Time-reversed infinite impulse response digital filtering

A digital filter includes a first memory device, a filter with at least one pole, i.e., a pole filter, coupled to the first memory device, and a second memory device coupled to the pole filter. The first memory device sequentially stores a digital signal in a forward sequence, and the pole filter sequentially reads the stored digital signal from the first memory device in a reverse sequence, thereby creating a time-reversed signal. The time-reversed signal is filtered by the pole filter, and is then stored as a filtered time-reversed signal in the second memory device in the forward sequence. A data signal is generated by the second memory device by reading the filtered time-reversed signal from the second memory device in the reverse sequence.

BACKGROUND OF THE INVENTION 
The present invention relates to discrete-time or digital filters. Even 
more particularly, the present invention relates to a time-reversed 
infinite impulse response digital filter device and method, wherein 
time-domain asymmetric signal can be filtered by the time-reversed 
infinite impulse response filter so as to generate a time-domain 
symmetric: signal or signal having linear phase, in response thereto. 
In a receive signal path, e.g., in a cellular telephone, a digital filter, 
or discrete-time filter, can be utilized to filter a received signal and 
to implement a response desired for further processing of the received 
signal, e.g., in a digital signal processor. Unfortunately, before the 
received signal is filtered by the discrete-time filter, the received 
signal is typically filtered by an analog filter in order to band-limit 
the received signal by filtering out undesirable frequency bands, e.g., 
frequency bands other than a desired cellular frequency band, such as 
adjacent cellular frequency bands. Problematically, the analog filter 
distorts the received signal, and generates a time-domain asymmetric 
signal. 
As used herein, the term time-domain asymmetric signal, refers to a signal 
that is the output signal of an asymmetric electrical circuit. The 
asymmetric electrical circuit is a circuit that, in response to an 
electrical impulse or delta function, generates an output signal, i.e., 
impulse response, that is not symmetric about any point in time. Typically 
the impulse response has more energy following its time domain peak than 
proceeding it. In other words, the impulse response builds more quickly 
than it decays. For example, the impulse response may be an oscillating 
signal that is generated in response to the electrical impulse, and that 
dissipates (or decays) over time. As a result, the oscillating signal has 
a time-domain energy distribution wherein very little energy is 
distributed before its peak, and a greater amount of energy is distributed 
after its peak. This type of energy distribution can also be described as 
lacking linear phase. 
Because analog filters are infinite impulse response filters, they are one 
type of asymmetric electrical circuit. Thus, the output signal generated 
by the analog filter, mentioned above, is referred to herein as the 
"asymmetric" signal, i.e., a signal that has been asymmetrically distorted 
by the analog filter. 
Two types of discrete-time, or digital, filters are characterized based on 
their response to an electrical impulse, or delta function: finite impulse 
response filters (or FIR filters) and infinite impulse response filters 
(or IIR filters). The infinite impulse response filter exhibits asymmetric 
distortion (or lack of linear phase) similar to that of the analog filter 
described above. Because the IIR filter tends to asymmetrically distort 
the asymmetric signal, described above, and therefore tends to compound 
the asymmetric distortion caused by the analog filter, the IIR filter has, 
heretofore, not been a preferred filter for use in the cellular telephone 
receive signal path, where a symmetric output signal (i.e., an output 
signal free from asymmetric distortion, or having linear phase) is 
desired. 
In theory, the FIR filter, however, can be designed to completely restore 
the distortions made by the analog filter, i.e., can be designed to 
generate a symmetric signal in response to the asymmetric signal. 
Problematically, in order to achieve this design an infinite number of 
unit delays and "taps" must be utilized, and an infinite amount of time is 
needed for an output signal, completely free from distortion, to be 
generated. Thus, as a practical matter, the FIR filter cannot completely 
restore the distortions made by the analog filter. 
Instead, FIR filters used in receive signal paths where a symmetric output 
signal is desired are typically designed to restore the distortions made 
by the analog filter to within a prescribed tolerance. As a result, the 
FIR filter can be designed with, e.g., eight unit delays and sixteen 
"taps" which is considered reasonable for, e.g., the processing of 
cellular telephone signals. Additional delays and taps may be utilized to 
increase the symmetry of the FIR filter's impulse response. 
Improved or complete restoration of the distortions introduced by the 
analog filter, without the need for additional delays and taps, is however 
very desirable. 
Thus, improvements are needed in the filtering of electrical signals so as 
to, e.g., restore distortions introduced into the electrical signal by an 
asymmetric electrical circuit, such as an analog filter. 
SUMMARY OF THE INVENTION 
The present invention provides a time-reversed filter, and a method of 
using the same, wherein, e.g., an asymmetric signal can be filtered by the 
time-reversed filter so as to generate a symmetric signal in response 
thereto. 
The invention can be characterized as a device for digitally filtering a 
digital signal. The device includes (1) a first memory device, (2) a pole 
filter having a pole and being coupled to the first memory device and (3) 
a second memory device coupled to the pole filter. Note that the term pole 
filter, as used herein should not be confused with the term all-pole 
filter, which implies a filter having no zeros. The term pole filter means 
a filter having at least one pole, and possibly having one or more zeros. 
The first memory device stores the digital signal in a forward sequence 
such that a first sample within the digital signal is stored first within 
the first memory device, and a last sample within the digital signal is 
stored last within the first memory device. The first sample can be the 
first sample in a burst-wise digital signal, as is used for example in 
time-division multiple access (TDMA) encoded cellular telephone signals, 
or may be any sample within a continuous stream of digital samples or 
within a burst of digital samples. Similarly, the last sample may be the 
last sample in a burstwise digital signal, or may be any sample within a 
continuous stream of digital samples or within a burst of digital samples. 
The pole filter, which may be a digital infinite impulse response (IIR) 
filter, reads the stored digital signal from the first memory device in a 
reverse sequence such that the last sample is read first from the first 
memory device, and the first sample is read last from the first memory 
device, thereby creating a time-reversed signal. The time-reversed signal 
is digitally filtered by the pole filter, and is then stored in a second 
memory device in the forward sequence. Finally, the time-reversed signal, 
having been filtered by the pole filter, is read from the second memory 
device in the reverse sequence. A substantially symmetric data signal is 
generated by the second memory device in response to the reading of the 
filtered time-reversed signal in the reverse sequence. 
The invention may also be characterized as a method of digitally filtering 
a digital signal. The method includes (a) storing sequentially the digital 
signal in a forward sequence such that a first sample within the digital 
signal is stored first, and a last sample within the digital signal is 
stored last; and (b) filtering the digital signal in a reverse sequence 
such that the last sample is filtered first, and the first sample is 
filtered last. The filtering may be performed using any filter having a 
pole, such as a digital IIR filter or an analog filter having a pole. The 
method also includes (c) storing the digital signal, having been filtered, 
in the forward sequence; and (d) generating a data signal by retrieving 
the stored filtered digital signal in the reverse sequence.

DETAILED DESCRIPTION OF THE INVENTION 
The following description of the presently contemplated best mode of 
practicing the invention is not to be taken in a limiting sense, but is 
made merely for the purpose of describing the general principles of the 
invention. The scope of the invention should be determined with reference 
to the claims. 
Referring first to FIG.. 1, a block diagram is shown of a receive signal 
path 10 for use in a cellular telephone, wherein a digital filter 12 made 
in accordance with the present invention is utilized to restore 
distortions that are introduced by an analog filter 14. 
An antenna 16 is shown coupled to a first input of a multiplier (or 
multiplexer) 18, and a down conversion frequency generator 20 is shown 
coupled to a second input of the multiplier 18. In practice, the 
multiplier 18 and down conversion frequency generator 20 carry out a 
demodulation process from carrier frequencies to baseband, but are 
described here conceptually. Typically, multiple demodulation stages 
consisting of multiple multipliers and down conversion frequency 
generators, and multiple intermediate filters are utilized. As shown, a 
carrier frequency cellular signal is received into the antenna 16 and 
demodulated to a lower intermediate frequency (IF) signal, which passes 
through the analog filter 14. Subsequently, the IF signal is reduced to 
baseband by first and second demodulation circuits 19, 21, 19', 21'. Such 
multiple demodulation stages are known in the art and are therefore not 
described in detail herein. 
An output of the multiplier is coupled to the analog filter 14, which is 
coupled to the first and second demodulation circuits 19, 21, 19', 21'. 
The first and second demodulation circuits 19, 21, 19', 21' include first 
and second multipliers 19, 19', which are coupled to the analog filter 14 
and to analog to digital converters 22, 22'. The multipliers 19, 19' are 
also coupled to first and second demodulation frequency generators 21, 
21', so as to combine first and second demodulation signals with the 
output of the analog filter 14, as is known in the art. The first 
demodulation circuit is coupled to the analog to digital converter (ADC) 
22, and the ADC 22 is coupled to the digital filter 12. The digital filter 
12 is coupled to a signal processor 24. Together the antenna 16, the 
multiplier 18, the analog filter 14, the first demodulation circuit 19, 
21, the ADC 22, the digital filter 12, and the signal processor 24 
comprise an "I" channel of the receive signal path 10. 
A "Q" channel of the receive signal path 10 consists of the antenna 16, the 
multiplier 18, the analog filter 14, the second demodulation circuit 19', 
21', another ADC 22', another digital filter 12', and the signal processor 
24. The "Q" channel functions in a manner similar to the "I" channel 
except that it is orthogonal to the "I" channel, because the second 
demodulation signal is 90.degree. out of phase from the first demodulation 
signal. Therefore, only the "I" channel is described hereinbelow. The use 
of orthogonal "I" and "Q" channels is well known in the cellular 
communications art. 
In operation, the antenna 16 receives a cellular signal 6 via radio 
frequency air waves. The cellular signal 6 is generated at a base station 
27 as is known in the art of cellular telephone communications. The 
cellular signal 6 is a burstwise signal, i.e., it is transmitted by the 
base station 27 in a plurality of signal bursts consisting of finite 
blocks of information. Each of the plurality of bursts is received by the 
antenna 16, and is passed into the receive signal path 10 as it is 
received. 
The cellular signal 6 is passed by the antenna 16 to the multiplier 18 
where it is multiplied by a down conversion frequency signal from the down 
conversion frequency generator 20. The receiver signal is then passed to 
the analog filter 14. As is known in the art, there can be, and generally 
are, several multipliers, several down conversion frequency generators, 
and several analog filters within the receive signal path 10. The 
multiplier 18, the down conversion frequency generator 20 and the analog 
filter 14 of FIG. 1 are shown merely as an example of the multipliers, 
down conversion frequency generators and analog filters that are commonly 
used in the receive signal path 10 of a cellular telephone. 
Unfortunately, in addition to filtering unwanted signals from the receiver 
signal, such as adjacent-band cellular signals, the analog filter 14 
causes asymmetric distortion in the receiver signal and generates an 
asymmetric signal, as defined hereinabove. 
The asymmetric signal is demodulated by the first demodulator 19, 21 and 
digitized, i.e., sampled, as is known in the art, by the ADC 22. The ADC 
22 generates an asymmetric data signal (or digital signal) in response to 
the demodulating and sampling of the asymmetric signal. The asymmetric 
data signal includes several binary samples, which include voltages that 
represent binary digits (i.e., bits). The binary samples correspond to the 
time-domain voltage amplitudes of the asymmetric signal and consist of, 
e.g., sixteen bit words. 
The asymmetric data signal is passed into-the digital filter 12, which in 
the preferred embodiment is referred to as a time-reversed infinite 
impulse response filter 12. The time-reversed infinite impulse response 
filter 12 includes a first memory device 26, namely a first 
last-in-first-out (LIFO) buffer 26, an infinite impulse response (IIR) 
filter 28, and a second memory device 29, namely a second 
last-in-first-out (LIFO) buffer 29. The first LIFO buffer 26 receives each 
sample of the asymmetric data signal from the ADC 22 as each sample is 
generated by the ADC 22. The first LIFO buffer 26 stores each sample at a 
memory location within the first LIFO buffer 26. 
After the last sample within one Of the plurality of signal bursts is 
stored in the first LIFO buffer 26, the last sample is passed from the 
first memory to the IIR filter 28. Next, the second-to-last sample is 
passed to the IIR filter 28, then the third-to-last sample is similarly 
passed, and so on until the first sample is passed to the IIR filter 28. 
Thus, the samples of each signal burst are passed to the IIR filter 28 in 
a reverse sequence, i.e., in reverse of the order in which they are 
received into the first LIFO buffer 26. The samples, in the reverse 
sequence, are referred to herein as a time-reversed signal. Next, the IIR 
filter 28 filters the time-reversed signal and causes asymmetric 
distortion of the time-reversed signal. 
Note that while a digital IIR filter 28 is a preferred filter for use with 
the invention, many other types of filters with one or more poles 
(referred to herein as pole filters) should be understood to be within the 
scope of the invention. For example, an analog filter can be used 
(provided a digital to analog converter is interposed between the first 
LIFO buffer 26 and the analog filter, and, similarly, a analog to digital 
converter is interposed between the analog filter and the second LIFO 
buffer 29). Alternatively, a digital lattice structure with feedback can 
also be used as the pole filter. 
After having been filtered by the IIR filter 28, the time-reversed signal 
is passed into the second LIFO buffer 29. After the last sample within one 
of the plurality of signal bursts of the filtered time-reversed signal is 
stored within the second LIFO buffer 29, the samples are read from the 
second LIFO buffer 29 in the reverse sequence, i.e., last sample, then 
second-to-last sample, etc., and a substantially symmetric data signal is 
generated in response thereto. This process, i.e., reversing, IIR 
filtering and then re-reversing, is repeated for each of the plurality of 
signal bursts within the asymmetric data signal. 
Note that in the event a the digital filter 12 is used in a system that 
does not utilize a burstwise incoming signal such as the cellular signal 
6, the incoming signal will need to be segmented or blocked so that 
discrete bursts or blocks of the signal can be stored in the first LIFO 26 
and recalled using the reverse sequence, etc. 
The IIR filter 28, as mentioned above, causes asymmetric distortion in the 
asymmetric data signal. However, because each burst (or block) of the 
asymmetric data signal is received into the IIR filter 28 in reverse, 
i.e., last sample first etc., filtered by the IIR filter 28 and then 
re-reversed, the asymmetric distortion caused by the IIR filter 28 
surprisingly tends to cause the asymmetric data signal to become 
symmetric. By adjusting the asymmetric distortion caused by the IIR filter 
28, the IIR filter 28 can be made to generate a substantially symmetric 
data signal. (A symmetric signal is a signal produced by a filter or 
combination of filters that have a substantially symmetric energy 
distribution about a time-domain peak in the filter's impulse response.) 
Finally, the substantially symmetric data signal is passed to the signal 
processor 24, and is processed therein, along with a similar symmetric 
data signal from the "Q" channel, as is known in the art of cellular 
telephone communications. 
In this way the time-reversed infinite impulse response filter 12 restores 
the asymmetrio distortion caused by the analog filter 14 using a digital 
filter having a sharply reduced number of "taps". 
Note that the digital filter of the present invention need not be used to 
restore symmetry or linear phase to a time-domain asymmetrically distorted 
signal. The restoration of symmetry is described herein only in the 
context of describing the function of the preferred embodiment of the 
digital filter. Numerous other applications of the invention are 
contemplated and can be achieved through the selection of the pole(s) and 
possible zero(s) of the pole filter. Such applications include satellite 
communications modems, telephone line modems, terrestrial microwave 
point-to-point modems and numerous other applications. 
Referring next to FIG. 2, a graph is shown of the amplitude frequency 
response to the analog filter 14 of FIG. 1, wherein frequency offset 
relative to a carrier frequency is shown on a horizontal axis and 
amplitude is shown on a vertical axis. As can be seen, the amplitude 
frequency response has a generally rounded amplitude (frequency-domain) 
profile 30 between approximately -15 and +15 kHz from the center frequency 
of the analog filter (i.e., the intermediate frequency (IF)). 
Referring to FIG. 3, a graph is shown of group delay frequency response to 
the analog filter 14 of FIG. 1, wherein frequency offset relative to the 
carrier frequency is shown on a horizontal axis and group delay is shown 
on a vertical axis. Pronounced peaks 32, 33 in group delay, i.e., around 
40 .mu.S, can be seen at approximately -15 and +15 kHz from the center 
frequency of the analog filter. These peaks are indicative of the 
asymmetric (time-domain) distortion in the asymmetric signal. 
Ideally, both the amplitude (FIG. 2) and group delay (FIG. 3) responses 
would be constant across the frequency band of interest, i.e., between 
approximately -15 and +15 kHz from the center frequency of the analog 
filter. One useful statistic used in interpreting the analog filter's 
response in a digital modulation environment is the RMS (square-root means 
of the squares) phase error that the filter imposes on ideally modulated 
signals, as observed at an output of the receive signal path. For example, 
for QPSK systems, such as are commonly used in cellular telephone 
communications, the RMS phase error should be kept to a maximum of 
approximately two degrees. 
In order to improve the RMS phase error, two improvements to the frequency 
response can be made. The first is to flatten the amplitude response. 
Typically, this can be achieved through the selection of appropriate pole 
and zero locations in a `z`-domain filter model. Poles can be placed near 
regions requiring more gain, and zero's near those requiring more 
attenuation. 
With IIR filters, flexibility in location of poles is limited by the need 
for stability. This limitation heretofore tended to correspond in the 
time-domain to asymmetry in the filter's impulse response, with more 
energy coming after the peak--thus, tending to exaggerate the asymmetry 
caused by analog filters such as the analog filter 14 of FIG. 1, instead 
of improving it. Viewed another way, this corresponds to increases in 
group delay at higher frequencies. 
The second desired improvement in frequency response is to flatten the 
group delay (giving the filter fixed delay as a function of frequency, a 
property called linear phase). 
To have flexibility in control of group delay has heretofore implied the 
use of an FIR (finite impulse response) filter where only zeros (and no 
poles) are available for placement. Stability is not a problem, so zero 
location is flexible and group delay as well as amplitude response can be 
successfully manipulated. To equalize for the effect of analog filters 
like those in FIG. 1, the impulse response of a properly designed FIR 
filter will be asymmetric with more energy preceding the peak. 
Problematically, as described above, such FIR filters cannot completely 
correct for the asymmetric distortion caused by, e.g., the analog filter 
of FIG. 1. 
The invention consists of a filter having a pole, or pole filter, such as 
an digital IIR filter, which achieves the goal of placing more energy 
before the peak of the impulse response, or equivalently, creating a 
reduction in group delay with frequency. In the z-domain, this corresponds 
to enabling the placement of poles outside the unit circle. FIGS. 6 and 7 
illustrate preferred amplitude and group delay characteristics, 
respectively, for the digital filter 12 of the present invention. 
The analog filter 14 depicted in FIG. 1, and with characteristics shown in 
FIGS. 2 and 3, results in 3.09 degrees RMS phase error. By further 
filtering the asymmetric signal from the analog filter 14 with the digital 
filter 12 of the present invention, RMS phase error is reduced to 1.03 
degrees. 
Referring next to FIG. 4, a schematic view is shown of the IIR filter 28 of 
FIG. 1. 
The time-reversed burst signal is symbolized as i[-n], and the filtered 
time-reversed burst signal, i.e., the time-reversed burst signal having 
been filtered by the signal, is symbolized as o[-n]. The independent 
variable -n represents the sequence of samples within each of the 
plurality of bursts, having been read from the first LIFO in reverse 
order--hence the "-" sign. 
The asymmetric data signal, having been time reversed, i.e., the 
time-reversed burst signal 40, enters a summer 36. The summer 36 generates 
the filtered time-reversed burst signal (i.e., the time-reversed burst 
signal having been filtered by the IIR filter) in response to the 
asymmetric data signal 40 and to a feed back signal 42. In order to 
generate the filtered time-reversed burst signal 38, the feed back signal 
42 is added to the time-reversed burst signal 40 by the summer 36. The 
filtered time-reversed burst signal 38, in addition to serving as an 
output 44 to the IIR filter 28, is passed from the summer 36 to a unit 
delay circuit 46. The unit delay circuit 46 delays the symmetric signal 38 
for a prescribed time period, e.g. 20.58 .mu.s, and then passes the 
filtered time-reversed burst signal, having been delayed 48, to a gain 
circuit 50. The gain circuit 50 attenuates the delayed filtered 
time-reversed burst signal 48 by an attenuation factor, k, and generates 
the feedback signal 42 in response thereto. As mentioned above, the 
feedback signal 42 is added to the time-reversed burst signal 40 in order 
to generate the filtered time-reversed burst signal 38. 
The z-transform for the IIR filter 28 in combination with the first and 
second LIFOs 26, 29, i.e., the time-reversed infinite impulse response 
filter 12, can be expressed as follows: 
##EQU1## 
which has a single pole at: 
##EQU2## 
and a region-of-convergence at: 
##EQU3## 
This corresponds in the discrete-time domain to the following impulse 
response: 
EQU h[n]=u[-n](k).sup.-n 
which is illustrated in FIG. 5. Note that the impulse response has more 
energy distributed before its peak 52 than after its peak 52. In order to 
ensure stable operation (i.e., that the region-of-convergence contains the 
unit circle), k is limited to lie in the range -1&lt;k&lt;1. 
In order to equalize the response depicted in FIG. 2, there is a need to 
attenuate the response in the low frequency offset areas, and to amplify 
it at higher frequency offsets. Consequently, it has been found that a 
high pass filtering structure is desirable, meaning that k should be 
negative, and lie in the range -1&lt;k&lt;0. 
Referring to FIG. 8, a graph is shown of the amplitude frequency response 
of a combination of the analog filter 14 and the time-reversed infinite 
impulse response filter 12 of FIG. 1, wherein frequency offset relative to 
a carrier frequency is shown on a horizontal axis and amplitude is shown 
on a vertical axis. As can be seen, the amplitude frequency response 30' 
is significantly flatter than the amplitude frequency response 30 of the 
analog filter 14 alone, as shown in FIG. 2. Thus, the first desired 
improvement--flattening the amplitude frequency response--is achieved by 
the invention. 
Similarly, referring to FIG. 9, a graph is shown of the group delay 
frequency response to the combination of the analog filter 14 and the 
time-reversed infinite impulse response filter 12 of FIG. 1, wherein 
frequency offset relative to a carrier frequency is shown on a horizontal 
axis and group delay is shown on a vertical axis. As can be seen, the 
group delay frequency response 32', 33' of the analog filter 14 in 
combination with the time-reversed infinite impulse response filter 12 is 
also significantly flatter than the group delay frequency response 32, 33 
of the analog filter 14 alone, as shown in FIG. 3. Thus, the second 
desired improvement--flattening the group delay frequency response--is 
achieved by the invention. 
While the invention herein disclosed has been described by means of 
specific embodiments and applications thereof, numerous modifications and 
variations could be made thereto by those skilled in the art without 
departing from the scope of the invention as set forth in the claims.