Abstract:
A method and system for correcting quantization loss of a signal during analog to digital to analog signal conversion, wherein the quantization loss is a function of (sin x)/x, is disclosed. The system and method of the present invention includes utilizing a continuous function polynomial to represent a (sin x)/x function, and applying an inverse function of the continuous function polynomial to the signal to provide a correction for the quantization loss.

Description:
FIELD OF THE INVENTION 
     The present invention relates to analog to digital to analog signal conversion, and more particularly to correcting quantization loss during such a conversion. 
     BACKGROUND OF THE INVENTION 
     FIG. 1 is a block diagram illustrating a conventional analog to digital to analog signal conversion system  10 . As is shown, an analog signal  20  is inputted into an analog to digital converter  12 , where the analog signal is converted into a digital discrete time signal  22 . A digital system  14  utilizes the digital signal  22  in such a manner that is well known to those skilled in the art, e.g., for storage, delay, etc., and is not germane to the present invention. The digital discrete time signal  22  is converted back into a reconstructed analog signal  24  by a digital to analog converter  16 . The resulting reconstructed analog signal  24  can be passed through a low pass filter  18  to filter out quantization noise before the signal  24  is finally outputted as an analog signal  26 . 
     Ideally, the analog signal out  26  is identical to the analog signal in  20 . Quantization losses, however, inherently occur during the conversion process. In particular, quantization losses occur when the analog signal  24  is being reconstructed from the digital discrete time signal  22 . This quantization loss is known as a “(sin X)/x loss,” and is reflected as a diminished energy amplitude for the signal. 
     The (sin x)/x loss is a function of how often the digital signal  22  is sampled in order to reconstruct the analog signal in  20 . At lower frequency bandwidths of the digital signal  22 , it is not difficult to select a sampling frequency that allows the digital to analog converter  16  to collect enough data points to accurately reconstruct the analog signal in  20 . Thus, at lower frequency bandwidths, the quantization loss is negligible. 
     At higher frequency bandwidths, however, the sampling frequency is not high enough to accurately capture the true nature of the signal. For instance, if the sampling frequency is only twice the upper frequency of the high frequency bandwidth, the converter  16  will attempt to reproduce the upper frequency based on only two sampled data points. The converter  16  can only estimate the nature of the signal between the two points, and oftentimes underestimates the amplitude peak of the signal. Consequently, the quantization loss can be significant and the high frequency bandwidth of analog signal out  26  will not be an accurate reproduction of the same bandwidth of the analog signal in  20 . 
     Accordingly what is needed is a system and method for correcting quantization loss during analog to digital to analog signal conversion. The system and method should provide an acceptable degree of accuracy of correction, but not require excessive processing power. The present invention addresses such a need. 
     SUMMARY OF THE INVENTION 
     The present invention provides a method and system for correcting quantization loss of a signal during analog to digital to analog signal conversion, wherein the quantization loss is a function of (sin x)/x. The system and method of the present invention includes utilizing a continuous function polynomial to represent a (sin x)/x function, and applying an inverse function of the continuous function polynomial to the signal to provide a correction for the quantization loss. 
     By utilizing a continuous function polynomial, such as McLaurin&#39;s polynomial, to represent the (sin x)/x function, the correction for the quantization loss can be calculated quickly and easily, thereby expending little processing power. Moreover, a filter that creates an inverse function of the continuous function polynomial leaves a small footprint because it does not require significant hardware resources. If a sampling frequency is at least four times an upper frequency of a desired frequency bandwidth, only the first two members of the polynomial need be used to accurately approximate the quantization loss. Accordingly, under these conditions, the correction is accurately provided by creating an inverse function of only the first two members of the polynomial. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram illustrating a conventional conversion system. 
     FIG. 2 is a block diagram illustrating a system that can be used in a preferred embodiment of the present invention. 
     FIG. 3 is a flowchart illustrating a process for correcting quantization losses. 
    
    
     DETAILED DESCRIPTION 
     The present invention relates to a method and system for correcting quantization loss during analog to digital to analog signal conversion. The following description is presented to enable one of ordinary skill in the art to make and use the invention and is provided in the context of a patent application and its requirements. Various modifications to the preferred embodiment and the generic principles and features described herein will be readily apparent to those skilled in the art. Thus, the present invention is not intended to be limited to the embodiment shown but is to be accorded the widest scope consistent with the principles and features described herein. 
     As discussed above, the quantization loss of a signal is a function of the sampling frequency. In particular, the quantization loss is represented by the formula: 
     
       
         Quantization Loss ( dB )=20 log [(sin x)/x] 
       
     
     where, x=π(F v /F r ). F v  is defined as the frequency of interest, which is generally the highest frequency of a particular signal bandwidth, and F s  is defined as the sampling frequency. (IEEE Std 746-1984). When the ratio F v /F s  approaches zero, i.e., the sampling frequency is much larger than the frequency of interest, the loss is negligible. Nevertheless, as the ratio F v /F s  approaches 0.50, i.e., as the frequency of interest approaches one-half the sampling frequency (Nyquist&#39;s criteria), the quantization loss becomes so significant that reproduction of the high frequency bandwidth is difficult without some correction mechanism. 
     FIG. 2 is a block diagram illustrating an analog-digital-analog conversion system  10 ′ which can be utilized by the preferred embodiment of the present invention. As is shown, the digital signal  22 ′ passes through a correction mechanism  30  before the signal  22 ′ is converted back into an analog signal  24 ′. The correction mechanism  30  distorts the digital signal  22 ′ to compensate for the quantization losses which will occur during digital to analog conversion. When the distorted digital signal  22 ″ is converted, the resulting reconstructed analog signal  24 ′ will not exhibit quantization losses, but rather, it will reflect accurately the incoming analog signal  20 ′. 
     The distortion directly offsets the quantization losses at a particular signal frequency and sampling frequency. Because the quantization loss is a function of (sin x)/x, the distortion is determined by an inverse (sin x)/x function. Accordingly, the correction mechanism  30  can provide the appropriate distortion to the digital signal  22 ′ by implementing a filter to create an inverse (sin x)/x function. While this is straightforward in principle, in practice, it requires significant hardware resources to implement such a filter. At a time when computer chips are becoming smaller and smaller, the footprint of such a filter is detrimental. Moreover, the calculation requires significant processing power and time, which adversely impacts performance. 
     The method and system of the present invention takes another approach for approximating the quantization loss at a given frequency of interest and sampling frequency. Because (sin x)/x is a continuous function, it can be represented by a mathematical formula in the form of a finite polynomial. 
     FIG. 3 is a flowchart illustrating a process for correcting quantization losses according to a preferred embodiment of the present invention. In step  102 , the (sin x)/x function is represented by a continuous function polynomial. Then, in step  104 , the correction mechanism  30  implements a filter that creates an inverse function of the continuous function polynomial in order to provide the (sin x)/x correction. The inverse function is then applied to the signal, in step  106 , to correct for quantization losses. 
     In a preferred embodiment, the continuous function polynomial is a McLaurin&#39;s polynomial of the form: 
     
       
         (sin  x )/ x =1−( x   2 /3!)+( x   4 /5!)−( x   6 /7!)+. . .  
       
     
     where x=π(F v /F s ). After substituting π and calculating factorials, the formula simplifies to: 
     
       
         (sin  x )/ x =1−1.64 f   2 +0.81 f   4 −0.19 f   6 +. . . 
       
     
     where f=F v /F s . 
     By calculating the first few members of the continuous function polynomial, the quantization loss can be approximated accurately and easily for a given frequency of interest and sampling frequency. Moreover, if the sampling frequency is much larger than the frequency of interest, e.g., the sampling frequency is at least four times the frequency of interest, then the quantization loss is accurately derived (to within 0.5%) using only the first two members of the polynomial. Thus, for f≦0.25, the approximation for (sin x)/x simplifies to: 
     
       
         (sin  x )/ x =1−1.64 f   2   
       
     
     which ensures accuracy to within 0.5%. 
     In the preferred embodiment of the present invention, the correction mechanism  30  implements a filter that creates an inverse function of the first two members of the continuous function polynomial, instead of creating an inverse (sin x)/x function, to correct the quantization loss. Because the simplified polynomial is far less complicated than the (sin x)/x function, implementation of the inverse function requires less hardware resources. Moreover, by using the first few members of the continuous function polynomial to approximate (sin x)/x, quantization losses can be calculated quickly, thereby reducing processing time and resources. Thus, the preferred embodiment of the present invention provides a small footprint and high performance along with conversion accuracy. 
     The present invention has been described in accordance with the embodiments shown, and one of ordinary skill in the art will readily recognize that there could be variations to the embodiments. For example, although correction is preferably performed before the digital signal  22 ′ is converted back to an analog signal, those skilled in the art would readily appreciate that correction could also be performed after digital to analog conversion. Any variations would be within the spirit and scope of the present invention. Accordingly, many modifications may be made by one of ordinary skill in the art without departing from the spirit and scope of the appended claims.