, with reference to figure 2, it should be stated that the halving of the sampling rate is only carried out after filtering. This sequence for the process as in figure 2 should to be adhered to strictly; according to the invention, however, the half-band filter can be divided into two halves, each of which is supplied from the start with every second sample of the input signal.
This means nothing else, however, than that the halving of the sampling rate can take place directly at the filter in pu~, as is shown in the block schematic diagram in figure 1.
Accordingly, the detailed circuits in figures 3 and 4 incorporate an input-side demultiplexer switch which supplies, on the one hand, the upper branch, and on the other, the lower ~2B13~3~
~anch with the input signal s(kT), in each instance in time at the sampling rate fA' = fA/2.
Both figure 3 and figure 4 show, as an example, a realisation for a filter length of N = 11. Accordingly, the lower branch incorporates a delay element of the time delay (N-3).T/2 = 4T, whereas the upper branch incorporates a chain of five delay elements of time delay 2T. Fi~ure 3 shows two realisations, namely, for a modulation phase angle ~ O = O and ~o = ~ for m = O and m = 2, respectively.
The output signal of the delay element of the lower ~ranch is weighted (multiplied) at h(5) = 1/2 (multiplied) and thus yields the imaginary portion si(2kT) of the output signal. At m = 2, this is weighted at -1/2. The further processing of the upper branch now takes place such that (N+~)/4, i.e., three differential signals are formed:
The first differential signal equals the input signal of the first, minus the ouput signal of the last delay element;
The second differential signal equals the lnput signal of the second, minus the output signal of the second last delay element;
The third differential signal equals the input signal of the third, minus the output signal of the third last, which is to say, the middle delay element.
Next, these differential signals are weighted ~multiplied), and summed and thereby yield the real component of the output 7 ~ 313~3~
c;gnal s(2kT). The weighting is effected according to the following tables.
Examples for N=ll and ht-l) = h(l) for 1 = o, 1, ...5, according to the prototype half-band filter according to the frequency range in figure 2a:
Table 1:
m=O (m=2 with in each instance a changed sign of the complex coefficients h = Ra(h)+jJm(h)) Re(h) O O O h(O) 0 0 0 Jm(h) -h(5) h(3) -h(l) h(l) -h(3) h~5) Table 2:
m=1 (m=3 with in each instance a changed sign of the complex coefficients) Re(h) h(5)-h(3) h(l) -h(l) h(3) -h(5) ... . . ....
Jm(h) O h(O) O O O
The realisation as in figure 4 takes place in the same manner as in the one shown in figure 3; the sole difference is in the other null phase value ~0 = m .Tr/~ with m = 1 and 3, the ~ly consequences of which are a diffe~rent welghting and an exchange of the filter branch outputs.
Figure 5 shows the block schematic diagram for the reversed use of the half-band filter as in figure 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 circuit shown above, which results in a reversal of the directions of all the arrows and the replacement of a branch by an adder, and vice versa, as well as the replacement of a demultiplexer by a multiplexer. In a similar manner, the example circuit in figure 6 is derived from figure 3, and the circuit in figure 7 is derived from that in figure 4.
Claims (9)
1. A non-recursive half-hand filter with complex coefficients for processing a real value input signal s(kT) by halving the sampling frequency fA = 1/T and for converting this real value input signal s(kT) into a complex value output signal s(2kT), wherein its complex coefficients h(1) with 1 = -(N-1)/2 to (N-1)/2 and the unstraight filter length N, have alternating purely real and purely imaginary values, therefore no complex values in the fullest sense; wherein the impulse response of a half-band filter h(1) with exclusively real values, with the characteristics h(1) =
h(-1) for all ¦1¦ ? (N-1)/2 and h(1) = 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 give h(1) = h(1) ej(? 2.pi.1fA/4fA) + ?0) = j?1 ? ej?0 ? h(1);
and wherein the null phase ?0 of this complex carrier is an integer multiple m of .pi./2 ( 0 = m ..pi./2 with m = 0, 1,
h(-1) for all ¦1¦ ? (N-1)/2 and h(1) = 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 give h(1) = h(1) ej(? 2.pi.1fA/4fA) + ?0) = j?1 ? ej?0 ? h(1);
and wherein the null phase ?0 of this complex carrier is an integer multiple m of .pi./2 ( 0 = m ..pi./2 with m = 0, 1,
2, 3, ...).
2. A non-recursive half-band filter with complex coefficients for processing a complex value input signal s(2kT) and for doubling the sampling frequency fA' = 1/2T to fA =2fA' and to convert this complex value input signal s(2kT) into a real value output signal s(kT), wherein its complex coefficients h(1) with 1 = -(N-1)/2 to (N-1)/2 and the unstraight filter length N have alternating purely real and purely imaginary values, therefore no complex values in the fullest sense; wherein the impulse response of a half-band filter h(1) with exclusively real values, with the characteristics h(1) = h(-1) for all ¦1¦ ? (N-1)/2 and h(1) = 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 give h(1) = h(1) . ej(? 2.pi.1fA/4fA) + ?
0) = j?1 . ej?0 ? h(1); and wherein the null phase ?0 of this complex carrier is an integer multiple m of .pi./2 (?0 =
m . .pi./2 with m = 0, 1, 2, 3, ...).
2. A non-recursive half-band filter with complex coefficients for processing a complex value input signal s(2kT) and for doubling the sampling frequency fA' = 1/2T to fA =2fA' and to convert this complex value input signal s(2kT) into a real value output signal s(kT), wherein its complex coefficients h(1) with 1 = -(N-1)/2 to (N-1)/2 and the unstraight filter length N have alternating purely real and purely imaginary values, therefore no complex values in the fullest sense; wherein the impulse response of a half-band filter h(1) with exclusively real values, with the characteristics h(1) = h(-1) for all ¦1¦ ? (N-1)/2 and h(1) = 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 give h(1) = h(1) . ej(? 2.pi.1fA/4fA) + ?
0) = j?1 . ej?0 ? h(1); and wherein the null phase ?0 of this complex carrier is an integer multiple m of .pi./2 (?0 =
m . .pi./2 with m = 0, 1, 2, 3, ...).
3. A non-recursive half-band filter as defined in claim 1, wherein each second sampling value of the input signal s(kT) is routed into a chain of (N-1)/2 delay elements of delay time 2T; wherein in each instance differential signals are formed from output signals of the last delay element minus the input signal of the first delay element equal to the first differential signal, output signal of the second last delay element minus the input signal of the second delay element equal to the second differential signal, output signal of the third-last delay element minus input signal of the third delay element equal to the third differential signal, and so on; wherein these differential signals are subjected to weighting (multiplication) with a value h(1) of the impulse response, summed, and then yield the real or imaginary part of the filter output signal s(2kT); wherein in a second branch there is a delay element with the time delay T. (N-3)/2, into which is routed each second sample that is shifted relative to the above sampling values the output signal of which, weighted with the value h(0) yields the imaginary or the real part of the filter output signal s(2kT).
4. A non-recursive half-band filter as defined in claim 3 with N=11 and m=1, wherein the first differential signal is weighted with -h(5), the second differential signal with h(3) and the third with -h(1); wherein h(0) = 1/2; and wherein the sum of the differential signals yields the real part sr(2kT) and the signal weighted with h(0) yields the imaginary part si(2kT)
5. A non-recursive half-band filter as defined in claim 3 with n=11 and m=3 wherein the first differential signal is weighted with h(5) the second with -h(3) and the third with h(1); wherein h(0) = -1/2; and wherein the sum of the differential signals yields the real part sr(2kT) and the signal weighted with h(0) yields the imaginary part si(2kT)
6. A non-recursive half-band filter as defined in claim 3 with n=11 and m=0 wherein the first differential signal is weighted with h(5) the second with -h(3) and the third with h(1); and wherein the sum of the differential signals yields the imaginary part si(2kT) and the signal weighted with h(0)=1/2 yields the real part sr(2kT).
7. A non-recursive half-band filter as defined in claim 3, with n=11 and m=2, wherein the first differential signal is weighted with -h(5), the second with h(3), and the third with -h(1); and wherein the sum of the differential signals yields the imaginary part si(2kT) and the signal weighted with h(0) = -1/2 yields the real part sr(2kT).
8. A non-recursive half-band filter as defined in claim 2, wherein a chain of (N-1)/2 delay elements of time delay 2T
is provided; wherein the real part sr(2kT) weighted with a value of h(1) of the impulse response is passed to the first delay element of this chain and subtracted from the output signal of the last delay element of this chain, which differential yields every second sample of the real filter output signal s(kT); wherein on the transverse signal of this delay element chain, momentary values of the real part sr(2kT) of the filter input signal additionally weighted with a value h(1) of the impulse response are added at the further points; and wherein a further delay element of time delay T.(N-3)/2 is provided, in the input of which is inserted the imaginary part si(2kT) that is weighted with h(0), and the output of which yields each time shifted second sample of the real filter output signal s(kT).
is provided; wherein the real part sr(2kT) weighted with a value of h(1) of the impulse response is passed to the first delay element of this chain and subtracted from the output signal of the last delay element of this chain, which differential yields every second sample of the real filter output signal s(kT); wherein on the transverse signal of this delay element chain, momentary values of the real part sr(2kT) of the filter input signal additionally weighted with a value h(1) of the impulse response are added at the further points; and wherein a further delay element of time delay T.(N-3)/2 is provided, in the input of which is inserted the imaginary part si(2kT) that is weighted with h(0), and the output of which yields each time shifted second sample of the real filter output signal s(kT).
9. A non-recursive half-band filter as defined in claim 8, with m=0 and N=11, wherein the weighting of the applied momentary values of the real part sr(2kT) of the filter input signal is effected as follows:
at the input of the first delay element h(5) at the input of the second delay element -h(3) at the input of the third delay element h(1) at the input of the fourth delay element -h(1) at the input of the fifth delay element h(3) at the output of the fifth delay element -h(5) and wherein h(0) = 1/2.
at the input of the first delay element h(5) at the input of the second delay element -h(3) at the input of the third delay element h(1) at the input of the fourth delay element -h(1) at the input of the fifth delay element h(3) at the output of the fifth delay element -h(5) and wherein h(0) = 1/2.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA000555352A CA1281382C (en) | 1987-12-24 | 1987-12-24 | Non-recursive half-band filter |
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| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA000555352A CA1281382C (en) | 1987-12-24 | 1987-12-24 | Non-recursive half-band filter |
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| CA1281382C true CA1281382C (en) | 1991-03-12 |
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1987
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