Patent Publication Number: US-6661363-B2

Title: System and method for adaptive sigma-delta modulation

Description:
TECHNICAL FIELD AND BACKGROUND ART 
     The invention generally relates to signal processing, and more particularly, to analog to digital conversion using sigma-delta modulation. Sigma-delta (Σ-Δ) modulation is a widely used and thoroughly investigated technique for converting an analog signal into a high-frequency digital sequence. See, for example, “Oversampling Delta-Sigma Data Converters,” eds. J. C. Candy and G. C. Temes, IEEE Press, 1992, (hereinafter Candy) and “Delta-Sigma Data Converters,” eds. S. R. Northworthy, R. Schreier, G. C. Temes, IEEE Press, 1997, both of which are hereby incorporated herein by reference. 
     In Σ-Δ modulation, a low-resolution quantizer is incorporated within a feedback loop configuration in which the sampling frequency is much higher than the Nyquist frequency of the input signal (i.e., much higher than twice the maximum input frequency). In addition, the noise energy introduced in the quantizer is shaped towards higher frequencies according to a so called “noise-transfer-function” NTF(z), and the signal passes the modulator more or less unchanged according to a so called “signal-transfer-function” STF(z). 
     FIG.  1 ( a ) depicts a simple first order Σ-Δ modulator for a discrete time system having a subtraction stage  101 , an accumulator  102  (including an integrator adder  103  and a delay line  104 ), a one-bit quantizer  105 , and a 1-bit digital-to-analog converter (DAC)  106 . In normal operation, an input signal x(n) within the range [−a, +a] is converted to the binary output sequence y 0 (n) ε±1. Quantizer  105  produces a+1 for a positive input and a−1 for a negative input. The output from quantizer  105  is fed back through DAC  106  and subtracted from input signal x(n) by subtraction stage  101 . Thus, the output of subtraction stage  101  represents the difference between input signal x(n) and the quantized output signal y 0 (n). As can be seen from FIG.  1 ( a ), the output of accumulator  102  represents the sum of its previous input and its previous output. Thus, depending on whether the output of the accumulator  102  is positive or negative, the one-bit quantizer  105  outputs a+1 or a−1 as appropriate. Herein, and in the appended claims, analog (physical) and digital representations of signals are distinguished from each other by labeling digital one or multi-bit signals with the subscript “0”. 
     In FIG.  1 ( b ), a linear model of FIG.  1 ( a ) is shown, and similarly includes a subtraction stage  107 , and an accumulator  111  (including an integrator adder  112  and a delay line  113 ). Quantizer  105  is replaced by an adder  108  and a noise source  109 . To convert signal y(n) to y 0 (n), a comparator  110  for detection of the sign of y(n) is included. The basic relationship between the z-transforms of system input x(n), quantizer noise γ a (n), and the two-level output sequence y(n) is: 
     
       
           Y ( z )= z   −1   X ( z )+(1 −z   −1 )Γ a ( z )  (1) 
       
     
     The signal transfer function and noise-transfer function can be identified as STF(z)=z −1  and NTF(z)=(1−z −1 ), respectively. 
     Quality of digital representation can be described by the signal-to-noise ratio          SNR   =     101                   log   10                     S   N         ,                   
     where S is the signal power and N is the noise power within a given bandwidth B. Regarding equation (1), the noise power N depends on both the noise transfer function NTF(z) and the overall amount of noise Γ a (z) added in the quantization stage. To improve the SNR, two approaches can be pursued: 
     (a) for a given overall noise power Γ a (z), i.e., for given quantizer levels ±a, modify the NTF(z) to remove more noise power from the base band by improved noise shaping, and 
     (b) for a given NTF(z), try to reduce the overall noise power introduced to the system. 
     Approach (a) can be achieved, for example, by increasing the order of the sigma-delta modulator, as described by Candy. For higher order modulators, the noise transfer function becomes NTF(z)=(1−z −1 ) k , which means an enhanced noise-shaping effect. For examples of approach (b) see Zierhofer C. M., “Adaptive sigma-delta modulation with one-bit quantization,” IEEE trans. CAS II, vol. 47, No. 5, May 2000 (hereinafter Zierhofer), and U.S. patent application for Adaptive Sigma-delta Modulation with One-bit Quantization, Ser. No.: 09/496,756, filed Feb. 3, 2000 (hereinafter U.S. patent application Ser. No.: 09/496,756), both of which are incorporated herein by reference, where a sigma-delta modulator is employed within a feedback loop configuration. It is shown that the input signal of this modulator applies within a reduced range, and thus the two levels of the quantizer can be reduced. As a consequence, less noise power is introduced to the system, and the SNR is improved. 
     SUMMARY OF THE INVENTION 
     In accordance with one embodiment of the invention, a method and system for an adaptive sigma-delta (Σ-Δ) modulator with one bit quantization that improves the signal-to-noise (SNR) of a Σ-Δ modulator is presented. A difference signal is produced representing the difference between an analog input signal x(n) and a first adaptive feedback signal z(n), the amplitude of the analog input x(n) within a first range [−a, +a]. The difference signal is accumulated to produce a first accumulated signal representing an accumulation of the difference signal. The first accumulated signal is then quantized, creating a first quantized digital signal y 0 (n). Based on the first quantized digital signal y 0 (n) a first digital output signal z 0 (n) is produced, such that the instantaneous magnitude of the first analog feedback signal z(n) is substantially kept within the first range [−a, +a] and greater than the analog input signal&#39;s x(n) instantaneous magnitude. A digital-to-analog conversion is performed on the first digital output signal z 0 (n) to produce the first adaptive feedback signal z(n). 
     In related embodiments of the invention, the first quantized digital signal y 0 (n) produced includes a two-level digital output sequence. The two-level digital output may include values of +1 and −1. Producing the first digital output signal z 0 (n) may include multiplying the first quantized digital signal y 0 (n) by a step size c 0 (n). The step size c 0 (n) may be based on a set Y of code words, where Y={y 0 (n), y 0 (n−1), y 0 (n−2) . . . y 0 (n−n x )}, n x  being a predetermined integer. Determining the step size c 0 (n) may include increasing the step size c 0 (n) if a majority of the code words are equal, or decreasing the step size c 0 (n) if the code words alternate. The step size c 0 (n) may be non-linear. Multiplying the first quantized digital signal y 0 (n) by a step size c 0 (n) may include using a look-up-table RAM. 
     Other related embodiments of the invention may further comprise high pass filtering the analog input signal x(n). In one embodiment of the invention, high pass filtering may include subtracting from the analog input signal x(n) a feedback signal based on the digital output signal z 0 (n). In another embodiment of the invention, high pass filtering may include producing a second accumulated signal az 0 (n) representing the accumulation of the first digital output signal z 0 (n). The second accumulated signal az 0 (n) is multiplied by a factor θ to produce a multiplied signal w 0 (n). The analog input signal x(n) is then produced by subtracting from an analog pre-input signal x p (n) the second analog feedback signal w(n). In yet another embodiment of the invention, high pass filtering the analog input signal x(n) may include producing a second accumulated signal az 0 (n) representing the accumulation of the first digital output signal z 0 (n). A digital signal w 0 (n) is produced using an embedded numeric multibit sigma-delta modulator, the digital signal w 0 (n) representing the accumulated signal az 0 (n) multiplied by a factor θ such that the digital signal w 0 (n) includes low frequency components of the analog input signal x(n). The digital signal w 0 (n) is then added to the first digital output signal z 0 (n). In producing digital signal w 0 (n), the digital signal w 0 (n) may be represented by a smaller number of bits as compared to the accumulated signal az 0 (n), with information contained in the bit difference being retained in the temporal fine structure of w 0 (n). 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention will be more readily understood by reference to the following detailed description taken with the accompanying drawings, in which: 
     FIG.  1 ( a ) is a block diagram of a prior art first order Σ-Δ modulator for a discrete time system; 
     FIG.  1 ( b ) is a block diagram of a prior art first order Σ-Δ modulator for a linear model, where the quantizer is replaced by an adder and a noise source; 
     FIG. 2 is a block diagram of an adaptive sigma-delta (Σ-Δ) modulator with one bit quantization that improves the signal-to-noise (SNR) of a Σ-Δ modulator in accordance with one embodiment of the invention; 
     FIG.  3 ( a ) shows representative waveforms for an adaptive sigma-delta (Σ-Δ) modulator in accordance with one embodiment of the invention; 
     FIG.  3 ( b ) shows representative waveforms for an adaptive sigma-delta (Σ-Δ) modulator in accordance with one embodiment of the invention; 
     FIG. 4 shows SNR-simulation results representative of various types of analog-to-digital converters; 
     FIG. 5 shows SNR-simulation results representative for adaptive sigma-delta (Σ-Δ) modulators with ideal and non-ideal digital-to-analog converters in accordance with one embodiment of the invention; 
     FIG. 6 is a block diagram of an adaptive sigma-delta (Σ-Δ) modulator with one bit quantization that includes a high pass filter stage in accordance with one embodiment of the invention; 
     FIG. 7 is a block diagram of an adaptive sigma-delta (Σ-Δ) modulator with one bit quantization having a high pass filter stage that includes a numeric multibit sigma-delta modulator in accordance with one embodiment of the invention; 
     FIG. 8 shows representative waveforms for the system depicted in FIG. 7; and 
     FIG. 9 shows SNR-simulation results pertaining to the signal-to-noise ratio for the system depicted in FIG.  7 . 
    
    
     DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS 
     A method and system for an adaptive sigma-delta (Σ-Δ) modulator with one bit quantization that improves the signal-to-noise (SNR) of a Σ-Δ modulator is presented. A block diagram of the system in accordance with one embodiment of the invention is shown in FIG.  2 . The output y 0 (n) and two level feedback signal y(n) ε±a of a standard Σ-Δ modulator of 1 st  order is replaced by a multilevel output signal z 0 (n) and feedback signal z(n), respectively. The instantaneous magnitude of the multilevel feedback signal z(n) is kept within the range [−a&lt;z(n)&lt;a] and greater than the instantaneous magnitude of the input signal x(n) by an adaptation stage  205 . Compared to the-non-adaptive modulator, the noise source  107  in FIG.  1 ( b ) is thus reduced, and the SNR is considerably enhanced. The adaptation algorithm of the adaptive Σ-Δ presented can be fully exploited, if the input x(n) is a zero-mean signal, or if the dc-component of x(n) is at least close to zero. If x(n) contains a considerable dc-component, the input dynamic range is reduced. Accordingly, an implementation of a high-pass filter stage is also presented. 
     FIG. 2 can be separated into an analog and a digital section. An analog input x(n) having range [−a, +a], is inputted into a subtraction stage  208 . The analog output of the subtraction stage is then inputted into an accumulator  209  (including an integrator adder  210  and a delay line  211 ) and then quantized  202 . The digital output of the quantizer  202 , which may be, but is not limited to, y 0 (n) ε±1, is multiplied  207  by a step size sequence c 0 (n) to create output signal z 0 (n). Output signal z 0 (n) is passed through a digital-to-analog converter (DAC)  204  to create the analog feedback signal z(n). 
     Step size sequence c 0 (n) is generated in an adaptation stage  205 . The step size c 0 (n) at a particular instant is controlled by a set of code words [y 0 (n), y 0 (n−1), y 0 (n−2), . . . ] which represent the instantaneous value of y 0 (n), and a particular (finite) number of previous code words y 0 (n−1), y 0 (n−2) . . . The primary intention of adaptation stage  205  is to keep the instantaneous magnitude of z(n) greater than the instantaneous magnitude of input signal x(n), 
     
       
         | z ( n )|&gt;| x ( n )|.  (2) 
       
     
     The way the adaptation stage works is intuitively clear. Step size c 0 (n) needs to be increased, if the set [y 0 (n), y 0 (n−1), y 0 (n−2), . . . ] contains many equal code words. In this case, |x(n)| tends to exceed |z(n)|, which violates condition (2). On the other hand, c 0 (n) needs to be decreased, if the set [y 0 (n), y 0 (n−1), y 0 (n−2), . . . ] shows an alternating pattern of code words. 
     Using, for example, a 9-bit DAC  204  for the system shown in FIG. 2, the input range [−a, +a] is subdivided into 511 equally spaced discrete signal levels. Thus, the digital signals z 0 (n) and c 0 (n) are composed of 9 bits, and 8 bits, respectively. In accordance with one embodiment of the invention, an adaptation algorithm for a system of 1st order, with constant α chosen to be          α   =     16   15       ,                   
     is provided in Tab. 1. 
     As shown in Tab. 1, the step size is increased by approximately a factor α 3 , if five consecutive code words are equal, and decreased by about a factor 
     
       
         
           
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Code 
                 Step Size Multiplier 
               
               
                   
               
             
            
               
                 y 0 (n) = y 0 (n − 1) = y 0 (n − 2) = y 0 (n − 3) = y 0 (n − 4) 
                 c 0 (n) = round(α 3 c 0 (n − 1)) 
               
               
                 y 0 (n) = −y 0 (n − 1) = y 0 (n − 2) = −y 0 (n − 3) 
                 c 0 (n) = round(α −1 c 0 (n − 1)) 
               
               
                 Other combinations 
                 c 0 (n) = c 0 (n − 1) 
               
               
                   
               
            
           
         
       
     
     α −1 , if four consecutive code words have alternating signs. Since step sizes c 0 (n) have a limited resolution of 8-bits, the products α 3 c 0 (n−1) and α −1 c 0 (n−1) cannot be implemented exactly, but have to be rounded to the next integer. For small step sizes, the deviations due to rounding are considerable, but this type of imperfection is not essential for the system performance. While a multiplier can be used to calculate c 0 (n), in various embodiments of the invention, a look-up-table RAM  206  is utilized instead, where all possible step sizes c 0 (n) are stored as 8-bit integers, for example. The minimum and maximum step sizes are then c 0,min =1 and c 0,max =255, respectively. The computation of product α 3 c 0 (n) may be achieved by simply increasing the instantaneous RAM-address index by 3. Similarly, for product α −1 c 0 (n), the instantaneous address index is decreased by 1. 
     Sample waveforms for an adaptive Σ-Δ modulator implementing the adaptation algorithm described in Tab. 1 are shown in FIGS.  3 ( a ) and  3 ( b ). The first trace  301  in FIG.  3 ( a ) depicts an example of an input signal x(n). The second trace  302  in FIG.  3 ( a ) shows the full wave rectified version |x(n)| together with the magnitude |z(n)| of the DAC-output signal. Condition (2) is fulfilled for almost all samples, but a close examination shows that there is a violation at instant n=1643. This remains singular, however, and has negligible impact on the overall performance of the system. The third trace  303  in FIG.  3 ( a ) illustrates the full DAC-output signal z(n). In FIG.  3 ( b ), the system behavior is shown, if signal x(n) is attenuated by 40 dB, i.e. for x(n)/100. Traces  304 ,  305 , and  306  represent an input signal x(n), the full wave rectified version |x(n)| together with the magnitude |z(n)| of the DAC-output signal, and the full DAC-output signal z(n), respectively. As expected, the quantization of signals |z(n)| and z(n) appears more pronounced, and the digital sequence c 0 (n) varies between 1 and 3. 
     The examples FIGS.  3 ( a ) and  3 ( b ) also demonstrate that the step-size adaptation algorithm works instantaneously, that is, step size multiplier c(n) tracks the individual maxima and minima of input x(n). Signal c(n) can directly be used to estimate the instantaneous power of the input signal, which is advantageous, for example, in signal processing applications for automatic gain control (AGC) for speech signals. Adaptive Σ-Δ modulation schemes typically use comparatively slow adaptation algorithms, where time constants in the range of tens of milliseconds are involved (usually referred to as “syllabic compression”). An example is Chakravarthy, C. V., “An amplitude-controlled adaptive delta sigma modulator,” Radio &amp; Electronic Engineer (London), vol. 49, pp. 49-54, January 1979, which is hereby incorporated by reference. Systems like this cause gross errors in case of sudden increase of the amplitudes of the input signals and are not practical in signal processing applications, where a permanent accurate representation of the input signal is of importance. Additionally, the use of non-linear step sizes makes the adaptive algorithm more responsive to changes in input amplitudes compared to prior art adaptive algorithms with a constant step size, such as described in Jaggi, M. P., “Instantaneously Adaptive Delta Sigma Modulator” Can. Elect. Eng. 1, Vol. 11 No. 1, 1986, which is herein incorporated by reference. This is important, for example, in keeping the instantaneous magnitude of the first analog feedback signal greater than the input signal&#39;s instantaneous magnitude. 
     In FIG. 4, the SNR of various types of analog-to-digital converters are shown as a function of the input signal&#39;s power. The input x(n), within the range [−1&lt;x(n)&lt;1] (i.e., a=1), is a periodic zero-mean noise sequence composed of 10000 samples, and the bandwidth is B=10 kHz. Within this bandwidth, amplitudes and phases of the spectral lines are randomized. Different values of signal power are obtained by proportional amplification of this signal. The input power is referenced to the power level of a dc-signal with amplitude a=1. At the maximum input power level shown in FIG. 4 (i.e., at −9.45 dB), the maximum signal amplitude reaches 0.99, which is just below the limit of the reference level a=1. The sampling rate for all systems is            1   T     =     1                 MHz       ,                   
     and the SNRs are computed within B=10 kHz. 
     Curve  401  depicts the SNR of an ideal adaptive sigma-delta modulator in accordance with one embodiment of the invention, where the adaptation algorithm of Tab. 1 and a 9-bit DAC is used (a=1). Reducing the input power from the maximum level to lower levels, the SNR tends to remain constant. For input levels smaller than about −50 dB, the SNR is decreasing. Curve  402  is the SNR of an ideal standard sigma-delta modulator of 1 st  order with y a (n) ε±1. The maximum SNR is obtained at the maximum input power level, and a decrease of input power results in a decrease of the SNR. Comparing curves  401  and  402  clearly demonstrates the benefit of the adaptive sigma-delta modulator. Curve  402  is very similar to the segment of curve  401  having input levels smaller than about −50 dB, shifted to the right by about 48 dB. This shift reflects the additional 8 bits of signal z(n) as compared to y a (n). For input levels smaller than about −50 dB, the adaptive sigma-delta modulator operates in a manner similar to a standard sigma-delta modulator, since the feedback-signal z(n) is a two-level signal, z(n) ε±1/256. Note that for high input levels, the SNR of the adaptive modulator is not substantially higher than for the standard modulator. However, the input dynamic range has been expanded by approximately 48 dB. Curve  403  depicts the SNR of a sigma-delta-modulator of 2 nd  order. Obviously, at lower input levels, the 2 nd  order system is outperformed by the adaptive modulator. Curves  404  and  405  depict the SNR&#39;s of Pulse Code Modulation (PCM) systems with 13 and 14 bit resolutions, respectively. Whereas the 14-bit PCM system is superior to the adaptive sigma-delta modulator for all input levels, the 13-bit PCM system is inferior at least at low-level input signals. 
     In accordance with one embodiment of the invention, the adaptive sigma-delta modulator includes a multi-bit DAC  204  in the feedback loop, as shown in FIG. 2, since the feedback signal z(n) is a multi-level signal. In preferred embodiments of the invention, the specifications of this DAC  204  is chosen to have minimal effects on the SNR of the adaptive sigma-delta modulator. The non-ideal DAC can be regarded as an ideal converter plus a noise source, with the transfer function of this additional noise source being −NTF(z)=−z −1 . Thus, this noise (multiplied by −1) is directly added to the input signal and enhances the noise energy in the base band. For the DAC  204 , the following assumptions can be made: 
     (1) The distribution of DAC-errors is symmetric around zero. 
     (2) Each discrete DAC-levels is implemented by means of superposition of binary weights. 
     (3) Each binary weight itself is composed of elementary unit components. This implies, e.g., that for a particular binary weight which is composed of K unit components, the nominal value is increasing proportional to K, whereas the error is increasing proportional only to {square root over (K)}. 
     For example, for a 9-bit DAC, the pattern c 0 (n)=[1 0 0 1 0 1 0 1] contains the binary weights 128, 16, 4, and 1, and hence the nominal magnitude of the resulting DAC-level is          149   255          a   .                     
     However, the single weights can only be realized as          128        (     1   ±     Δ     128         )       ,                16        (     1   ±     Δ     16         )       ,                4        (     1   ±     Δ     4         )       ,                   
     and (1±Δ), which results in a deviation from the nominal value. 
     FIG. 5 shows the SNRs of an adaptive sigma-delta modulator with an ideal 9-bit DAC (Δ=0)  501 , and non-ideal DACs for Δ=1%  502 , Δ=3%  503 , and Δ=5%  504 , where Δ denotes the maximum deviation of the implemented level number K=1 from the nominal value. In all cases, the contribution of the dc-offset error is omitted, and the input signal is the same as for FIG. 4 (N=10000). FIG. 5 shows that the SNR reduction will be within acceptable limits with parameters Δ better than about Δ≈1%. 
     The adaptive sigma-delta modulator shown in FIG. 2 yields its optimum performance if the input x(n) is a zero-mean signal, or if the dc-component of x(n) is at least close to zero. However, if x(n) contains a considerable dc-component, the input dynamic range is reduced. Consider, for example, an input signal with a dc-component and comparatively very small ac-component. The step size reduction will be governed by the dc-component and not the ac-component, and thus it remains too large. Dc-components can be introduced from either the input signal directly, or by offset-errors due to non-ideal components of the modulator itself. In practical implementations, offset errors in the DAC might occur. In any case, a high-pass filter removes most problems involved with dc- or very low frequency components. 
     In accordance with one embodiment of the invention, an implementation of a high-pass filter is implemented as shown in FIG.  6 . Here, the adaptive sigma-delta modulator  203  of FIG. 2 is embedded in a feedback loop system, which represents a typical digital high-pass filter. The digital output z 0 (n) of the modulator is first accumulated  601 , resulting in signal az 0 (n), and then multiplied  602  by a factor θ, resulting in signal w 0 (n). Digital-to-analog conversion of w 0 (n) by means of a second DAC  603  results in signal w(n), which is subtracted  604  from the input x(n). The overall signal-transfer-function ST HP (z) of such a system is given by 
     
       
           STF   HP ( z )= STF ( z ) HP ( z ),  (3) 
       
     
     with the standard Σ-Δ signal-transfer-function, i.e., STF(z)=z −1 , and the high-pass transfer-function          HP        (   z   )       =         1   -     z     -   1           1   -       (     1   -   θ     )          z     -   1             .                     
     There is a zero at z=1 corresponding to a frequency f=0, and a pole at z=1−θ. Using, for example, a sigma-delta rate of 1 MHz and        θ   =     1   1024                     
     results in a 3-dB cut off frequency of about 150 Hz. 
     The system shown in FIG. 6 is not well suited for practical implementation, since a 19-bit DAC is involved. Therefore, in accordance with various embodiments of the invention, means to circumvent such a high precision DAC are implemented. In accordance with one embodiment of the invention, inserting a numeric version of a multibit sigma-delta modulator  701  into the system results, for example, in the system as shown in FIG.  7 . Assuming          θ   =     1   1024       ,                   
     signal az 0 (n) comprises 19 bits. Following the general rules of sigma-delta modulation, this signal is converted to output signal 1024w d0 (n), which represents the input signal az 0 (n), delayed by one clock period. In the present application, signal w 0 (n)=w d0 (n+1) is used as a high-pass filter feedback signal, which represents the non-delayed input az 0 (n), multiplied by 1/1024. In the present embodiment, two identical 9-bit numeric quantizers  702  and  703  with a transfer characteristics of mid-tread-type are employed. Possible output numbers are −255, −254, . . . , −2, −1, 0, 1, 2, . . . 254, 255. Note that using the numeric multibit sigma-delta modulator provides a very efficient method to implement the constant-factor multiplication 1/1024. 
     Signal w 0 (n) is the sigma-delta version of signal              az   0          (   n   )       1024     .                   
     Thus, the number of bits has been reduced from 19 in signal az 0 (n) to 9 in signal w 0 (n). However, following the principles of multibit sigma-delta modulation, although there is a difference of 10 bits both signals contain almost the same information. Information contained in the 10 bit difference is preserved in the temporal fine structure of w 0 (n). Additionally, in this example, since both signals z 0 (n) and w 0 (n) are composed of 9 bits, the 2 DACs of FIG. 6 can be replaced by a single DAC  704 , controlled by the sum  705  of signals z 0 (n) and w 0 (n). FIG. 8 depicts typical waveforms of the system in FIG.  7 . The first trace  801  shows an input signal x(n) with a step-like transition after 2 ms. The second trace  802  depicts the resulting 9-bit signal w(n), which shows the typical low-pass character. With a cut off frequency of about 150 Hz, the time is about        τ   =         1     2                 π                 150          s     ≈     1                   ms   .                         
     As stated above, information is contained in the temporal fine structure of the signal. The third trace  803  shows the difference signal x(n)−w(n) , which represents the high-pass filtered version of x(n). 
     In FIG. 9, the SNR obtained with the system depicted in FIG. 7, using an ideal high-pass filter, is compared with the SNR obtained with the system depicted in FIG. 8, which uses a high-pass filter with numeric multibit sigma-delta modulator (employing 9-bit quantizers of mid-tread type), as waveforms  901  and  902 , respectively. The same input signal as specified in FIGS. 4 and 5, with N=20000, is used. For both systems, the noise power in the signal band is derived from the difference between the system output signal and a reference signal. The reference signal is the high-pass filtered version of x(n), assuming the ideal transfer function (3). The additional noise introduced by the numeric multibit sigma-delta modulator results in a loss of SNR at low input power levels. The input dynamic range is reduced by approximately 6 dB. The decrease of SNR at high input levels is due to a clipping effect and appears, if the sum w 0 (n)+z 0 (n) exceeds the range [−255, 255]. The two SNR-curves  901 ,  902  are obtained assuming DAC&#39;s with Δ=1%. 
     Alternative embodiments of the invention may be implemented as a computer program product for use with a computer system. Such implementation may include a series of computer instructions fixed either on a tangible medium, such as a computer readable media (e.g., a diskette, CD-ROM, ROM, or fixed disk), or fixed in a computer data signal embodied in a carrier wave that is transmittable to a computer system via a modem or other interface device, such as a communications adapter connected to a network over a medium. The medium may be either a tangible medium (e.g., optical or analog communications lines) or a medium implemented with wireless techniques (e.g., microwave, infrared or other transmission techniques). The series of computer instructions embodies all or part of the functionality previously described herein with respect to the system. Those skilled in the art should appreciate that such computer instructions can be written in a number of programming languages for use with many computer architectures or operating systems. Furthermore, such instructions may be stored in any memory device, such as semiconductor, magnetic, optical or other memory devices, and may be transmitted using any communications technology, such as optical, infrared, microwave, or other transmission technologies. It is expected that such a computer program product may be distributed as a removable medium with accompanying printed or electronic documentation (e.g., shrink wrapped software), preloaded with a computer system (e.g., on system ROM or fixed disk), or distributed from a server or electronic bulletin board over the network (e.g., the Internet or World Wide Web). 
     Although various exemplary embodiments of the invention have been disclosed, it should be apparent to those skilled in the art that various changes and modifications can be made which will achieve some of the advantages of the invention without departing from the true scope of the invention. These and other obvious modifications are intended to be covered by the appended claims.