|Publication number||USRE42809 E1|
|Application number||US 11/094,754|
|Publication date||Oct 4, 2011|
|Priority date||Sep 1, 2000|
|Also published as||DE60137515D1, EP1314107A1, EP1314107A4, EP1314107B1, US6542914, WO2002019150A1|
|Publication number||094754, 11094754, US RE42809 E1, US RE42809E1, US-E1-RE42809, USRE42809 E1, USRE42809E1|
|Inventors||Peter J. Pupalaikis|
|Original Assignee||Lecroy Corporation|
|Export Citation||BiBTeX, EndNote, RefMan|
|Patent Citations (41), Non-Patent Citations (28), Classifications (9), Legal Events (6)|
|External Links: USPTO, USPTO Assignment, Espacenet|
This application claims the benefit of U.S. Provisional Application Ser. No. 60/229,856, filed Sep. 1, 2000, the entire contents of which are incorporated herein by reference.
This invention related generally to the digital manipulation of a continuous time domain sample that is to be sampled in a digital oscilloscope, and more particularly to a digital filter that is capable of increasing the bandwidth of the sampling system beyond the bandwidth range achievable in an analog system.
The present state of the art deals with an attempt to increase bandwidth based upon the assumption that only analog manipulation techniques for modifying a signal to improve the bandwidth characteristics of an apparatus are possible. Other digital techniques are seen as manipulations of the signal that change the output result of the system. This results in a design methodology in which analog design engineers painstakingly design to the best of their ability analog circuitry that has high bandwidth, flat frequency response, good pulse response and is noise-free.
In many cases, these designs are extremely complicated, particularly in the design of a digital oscilloscope. Some reasons for this difficulty are:
What makes things worse is that even upon observing and confirming the existence of problems with the bandwidth, flatness, pulse-response, and noise performance in the system, little can often be done to rectify the situation. This is because circuits designed to fix such problems are often not practically realizable.
Therefore, it would be beneficial to provide an improved Digital Signal Processing (DSP) method and apparatus capable of surgically dealing with lack of bandwidth, while offering some additional control of the pulse-response and flatness, and thereby decreasing the overall noise of the system as well.
It is therefore an object of the invention to provide an improved Digital Signal Processing (DSP) method and apparatus capable of surgically dealing with lack of bandwidth, while offering some additional control of the pulse-response and flatness.
Another object of the invention is to provide an improved Digital Signal Processing (DSP) method and apparatus capable of surgically dealing with lack of bandwidth, while offering some additional control of the pulse-response and flatness, and in which the overall noise of the system can be decreased.
A still further object of the invention is to provide an improved Digital Signal Processing (DSP) method and apparatus capable of surgically dealing with lack of bandwidth, while offering some additional control of the pulse-response and flatness by increasing the bandwidth in a very controlled manner.
Still other objects and advantages of the invention will in part be obvious and will in part be apparent from the specification and the drawings.
Generally speaking, in accordance with the invention, consider a system whose frequency response is shown in
What this system needs in order to meet its bandwidth specification is a slight boost in the frequencies in the range of 1.8 to 2 GHz. While such a boost might appear to be an easy solution, this is quite a difficult problem to solve in hardware. First, the frequencies of the hardware are quite high. Second, it is difficult to increase only the frequency response in this small band without ruining the performance of the unit.
Therefore, in accordance with the invention, a filter as specified in
After applying a filter with a response characteristic as shown in
Therefore, the invention focuses on providing a filter that meets the specifications shown in
In addition to increasing the bandwidth of the system to 2 GHz, the filter in accordance with the invention has an effect on the noise floor and roll-off. As is shown in
The invention accordingly comprises the several steps and the relation of one or more of such steps with respect to each of the others, and the apparatus embodying features of construction, combination(s) of elements and arrangement of parts that are adapted to effect such steps, all as exemplified in the following detailed disclosure, and the scope of the invention will be indicated in the claims.
For a more complete understanding of the invention, reference is made to the following description and accompanying drawings, in which:
A detailed description of the invention will now be provided, making reference to the figures. In accordance with the invention, given a desired ideal filter specification as shown in
First, applicant introduces some explanation of notation regarding the filter specification:
TABLE 1 Abbreviation Name for Description Fsbs Stop-band start The frequency at which frequency-the the boost begins. first edge of the stop-band Fpbs Pass-band start The frequency at which frequency-the the boost levels off to first edge of the boost amount the pass-band Fpbe Pass-band end The frequency at which frequency-the the boost begins to last edge of the drop off. pass-band. Fsbe Stop-band end The frequency at which frequency-the the final attenuation is last edge of the achieved. stop-band Fe The final Usually the Nyquist frequency rate at the lowest specified sample rate used with this filter. B The boost The gain of the filter in amount the boost region. A The The attenuation of the attenuation filter in the attenuation amount. region.
This notation is shown diagrammatically in
In order to implement the ideal filter specification of
In order to realize such a filter, it is natural to consider a filter consisting of a pair of complex conjugate poles, and zeros—a fourth order system. The poles and zeros are described in order of their distance from the origin in the s plane:
TABLE 2 Pole or Zero Frequency Purpose Zero 0 Around Fsbs To begin the boost ramp Pole 0 Between Fpbs To finish the boost and Fpbe ramp. Pole 1 Between Fpbs To sustain the boost and Fpbe (but and begin the boost after Pole 0) drop. Zero 1 Around Fsbe To stabilize the boost drop and to begin the attenuation region. This zero is also instrumental in the design of the resulting digital filter.
The s domain transfer function of such a filter is described using the following equation:
Where N is 2 for this case.
The filter in Equation 1 could be implemented in analog circuitry, but for the current purpose, it will be considered as an analog prototype filter that will be converted to a digital filter later in the design process.
The design of the analog prototype filter boils down to the determination of 8 values that will best meet the filter design criteria. These values are:
Qz0Qz1Qp0Qp1 Equation 2
Considering the filter in Equation 1, it is clear that the magnitude response of such a system is described as follows:
The solution of the variables in Equation 2 is performed by finding these set of variables in which the magnitude response given by Equation 3 best matches the filter design criteria in the least-squares sense. Unfortunately, Equation 3 represents a non-linear function of these variables therefore requiring methods of non-linear fitting. The method used for this invention is the Levenberg-Marquardt algorithm. In order to use this algorithm, several items must be provided. First, the partial derivatives of must be provided with respect to each of the variables being solved for:
Next, the vectors containing the frequencies and the responses desired must be created:
Note the introduction of a frequency Fhs. It is useful to control this frequency in the specification, as can be seen later. For now, assume it is three quarters of Fsbs.
The specified frequencies given are calculated specifically to provide enough points in the flat region to ensure flatness, enough points in the boost ramp region to provide a controlled boost. In short, these are the most important points in the design and therefore there are more points specified in this region.
Next, the response vector is calculated. The response vector contains the desired response at each of these frequency points. If controlling of flatness in the system is desired, frequencies and responses may be contrived which are the negative of the actual, unboosted response. I have chosen to use the specification as shown in
The response vector is generated as:
Mspec=Mdes(fspec) Equation 10
Finally, the Levenberg-Marquardt algorithm requires a vector of guesses at the values of the variables being solved for. Since Table 2 outlined the approximate pole and zero locations, it seems reasonable to use these approximations in the fitting algorithm. It also seems reasonable to select Q values that are somewhat high because of the sharp changes in the filter criteria. Therefore, the guess vector becomes:
Note the order of the values in this vector. This vector supplies the initial guesses at the variables, and is altered on each iteration of the Levenberg-Marquardt algorithm until convergence is achieved. Therefore, subsequently, the values of this guess vector are assumed to correspond to the following variables being solved for:
Upon determining these guesses, all that is necessary is to run multiple iterations of the Levenberg-Marquardt algorithm. Typically, iteration is halted once the mean-squared error has become small enough. An iteration of the Levenberg-Marquardt algorithm is shown here:
For the filter specified in
The poles and zero locations of this filter are the roots of the four equations of the following form:
This form is shown in
The response of this filter can be compared to the filter specifications and is shown in
Some observations regarding
The method chosen here for conversion to a digital filter is the bilinear transformation. In order to make this transformation, the pole and zero locations must be pre-warped to account for the nonlinear frequency mapping enforced by the bilinear transform:
Where fs is the sampling rate of the system (in GS/s, in this case) and α is the pole or zero being pre-warped. Note that at this point in the filter design, the sampling rate of the system must be known. In the implementation of this invention in a digital oscilloscope where the sample rate is variable, the design steps starting with the application of Equation 15 are performed dynamically within the oscilloscope itself as the sample rate is changed.
After the pre-warping, the new f and Q values are calculated. These are calculated as follows:
Note that only one of the complex conjugate pairs of the two sets of poles and zeros need be considered in Equation 16.
In order to convert this prototype into a digital filter, the transfer function described in Equation 1 must be factored in s and placed in the following form:
Once this is done, the filter coefficients are calculated as:
Finally, the analog filter coefficients are converted to digital filter coefficients using the bilinear coefficient formulae:
For a sample rate of 50 GS/s, the filter coefficients for the design specified are calculated as:
The final Z domain transfer function is of the form:
The affect of this filter on the overall magnitude response of the system is shown in
Where xk is the data sampled by the digitizing system and yk is the data at the output of the boost filter. The filter implementation is Infinite Impulse Response (IIR).
1.1 Noise Boost in the Pass-band
The application of this filter will boost, along with the signal, any noise contained in the boost ramp region and boost region. Therefore, before application of this filter, the unboosted system noise profile must be analyzed to determine the applicability of this filter. It is important to note that while boosting noise in these regions, the filter also attenuates noise beyond the pass-band of the system. This may or may not result in an overall noise performance improvement, depending on the noise sources.
In order to check this, a 260.6 mV rms sine wave at 2 GHz is applied using an RF signal generator and a sine-fit is applied. The sinefit tells the Signal to Noise Ratio (SNR) and also the Effective Number of Bits (ENOB).
1.2 Nyquist Limitation and Stabilizing Zero Placement
As with all digital filters, there is the limitation of sample rate on the system. Because of Nyquist's criteria, a digital filter can perform like an analog system up to ½ the sample rate of the system, after which the filter response repeats over and over. In other words, frequencies above ½ the sample rate appear as aliases at frequencies under ½ the sample rate. Even worse, the conversion of the analog prototype filter has big problems using the bilinear transformation if any pole or zero locations are above the Nyquist rate.
FcFe in the filter specification must appear at or below the Nyquist rate. The determination of the location of the stabilizing zero is performed mostly through the specification of the attenuation at Fsbe. The fact that Fsbe in this design has attenuation A specified constrains the stabilizing zero to appear at a frequency between and Fsbe and Fe. While this causes the objectionable decrease in attenuation in the attenuation region, it does help in the realization of the digital filter. The attenuation of Fsbe may be decreased, which will move the stabilizing zero higher in frequency, but the design must keep this zero below the Nyquist rate of the system, otherwise the filter design will fail.
1.3 Filter Startup
Any system employing memory will take some time to stabilize after the signal appears for the first time. This is not generally a problem, and most designs handle this by waiting some time for signals to stabilize. In the case of a digital oscilloscope, this is accomplished through pre-trigger hold-off. When the acquisition system is armed and acquiring, the trigger is held off until enough time has passed for everything to stabilize. In the case of a digital oscilloscope utilizing the present invention, there is an additional problem. There will be some stabilization time associated with the filter, and the system does not get to see the waveform until the point in time that the waveform has been acquired and is being read-out of the acquisition system memory. This means that additional samples must be acquired at the beginning of the waveform so that when the filter is applied, the system will have stabilized prior to the point at which the waveform comes on-screen. The points prior to the left edge of the oscilloscope screen where the filter is stabilizing are simply discarded. Since the filter-taps are loaded with zeros (or more commonly, the first point in the input waveform), the first point entering the filter looks like a step. Therefore, the startup time can be estimated by examining the impulse response of the filter.
The impulse response for the design example provided is shown in
In order to determine the precise startup time required,
1.4 The Importance of the Stabilizing Zero in the Digital Filter Design.
The final zero in the system, the stabilizing zero, has the effect of leveling off the attenuation of the system. Some might consider this objectionable, preferring the attenuation to continue, and thus gaining the maximum noise attenuation. This is possible, however some problems arise in the removal of this zero.
First, Equation 15, which provides the pre-warping equation, works only with an equal number of poles and zeros in the system. This is a huge benefit, because Equation 15 provides a digital filter whose frequency domain performance matches almost identically the analog filter performance, even when the frequencies of interest are up near the Nyquist rate of the system. Therefore, removal of the stabilizing zero will cause difficulties in matching the digital filter to the analog prototype filter.
Second, keep in mind that digital filters repeat above the Nyquist rate—they first fold about the Nyquist frequency up to the sample rate of the system, after which the filter image repeats over and over again. This means that a boost at a frequency at 2 GHz in a system sampling at 8 GS/s, for example, will also boost frequencies at 6 GHz. Typically, only noise is present at this frequency, but the boosting of this noise in conjunction with the obliteration of all other noise might cause artifacts which are objectionable.
1.5 The Placement of Fhs.
Equation 8 makes reference to a frequency in the design specification called Fhs. Note in
1.6 The Compromise Between Noise Performance and Pulse-response.
In this design in accordance with the invention, there are particular frequency response specifications that have significant implications with regard to the time-domain performance and to the noise reduction. Out to the boost ramp region, any smoothening of the system roll-off will improve the pulse response and reduce overshoot while simultaneously reducing noise. Beyond this point, a trade-off must be made regarding noise and pulse-response. Since the purpose of this invention is the boosting of the bandwidth of the system, the implication is that the system frequency response is rolling off around the boost point, and the roll-off rate is generally increasing. In situations like this, the pulse response will only be worsened unless the filter prior to the boost point has already applied some attenuation. In any case, the roll-off rate of the system can be controlled to some extent.
It has already been noted that the stabilizing zero does not allow the system to reach it's full potential with regard to noise performance. This zero does, however, provide the benefit of controlling the roll-off in the boost drop region. This is a good thing with regard to pulse response. It can be shown that with proper selection of the frequency and response for the frequencies Fpbe and Fsbe, and the attenuation at Fe, the trade-off between system roll-off (and thus pulse-response performance) and noise attenuation outside of the pass-band can be made. The placement of these frequencies and their responses must be made with care, however, in order to maintain the ability to fit the analog filter to the design criteria and to provide an analog filter which is realizable as a digital filter with the given sample rate constraints.
1.7 The Inverse Response Specification
The filter design specification as shown in
In the implementation of any high order filter, it is sometimes useful to separate the filter into sections, effectively cascading sections of lower order. Usually the sections are second order biquad sections. This was not deemed necessary for this design in accordance with the invention. The method of separating this filter into multiple, lower order, cascaded sections is well known by those practiced in the art of digital signal processing.
It will thus be seen that the objects set forth above, among those made apparent from the preceding description, are efficiently attained and, because certain changes may be made in carrying out the above method and in the construction(s) set forth without departing from the spirit and scope of the invention, it is intended that all matter contained in the above description and shown in the accompanying drawings shall be interpreted as illustrative and not in a limiting sense.
It is also to be understood that the following claims are intended to cover all of the generic and specific features of the invention herein described and all statements of the scope of the invention which, as a matter of language, might be said to fall therebetween.
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|International Classification||G01R13/02, G06F17/10, G01R13/22, H03H17/02|
|Cooperative Classification||G01R13/0272, H03H17/02|
|European Classification||G01R13/02C6, H03H17/02|
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