Abstract:
Precision of multi-stage digital signal processing is increased by preserving least significant bits of one or more output samples of a particular processing stage, having finite word widths, while avoiding the loss of most significant bits. The technique is applicable to one or more stages of multi-stage digital signal processing, thereby increasing precision therein and the signal-to-noise ratio. A plurality of output samples are calculated using a plurality of input samples, and the dynamic range of one or more of the output samples is decreased if the output sample can be represented in a smaller dynamic range without losing a significant bit. The input samples of a particular stage, obtained from the output samples of a previous stage, may further be normalized so that the input samples are represented in the same dynamic range before being processed.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Application No. 60/298,347, filed Jun. 15, 2001, which is herein incorporated in its entirety by reference. 

   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The invention relates to the processing of digital signals and, in particular, to methods and apparati for improving the performance of multi-stage digital signal processing having a finite precision. 
   2. Description of Background Art 
   Sophisticated algorithms for processing blocks of digital signals to improve system performance are widely applied in communications systems, image/sound/video processing systems, and storage systems. For example, in some communications systems, digital data is modulated and processed through several stages of interpolation/decimation finite/infinite impulse-response filters, e.g., at both the transmitter and the receiver. Several digital signal processing algorithms (or a sequence of such algorithms) are executed in multiple stages, where each stage employs finite word lengths for its input and output. The use of multiple stages provides increased efficiency in speed, power consumption, memory usage, and cost. 
   Various xDSL systems, such as asymmetric DSL, use discrete multi-tone (DMT) modulation to modulate digital data over a transmission medium, such as carrier lines. By applying an Inverse Fast Fourier Transform (IFFT) to the data signal, DMT provides for efficient frequency division multiplexing of the digital data. The output of the IFFT is further treated by multi-rate digital filters before the interface to an analog front-end. The analog front-end transforms the final digital data into a continuous-time waveform suitable for transmission over the transmission medium. At the receiver, the received continuous-time waveform is sampled and digitized. The digitized signal is further processed through several multi-rate digital filters and demodulated by a Fast Fourier Transform (FFT) before being passed to an estimation device. Typically, the IFFT and FFT are performed in multiple stages by algorithms such as a decimation-in-time (DIT) algorithm or a decimation-in-frequency (DIF) algorithm. 
     FIG. 1  illustrates a general representation of multi-stage processing of digital signals. The multi-stage processing comprises a series of stage processes  100 , in which a set of digital data is transformed from a set of input samples  110  into a set of output samples  120 . The N stage processes  100  typically include multiple stages of several constituent processing blocks (e.g., FFT, IFFT, multi-stage interpolation/decimation filters). The input samples  110  and output samples  120  comprise digital data expressed as a set of binary numbers. 
   One or more memory segments  130  may be provided for each stage process  100 , coupling adjacent stage processes  100  so that the digital signal can be communicated from one stage to the next.  FIG. 1  illustrates the flow of the digital signal as the data are stored in and then retrieved from each memory segment  130 . For example, in the first stage  100 (n−1), several samples from the input block may be arithmetically combined, multiplied with coefficients, or otherwise manipulated, and further combined to produce a block of output samples  120 (n−1). This block of output samples  120 (n−1) from the first stage serves as the input samples  110 ( n ) to the second stage for similar processing, and so on. In this way, the output samples  120 (n−1) of an (n−1) th  stage are used as the input samples  110 ( n ) of an n th  stage, and the output samples  120  of the n th  stage are used as the input samples  110 (n+1) of an (n+1) th  stage. For the last stage N, an (N+1) th  memory segment  130 (N) may also be provided for storing the output samples  120 (N). 
   At each stage n, the output samples  120 ( n ) are represented by a finite word width, b(n+1), which is determined by the word width allocated to each memory segment  130 (n+1) of the (n+1) th  stage. In many cases, the same memory can be used to store both the input and output of a stage, as is typically done in IFFT/FFT implementations. The output samples  120 ( n ) of the n th  stage are stored in the memory segments  130 (n+1) for the (n+1) th  stage. Accordingly, the word width b(n+1) in the memory segments  130 (n+1) for the (n+1) th  stage determines the word width allocated for storing the output samples of the n th  stage. Moreover, this word width b(n+1) corresponds to the word width for the input samples of the (n+1) th  stage. 
   The arithmetic operations in an arbitrary n th  stage of processing could require a larger word width to represent the output samples in the same dynamic range than that allocated for the output samples. In conventional processing, when this happens, the most significant bits are retained at the output of the stage while the least significant bits are lost. This “rounding” of the binary numbers leads to a loss of precision. If the number of stages is large, as it is in a typical digital signal processing application, the aggregate effect of lost least significant bits could lead to a substantial error in the digital signal due to the finite precision in the digital signal processing. This error decreases the signal-to-noise ratio (SNR), which decreases the ability of the system to transmit data and, ultimately, decreases data transfer rates. Generally, a low SNR causes frequent errors necessitating retransmission of data, thereby decreasing the system efficiency and overall transfer rate. 
   A signal value can be represented digitally by a finite number of bits in various dynamic ranges. In two&#39;s complement notation, for example, four binary bits represent a signal value of “6” as “0110” in the dynamic range [−8,7], or “0011” in the dynamic range [−16,14]. However, the signal value of “6” cannot be represented in the dynamic range [−4,3.5] without losing a significant digit (i.e., the leftmost “1” other than the sign bit). In another example, using four bits and two&#39;s complement notation, the signal value “7” cannot be represented in the dynamic range [−16, 14] without a loss of precision. Representing the signal value “7” in this case requires that the value be rounded up to 8 (expressed as “0100”) or down to 6 (expressed as “0011”). It is thus apparent that precision decreases as the dynamic range in which a signal value is represented increases (assuming there are not enough bits to represent the specified dynamic range). 
   At a particular stage of the processing, the output samples of the arithmetic computations typically need to be stored in memory, which is often constrained to be a certain number of bits. The signal processors may have the ability to express the output samples with a greater number of bits than are allocated for the samples in memory, so often the output samples must be truncated before being stored in memory. A conventional approach is to represent the output signal in the largest dynamic range in which the output samples can possibly be, without considering the actual value of the output samples. From the signal processing perspective, as demonstrated above, it is desirable to represent the output signal value in the smallest dynamic range possible without losing significant digits. This maximizes precision and thereby increases the overall SNR in the system by minimizing the quantization noise resulting from the loss of least significant bits. For example, if the digital output at a particular stage of processing has five bits, b4b3b2b1b0, but only four bits can be stored in the memory, the conventional approach is to take bits b4b3b2b1 regardless of the value of the output signal. However, it is advantageous to keep bits b3b2b1b0, provided that there is no loss of significant digits—i.e., bit b4 is not significant, like a leading “0” bit or a sign extension. 
   What is needed therefore are techniques for increasing the precision of digital signal processing by preserving the least significant bits in the output samples of a multi-stage digital signal processing block having finite word widths, while avoiding the loss of most significant bits. 
   SUMMARY OF THE INVENTION 
   Embodiments of the invention increase the precision of digital signal processing by preserving the least significant bits in output samples having finite widths, while avoiding the loss of most significant bits. In one embodiment, a method comprises calculating a plurality of output samples using a plurality of input samples, and decreasing the dynamic range of one or more of the output samples if the output sample can be represented in a smaller dynamic range without losing a significant bit. In an aspect of an embodiment, precision is maintained at the output of one or more stages by selecting for one or more of the output samples a dynamic range that is the larger of (1) the smallest dynamic range in which the output sample can be represented without losing a significant bit, and (2) the largest dynamic range selected for any previously processed output sample. In another aspect of an embodiment, the input samples of a particular stage, obtained from the output samples of a previous stage, are normalized so that the input samples are represented in the same dynamic range before being processed. 
   The embodiments may advantageously be applied to one or more stages of multi-stage digital signal processing, thereby increasing precision in the one or more stages. The increased precision achieved for several stages may significantly increase the precision of the overall multi-stage digital signal processing, thereby increasing SNR. 
   In one embodiment, a device includes a calculation module adapted to compute a set of output samples using a set of input samples, and a post-calculation module operatively coupled thereto, the post-calculation module adapted to decrease the dynamic range of at least one of the output samples for at least one stage if the output sample can be represented in a smaller dynamic range without losing a significant bit. Other embodiments of the device include a pre-calculation module for normalizing the dynamic ranges of the input samples, memory for storing processed and shifted output samples, and/or a dynamic range summer for tracking a cumulative decrease in dynamic range for the multiple stages. In some embodiments, the stages of processing and selection of dynamic ranges are performed by the same calculation and post-calculation modules. In other embodiments, a plurality of calculation and post-calculation modules are coupled together, each pair performing one or more of the various stages. The methods and devices described herein may be advantageously implemented within a DSL modem to enhance the performance thereof. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a diagram of a general multi-stage digital signal processing scheme. 
       FIG. 2  is a diagram that depicts the decreasing of the dynamic range of an output sample. 
       FIG. 3  is a schematic of a digital signal processor according to an embodiment of the invention. 
       FIG. 4  is a flow chart of the multi-stage digital signal processing according to an embodiment of the present invention. 
       FIG. 5  is a flow chart of a preprocessing method according to an embodiment of the present invention. 
       FIG. 6  is a flow chart of the postprocessing method according to an embodiment of the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   An embodiment of the present invention is advantageously applied to multi-stage digital signal processing, such as the process illustrated in  FIG. 1 . This signal processing involves one or more stages wherein the digital signal is expressed in memory segments  130  having a finite number of bits. Expressing the signal in finite memory can lead to error, or noise, caused by the loss of least significant bits (i.e., “rounding off” due to truncation); however, the present invention reduces this error. 
   The following table illustrates an example to demonstrate one advantage of the present invention. A typical stage in multi-stage digital signal processing involves a large set of data; however, only one sample is described here for explanation purposes. 
                                                             Input (stage n)   Output (stage n)                                        Word width   b(n) = 4 bits   b(n + 1) = 5 bits           Signal value (decimal)   6   2.75           Dynamic range   [−8,7]   [−4,3.75]           Sample (binary)   0110   01011                        
In this example, a particular stage of processing, n, receives an input of “6”. If this input sample is stored in two&#39;s complement notation using 4 bits in the dynamic range [−8,7], it would be expressed with the bits “0110”. Using the input sample, and perhaps other input samples, the stage processing produces an output sample of “2.75”. In this example, a 5-bit word width is allocated for the output samples of this stage. In conventional signal processing, this output sample might be stored as “00101” (rounded to “2.5”) in a dynamic range of [−8,7.5]; thus, the least significant bit is discarded and the signal is degraded. However, in accordance with an embodiment of the invention, the dynamic range of the output sample is decreased to [−4,3.75], and in that dynamic range the output sample is represented as “01011”. Because more least significant bits (LSBs) are retained in the output sample, the output sample contains a more accurate representation of the digital signal. It is thus desirable in a multi-stage digital signal process to represent the digital signal values in a smaller dynamic range without losing significant digits. Accordingly, a mechanism for tailoring the dynamic range of the digital signal being processed facilitates this desirable outcome.
 
     FIG. 2  illustrates how the dynamic range of an output sample can be changed to improve calculation precision in a stage of processing in accordance with an embodiment of the present invention. In this example, a signal value  210  of “5.6875” (e.g., for a given j th  sample of the n th  stage) is depicted in binary form with nine bits. The location  220  allocated for storing the signal value  210  is six bits wide; therefore, the signal value  210  must be truncated. The pre-shifted output sample  230  has a dynamic range of [31.5,0], where the numbers below each bit show the value of the bit for the chosen dynamic range. In the pre-shifted output sample  230 , the two leftmost bits stored in location  220  are not significant, as they are leading “0”s. (Note that there is no sign bit in this example.) The three rightmost LSBs would not fit in the location  220  and are thus discarded, in conventional processing. Therefore, the value actually stored in location  220  is “5.5”, which leads to a 3.3% error from the true signal value  210  due to truncation. 
   In accordance with an aspect of the invention, the dynamic range of the output sample is shifted to increase the precision of the stored data. In this example, the dynamic range of the location  220  is changed from [31.5,0] to [7.875,0], as shown by the numbers below each bit. Decreasing the dynamic range results in a higher precision output sample  240 . Shifted output sample  240  represents the value “5.625”, which has the significantly lower error of 1.1% compared to the pre-shifted output sample  230 . Multiplied by a large number of samples J and a large number of stages N, the resulting increase in precision for a multi-stage digital signal processor can be significant. Increasing precision increases the SNR, which allows for a greater amount of data to be transferred over a data line. 
   Accordingly, principles of the present invention increases the precision of intermediate output samples in multi-stage digital signal processing. The techniques described herein can generally be used in any environment where data samples are processed over multiple stages.  FIG. 3  depicts in block diagram form an embodiment of the present invention implemented in a sample digital signal processing system  300 . System  300  includes a memory  310 , a pre-calculation module  320 , a calculation module  330 , a post-calculation module  340 , a dynamic range summer  350 , a final stage processor  360 , and a central processing unit (CPU)  370 . The pre-calculation module  320 , calculation module  330 , post-calculation module  340 , dynamic range summer  350 , and final stage processor  360  can each be implemented as hardware, software, firmware, or some combination thereof. For example, these modules can be implemented as software instructions executing on one or more digital signal processors (DSP) or other suitable processing environment (e.g., CPU  370 ). Alternatively, these modules can be implemented as one or more application specific integrated circuits (ASIC) or other high-speed silicon process. 
   Memory  310  can be any conventional memory device, such as random access memory (RAM) or flash memory. Alternatively, memory  310  can be a cache memory internal to CPU  370 . Memory  310  is adapted to store samples used for calculation and processing by system  300 . Memory  310  may store other information as well, such as any coefficients used for the calculations. Memory  310  is coupled to the pre-calculation module  320  for providing sample values thereto, and is further coupled to the post-calculation module  340  for receiving values therefrom and storing them. Memory  310  is also coupled to the dynamic range summer  350  and the final stage processor  360 . Pre-calculation module  320  performs operations on the sample values that it receives from memory  310 , as described in more detail below with respect to  FIGS. 4 and 5 . Pre-calculation module  320  is further adapted to provide at least one input sample to calculation module  330 . Calculation module  330  is any function-applying device, such as a combination of a multiplier, adder, subtracter, divider, or some combination thereof. Generally, calculation module  330  implements the functionality of one or more stages of the multi-stage digital signal processing, such as modulation or filtering stages. Calculation module  330  provides at least one output sample to post-calculation module  340 . Post-calculation module  340  performs operations on the output samples received from calculation module  330 , described in more detail with reference to  FIGS. 4 and 6 . Memory  330  is adapted to store output samples received from the post-calculation module  340 . 
   The methods and devices described herein may be advantageously implemented within a DSL modem to enhance the performance thereof. In one embodiment, system  300  shown in  FIG. 3  is implemented in a digital signal processor (DSP) chip of a DSL modem. Because DSL modems often employ many signal processing techniques that use multiple stages of processing, such as discrete multi-tone (DMT) modulation, the system  300  advantageously increases the precision, SNR, and, ultimately, data transfer rates. 
   An algorithmic overview of the operation of system  300 , in accordance with one embodiment of the present invention, is shown in the flow chart of  FIG. 4 . The following table describes variables used in the discussion of this embodiment. 
                               Variable   Brief Description                   n   A value that represents the current stage, which is           included in a group of stages ranging from 1 to N       j   A value that represents the current sample, which is           included in a group of samples ranging from 1 to J       input sample(n,j)   A value that represents the j th  input for the           n th  stage process 100       output sample(n,j)   A value that represents the j th  output for the           n th  stage process 100       dr — shift(j),   For a given stage, a value that represents the decrease       where j = 1 to J   in dynamic range in which the j th  output sample           is represented relative to the dynamic range in which           the j th  input sample is represented       min — dr — shift(n),   A value that represents the smallest decrease in       where n = 1 to N   dynamic range of the output samples of the n th             stage-i.e., the smallest dr — shift(j) for the n th  stage       dr — cnt   A value that represents the cumulative of the minimum           dynamic range changes for stages 1 through N                    
For each of the J output samples  120  within a memory segment  130 , the variable array dr_shift(j), where j=1 to J, is allocated to record the decrease in dynamic range of the corresponding output sample  120  from the input sample  110  of that stage n. For each of the N stages, a variable min_dr_shift(n), where n=1 to N, records the smallest decrease in dynamic range of the output samples  120  for each stage. Variable dr_cnt records the cumulative minimum dynamic range change for all stages 1 through N. The variable arrays dr_shift(j) and min_dr_shift(n) and the variable dr_cnt may each be stored in memory  310  or located in a register of, e.g., the post-calculation module  340  or dynamic range summer  350  of system  300 .
 
   In the present context, a decrease in dynamic range is defined to be a positive change or shift in dynamic range. More specifically, the leftmost bits that are not significant (such as leading “0”s and sign extensions) are effectively shifted out of the process and therefore ignored. For each leftmost bit that is shifted out, one more rightmost bit can be shifted in. This act of shifting to eliminate the non-significant leftmost bits effectively reduces the dynamic range of the data sample. 
   The operation of system  300  can be described as applied to the general multi-stage processing illustrated in  FIG. 1 . In such a process, recall that input samples  110  of a particular stage are derived from the output samples  120  of the previous stage, which were derived using the input samples  110  of that previous stage according to the stage process  100  of that previous stage. In one embodiment, the input samples  110  are read from the memory  310 , and the output samples  120  are stored on the memory  310 . Because the stage processes  100  can be performed sequentially in time, some or all of the memory segments  130  can overlap or share memory locations in memory  310 . 
     FIG. 4  depicts an example process  400  in accordance with an embodiment of the present invention. This method can be performed, for example, by the system  300  of  FIG. 3 , although other embodiments and configurations will be apparent in light of this disclosure. As shown, the process  400  begins with the first stage, where n=1. At the beginning of the process  400 , the variable arrays min_dr_shift(n) for n=1 to N and dr_shift(j) for j=1 to J are initialized, e.g., set to 0. 
   The input samples  110  for the current stage are first read  410  from a memory  310 , or other storage facility. The input samples  110  may be read by the pre-calculation module  320 , which is coupled to the memory  310 . Before the stage process  100  for the n th  stage is performed, the dynamic range of each of the input samples  110  is normalized  420 . The normalizing  420  may be trivial for the first stage, assuming that the input samples  110  provided thereto have the same dynamic range; however, the input samples  110  of subsequent stages may not have the same dynamic range. 
   Normalizing  420  the input samples  110  is necessary for a typical system  300  in which the calculation module  330  requires all of the input samples  110  to be expressed in the same dynamic range. A calculation module  330  could be programmed or otherwise configured to compensate internally for any differences in dynamic ranges among the input samples  110 . In an embodiment, for example, the calculation module  330  keeps track of the cumulative or relative dynamic range changes for each of the input samples  110 . With this information, the calculation module  330  adjusts the dynamic range of the input sample  110  as needed within the calculation module  330 . In such a system, it is not necessary to normalize the dynamic ranges of the input samples  110  before providing them to the calculation module  330 , and any additional rounding due thereto is eliminated. 
   Normalizing  420  the input samples  110  is performed in a pre-calculation module  320 . One embodiment of a normalizing procedure  420  is shown in detail in  FIG. 5 , which depicts an embodiment of the pre-calculation normalizing  420  of process  400 . Generally, the normalizing  420  is performed by pre-calculation module  320 , which may be integrated into calculation module  330 . Normalizing  420  the input samples  110  before transmitting them to the calculation module  330  ensures that the dynamic ranges are the same for all of the values used in the calculation module  330 . In one embodiment of the pre-calculation normalizing  420 , the following actions  510  through  530  are performed for each j th  input sample  110  (e.g., for j=1 to J). 
   It is assumed that the inputs for the first stage are provided in the same dynamic range, so, for n=1, the normalizing process  420  is skipped. In stages other than the first stage, for each j th  input sample  110 , the pre-calculation module  320  retrieves  510  the dynamic range change of the current input sample  110 . This value can be read from the array dr_shift(j) associated with the previous stage, as the dynamic range of the input samples  110  of the present stage corresponds to the dynamic range of the output samples  120  of the last stage. 
   The pre-calculation module  330  then determines  520  if the decrease in dynamic range of the current input sample, dr_shift(j), is greater than the smallest decrease in dynamic range of all the input samples, min_dr_shift. If not, then the input sample  110  does not need to have its dynamic range increased, and the pre-calculation module  330  processes the next input sample  110  by incrementing j and repeating action  510 . If dr_shift(j) is greater than min_dr_shift, the dynamic range of the input sample  110  is increased  530  by the amount of the difference (dr_shift(j)−min_dr_shift) so that the input sample  110  is expressed in the largest dynamic range associated with the set of input samples  110 . The dynamic range of an input sample  110  can be increased  530 , for example, by deleting a particular number of least significant bits and sign extending the most significant bit by the same amount. This process results in setting the dynamic range of each of the input samples  110  to be the same—specifically, to be the largest dynamic range in the set of input samples  110 . Setting the dynamic ranges of the input samples  110  to a larger dynamic range would decrease precision, whereas setting the dynamic ranges to a lower dynamic range would result in the loss of significant bits in one or more of the input samples  110 . 
   Referring back to  FIG. 4 , the calculation module  330  receives the input samples  110  (now normalized to have the same dynamic range) from the pre-calculation module  320  and then calculates  430  a set of output samples  120  using the set of input samples  110 . The calculation module  330  is adapted to perform any programmed operation, mathematical or other, on the output samples  100  according to the stage process  100  that corresponds to the current stage. As explained above, this stage process  100  may include, for example, several steps of adding the samples and/or multiplying them by coefficients to implement, e.g., a modulation, FFT or IFFT, or filtering process. Digital signal processing typically includes a combination of these stage processes  100 . 
   The calculation module  330  then transmits the output samples  120  to the post-calculation module  340 . The post-calculation module  340  determines the dynamic range in which each output sample  120  is to be represented (e.g., when writing the output samples  120  to memory  310 ) and then adjusts  440  the dynamic ranges of the output samples  120  accordingly. The post-calculation module  340  also records the dynamic range change relative to input dynamic range (or otherwise tracks the dynamic range) for each output sample  120  in the variable dr_shift(j), and records the smallest dynamic range decrease among the output samples  120  in the variable min_dr_shift(n). Rather than recording the dynamic range changes for the output samples  120 , the post-calculation module  340  can alternatively track dynamic ranges, e.g., by recording a value that corresponds to the actual dynamic range for each output sample  120 , perhaps using a look-up table to determine the dynamic range from that value. 
   As previously explained, given a finite word width, it is advantageous to represent the output samples  120  in a lower dynamic range to achieve greater precision—provided that no significant bits are lost. Accordingly, the post-calculation module  340  adjusts  440  the dynamic ranges of the output samples  120  according to an algorithm for achieving this increased precision. Preferably, the post-calculation module  340  lowers the dynamic range of one or more output samples  120  if possible without losing any of the output samples&#39; significant bits. Depending on the notation scheme, lowering the dynamic range may be accomplished by bit shifting the word and dropping nonsignificant leftmost bits. The post-calculation module  340  can be programmed or otherwise configured to perform a number of algorithms for adjusting  440  the dynamic ranges, depending on the design parameters. An embodiment of the post-calculation adjusting  440  is illustrated in greater detail in  FIG. 6 . 
     FIG. 6  depicts an embodiment of the post-calculation dynamic range adjustment  440  of process  400 . An advantage provided by the dynamic range adjustment  440  is to decrease the dynamic range of one or more output samples  120  before storing  450  the output samples  120  in memory  310  for processing in a subsequent stage. In this process  440 , post-calculation module  340  performs the following actions  610  through  640  for each j th  input sample  110  (e.g., for j=1 to J). 
   In action  610 , the post-calculation module determines the minimum dynamic range in which the output sample  120  can be represented without losing significant bits (e.g.,  FIG. 2 ). The difference between the current dynamic range and this minimum dynamic range, for the j th  output sample  120 , is stored in the variable dr_shift(j). In one embodiment, the dynamic range change relative to the input can be determined from the calculation: 
                 min_dr   ⁢   _shift     =       log   2     ⁡     [         D   o     ⁡     (     n   ,   j     )           D   i     ⁡     (   n   )         ]         ,           (   1   )             
 
where D o (n,j) denotes the smallest dynamic range in which the j th  output sample  120  of the n th  stage can be represented without losing a significant bit, and D i (n) denotes the dynamic range in which the input samples  110  of the n th  stage are actually represented. D o (n,j) can be determined by any conventional dynamic range calculation method, such as overflow/underflow checking or level detection checking. After min_dr_shift is determined, the post-calculation module  340  then performs action  620 .
 
   In action  620 , the post-calculation module  340  decreases the dynamic range of the output sample  120  by the amount min_dr_shift. After action  620 , the post-calculation module  340  performs action  630 . In action  630 , the post-calculation module  340  records the change in dynamic range for the output sample  120  by setting the variable dr_shift(j) to equal min_dr_shift. 
   After action  630 , the system  300  increments j and, if there are still output samples  120  to process (i.e., if j is not larger than J), the post-calculation module  340  determines  640  if the dynamic range of the next output sample can be decreased by min_dr_shift without losing significant bits. If so, action  620  follows; otherwise, action  610  follows. The post-calculation module  340  performs the actions according to the flow described with reference to  FIG. 6  until there are no more output samples  120  to process for the current stage (i.e., j&gt;J). 
   In the embodiment of the post-calculation adjusting process  440  shown in  FIG. 6 , the post-calculation module  340  selects for each of the output samples  120  a dynamic range that is the larger of (1) the smallest dynamic range in which the output sample can be represented without losing a significant bit, and (2) the largest dynamic range selected for any previously processed output sample. This embodiment optimizes precision by selecting the lowest feasible dynamic range for a given set of output samples  120 . The embodiment further avoids selecting (for a given output sample  120 ) a lower dynamic range than the largest dynamic range already selected for a previously processed output sample  120 . This is advantageous because, in the case where pre-calculation normalizing  420  is performed, selecting a lower dynamic range would be wasteful. Were the embodiment to select for one output sample  120  a dynamic range lower than another output sample&#39;s dynamic range, the pre-calculation module  320  would have to increase the dynamic range of the former output sample  120  during the normalizing process  420 . 
   Additionally, it is apparent that this embodiment of the method will not select for each of the output samples  120  a dynamic range that is smaller than the smallest dynamic range in which the output sample can be represented without losing a significant bit. By selecting a dynamic range in this manner, the embodiment advantageously avoids losing significant bits when, e.g., storing the computed output samples  120  to a memory  310  for processing in the next stage. Accordingly, the embodiment automatically avoids overflow errors that occur when significant bits of an output sample  120  are lost. 
   Referring back to  FIG. 4 , the post-calculation module  340  stores  450  the shifted output samples  120  in the memory  310  for processing as input samples  110  for the next stage. In an alternative embodiment, the output samples  120  from the post-calculation module  340  may be sent directly to the pre-calculation module  320  or calculation module  330  as input samples  110  of the next stage process  100 . The dynamic range change for each of the stored output samples  120  is also recorded in the variable dr_shift(j). After all of the output samples  120  have been processed, the smallest decrease in dynamic range for that stage&#39;s set of output samples  120  is recorded  460  in the variable min_dr_shift(n). 
   The system  300  repeats processes  410  through  460  for each stage n, from n=1 to N. After the stage processes  100  are completed, the system  300  may apply  470  the final stage correction to the output samples  120  of the last stage N. The final stage correction is performed to ensure that the output samples  120  of the last stage N are expressed in the same dynamic range. This normalization can be performed in the same way as the pre-calculation normalization process  420  (shown in  FIG. 5  and discussed in reference thereto). After applying  470  the final stage correction to the output samples  120 , the variable dr_cnt represents the cumulative minimum decrease in dynamic range over the N stages. Therefore, the dynamic range of each final output sample  120  is the dynamic range of the original input samples  110  of the first stage decreased by the amount dr_cnt. If desired, the dynamic ranges of the set of output samples  120  of the last stage may be returned to the original dynamic range of the first stage&#39;s input samples  110  by right-shifting the last stage&#39;s output samples  120  according to dr_cnt. After applying  470  the final stage correction to the final output samples  120 , the final stage processor  360  stores  480  the corrected output samples  120  into memory  310 . The processed digital signal can then be, e.g., sent to a D/A converter for transmission over a DSL line. 
   The foregoing description of the embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Persons skilled in the relevant art can appreciate that many modifications and variations are possible in light of the above teaching. It is therefore intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.