Abstract:
Methodologies for designing and assembling an analog Nyquist filter require a filter unit which includes a low pass filter in cascade with at least one tapped delay filter. A Signal Generator is used to generate a test pattern for input into the filter unit in order to create a reaction signal from the filter unit. This reaction signal is then compared with a desired Nyquist response. Based on this comparison, amplifier gains for taps in the tapped delay filter are weighted to establish a transfer function in the filter unit. In operation the transfer function shapes analog input signals with the desired Nyquist response for use as an output from the Nyquist filter.

Description:
This application is a continuation-in-part of application Ser. No. 14/200,592 filed Mar. 7, 2014, which is currently pending. The contents of application Ser. No. 14/200,592 are incorporated herein by reference. 
    
    
     FIELD OF THE INVENTION 
     The present invention pertains generally to analog filters which can be used to filter analog or digital signals having a predetermined symbol rate. More particularly, the present invention pertains to Nyquist filters which are designed and assembled to filter signals using only hardware components. The present invention is particularly, but not exclusively, useful for designing filters having a Nyquist response, wherein samples are taken from an analog input signal by a tapped delay filter (equalizer) and are weighted to create the filter, based on a comparison between the filter&#39;s response to a known test pattern and a desired Nyquist response. 
     BACKGROUND OF THE INVENTION 
     All telecommunication systems have the objectives of transmitting signals with minimal signal loss, and with the highest possible signal to noise ratio (SNR). Also, in order to simplify the transmitting system&#39;s architecture and reduce its power requirements, it is desirable that signal transmissions be successfully accomplished with limited bandwidth requirements. To further these purposes, filters are often used in transmission systems to reduce bandwidth requirements and to minimize unwanted noise (interference) during a signal transmission. 
     In the context of the present invention, it is to be understood that many data signals are digitally generated as a stream of bits. Accordingly, digital signal processing techniques are typically considered. It is known, however, that digital processing techniques are generally impractical at high data rates. 
     With a view toward processing digital data signals at high data rates, it is to be appreciated that these signals can be characterized as a sequence of symbols which are akin to a frequency. In this characterization, each symbol includes an integer number of bits, and each symbol in the sequence has a same duration time, T. A transmitting device can then put symbols onto a channel at a symbol rate, Rs, with T=1/Rs. An important consequence of this is that by using techniques well known in the pertinent art, digital signals can be effectively processed as analog signals. 
     As implied above, many telecommunications systems incorporate low pass filters for the purpose of limiting a system&#39;s bandwidth requirements. As the name implies, low pass filters are designed to pass signals having frequencies which are below a predetermined stop bandwidth frequency. Because the signals are sinusoidal in nature, it happens that the signal level will begin to noticeably diminish at the higher frequencies in a bandwidth which are near, but below, the stop bandwidth frequency. 
     It is, of course, desirable to effectively use all frequencies in a bandwidth. To do this, a particular type of filter which comes closest to maximizing the useful bandwidth response is a low pass filter which is commonly referred to as a “Nyquist filter.” Ideally, a Nyquist filter will reduce the required bandwidth for transmitting a data signal, and it will do so without degrading the signal. 
     With the above in mind, it is an object of the present invention to present methodologies which employ empirical, analog signal processing techniques for designing and assembling tapped delay filters that provide a Nyquist filter response in a telecommunications system. In another aspect, it is an object of the present invention to provide methodologies using analog techniques for equalizing a simple low pass filter to manufacture an analog Nyquist filter having sophisticated performance characteristics. Another object of the present invention is to provide methodologies for use in the manufacture of analog Nyquist filters that rely solely on altering the hardware characteristics of filter components. It is also an object of the present invention to design a Nyquist filter that is capable of processing digital signals at high data rates. Still another object of the present invention is to provide methods for designing and assembling analog Nyquist filters which are easy to implement, are simple to modify in order to accommodate specific implementations, and are cost effective. 
     SUMMARY OF THE INVENTION 
     In overview, the present invention is directed to methods and systems for designing and assembling an analog Nyquist filter that will provide a low pass filter function. In particular, the methodologies of the present invention involve empirically setting the amplifier gains (i.e. tap weights) for a tapped delay filter, and to thereby configure a filter unit which will elicit an operational analog Nyquist response with a low pass filter function during the transmission of a data signal. The desired output spectrum, Hrc(α,f), for this analog Nyquist response is a raised cosine function, wherein α is a bandwidth factor and f is frequency. Operationally, the present invention configures the filter unit by creating a transfer function, RC(α,f), for the filter unit, which can be mathematically characterized as a sin c corrected, raised cosine function. More specifically, this sin c corrected, raised cosine filter function, RC(α,f), results by correcting the analog Nyquist response Hrc(α,f), i.e. the filter unit output, with a sin c function, sin c(πf/Rs). Thus, as a mathematical expression:
 
 RC (α, f )=sin  c (π f/Rs ) −1   Hrc (α, f ).
 
     For the present invention, when using a Pseudo-Random Bit Stream (PRBS) as an input, the reaction signal of the filter unit will have a beginning roll-off frequency, F R1 , and an ending roll-off frequency, F R2 . Mathematically expressed:
 
 F   R1 =(1−α) Rs/ 2; and
 
 F   R2 =(1+α) Rs/ 2.
 
In the above expressions, Rs is the symbol rate of the data signal being filtered, and α is a bandwidth factor which depends on the number of taps, N, that are used for the tapped delay filter. For example, α=0.25 will typically correspond with a tapped delay filter having seven taps, N=7. Further, F R2  is preferably established such that, F R2 =(1+α)Rs/2=0.625Rs.
 
     With the above in mind, a method for designing a Nyquist filter with a low pass filter function in accordance with the present invention requires initially providing a filter unit. Structurally, the filter unit includes a low pass analog filter which is connected in cascade with at least one, or two tapped delay filters. As is well known in the pertinent art, each tap of the tapped delay filter will have a dedicated amplifier. For the present invention, the tapped delay filter may be a Finite Impulse Response (FIR) filter, or an Infinite Impulse Response (IIR) filter, or it may be a Feed Forward Equalizer (FFE) filter. Insofar as the low pass analog filter is concerned, for testing and design purposes, the present invention envisions the use of any type filter well known in the pertinent art, such as a lossy transmission line, an LC filter, or a linear phase low pass filter. Preferably, in order to reduce the required number of taps for testing and design, the filter that is used will have a 3 dB bandwidth that is typically 0.7Rs. 
     Using the filter unit, the next step in the methodology of the present invention is to generate a test pattern for use as an input into the filter unit. Specifically, for this purpose the test pattern can be an analog signal having a predetermined symbol rate (Rs), such as a Pseudo-Random Bit Stream (PRBS). Alternatively, another test pattern type that can be used is an impulse function. In this latter case, an impulse function can be approximated by a narrow pulse, as long as the pulse width is less than the tap delay, τ. In each case, the test pattern is used to create a reaction signal which is then compared with the desired Nyquist response. In this comparison, the reaction signal can be evaluated using an eye diagram (pattern) in a manner well known in the art. 
     Based on the comparison between the reaction signal and the desired Nyquist response, tap weights for the taps of the tapped delay filter are determined. For the case where an impulse function (waveform) is used for design purposes, there is the added advantage that the reaction signal sequentially reads out the tap weights in the time domain. This allows a simple method for individually adjusting the tap weights so that the reaction signal matches the impulse response of the desired sin c-corrected raised cosine filter function RC(α,f). In any event, it is with the tap weights that a gain is identified for each dedicated amplifier in the tapped delay filter. Consequently, the gain for each tap of the tapped delay filter is based on the comparison between the desired Nyquist response and the reaction signal, and this comparison is used to alter the tapped delay filter to establish a transfer function, RC(α,f), for the filter unit. 
     In some cases the transfer function for the analog low pass filter, CH(f), of the filter unit, in the frequency domain, is well defined or can be measured. Eq(f) is an equalizer function in the frequency domain for the tapped delay filter of the Nyquist filter which can then be expressed as the ratio of the transfer function RC(α,f) of the filter unit to the transfer function of the analog low pass filter, CH(f) in the filter unit:
 
 Eq ( f )= RC (α, f )/ CH ( f ).
 
It will also be appreciated that in the time domain a corresponding impulse response, eq, can then be expressed for the tapped delay filter as the Inverse Fast Fourier Transform (IFFT) of the equalizer function Eq(f) from the frequency domain:
 
 eq=IFFT ( Eq ( f )).
 
It is this impulse response, eq, that is then sampled an n number of times for each symbol in order to establish amplifier gains as weight taps in the tapped delay filter.
 
     In sum, the gain for each tap of the tapped delay filter is adjusted so that the product of the transfer function of the tapped delay filter, Eq(f), and the transfer function of the low pass filter response, CH(f), equals the sin c-corrected raised cosine transfer function, RC(α,f) of the filter unit:
 
 RC (α, f )= Eq ( f ) CH ( f ).
 
The resultant filter unit can then be used as a Nyquist filter.
 
     For a preferred embodiment of the present invention, which will give the best signal to noise ratio, albeit at the expense of more hardware, the filter unit includes a first tapped delay filter and a second tapped delay filter. In combination, the first tapped delay filter is connected to an input of the low pass filter and the second tapped delay filter is connected to an output of the low pass filter. Further, the second tapped delay filter is matched to the first tapped delay filter. For this particular combination, each filter has a transfer function that is equal to the square root of Eq(α,f). Consequently, the cascaded function of all three filters will equal RC(α,f), for the sin c-corrected raised cosine filter. In one alternate embodiment of the present invention, a single tapped delay filter is connected to an input of the low pass filter. In another alternate embodiment of the present invention, a single tapped delay filter is connected to an output of the low pass filter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The novel features of this invention, as well as the invention itself, both as to its structure and its operation, will be best understood from the accompanying drawings, taken in conjunction with the accompanying description, in which similar reference characters refer to similar parts, and in which: 
         FIG. 1  is a schematic presentation of components in a filter unit that is to be designed and assembled as a Nyquist filter in accordance with the present invention; 
         FIG. 2  shows a frequency response of a “Sin c Corrected Raised Cosine Filter” which is used as a desired Nyquist Response for the present invention; 
         FIG. 3  shows a desired frequency spectrum of the output reaction signal of a Nyquist filter of the present invention, with the output shown superposed on pertinent frequency functions used for the design of the output; 
         FIG. 4A  is an operational flow chart showing the interactive tasks that are necessary for modifying the transfer functions shown in  FIG. 4B  during the design and assembly of an analog Nyquist filter in accordance with the methodologies of the present invention; 
         FIG. 4B  is a schematic presentation of a transfer function interaction between inter-operative components of the present invention during the design and assembly of an analog Nyquist filter in accordance with the present invention; 
         FIG. 5  is an exemplary Non-Return to Zero (NRZ) bit stream (time domain) to be used by the present invention as a test pattern (for design and assembly purposes), or as an input (for operational purposes); 
         FIG. 6  is an exemplary eye diagram (pattern), of a filter output for use in evaluating the operational design and assembly of the present invention; 
         FIG. 6A  shows exemplary input and output spectra for a filter unit with a Pseudo-Random Bit Stream (PRBS) input, showing bandwidth reduction using a 3-pole analog low pass filter, having a 3-dB bandwidth of 66% of the symbol rate Rs; 
         FIG. 7  is a generalized Nyquist filter in accordance with the present invention shown connected with peripheral components for operational use; 
         FIG. 8A  shows an embodiment of a Nyquist filter designed by the present invention which has no equalizer downstream from the low pass filter; 
         FIG. 8B  shows an embodiment of a Nyquist filter designed by the present invention which has no equalizer upstream from the low pass filter; and 
         FIG. 8C  shows an embodiment of a Nyquist filter designed by the present invention which has equalizers both upstream and downstream from the low pass filter. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Referring initially to  FIG. 1 , a filter unit in accordance with the present invention is shown and is generally designated  10 . As shown, the exemplary filter unit  10  includes a tapped delay filter  12  and an analog low pass filter  14  which are connected in cascade with each other. In this combination, the tapped delay filter  12  may be of any type that is well known in the pertinent art, such as a Finite Impulse Response (FIR) filter, an Infinite Impulse Response (IIR) filter, or a Feed Forward Equalizer (FFE) filter. Insofar as the analog low pass filter  14  is concerned, it may also be of any type filter well known in the pertinent art, such as a lossy transmission line, an LC filter, or a linear phase low pass filter. 
     As appreciated by the present invention, the filter unit  10  inherently creates a transfer function RC(α,f) that will shape an input, e.g. a test pattern (data signal)  16 , into a reaction signal (useable output)  18 . The present invention, however, is used to specifically shape the transfer function RC(α,f) to produce a desired or predetermined output signal  18 . Specifically, in accordance with the present invention, this is done by properly configuring the tapped delay filter  12 . 
     For discussion purposes, a typical input filter unit  10  may include a tapped delay filter  12  having an N number of taps. Operationally, an n number of these N taps will be used to sequentially sample each symbol in the data signal  16 . As envisioned for the present invention, the number n is a real number that will preferably be less than about 2 (e.g. n=1.7). In any case, n must be greater than 1 (n&gt;1). For disclosure purposes, and as shown in  FIG. 1 , N=7 for taps  20   a - 20   g . Between all adjacent taps  20  there will be a same duration time, τ, where τ=(nRs) −1 , and each tap  20  will have a dedicated amplifier (e.g. w 1  . . . w 7 ). As intended for the present invention, a preferred transfer function RC(α,f) for the filter unit  10  will have a frequency response  22  substantially as shown in  FIG. 2 . To achieve this response  22 , however, requires a correction of the filter unit  10 . 
     In order to design the transfer function RC(α,f) for the filter unit  10 , it is to be appreciated that, with a PRBS input to the filter unit  10 , the output spectrum of the reaction signal  18  in the frequency domain will be a raised cosine function, Hrc(α,f). 
     Mathematically, the various transfer functions involved with the present invention, and their relationships with each other, are set forth below. For each of these mathematical expressions, α is a bandwidth factor, Rs is a symbol frequency, and f is frequency. In  FIG. 3 , it will be appreciated that each transfer function can be considered across three frequency regions. In this case, the regions are defined relative to a beginning roll-off frequency, F R1 , and an ending roll-off frequency, F R2  which, as indicated above, are respectively expressed as:
 
 F   R1 =(1−α) Rs/ 2; and
 
 F   R2 =(1+α) Rs/ 2.
 
More specifically, a first region can be defined for frequencies f below F R1 , a second region can be defined for frequencies f between F R1  and F R2 , and a third region can be defined for frequencies f greater than F R2 . In detail, these regions are respectively defined, in order, as:
 
     
       
         
           
             
               
                 
                   f 
                   &lt; 
                   
                     
                       ( 
                       
                         1 
                         - 
                         α 
                       
                       ) 
                     
                     · 
                     
                       Rs 
                       2 
                     
                   
                 
               
               
                 
                   
                     Hrc 
                     ⁡ 
                     
                       ( 
                       
                         α 
                         , 
                         f 
                       
                       ) 
                     
                   
                   = 
                   1 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         1 
                         - 
                         α 
                       
                       ) 
                     
                     · 
                     
                       Rs 
                       2 
                     
                   
                   ≤ 
                   f 
                   ≤ 
                   
                     
                       ( 
                       
                         1 
                         + 
                         α 
                       
                       ) 
                     
                     · 
                     
                       Rs 
                       2 
                     
                   
                 
               
               
                 
                   
                     Hrc 
                     ⁡ 
                     
                       ( 
                       
                         α 
                         , 
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                     · 
                     
                       [ 
                       
                         1 
                         + 
                         
                           cos 
                           ⁡ 
                           
                             [ 
                             
                               
                                 π 
                                 α 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     f 
                                     Rs 
                                   
                                   - 
                                   
                                     
                                       1 
                                       - 
                                       α 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   f 
                   &gt; 
                   
                     
                       ( 
                       
                         1 
                         + 
                         α 
                       
                       ) 
                     
                     · 
                     
                       Rs 
                       2 
                     
                   
                 
               
               
                 
                   
                     Hrc 
                     ⁡ 
                     
                       ( 
                       
                         α 
                         , 
                         f 
                       
                       ) 
                     
                   
                   = 
                   0 
                 
               
             
           
         
       
     
     Of particular importance here are the values for the output spectrum of the reaction signal Hrc(α,f) for each of the regions. The result here is plot  24  of the reaction signal shown in  FIG. 3 . 
     Still referring to  FIG. 3 , the input spectrum  26 , transfer function  22  and the plot of desired output reaction signal  24  can be expressed as: 
     Input Spectrum (the sin c Function  26 ) 
     
       
         
           
             
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               
                 c 
                 ⁡ 
                 
                   ( 
                   
                     π 
                     ⁢ 
                     
                       f 
                       Rs 
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 sin 
                 ⁡ 
                 
                   ( 
                   
                     π 
                     · 
                     
                       f 
                       Rs 
                     
                   
                   ) 
                 
               
               
                 ( 
                 
                   π 
                   · 
                   
                     f 
                     Rs 
                   
                 
                 ) 
               
             
           
         
       
     
     Filter Unit Transfer Function (Frequency Response  22 ) 
     
       
         
           
             
               RC 
               ⁡ 
               
                 ( 
                 
                   α 
                   , 
                   f 
                 
                 ) 
               
             
             = 
             
               
                 Hrc 
                 ⁡ 
                 
                   ( 
                   
                     α 
                     , 
                     fn 
                   
                   ) 
                 
               
               
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   c 
                   ⁡ 
                   
                     ( 
                     
                       π 
                       ⁢ 
                       
                         f 
                         Rs 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Output Spectrum (Reaction Signal  24 ) 
               Hrc   ⁡     (     α   ,   f     )       =     sin   ⁢           ⁢     c   ⁡     (     π   ·     f   Rs       )       ⁢     RC   ⁡     (     α   ,   f     )               
It is to be appreciated that for purposes of the present invention, the frequency responses shown in  FIG. 3  are exemplary of a test filter unit  10 . For this example, α=0.25, which will typically correspond with a tapped delay filter having seven taps, N=7, and n=2, so that τ=(nRs) −1 =(2Rs) −1 . Also, F R2 =(1+α)Rs/2=0.625Rs.
 
     An operation of the present invention is perhaps best appreciated by cross referencing  FIG. 4A  with  FIG. 4B . For the design and testing of the filter unit  10  in accordance with the present invention, a user of the invention will normally follow the task functions indicated by an exemplary method diagram which is generally designated  28  in  FIG. 4A . According to the diagram  28 , block  30  indicates that a test pattern  16  needs to be inputted into the filter unit  10 . For this purpose, the test pattern  16  will typically be a bit stream  32  similar to the one shown in  FIG. 5 . Further,  FIG. 4B  indicates the spectrum of the input test pattern  16  will be a sin c function  26 . In general, the test pattern (data signal)  16  can be any PRBS input, such as a Non-Return-to-Zero (NRZ) signal, a Pulse-Amplitude-Modulation (PAM) signal, a Duo-Binary (DB) signal, or a signal from a symbol generator. Importantly, in each instance, the input signal will have a fixed (predetermined) symbol rate, Rs, which is typically greater than one gigabaud. 
       FIG. 4A  indicates that the test pattern  16  is passed through the filter unit  10 , where it is influenced by the transfer function, RC(α,f) for output from the filter unit  10  as the reaction signal  18  having a spectrum Hrc(α,f) (see  FIG. 4B ). Inquiry block  34  of the diagram  28  further indicates that the desired Nyquist frequency response  22  is provided for comparison with the transfer function RC(α,f) of the filter unit  10 . In the event there is an inaccurate, incomplete or otherwise disparate comparison between RC(α,f) and the desired Nyquist response  22 , Hrc(α,f), inquiry block  34  indicates that the next operational action is for an adjustment of the taps  20  on the tapped delay filter  12  (see block  38 ). In effect, these adjustments change the transfer function RC(α,f). 
     As a practical matter, the comparison required by inquiry block  34  can be accomplished using an oscilloscope (not shown) to produce an eye diagram  44  (see  FIG. 6 ) of a type well known in the pertinent art. In particular, the eye diagram  44  should be taken at a point  46  which is located off the output of filter unit  10  (see  FIG. 1 ). Using the eye diagram  44 , the user can then adjust the amplifier gains (tap weights) of the taps  20  on tapped delay filter  12 . The intended result will then be an output similar to the output spectrum  47  shown in  FIG. 6A . 
     In accordance with the above, the purpose of the present invention is to design and assemble a filter unit  10  for use in a data transmission system, such as the one shown in  FIG. 7  and generally designated  48 . For the exemplary case shown in  FIG. 7 , the filter unit  10  is shown receiving an analog input signal  50 . In this case, the filter unit  10  includes a tapped delay filter  12 ′, which is connected upstream from the low pass filter  14 , and it also includes a tapped delay filter  12 ″, which is connected downstream from the low pass filter  14 .  FIG. 7  also shows that the present invention envisions employing a decision circuit  52 , and possibly a Decision Feedback Equalization (DFE) circuit  54  for enhancing the performance of the decision circuit  52  that converts the filtered analog signal  50  into an output bit stream  56 . Further,  FIG. 7  indicates that the system  48  can be monitored and periodically evaluated with an eye diagram  44  taken at the point  46 . 
     Alternate embodiments for the system  48  can selectively include filter units  10   a ,  10   b  or  10   c , which are respectively shown in  FIGS. 8A-C . Specifically, in  FIG. 8A  the filter unit  10   a  is shown to have a tapped delay filter  12 ′ which has been designed, as disclosed above, with a transfer function Eq(α,f). Accordingly, the tapped delay filter  12 ″ for the alternate embodiment shown in  FIG. 8A  has a transfer function H(f)=1 (i.e. there essentially is no tapped delay filter  12 ″). On the other hand, in  FIG. 8B , it is the tapped delay filter  12 ′ that has a transfer function H(f)=1, with the tapped filter  12 ″ being designed as disclosed above with the transfer function Eq(α,f). In  FIG. 8C , however, both of the tapped delay filters  12 ′ and  12 ″ are functional, and they are both designed as square root functions, i.e. the square root of Eq(α,f), which are to be used, in cascade, in a manner well known in the pertinent art to establish the transfer functions Eq(α,f), and RC(α,f) for the filter unit  10 . 
     While the particular Method for Designing an Analog Nyquist Filter as herein shown and disclosed in detail is fully capable of obtaining the objects and providing the advantages herein before stated, it is to be understood that it is merely illustrative of the presently preferred embodiments of the invention and that no limitations are intended to the details of construction or design herein shown other than as described in the appended claims.