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A Most Efficient Digital Filter
The 2Path Recursive Allpass Filter
fred harris
San Diego State University
fred.harris@sdsu.edu
Many of us are introduced to digital recursive filters as mappings from analog prototype structures
mapped to the sample data domain by the bilinear ZTransform. These digitals filters are normally im
plemented as a cascade of canonic second order filters that independently form its two poles and two
zeros with two feedback and two feed forward coefficients respectively. We discuss here an alternative
recursive filter structure based on simple recursive allpass filters that use a single coefficient to form
both a pole and a zero or to form two poles and two zeros. An allpass filter has unity gain at all fre 
quencies and otherwise exhibits a frequency dependent phase shift. We might then wonder that if the
filter has unity gain at all frequencies, how it can form a stop band. We accomplish this by adjusting
the phase in each path of a 2path filter to obtain destructive cancellation of signals occupying specific
spectral bands. Thus the stopband zeros are formed by the destructive cancellation of components in
the multiple paths rather than as explicit polynomial zeros. This approach leads to a wide class of very
efficient digital filters that require only 25% to 50% of the computational workload of the standard cas
cade of canonic second order filters. These filters also permit the interchange of the resampling and fil
tering to obtain further workload reductions.
Introduction:
Allpass networks are the building blocks of every digital filter. Allpass networks exhibit unity gain at
all frequencies and a phase shift that varies as a function of frequency. Allpass networks have poles
and zeros that occur in (conjugate) reciprocal pairs. Since allpass networks have reciprocal polezero
pairs, the numerator and denominator are seen to be reciprocal polynomials. If the denominator is an
Nth order polynomial PN(Z), the reciprocal polynomial in the numerator is ZNPN(Z1). It is easily seen
that the vector of coefficients that represent the denominator polynomial is reversed to form the vector
representing the numerator polynomial. A cascade of allpass filters is also seen to be an allpass filter.
A sum of allpass networks is not allpass and we use these two properties to build our class our filters.
Every allpass network can be decomposed into a product of first and second order allpass networks,
thus it is sufficient to limit our discussion to first and second order filters which we refer to as typeI
and typeII respectively. In this paper we limited our discussion to first and second order polynomials
in Z and Z2. The transfer functions of typeI and typeII allpass networks are shown in equation 1 with
the corresponding polezero diagrams shown in figure 1.
II Type
Type
:
)(
)1(
)(,
)(
)1(
)(
I:
)(
)1(
)(,
)(
)1(
)(
2
2
1
4
4
2
2
1
2
2
21
2
2
21
2
2
2
2
11
αα αα
αα αα α
α
α
α
++ ++
=
++ ++
=
+
+
=
+
+
=
ZZ
ZZ
ZH
ZZ
ZZ
ZH
Z
Z
ZH
Z
Z
ZH
(1)
Figure 1. PoleZero Structure of TypeI and TypeII AllPass Filters
of Degrees 1 and 2
Note that the single sample delay with Ztransform Z1 (or 1/Z) is a special case of the typeI all pass
structure obtained by setting the coefficient α to zero. Linear phase delay is all that remains as the pole
of this structure approaches the origin while its zero simultaneously approaches infinity. We use the
fundamental allpass filter (Z1) as the starting point of our design process and then develop the more
general case of the typeI and typeII allpass networks to form our class of filters.
A closed form expression for the phase function of the type1 transfer function is obtained by evaluat
ing the transfer function on the unit circle. The result of this exercise is shown in (2).
2,1,)
2
(tan
)1(
)1(
atan2
=
−
+
−=
MM
θ
α
α
φ
, (2)
The phase response of the first and second order allpass typeI structures is shown in figure 2a and 2b.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0Phase Shift of TypeI AllPass Network of Order 1
Normalized Frequency (f/f
s
)
Normalized Phaseshift (theta/(2
π
)
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
Figure 2a. Phase Response of TypeI AllPass Filter, First Order Polynomial in Z1,
as Function of Coefficient α, (α = 0.9, 0.8,….,0 ,….., 0.8, 0.9)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0Phase Shift of TypeI AllPass Network of Order 2
Normalized Frequency (f/f
s
)
Normalized Phaseshift (theta/(2
π
)
0.8
0.0
0.8
0.8
0.0
0.8
Figure 2b. Phase Response of TypeII AllPass Filter, First Order Polynomial in Z2,
as Function of Coefficient α, (α = 0.9, 0.8,….,0 ,….., 0.8, 0.9)
Note that for the first order polynomial in Z1, the transfer function for α = 0 defaults to the pure delay,
and as expected, its phase function is linear with frequency. The phase function will vary with α and
this network can be thought of, and will be used as, the generalized delay element. We observe that the
phase function is anchored at its end points (0 degrees at zero frequency and 180 degrees at the half
sample rate) and that it warps with variation in α. It bows upward (less phase) for positive α and bows
downward (more phase) for negative α. The bowing phase function permits us to use the generalized
delay to obtain a specified phase shift angle at any frequency. For instance, we note that when α = 0,
the frequency for which we realize 90 degrees phase shift is 0.25 (the quarter sample rate). We can de
termine a value of α for which the 90 degree phase shift is obtained at any normalized frequency such
as at normalized frequency 0.45 (α = 0.8) or at normalized frequency 0.05 (α = 0.73).
Implementing and Combining AllPass Networks:
While the typeI allpass network can be implemented in a number of architectures we limit the discus
sion to the one shown in figure 3. This structure has a number of desirable implementation attributes
that are useful when multiple stages are placed in cascade. We observe that the single multiplier resides
in the feedback loop of the lower delay register to form the denominator of the transfer function. The
single multiplier also resides in the feed forward path of the upper delay register to form the numerator
of the transfer function. The single multiplier thus forms all the poles and zeros of this allpass network
and we call attention to this in the equivalent processing block to the right of the filter block diagram.
α
α
z
z
 M
 M

X ( Z )
X ( Z )
Y ( Z )
Y ( Z )
H ( Z )
M
Figure 3. Single Coefficient Type1 AllPass Filter Structure
TwoPath Filters
While the Mth order allpass filter finds general use in an Mpath polyphase structure, we restrict our
discussion in this paper to twopath filters. We first develop an understanding of the simplest two path
structure and then expand the class by invoking a simple set of frequency transformations. The struc 
ture of the two path filter is presented in figure 4. Each path is formed as a cascade of allpass filters in
powers of Z2. The delay in the lower path can be placed on either side of the allpass network. When
the filter is implemented as a multirate device the delay is positioned on the side of the filter operating
at the higher of the two rates where it is absorbed by the input (or output) commutator.
Figure 4. Two Path Polyphase Filter
This deceivingly simple filter structure offers a surprisingly rich class of filter responses. The 2path
allpass structure can implement halfband lowpass and highpass filters, as well as Hilbert transform
filters that exhibit minimum or nonminimum phase response. The twopath filter implements standard
recursive filters such as the Butterworth and the Elliptic filter. A MATLAB routine, tony_des2 that
computes the coefficients of the twopath filter and a number of its frequency transformed variants is
available from the author via an email request. Also, as suggested earlier, the halfband filters can be
configured to embed a 1to2 upsampling or a 2to1 downsampling operation within the filtering
process.
The prototype halfband filters have their 3dB band edge at the quarter sample rate. Allpass fre
quency transformations applied to the 2path prototype form arbitrary bandwidth lowpass and high
pass complementary filters, and arbitrary center frequency passband and stopband complementary
filters. Zero packing the time response of the 2path filter, another trivial allpass transformation,
causes spectral scaling and replication. The zeropacked structure is used in cascade with other filters
in iterative filter designs to achieve composite spectral responses exhibiting narrow transition band
widths with loworder filters.
The specific form of the prototype halfband 2path filter is shown in figure 5. The number of poles (or
order of the polynomials) in the two paths differ by precisely one, a condition assured when the num
ber of filter segments in the lower leg is equal to or is one less than the number of filter segments in the
upper leg. The structure forms complementary low pass and high pass filters as the scaled sum and dif
ference of the two paths.
Figure 5. TwoPath AllPass Filter
The transfer function of the 2path filter shown in figure 5 is shown in equation (3).
2 1 2
0 1
2 2
0
2
2
2
( ) ( ) ( )
( ) ( ), 0,1
,
1 ( , )
( )
,( , )
Ki
k
H Z P Z Z P Z
P Z H Z i
i i k
i k Z
H Z
i k Z i k
αα
−
=
= ±
= =
+
=+
∏
(3)
In particular, we can examine the simple case of two allpass filters in each path. The transfer function
for this case is shown in equation (4).
2 2
2 2
0 3
2 1
2 2 2 2
0 2 1 3
9 8 7 6 5 4 3 2 1
0 1 2 3 4 4 3 2 1 0
2 2 2 2
0 2 3 3
1 1
1 11
( )
b b
( )( )( )( )
Z Z
Z Z
H Z Z Z Z Z Z
b Z b Z Z b Z b Z b Z b Z b Z Z b
Z Z Z Z Z
α α
α α
α α α α
α α α α
+ +
+ +
= ±
+ + + +
± + ± + ± + ± + ±
=+ + + +
(4)
We note a number of interesting properties of this transfer function, applicable to all the 2path proto
type filters. The denominator roots are on the imaginary axis restricted to the interval ± 1 to assure sta
bility. The numerator is a linearphase FIR filter with a symmetric weight vector. As such the numera
tor roots must appear either on the unit circle, or if off and real, in reciprocal pairs, and if off and com 
plex, in reciprocal conjugate quads. Thus for appropriate choice of the filter weights, the zeros of the
transfer function can be placed on the unit circle, and can be distributed to obtain an equal ripple stop
band response. In addition, due to the one pole difference between the two paths, the numerator must
have a zero at ± 1. When the two paths are added, the numerator roots are located in the left half plane,
and when subtracted, the numerator roots are mirror imaged to the right half plane forming lowpass
and highpass filters respectively.
The attraction of this class of filters is the unusual manner in which the transfer function zeros are
formed. The zeros of the allpass subfilters reside outside the unit circle (at the reciprocal of the stable
pole positions) but migrate to the unit circle as a result of the sum or difference of the two paths. The
zeros appear on the unit circle because of destructive cancellation of spectral components delivered to
the summing junction via the two distinct paths, as opposed to being formed by numerator weights in
the feed forward path of standard biquadratic filters. The stop band zeros are a windfall. They start as
the maximum phase allpass zeros formed concurrently with the allpass denominator roots by a single
shared coefficient and migrate to the unit circle in response to addition of the path signals. This is the
reason that the two path filter requires less than half the multiplies of the standard biquadratic filter.
Figure 6 presents the polezero diagram for this filter. The composite filter contains 9 poles and 9 zeros
and requires two coefficients for path0 and two coefficients for path1. The tony_des2 design routine
was used to compute weights for the 9thorder filter with 60 dB equal ripple stop band. The passband
edge is located at a normalized frequency of 0.25 and the stopband edge that achieved the desired 60
dB stopband attenuation is located at a normalized frequency of 0.284. This is an elliptic filter with
constraints on the pole positions. The denominator coefficient vectors of the filter are listed here in de
creasing powers of Z:
Path0 Polynomial Coefficients:
Filter0 [1 0 0.101467517]
Filter2 [1 0 0.612422841]
Path1 Polynomial Coefficients:
Filter1 [1 0 0.342095596]
Filter3 [1 0 0.867647439]
The roots presented here represent the lowpass filter formed from the 2path filter. Figure 7 presents
the phase slopes of the two paths of this filter as well as the filter frequency response. We note that the
zeros of the spectrum correspond to the zero locations on the unit circle in the polezero diagram.
1.5 1 0.5 0 0.5 1 1.5
1.5
1
0.5
0
0.5
1
1.5
Roots of Prototype 2Path Filter
Figure 6 PoleZero Diagram of TwoPath, NinePole, FourMultiplier Filter
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
2.5
2
1.5
1
0.5
0Phase: Each P ath of 2Path Filter
Normalized Phase (
φ
/
π
)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
80
60
40
20
0
Spectrum, Filt er Formed as Scaled Sum of 2Paths
Normalized Frequency (f/f
s
)
Log Gain (dB)
π
Phase
α
0
,
α
2
= 0
Phase
α
1
,
α
3
= 0
Figure 7. TwoPath Phase Slopes and Frequency response 2Path, NinePole, FourMultiplier Filter
The first subplot in figure 7 presents two sets of phase responses for each path of the 2path filter. The
dashed lines represent the phase response of the two paths when the filter coefficients are set to zero. In
this case the two paths default to 2 delays in the top path and 3 delays in the bottom path. Since the two
paths differ by one delay, the phase shift difference is precisely 180 degrees at the half sample rate.
When the filter coefficients in each path are adjusted to their design values, the phase response of both
paths assume the bowed “lazy S” curve described earlier in figure 2b. Note that at low frequencies, the
two phase curves exhibit the same phase profile and that at high frequencies, the twophase curves
maintain the same 180degree phase difference. Thus the addition of the signals from the two paths
will lead to a gain of 2 in the band of frequencies with the same phase and will result in destructive
cancellation in the band of frequencies with 180degree phase difference. These two bands of course
are the pass band and stop band respectively. We note that the two phase curves differ by exactly 180
degrees at four distinct frequencies as well as the half sample rate: these frequencies corresponding to
the spectral zeros of the filter. Between these zeros, the filter exhibits stopband side lobes that, by de
sign, are equal ripple.
Linear Phase TwoPath HalfBand Filters
We can modify the structure of the 2path filter to form filters with approximately linear phase re 
sponse by restricting one of the paths to be pure delay. We accomplish this by setting all the filter coef
ficients in the upper leg to zero. This sets the allpass filters in this leg to their default responses of
pure delay with poles at the origin. As we pursue the solution to the phase matching problem in the
equalripple approximation we find that the allpass poles must move off the imaginary axis. In order
to keep real coefficients for the allpass filters, we call on the typeII allpass filter structure. The lower
path then contains first and second order filters in Z2. We lose a design degree of freedom when we set
the phase slope in one path to be a constant. Consequently when we design an equalripple group delay
approximation to a specified performance we require additional allpass sections. To meet the same
outofband attenuation and the same stop band bandedge as the nonlinear phase design of the previ
ous section, our design routine lineardesign, determined that we require two first order filters in Z2 and
three second order filters in Z2. This means that 8coefficients are required to meet the specifications
that in the nonlinear phase design only required 4 coefficients. Path0 (the delay only path) requires
16 units of delay while the allpass denominator coefficient vector list is presented below in decreasing
powers of Z which along with its single delay element form a 17the order denominator.
Path0 Polynomial Coefficients
Delay [zeros(1,16) 1]
Path1 Polynomial Coefficients
Filter0 [1 0 0.832280776]
Filter1 [1 0 0.421241137]
Filter2 [1 0 0.67623706 0 0.23192313]
Filter3 [1 0 0.00359228 0 0.19159423]
Filter4 [1 0 0.59689082 0 0.18016931]
Figure 8 presents the polezero diagram of the linear phase allpass filter structure that meets the same
spectral characteristics as those outlined in the previous section. We first note that the filter is nonmin
imum phase due to the zeros outside the unit circle. We also note the near cancellation of the right half
plane pole cluster with the reciprocal zeros of the nonminimum phase zeros. Figure 9 presents the
phase slopes of the two filter paths and the filter frequency response. We first note that the phase of the
two paths is linear; consequently the group delay is constant over the filter pass band. The constant
group delay matches the time delay to the peak of the impulse response which corresponds to the 16
sample time delay of the top path. Of course the spectral zeros of the frequency response coincide with
the transfer function zeros on the unit circle.
1.5 1 0.5 0 0.5 1 1.5
1.5
1
0.5
0
0.5
1
1.5
Figure 8. PoleZero Diagram of TwoPath, Thirty Three Pole, EightMultiplier Filter
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
8
6
4
2
0Phase:Twopath of Linear Phase IIR Filter
Normalized Phase (
φ
/
π
)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
80
60
40
20
0
Spectrum, Halfband Linear Phase IIR Filter
Log Gain (dB)
Normalized Frequency (f/f
s
)
π
Figure 9. TwoPath Phase Slopes and Frequency Response of 2Path, 33Pole, 8Multiplier Filter
Pass Band Response in TwoPath Half Band Filters
The allpass networks that formed the halfband filter exhibit unity gain at all frequencies. These are
lossless filters affecting only the phase response of signal they process. This leads to an interesting re
lationship between the passband and stopband ripple response of the halfband filter and in fact for
any of the twopath filters discussed in this paper. We noted earlier that the twopath filter presents
complimentary low pass and the high pass versions of the halfband filter, the frequency responses of
which are shown in figure 10 where pass band and stop band ripples have been denoted by δ1 and δ2 re
spectively.
00 . 2 5 0 . 5
f / f s
δ
1
δ
2
L o w  P a s s H i g h  P a s s
M a g
Figure 10. Magnitude Response of LowPass and HighPass HalfBand Filter
The transfer functions of the low pass and of the high pass filters are shown in (5), where P0(Z) and
P1(Z) are the transfer functions of the allpass filters in each of the two paths. The power gain of the
lowpass and highpass filters is shown in (6). When we form the sum of the power gains, the cross
terms in the pair of products cancel and we obtain the results shown in (7).
1
0 1
1
0 1
( ) 0.5 [ ( ) ( )]
( ) 0.5 [ ( ) ( )]
LOW
HIGH
H Z P Z Z P Z
H Z P Z Z P Z
−
−
= × +
= × −
(5)
2 1
1 1 1
0 1 0 1
2 1
1 1 1
0 1 0 1
 ( )  ( ) ( )
0.25 [ ( ) ( )] [ ( ) ( )]
 ( )  ( ) ( )
0.25 [ ( ) ( )] [ ( ) ( )]
LOW LOW LOW
HIGH HIGH HIGH
H Z H Z H Z
P Z Z P Z P Z Z P Z
H Z H Z H Z
P Z Z P Z P Z Z P Z
−
− − −
−
− − −
=
= × + × +
=
= × − × −
(6)
2 2 2 2
0 1
 ( )   ( )  0.25 [2  ( )  2  ( )  ]
1
LOW HIGH
H Z H Z P Z P Z+ = × × + ×
=
(7)
Equation (7) tells us that at any frequency, the squared magnitude of the low pass gain and the squared
magnitude of the high pass gain sum to unity. This is a consequence of the filters being lossless. En
ergy that enters the filter is never dissipated, a fraction of it is available at the low pass output and the
rest of it is available at the high pass output. This property is the reason the complementary lowpass
and highpass filters cross at their 3dB points. If we substitute the gains at peak ripple of the low pass
and high pass filters into (7), we obtain (8) which we can rearrange and solve for the relationship be 
tween δ1and δ2. The result is interesting. We learn here that the peak to peak inband ripple is approxi
mately half the square of the outofband peak ripple. Thus if the out of band ripple is –60 dB or 1part
in a 1000, then the inband peak to peak ripple is half of 1 part in a 1,000,000, which is on the order of
5 µdB (4.34 µdB). The halfband recursive allpass filter exhibits an extremely small in band ripple.
The inband ripple response of the twopath 9pole filter is seen in figure 11.
2 2
1 2
2 2
1 2 2
2
1 2
[1 ] [ ] 1
[1 ] 1 1 0.5
0.5
δ δ
δ δ δ
δ δ
− + =
− = − ≅ − ×
≅ ×
(8)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
8
7
6
5
4
3
2
1
0
1
2x 10
6
Magnitude response of two path filter
normalized frequency (f/fs)
log magnitude (dB)
Figure 11. InBand Ripple Level of Twopath, NinePole Recursive Filter
Transforming the Prototype Half Band To Arbitrary Bandwidth
In the previous section we examined the design of twopath halfband filters formed from recursive all
pass firstorder filters in the variable Z2. We did this because we have easy access to the weights of this
simple constrained filter, the constraint being stated in equation (5). If we include a requirement that
the stop band be equal ripple, the half band filters we examine are elliptic filters that can be designed
from standard design routines. Our program Tony_des2 essentially does this in addition to the fre
quency transformations we are about to examine. The prototype halfband filter can be transformed to
form other filters with specified (arbitrary) bandwidth and center frequency. In this section, elementary
frequency transformations are introduced and their impact on the prototype architecture as well as on
the system response is reviewed. In particular the frequency transformation that permits bandwidth
tuning of the prototype is introduced first. Additional transformations that permit tuning of the center
frequency of the prototype filter are also discussed.
LowPass To LowPass Transformation
We now address the transformation that converts a lowpass halfband filter to a lowpass arbi
trarybandwidth filter. Frequency transformations occur when an existing allpass sub network in a fil
ter is replaced by another allpass sub network. In particular, we present the transformation shown in
(9).
1 tan( / 2)
1 1 ; ; 2
1 tan( / 2)
b b
b
b S
f
bZ b
Z Z b f
θθ π
θ
−
+
⇒ = =
+ +
(9)
This is the generalized delay element we introduced in the initial discussion of firstorder allpass net
works. We can physically replace each delay in the prototype filter with the allpass network and then
tune the prototype by adjusting the parameter “b”. We have fielded many designs in which we perform
this substitution. Some of these designs are cited in the bibliography. For the purpose of this paper we
perform the substitution algebraically in the allpass filters comprising the twopath halfband filter,
and in doing so generate a second structure for which we will develop and present an appropriate ar
chitecture.
We substitute (9) into the first order, in Z2, allpass filter introduced in (2) and rewritten in (10),
2
1
2
2
1
( ) ( )
1
( )
Z b
ZbZ
Z b
ZbZ
G Z H Z
Z
G Z Z
αα
+
⇒+
+
⇒+
=
+
=+
(10)
After performing the indicated substitution and gathering terms, we find the form of the transformed
transfer function is as shown in (11).
22
1 2
1 2
2 2 2
1 2
1 2 (1 )
( ) ;
1 1
c Z c Z b b
G Z c c
Z c Z c b b
α α
α α
+ + + +
= = =
+ + + +
(11)
As expected, when
α,c and0,c0,b
21
⇒⇒⇒
the transformed allpass filter reverts back to the original
firstorder filter in Z2. The architecture of the transformed filter, which permits one multiplier to form
the matching numerator and denominator coefficient simultaneously, is shown in figure 12. Also
shown is a processing block G(Z) that uses two coefficients c1 and c2. This is seen to be an extension of
the onemultiply structure presented in figure 3. The primary difference in the two architectures is the
presence of the coefficient and multiplier c1 associated with the power of Z1. This term, formerly zero,
is the sum of the polynomial roots, and hence is minus twice the real part of the roots. With this coeffi
cient being nonzero, the roots of the polynomial are no longer restricted to the imaginary axis.
c
1
c
1
c
2
c
2
zz
zz
 1  1
 1  1


X ( Z )
Y ( Z )
G ( Z )
Figure 12. Block Diagram of General Second Order AllPass Filter
The root locations of the transformed, or generalized secondorder allpass filter, are arbitrary except
that they appear as conjugates inside the unit circle, and the poles and zeros appear in reciprocal sets as
indicated in figure 13.
e
e
e
e
r
r
r
r
j
θ
j
θ
 j
θ
 j
θ
Figure 13 Pole Zero Diagram of Generalized Second Order AllPass Filter
The twopath prototype filter contained one or more onemultiply first order recursive filters in Z2 and
a single delay. We effect a frequency transformation on the prototype filter by applying the lowpass
tolow pass transformation shown in (10). Doing so converts the onemultiply firstorder in Z2 allpass
filter to the generalized twomultiply secondorder allpass filter and converts the delay, a zeromulti
ply allpass filter to the generalized onemultiply firstorder in Z allpass filter. Figure 14 shows how
applying the frequency transformation affects the structure of the prototype. Note that the 9pole, 9
zero halfband filter, which is implemented with only 4 multipliers, now requires 9 multipliers to form
the same 9poles and 9zeros for the arbitrarybandwidth version of the twopath network. This is still
significantly less than the standard cascade of first and secondorder canonic filters for which the same
9pole, 9zero filter would require 23 multipliers.
Figure 15 presents the polezero diagram of the frequency transformed prototype filter. The 9poles
have been pulled off the imaginary axis, and the 9zeros have migrated around the unit circle to form
the reducedbandwidth version of the prototype. Figure 16 presents the phase response of the two paths
and the frequency response obtained by applying the lowpass to lowpass frequency transformation to
the prototype twopath, fourmultiply, halfband filter presented in figure 7. The lowpass to lowpass
transformation moved pass bandedge from normalized frequency 0.25 to normalized frequency 0.1.
Figure 14. Effect on Architecture of Frequency Transformation Applied to TwoPath HalfBand All
Pass Filter
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Roots of LP to LP Transformed TwoPath HalfBand Filter
Figure 15. PoleZero Diagram Obtained by Frequency Transforming HalfBand Filter to Normalized
Frequency 0.1.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
2.5
2
1.5
1
0.5
0Phase Slopes of 2Path Filter
Normalized phase,
φ
/(2
π
)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
80
60
40
20
0
Spectrum, 2Path Filter
Normalized Frequency, (f/f
s
)
Log Mag, (dB)
π
Figure 16. Phase Response of Two Paths and Frequency Response of 2Path HalfBand Filter Fre
quency Transformed to 0.1 Normalized Bandwidth
LowPass to BandPass Transformation
In the previous section we examined the design of twopath, arbitrarybandwidth lowpass filters
formed from recursive allpass first and second order filters as shown in figure 14. We formed this fil 
ter by a transformation of a prototype halfband filter. We now address the second transformation, one
that performs the lowpass to bandpass transformation. As in the previous section we invoke a fre
quency transformation wherein an existing allpass sub network in a filter is replaced by another all
pass sub network. In particular, we now examine the transformation shown in (12).
1 1 1 ; cos( ); 2
C
C C
S
f
cZ c
Z Z Z c f
θ θ π
−
⇒ − = =
−
(12)
This, except for the sign, is a cascade of a delay element with the generalized delay element we intro 
duced in the initial discussion of firstorder allpass networks. We can physically replace each delay in
the prototype filter with this allpass network and then tune the center frequency of the lowpass proto 
type by adjusting the parameter c. For the purpose of this paper we perform the substitution algebrai
cally in the allpass filters comprising the twopath predistorted arbitrarybandwidth filter, and in do
ing so generate yet a third structure for which we will develop and present an appropriate architecture.
We substitute (12) into the secondorder allpass filter derived in (11) and rewritten in (13).
( )
( 1)
2 2 2
2 2 2 ( )
( 1)
( ) ( )
( ) 2 (1 ) (1 )
(1 ) 2 (1 ) ( )
Z Z c
ZcZ
Z Z c
ZcZ
F Z G Z
b b Z b Z
b b Z b Z
α α α
α α α
−
⇒−
−
⇒−
=
+ + + + +
=+ + + + +
(13)
After performing the indicated substitution and gathering up terms, we find the form of the trans
formed transfer function is as shown in (14).
2 3 4
1 2 3 4
4 3 2
1 2 3 4
2 2
1 2
2 2
2
3 4
2 2
1
( )
2 (1 )(1 ) (1 )( (1 ) 2 )
1 1
2 (1 )(1 )
1 1
d Z d Z d Z d Z
F Z Z d Z d Z d Z d
c b b c b b
d d
b b
c b b b
d d
b b
α α
α α
α α
α α
+ + + +
=+ + + +
− + + + + +
= =
+ +
− + + +
= =
+ +
(14)
As expected, when we let
,cd and cd while0,d and d0,c
241231
⇒⇒⇒⇒
the weights default to those
of the prototype (arbitrarybandwidth) filter. The transformation from low pass to band pass generates
two spectral copies of the original spectrum, one each at the positive and negative tuned center fre
quency. The architecture of the transformed filter, which permits one multiplier to simultaneously form
the matching numerator and denominator coefficients, is shown in Figure 17. Also shown is a process
ing block F(Z) which uses four coefficients d1, d2, d3 and d4. This is seen to be an extension of the two
multiply structure presented in figure 14.
d
1
d
2
d
3
d
4
d
1
d
3
d
4
d
2
zz
z
zz
z
zz
 1  1
 1
 1  1
 1
 1  1




X ( Z )
Y ( Z )
F ( Z )
Figure 17. Block Diagram of General Fourth Order AllPass Filter
We have just described the lowpass to bandpass transformation that is applied to the secondorder
allpass networks of the twopath filter. One additional transformation that requires attention is the
lowpass to bandpass transformation that must be applied to the generalized delay or bandwidthtrans
formed delay from the prototype halfband filter. We substitute (12) into the firstorder allpass filter
derived in (9) and rewritten in (15).
( )
( 1)
2
2
1
( )
( 1) ( ) 1 (1 )
( ) ( 1) (1 )
Z Z c
ZcZ
bZ
E Z Z b
cZ bZ Z c c b Z bZ
Z Z c b cZ Z c b Z b
−
⇒−
+
=+
− + − − + − +
= =
− + − − − −
(15)
As expected, when
1,c
⇒
the denominator goes to (Z+b)(Z1) while the numerator goes to (1+bZ)(Z
1) so that the transformed allpass filter reverts back to the original firstorder filter. The distributed
minus sign in the numerator modifies the architecture of the transformed secondorder filter by shuf
fling signs in figure 13 to form the filter shown in figure 18. Also shown is a processing block E(Z)
which uses two coefficients e1 and e2.
e
1
e
1
e
2
e
2
zz
zz
 1  1
 1  1

X ( Z )
Y ( Z )
E ( Z )
Figure 18. Block Diagram of LowPass to BandPass Transformation Applied to LowPass to Low
Pass Transformed Delay Element
In the process of transforming the lowpass filter to a bandpass filter we convert the twomultiply sec 
ondorder allpass filter to a fourmultiply fourthorder allpass filter, and convert the onemultiply
lowpasstolowpass filter to a twomultiply allpass filter. The doubling of the number of multiplies
is the consequence of replicating the spectral response at two spectral centers of the real bandpass sys
tem. Note that the 9pole, 9zero arbitrary lowpass filter now requires 18 multipliers to form the 18
poles and 18zeros for bandpass version of the twopath network. This is still significantly less than
the standard cascade of first and secondorder canonic filters for which the same 18pole, 18zero filter
would require 45 multipliers. Figure 19 shows how the structure of the prototype is affected by apply
ing the lowpass to bandpass frequency transformation.
Figure 19. Effect on Architecture of LowPass to bandPass Frequency Transformation Applied to
TwoPath Arbitrary bandwidth AllPass Filter
Figure 20 presents the polezero diagram of the frequencytransformed prototype filter. The 9poles
defining the lowpass filter have been pulled to the neighborhood of the bandpass center frequency.
The 9zeros have also replicated, appearing both below and above the passband frequency. Figure 21
presents the phase response of the two paths and the frequency response obtained by applying the low
pass to bandpass frequency transformation to the prototype twopath, 9multiply, lowpass filter pre
sented in figure 14. The onesided bandwidth was originally adjusted to a normalized frequency of 0.1,
and is now translated to a center frequency of 0.22.
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
Roots of LP to BP Transformed TwoPath Filter
Figure 20. PoleZero Plot of TwoPath AllPass HalfBand Filter Subjected to LowPass to LowPass
and then LowPass to BandPass Transformations
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
5
4
3
2
1
0
1Phase Slopes of TwoPath IIR Filter
Normalized Phase,
φ
/(2*
π
)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
80
60
40
20
0
Spectrum, TwoPath IIR Filter
Normalized Frequency (f/f
s
)
Log Mag (dB)
π
π
Figure 21. Frequency Response of TwoPath AllPass Filter Subjected to LowPass to LowPass and
then LowPass to BandPass Transformations
Summary
We have presented a class of particularly efficient recursive filters based on twopath recursive allpass
filters. The numerator and denominator of an allpass filter have reciprocal polynomials with the coef
ficient sequence of the denominator reversed in the numerator. The allpass filters described in this pa 
per fold and align the numerator registers with the denominator registers so that the common coeffi
cients can be shared and thus form a pole and reciprocal zero with a single multiply. Coefficients are
selected via a design algorithm to obtain matching phase profiles in specific spectral intervals with 180
degree phase offsets in other spectral intervals. When the time series from the two paths are added, the
signals residing in the spectral intervals with 180 degree phase offsets are destructively cancelled.
From the transfer function perspective, the non minimum phase zeros in the separate numerators mi
grate to the unit circle as a result of the interaction of the numeratordenominator cross products result
ing when forming the common denominator from the separate transfer functions of the two paths. The
migration to the unit circle of the essentially free reciprocal zeros, formed while building the system
poles, is the reason this class of filters requires less than half the multiplies of a standard recursive fil
ter. The destructive cancellation of the spectral regions defined for the twopath halfband filter is pre
served when the filter is subjected to transformations that enable arbitrary band width and arbitrary
center frequencies. The one characteristic not preserved under the transformation is the ability to em
bed 1to2 or 2to1 multirate processing in the two path filter. The extension of the 2path filter struc
ture to an Mpath structure with similar computational efficiency is the topic of a second presentation.
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