Jitter Insertion System for Waveform Generation

Abstract

A test and measurement instrument for generating an analog waveform, comprising: an interpolator configured to receive a digital signal and output interpolated samples of the digital signal at a sampling rate; a filter modulation controller configured to output a first filter coefficient at a first time and a second filter coefficient at a second time; a convolver configured to generate a convolved signal by convolving the interpolated samples of the digital signal with the first filter coefficient and convolving the interpolated samples of the digital signal with the second filter coefficient; and a digital-to-analog converter configured to convert the convolved signal into an analog signal based on a fixed constant clock signal.

Description

[0001] priority

[0002] This disclosure claims the benefit of U.S. Provisional Application No. 62 / 801,028, filed on February 4, 2019, entitled “JITTER INSERTION SYSTEM FOR WAVEFORM GENERATION,” the entire contents of which are incorporated herein by reference. Technical Field

[0003] The present disclosure relates to systems and methods related to test and measurement systems, and in particular to a test and measurement system for inserting jitter into an interpolated digital signal prior to generating an analog signal. Background Art

[0004] Many test and measurement instruments, such as oscilloscopes, arbitrary waveform generators (AWGs), or bit error rate testers (BERTs), generate analog waveforms that can be transmitted to a device under test to provide information about the device under test to the user. It can often be beneficial to insert noise into the generated analog signal to test how well the device under test is performing. For example, it can be beneficial to insert jitter into the generated analog waveform.

[0005] Typically, test and measurement instruments apply edge jitter to the generated analog waveform by changing the sampling clock of the digital-to-analog converter (DAC). However, the disadvantage of this approach is that for very high-speed interleaved DAC systems, the design and execution of the calibration required for each DAC is complex and costly. In other words, changing the clock rate of the interleaved DAC can significantly multiply the cost of the calibration work.

[0006] Rather than changing the DAC's sampling clock, some test and measurement instruments introduce jitter into the signal by heavily oversampling the incoming digital signal in the digital signal processor (DSP), shifting the edges to new locations, then applying a bandwidth-limiting filter and resampling at the DAC sampling rate. However, in waveform generation systems where the DSP is expected to operate in real time at DAC sampling rates exceeding 200 gigasamples per second (GS / s), the oversampling approach is impractical due to the circuit speeds that the algorithms inherently require.

[0007] Embodiments of the present disclosure address these and other deficiencies of the prior art. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Various aspects, features and advantages of embodiments of the present disclosure will become apparent from the following description of embodiments with reference to the accompanying drawings, in which:

[0009] Figure 1is a block diagram of a portion of a test and measurement instrument for generating a waveform with inserted jitter according to some embodiments of the present disclosure.

[0010] Figure 2 is a block diagram of a portion of a test and measurement instrument for generating a waveform with inserted jitter according to other embodiments of the present disclosure.

[0011] Figure 3 is a comparison of an interpolated waveform and a waveform with inserted jitter according to some embodiments of the present disclosure.

[0012] Figure 4 is a flow chart illustrating a method for generating a dithering waveform according to an embodiment of the present disclosure. DETAILED DESCRIPTION

[0013] A test and measurement instrument is disclosed herein that can insert jitter into a waveform to be generated for and transmitted to a device under test without adjusting the sampling rate of one or more digital-to-analog converters or oversampling the digital signal in a DSP. That is, the test and measurement instrument can generate a jitter waveform while using a fixed constant clock for the digital-to-analog converter.

[0014] Embodiments of the present disclosure provide a device for introducing jitter into a digital signal by applying a filter whose coefficients change with time in the digital signal (such as but not limited to an all-pass filter). Specifically, and as discussed in more detail below, the digital signal can be filtered using an all-pass filter, and the filter controller can identify when to switch different coefficients of the filter with different group delays. Other criteria can be used to change the filter so that periodic long-term jitter can be introduced on multiple edges. However, embodiments of the present disclosure are not limited to modulation waveform edge transitions, and other types of modulation algorithms can be deployed by controlling the filter coefficients of the filter.

[0015] Figure 1 An example of a portion of a test and measurement instrument according to some embodiments of the present disclosure is illustrated. As will be appreciated by those skilled in the art, the test and measurement instrument may include additional components not shown, such as but not limited to a processor, memory, user input, etc.

[0016] An input (or input waveform) 100 is received at an interposer 102. The input 100 may be received from a memory (not shown) on a test and measurement instrument, or may be supplied by other means (such as through a cable connection to another device) or provided by a processor (not shown) on the test and measurement instrument. The input 100 may be any signal or sample representing a binary value, including, for example, a bit stream or a signal having high and low values. In some embodiments, the bit stream may include a pseudo-random bit stream pattern (PRBS).

[0017] Interpolator 102 converts or interpolates input 100 into samples at the sampling rate of digital-to-analog converter 104. The interpolated samples may be received at filter modulation controller 106 and convolver 108, each of which will be discussed in greater detail below.

[0018] The filter modulation controller 106 receives the filter coefficient array 110 from a memory (not shown). The filter coefficient array 110 may be stored, for example, in a lookup table so that the filter coefficient array may be easily accessed during waveform generation.

[0019] Filter coefficient array 110 is an array of filter coefficients of length M at a sampling rate fs. The sampling rate fs is the sampling rate of the output waveform 114 from DAC104. Each set of coefficients h 126 in the array has a different delay. The filter coefficients are generated by filter generator 112, which can be done before generating the output waveform 114 and stored in a lookup table in the memory of the test and measurement instrument as described above. In some embodiments, each filter has the same flat frequency response in the range of direct current (DC) to almost Nyquist. However, in other embodiments, the shape of the frequency response can also vary within the range of the filter index for the array.

[0020] Filter generator 112 may include filter array generator 116 and filter coefficient generator 130. Filter array generator 116 may be used to generate filter coefficient array 110 using filter coefficients generated by filter coefficient generator 130. The filter may vary in group delay, phase (f), and magnitude (f), where f is frequency. Filter array generator 116 receives minimum delay 118, maximum delay 120, length M 122, and sampling rate fs 124 from user input (not shown). User input may include a keyboard, mouse, trackball, touch screen, and / or any other control that may be used by a user to interact with a graphical user interface displayed on a display.

[0021] The filter array generator 116 may determine the delay t 128 and output the delay t 128, the length M 122, and the sampling rate fs 124 to the filter coefficient generator 130. The filter coefficient generator 130 may generate coefficients with different group delays. For example, in some embodiments, an all-pass filter coefficient h 126 may be generated with a flat frequency response. For periodic jitter, the frequency of the jitter modulation may be varied.

[0022] The filter coefficient array 110 may contain a set of one hundred or more filters whose group delay covers a time range within which the edges of the waveform will jitter. In some embodiments, the time range may be selected by the user and limited only by the length of each filter and the sampling rate fs 124. In other embodiments, the time range is automatically determined by the filter array generator 116 based on the filter lengths and the sampling rate fs 124.

[0023] Filter modulation controller 106 can determine when to update the filter coefficients in convolver 108 to shift each edge in input waveform 100 by a random group delay amount. That is, filter modulation controller 106 can control when a new set of coefficients from filter coefficient array 110 is transmitted to convolver 108 based on time.

[0024] exist Figure 1 , the filter modulation controller 106 may include a bit identifier 132 and a filter selector 134. The bit identifier 132 may determine when to update the coefficients in the convolver 108 based on the interpolated samples. For example, for random edge jitter, the bit identifier 132 may identify the time position at the center of each bit interval to determine the bit before the transition in the input waveform 100. When the center of each bit interval before the transition is determined, the filter selector 134 may select the filter coefficients to upload to the convolver 108.

[0025] Filter selector 134 selects which filter coefficients to transmit to convolver 108 from filter coefficient array 110. In some embodiments, filter selector 134 may use a random number generator to select which filter coefficients to use from the filter coefficient array. This may randomize the group delay to be updated to convolver 108 to prevent edge discontinuities that will jitter over time.

[0026] For periodic jitter, the bit identifier 132 can determine when to update the filter coefficients in the convolver 108 so that the group delay as a function of time is a sine wave, a ramp, a triangle, or any other shape as a function of time. Then, the filter selector 134 can select which filter coefficients from the filter coefficient array 110 to transmit to the convolver 108.

[0027] The convolver 108 convolves the interpolated waveform samples with the filter coefficients and introduces jitter into the interpolated waveform samples. The convolver 108 introduces jitter at the point in the system when the edges of the input waveform 100 are only shaped from the ideal shape to the shape caused by the interpolator 102. Therefore, the edges are different and can be moved by switching the filter bank delays.

[0028] In some embodiments, such as in a high-speed parallel pipeline system, the convolver 108 may have multiple finite impulse response (FIR) filters so that one filter may be computing the convolution to obtain one output sample while another filter is updating its coefficients and preparing to compute and output a subsequent waveform sample.

[0029] After the jitter is inserted into the waveform samples, the waveform samples may be further filtered by one or more filters 136. Typically, the test and measurement instrument may have one or more other waveform shaping filters 136 applied before generating the analog signal (output waveform) 114. These filters may include, for example, temperature compensation filters, bandwidth enhancement filters for correcting hardware response, emulation filters for emulating channels, continuous time linear equalization filters, or other general purpose filters such as for varying bandwidth. These filters may remain fixed in time when applied to the waveform.

[0030] Usually, and if Figure 1 As shown in , these filters can be applied after dither modulation by the convolver 108 because the filter modulation controller 106 can more accurately act on the interpolated waveform. That is, the filter modulation controller 106 can more accurately identify the appropriate time on the interpolated waveform rather than the filtered waveform to update the filter coefficients in the convolver 108. In addition, applying dithering by the convolver 108 before the filter can help minimize the amplitude discontinuities caused by phase shifts caused by switching the filter coefficients h 126 over regions where the derivative of the waveform is zero.

[0031] However, embodiments of the present disclosure are not limited to applying one or more filters 136 after introducing jitter into the interpolated waveform. In some embodiments, one or more filters 136 may filter the waveform before the convolver 108 and / or one or more filters 136 may be applied both before and after the convolver 108.

[0032] The output from the convolver 108 or the filter 136 is input to the DAC 104 to generate an analog signal (output waveform) 114 from the waveform samples. The DAC 104 receives a fixed constant clock signal 138, which can make the DAC 104 easier to calibrate during manufacturing. In some embodiments, the DAC 104 can be implemented by using multiple interleaved DACs or other topologies for combining the outputs from multiple DACs to reconstruct the output waveform.

[0033] Embodiments of the present disclosure are not limited to predetermining the filter coefficient array 110 and storing it in a lookup table. Rather, in some embodiments, the filter coefficient array may be generated in real time, such as Figure 2 as shown in .

[0034] exist Figure 2 1, the input waveform 100, interpolator 102, convolver 108, filter 136, DAC 104, fixed constant clock 138, and output waveform 114 operate in a manner similar to that described above with respect to Figure 1 Therefore, these components are given the same reference numerals and will no longer be compared with each other. Figure 2 Further detailed discussion.

[0035] exist Figure 2 1. In the embodiment of the present invention, the filter modulation controller 200 can generate filter coefficients in real time according to time or according to other criteria, including but not limited to logic decisions based on waveform parameters or based on external parameters. Similar to the filter modulation controller 106, the filter modulation controller 200 includes a bit identifier 202 for identifying a point in the input waveform 100 for loading new coefficients into the convolver 108.

[0036] The filter generator 204 can generate a filter based on a dithering or other modulation algorithm and then store the filter in the convolver 108 at a time identified by the bit identifier 202. This allows the filter generator 204 to generate the next set of filter coefficients while the convolver 108 convolves the interpolated waveform samples based on the previously loaded coefficients. The filter generator 204 can then begin transmitting or updating the coefficients in the convolver 108 based on the output from the bit identifier 202.

[0037] In some embodiments, Figure 1 and Figure 2 The filter generator 112 and 204 of can use filter length M, sampling rate fs and group delay t in seconds to generate all-pass filter. As mentioned above, filter coefficients can be returned in vector h 126. When using all-pass filter, filter generator 112 and 204 can first determine vector h 126 in frequency domain and its complex conjugate side in frequency domain. Then vector h 126 in frequency domain can be assigned as zero phase at all frequencies, and has a magnitude of 1.0 at all frequencies, except Nyquist, at which it is set to zero. Then, vector h 126 in frequency domain can be converted to time domain using inverse fast Fourier transform.

[0038] A plurality of vectors h 126 may be determined to generate an array of filter coefficients in filter array generator 116 or by filter generator 204. An array may be generated with filters delayed within a time range that may be set by a user.

[0039] As described above, bit identifiers 132 and 202 can identify when filter modulation controllers 106 and 200 transmit or update new filter coefficients to convolver 108. In some embodiments, this can be done by determining the first sample after the center of a bit interval. When the center is found, a new set of filter coefficients can be loaded into convolver 108, which can use the new coefficients to calculate a new output sample at that index.

[0040] The current set of coefficient filters in the convolver 108 will be used to calculate each subsequent output sample until it receives a new set of coefficient filters from the filter modulation controller 106 and 200. The filter modulation controller 106 and 200 may transmit the new set of coefficient filters at the first sample after the middle of the next bit interval. At this point, the filter modulation controller 106 and 200 will load the new set of filter coefficients into the convolver 108.

[0041] When random edge jitter is inserted, the filter coefficients in the convolver 108 are updated only by the filter modulation controllers 106 and 200 at bit intervals that have an edge transition at the end of the interval. Otherwise, the filter coefficients are not changed. This results in each edge being randomly shifted positively or negatively in time relative to the original edge transition.

[0042] Figure 3 1 illustrates examples of ideal interpolated waveforms and jittered waveforms before being converted to analog waveforms. Waveform 300 illustrates a linearly interpolated bit stream at a rate of 21 GS / s at a baud rate of 5 GBd. Waveform 302 is the output waveform from convolver 108 with jitter applied. Figure 3 As can be seen in FIG. 3 , the edges in waveform 302 are shifted in different directions by different delay amounts relative to the original edges.

[0043] Figure 4 A method for generating an analog waveform with jitter according to an embodiment of the present disclosure is illustrated.

[0044] Initially, in operation 400, user input may be received to specify the type of jitter to be inserted into the desired waveform, including but not limited to random edge jitter, periodic jitter, and time-varying intersymbol interference. The user may also input various parameters associated with each of these types of jitter or modulation selected. The user may also set parameters of the input waveform 100.

[0045] In operation 402, as discussed above, the input waveform 100 is interpolated into samples at the sampling rate of the DAC 104. Based on the interpolated samples, the filter modulation controller 106 or 200 may determine in operation 404 whether new filter coefficients need to be saved in the convolver 108. If the new filter does not need to be saved, the interpolated samples are convolved in the convolver 108 with the current filter settings in the convolver 108 in operation 406.

[0046] If the new filter is to be saved, the filter of the convolver 108 is updated with the new coefficients in operation 408 before the interpolated samples are convolved in operation 406. As discussed extensively above, the filter coefficients may be generated ahead of time and saved in memory or may be generated in real time.

[0047] As also described above, the convolved signal may then optionally be further filtered in operation 410. Figure 4 4, but in some embodiments, a filtering operation 410 is performed prior to operation 406. Finally, in operation 412, the DAC 104 converts the convolved and optionally filtered signal to an analog signal based on the fixed constant clock signal 138. This may allow jitter to be introduced into the generated waveform without the cost and complexity of modifying the clock of the DAC 104 to introduce jitter.

[0048] Aspects of the present disclosure may be operated on specially created hardware, firmware, digital signal processors, or on specially programmed computers including processors that operate according to programming instructions. The terms "controller" or "processor" used herein are intended to include microprocessors, microcomputers, application specific integrated circuits (ASICs), and dedicated hardware controllers. One or more aspects of the present disclosure may be embodied in computer-usable data and computer-executable instructions, such as embodied in one or more program modules executed by one or more computers (including monitoring modules) or other devices. Typically, program modules include routines, programs, objects, components, data structures, etc., which perform specific tasks or implement specific abstract data types when executed by processors in computers or other devices. Computer-executable instructions may be stored on computer-readable storage media (such as hard disks, optical disks, removable storage media, solid-state memory, random access memory (RAM), etc.). As will be appreciated by those skilled in the art, the functions of program modules may be combined or distributed as needed in various aspects. In addition, the functions may be embodied in firmware or hardware equivalents (such as integrated circuits, FPGAs, etc.) in whole or in part. Specific data structures may be used to more effectively implement one or more aspects of the present disclosure, and such data structures are considered within the scope of computer-executable instructions and computer-usable data described herein.

[0049] In some cases, the disclosed aspects may be implemented in hardware, firmware, software, or any combination thereof. The disclosed aspects may also be implemented as instructions carried or stored on one or more or computer-readable storage media, which may be read and executed by one or more processors. Such instructions may be referred to as computer program products. As discussed herein, computer-readable media means any medium that can be accessed by a computing device. As an example and not limitation, computer-readable media may include computer storage media and communication media.

[0050] Computer storage media means any medium that can be used to store computer-readable information. By way of example and not limitation, computer storage media may include RAM, ROM, electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technology, compact disk read-only memory (CD-ROM), digital video disk (DVD) or other optical disk storage devices, magnetic cassettes, magnetic tapes, magnetic disk storage devices or other magnetic storage devices, and any other volatile or non-volatile, removable or non-removable media implemented in any technology. Computer storage media does not include the signal itself and the temporary form of signal transmission.

[0051] Communication media refers to any medium that can be used for communication of computer-readable information. By way of example and not limitation, communication media may include coaxial cables, fiber optic cables, air, or any other medium suitable for communication of electrical, optical, radio frequency (RF), infrared, acoustic or other types of signals.

[0052] Example

[0053] Illustrative examples of the technology disclosed herein are provided below. Embodiments of the technology may include any one or more of the examples described below and any combination thereof.

[0054] Example 1 is a test and measurement instrument for generating analog waveforms, comprising: an interpolator configured to receive a digital signal and output interpolated samples of the digital signal at a sampling rate; a filter modulation controller configured to output a first filter coefficient at a first time and a second filter coefficient at a second time; a convolver configured to generate a convolved signal by convolving the interpolated samples of the digital signal with the first filter coefficient and convolving the interpolated samples of the digital signal with the second filter coefficient; and a digital-to-analog converter configured to convert the convolved signal into an analog signal.

[0055] Example 2 is the test and measurement instrument of Example 1, wherein the filter modulation controller is further configured to determine the first time and the second time by identifying a bit preceding a transition in the interpolated samples.

[0056] Example 3 is the test and measurement instrument of any of Examples 1 and 2, wherein group delays of the first filter coefficients and the second filter coefficients are different.

[0057] Example 4 is the test and measurement instrument of Example 3, wherein the filter modulation controller is configured to determine the first time and the second time so that the group delay as a function of time is a sine wave, a ramp, or a triangle.

[0058] Example 5 is the test and measurement instrument of any of Examples 1-4, wherein the digital-to-analog converter is configured to convert the convolved signal to an analog signal based on a fixed constant clock signal.

[0059] Example 6 is the test and measurement instrument of any of Examples 1-5, further comprising: a filter generator configured to generate first filter coefficients and second filter coefficients.

[0060] Example 7 is the test and measurement instrument of Example 6, wherein the filter modulation controller includes a filter generator.

[0061] Example 8 is the test and measurement instrument of Example 6 or 7, further comprising a user input, wherein the filter generator is configured to generate the first filter coefficient and the second filter coefficient based on the user input.

[0062] Example 9 is the test and measurement instrument of any of Examples 6-8, wherein the filter generator is configured to generate the first filter coefficient and the second filter coefficient based on the filter length, the sampling rate of the digital-to-analog converter, and the array length.

[0063] Example 10 is the test and measurement instrument of any of Examples 6-9, wherein the filter generator is further configured to generate the first filter coefficients and the second filter coefficients in real time.

[0064] Example 11 is the test and measurement instrument of any of Examples 1-10, further comprising: a memory configured to store a lookup table including first filter coefficients and second filter coefficients.

[0065] Example 12 is the test and measurement instrument of any of Examples 1-11, further comprising: one or more filters configured to filter the convolved signal before converting the convolved signal to an analog signal.

[0066] Example 13 is a method for introducing jitter into a generated analog waveform in a test and measurement instrument, comprising: receiving a digital signal; generating interpolated samples of the digital signal at a sampling rate; selecting a first set of filter coefficients at a first time and selecting a second set of filter coefficients at a second time; generating a convolved signal by convolving the interpolated samples of the digital signal with the first filter coefficient and convolving the interpolated samples of the digital signal with the second filter coefficient; and converting the convolved signal to an analog signal.

[0067] Example 14 is the method of Example 13, further comprising determining the first time and the second time by identifying a bit before a transition in the interpolated samples.

[0068] Example 15 is the method of any of Examples 13 or 14, wherein the group delays of the first filter coefficients and the second filter coefficients are different.

[0069] Example 16 is the method of any of Examples 13-15, generating the first filter coefficient and the second filter coefficient based on the filter length, the sampling rate of the digital-to-analog converter, and the array length.

[0070] Example 17 is the method of Example 16, further comprising: receiving at least one of a filter length, a maximum delay, and a minimum delay through a user input to generate a first filter coefficient and a second filter coefficient.

[0071] Example 18 is the method of any of Examples 16 or 17, wherein the first filter coefficients and the second filter coefficients are generated in real time.

[0072] Example 19 is the method of any of Examples 13-18, further comprising: filtering the convolved signal before converting the convolved signal to the analog signal.

[0073] Example 20 is the method of any of Examples 13-19, further comprising: storing a lookup table comprising the first filter coefficients and the second filter coefficients.

[0074] The previously described versions of the disclosed subject matter have many advantages that have either been described or will be apparent to those of ordinary skill. Even so, not all versions of the disclosed devices, systems, or methods require these advantages or features.

[0075] In addition, the written description mentions specific features. It should be understood that the disclosure in this specification includes all possible combinations of those specific features. Where specific features are disclosed in the context of a particular aspect or example, these features may also be used to the extent possible in the context of other aspects and examples.

[0076] Additionally, when the present application refers to a method having two or more defined steps or operations, the defined steps or operations may be performed in any order or simultaneously, unless the context excludes those possibilities.

[0077] Although specific examples of the present invention have been shown and described for purposes of illustration, it will be appreciated that various modifications may be made without departing from the spirit and scope of the invention.Accordingly, the present invention should not be limited, except as in the appended claims.

Claims

1. A test and measurement instrument for generating an analog waveform, comprising: an interpolator configured to receive a digital signal and output interpolated samples of the digital signal at a sampling rate; a filter modulation controller configured to output a first filter coefficient at a first time and to output a second filter coefficient at a second time; a convolver configured to generate a convolved signal by convolving the interpolated samples of the digital signal with the first filter coefficients and convolving the interpolated samples of the digital signal with the second filter coefficients; and a digital-to-analog converter configured to convert the convolved signal into an analog signal, The first filter coefficient and the second filter coefficient are generated based on the filter length, the sampling rate of the digital-to-analog converter, and the array length.

2. The test and measurement instrument of claim 1, wherein: The filter modulation controller is further configured to determine the first time and the second time by identifying a bit preceding a transition in the interpolated samples.

3. The test and measurement instrument of claim 1, wherein: The first filter coefficients and the second filter coefficients have different group delays.

4. The test and measurement instrument of claim 3, wherein: The filter modulation controller is configured to determine the first time and the second time such that the group delay as a function of time is a sine wave, a ramp, or a triangle.

5. The test and measurement instrument of claim 1, wherein: The digital-to-analog converter is configured to convert the convolved signal into an analog signal based on a fixed constant clock signal. 6 . The test and measurement instrument of claim 1 , further comprising a filter generator configured to generate the first filter coefficient and the second filter coefficient.

7. The test and measurement instrument of claim 6, wherein: The filter modulation controller includes the filter generator.

8. The test and measurement instrument of claim 6, further comprising a user input, wherein: The filter generator is configured to generate the first filter coefficients and the second filter coefficients based on the user input.

9. The test and measurement instrument of claim 6, wherein: The filter generator is also configured to generate the first filter coefficients and the second filter coefficients in real time.

10. The test and measurement instrument of claim 1, further comprising: A memory is configured to store a lookup table including the first filter coefficients and the second filter coefficients.

11. The test and measurement instrument of claim 1 , further comprising: One or more filters configured to filter the convolved signal before converting the convolved signal to an analog signal.

12. A method for introducing jitter into a generated analog waveform in a test and measurement instrument, comprising: Receiving digital signals; generating interpolated samples of the digital signal at a sampling rate; selecting a first set of filter coefficients at a first time and selecting a second set of filter coefficients at a second time; generating a convolved signal by convolving the interpolated samples of the digital signal with the first filter coefficients and convolving the interpolated samples of the digital signal with the second filter coefficients; and converting the convolved signal into an analog signal, The first filter coefficient and the second filter coefficient are generated based on the filter length, the sampling rate of the digital-to-analog converter, and the array length.

13. The method according to claim 12, further comprising: The first time and the second time are determined by identifying a bit preceding a transition in the interpolated samples.

14. The method according to claim 12, wherein: The first filter coefficients and the second filter coefficients have different group delays.

15. The method according to claim 12, further comprising: At least one of the filter length, maximum delay, and minimum delay is received through a user input to generate the first filter coefficient and the second filter coefficient.

16. The method according to claim 12, wherein: The first filter coefficients and the second filter coefficients are generated in real time.

17. The method according to claim 12, further comprising: The convolved signal is filtered before converting the convolved signal to an analog signal.

18. The method according to claim 12, further comprising: A lookup table including the first filter coefficients and the second filter coefficients is stored.

19. A computer program product having instructions which, when executed by a processor, cause the processor to perform the method of any one of claims 12 to 18.

Images