Abstract:
A method and apparatus for adaptively quantizing a digital signal prior to digital-to-analog conversion during pulse shaping in a pulse-amplitude modulated (PAM) system. In the method and apparatus, scaling constants applied to the input signal prior to digital-to-analog conversion change according to the time-varying periodic probability density function (pdf) of the input signal. Spectral components added to the signal due to the adaptive scaling of the method may be removed by performing adaptive amplification on the output signal of the DAC before transmission as a PAM signal.

Description:
CLAIM OF PRIORITY FROM A COPENDING PROVISIONAL PATENT APPLICATION 
     Priority is herewith claimed under 35 U.S.C. §119(e) from copending Provisional Patent Application No. 60/128,647, filed Apr. 9, 1999, entitled “Adaptive Quantization in A Pulse-Amplitude Modulated System,” by Giridhar D. Mandyam. The disclosure of this Provisional Patent Application is incorporated by reference herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to telecommunications systems and, more particularly, to a method and apparatus for adaptively quantizing a signal prior to digital-to-analog conversion during pulse shaping in a pulse amplitude modulated system. 
     BACKGROUND OF THE INVENTION 
     Pulse amplitude-modulated (PAM) systems involve the transmission of pulse-shaped impulses over a transmission channel. The amplitude of a particular impulse relates directly to the information carried in the impulse. In general, a PAM signal may be expressed by the following expression:                y        (   t   )       =         ∑     n   =     -   ∞         ∞            c   n          p        (     t   -   nT     )                   Equation  1.0                                
     where c n  represents the amplitude of the n th  pulse and p(t) is the pulse shaping waveform, In many modern digital communication systems pulse-shaping is performed digitally. There are several reasons for using digital pulse shaping. Because of matched filtering requirements at the receiver and spectral shaping requirements for electromagnetic compatibility, it is often easier to implement the desired pulse waveform digitally than in the analog domain. Additionally, because attainable sampling frequently affects the ease of design of both the analog-to-digital converter and image rejection filter, digital pulse shaping may be used to increase sampling rate and allow greater ease of design. 
     Modern digital communication systems may typically use channel aggregation or multicarrier modulation on a transmission link. Certain of these modern digital communication systems include the use of individual channel gain adjustment on the separate channels in order to provide enhanced performance for coherent reverse link reception and quality of service features. However, individual channel gain adjustment results in power adjustment of incoming data, which when combined with pulse shaping, can dramatically affect resultant signal power, and a transmitted signal on a multicarrier link may have a higher peak-to-average ratio in than in a single carrier system. A higher peak-to-average ratio generally requires higher precision and a digital-to-analog converter (DAC) using more bits. In a DAC the signal level must be adjusted to make use of the full conversion range. A higher signal level results in clipping noise. However, using a lower signal level to avoid clipping results in a waste of DAC precision with the quantization power causing a significant amount of quantization noise. As a result, the effective spurious free dynamic range (SFDR) of a signal may degrade in a multicarrier system if quantization effects or clipping distortion due to digital-to-analog conversion are not taken into account. A degradation in SFDR can negatively affect the adjacent channel power rejection (ACPR) in digital pulse-amplitude modulated communication system causing greater intra-channel interference. 
     SUMMARY OF THE INVENTION 
     The present invention provides a method and apparatus for adaptively quantizing a digital signal prior to digital-to-analog conversion during pulse shaping in a pulse-amplitude modulated (PAM) system. Implementation of the invention provides a performance advantage in terms of adjacent channel power rejection (ACPR). The method and apparatus utilizes the inherent cyclostationarity of an input pulse-amplitude modulated (PAM) transmission in an adaptive quantization method. In the adaptive quantization method, quantization constants applied to the input PAM signal prior to digital-to-analog conversion change according to the time-varying periodic probability density function (pdf) of the input for PAM signal. The adaptive quantization is optimized for the instantaneous pdf of the input PAM signal, rather than overall possible distributions of the pdf. Spectral components added to the PAM signal due to the adaptive scaling of the method and apparatus may be removed by performing adaptive amplification on the analog output signal of the DAC before transmission. The adaptive amplification prevents the need to transmit scaling data to the receiver. 
     In an embodiment of the invention, the method and apparatus is implemented in a transmitter of a PAM system transmitting at least one channel over an air interface to a receiver. In the embodiment, signals for at least one channel are digitally processed to generate an information signal and passed through a digital pulse-shaping filter. The output of the digital pulse-shaping filter is then adaptively quantized by processing the filter output through an adaptive scaler and digital-to-analog converter (DAC). The adaptive quantization is performed by determining scaling factors that are used to scale the output of the digital pulse-shaping filter before the signal is input to the DAC. The scaling factors are derived from the probability density function (pdf) of the signal at the output of the digital pulse-shaping filter. The scaling factors are derived by minimizing the mean-squared error between the DAC output signal and DAC input signal over each pdf that exists for the input signal. In order to remove any spectral distortion caused by scaling, the scaled signal may be rescaled after digital-to-analog conversion in the DAC; the resealing may be performed by multiplying the converted signal by 1/k(l) by a lowpass analog filter before transmission. 
    
    
     BRIEF DESCRIPTION OF THE FIGURES 
     FIG. 1 is a functional block diagram of a CDMA transmitter according to an embodiment of the invention; 
     FIG. 2 is a functional block diagram of a TDMA transmitter according to an embodiment of the invention; 
     FIG. 3A is a plot illustrating signal-to-quantization noise as a function of scaling constant according to an embodiment of the invention; and 
     FIG. 3B is a plot illustrating signal-to-quantization noise as a function of scaling constant according to an embodiment of the invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring now to FIG. 1, therein is illustrated a functional block diagram of a transmitter according to an embodiment of the invention. FIG. 1 shows one channel path of transmitter  100 , which may be, for example, implemented according to the standard specified in the TIA/EIA/IS-2000. 1-IS-2000.5 family of standards, published March 1999 (ballot version). Transmitter  100  comprises digital processor  102 , digital pulse shaping filters  104 ,  106 , adaptive scalers  108 ,  110 , digital-to-analog converters (DAC)  112 ,  114 , lowpass filters  116 ,  118 , intermediate frequency generator (IF)  128 , frequency shifters  120 ,  122 ,  126 , combiner  124 , transmit automatic gain control (TX AGC)  130 , baseband multiplier  132  and power amplifier (PA)  134  having output  136 . 
     In the embodiment of the invention, transmitter  100  may transmit aggregate channels having different relative gains for each channel. “Aggregate channel configuration” refers to a channel configuration where multiple physical channels are combined. For example, the cdma 2000  uplink may comprise a pilot channel, control channel, fundamental channel and up to two supplemental channel transmissions, all combined into one data stream at input  138  for transmission. 
     Digital processor  102  receives a signal X at input  138 . Digital processor  102  forms I and Q components of the signal X at  140   a  and  140   b , respectively. Each of the I and Q components of signal X is then filtered in digital pulse shaping filters  106  and  104 , respectively. Each signal at output  142   a  and  142   b  of digital pulse shaping filters  106  and  104 , respectively, is then adaptively scaled according to the embodiment of the invention in adaptive scalers  110  and  108 , respectively. The adaptive scaling is performed by determining scaling constants for each probability diversity function (pdf) of X. In the embodiment, a scaling constant is found for each pdf included in signal X. 
     The determination of the adaptive scaling constants may be illustrated by using the single channel case of the CDMA channel, for example, a pilot channel, transmitted singularly from transmitter  100 . The determinations of the adaptive scaling constants are identical for both the I and Q channels. The adaptive scaling constants may be determined for an aggregate channel configuration in an identical manner. This may be accomplished by observing the resultant signal distribution after channel aggregation and complex modulation, and determining the adaptive scaling constants based on the signal probability density function. 
     Given the output signal of DAC  112  and  114  has L discrete levels spaced Δ apart, the output signal DAC  112 ,  114  may be taken from the discrete levels {−(L−1)Δ/2, . . . , −Δ/2,Δ2, . . . ,(L−1) Δ/2}. Assume that the input signal is divided into intervals as xε{(−∞,−LΔ/2),(−LΔ/2,−(L+2)Δ/2), . . . ,((L−2)Δ/2,LΔ/2),(LΔ/2,∞)}. Thus each of the DAC  112 ,  114  output signal levels {−(L−1)Δ/2, . . . ,−Δ/2,Δ/2, . . . , (L−1)Δ2} corresponds to the input intervals {(−LΔ/2,−(L+2)Δ/2), . . . ,((L−2)Δ2,LΔ/2)}. The DAC  112 ,  114  rail output signal levels, {−(L−1)Δ/2,(L−1)Δ/2}, correspond to the input intervals {(−∞,−LΔ/2),(LΔ/2, ∞)}. Thus, if the time-varying symmetric pdf of x is denoted by p x (x,i), and the pdf of i is denoted by p i (i), then the mean-squared error between the DAC  112 ,  114  output signal r and the original signal x may be found by the following equation:                  E        {       (     r   -   x     )     2     }       =     2          ∫     -   ∞     ∞                       [         ∑     k   =   1         L   2     -   1                         ∫       (     k   -   1     )        Δ       k                 Δ                (     x   -         (       2      k     -   1     )        Δ     2       )     2            p   x          (     x   ,   i     )                          x           +       ∫         L                 Δ                  2     ∞              (     x   -         (     L   -   1     )        Δ     2       )     2            p   x          (     x   ,   i     )                          x           ]            p   i          (   i   )                          i                        Equation  2.0                                
     If x is a cdma 2000  transmitted signal, then the function p x (x,i) is not continuous over i. Therefore, the mean-squared error derivation of Equation 2.0 cannot be evaluated using a Riemann integral, since the Riemann integral may only be used on continuous functions. Thus, one cannot find an optimal quantizer in the Lloyd-Max sense for the output of this filter. However, due to the fact that the signal to be quantized in this case is cyclostationary (i.e., its statistics are invariant to a shift of the time origin by integral multiples of a constant), the Lebesgue integral may be evaluated. If the function p x (x,i) is in fact cyclostationary with period T, then it will satisfy the relationship p x (x,i)=p x (x,i+NT), where N is an integer. Moreover, assume that the function p x (x,i) changes k times over one period. In addition, assume one period may be broken down into the disjoint intervals {(A 0 ,A 1 ),(A 1 ,A 2 ), . . . ,(A k−1 , A k )}, where the following hold true: (1) A 0 &lt;A 1 &lt; . . . &lt;A k (2) A k −A 0 =T, and (3) p x (x,A j )=p x (x,A j +ε) for 0≦ε&lt;A j+1 −A j  and integer value j satisfying 0&lt;j &lt;k. In this case, the pdf of i for p x (x,i) may be written as                  p        (   i   )       =     1       A     j   +   1       -     A   j           ,                  A   j     ≤   i   &lt;     A     j   +   1                 Equation  3.0                                
     Now, the relationship of Equation 2.0 may be evaluated in a piecewise form. In the embodiment of FIG. 1, the number of pdf&#39;s can be determined from using values of oversampling ration, L, of 4, with a pulse-shaping filter of length of N=48 taps. In this case, where N is an integer multiple of L, the number of pdf&#39;s is equal to L/2 or 2. If these two pdf&#39;s are denoted as p x (x, 1) and p x (x, 2), then the mean-squared error may now be written as                  E        {       (     r   -   x     )     2     }       =         ∑     k   =   1         L   2     -   1                         ∫       (     k   -   1     )        Δ       k                 Δ                (     x   -         (       2      k     -   1     )        Δ     2       )     2            p   x          (     x   ,   1     )                          x           +       ∫         L                 Δ                  2     ∞              (     x   -         (     L   -   1     )        Δ     2       )     2            p   x          (     x   ,   1     )                          x         +       ∑     k   =   1         L   2     -   1                         ∫       (     k   -   1     )        Δ       k                 Δ                (     x   -         (       2      k     -   1     )        Δ     2       )     2            p   x          (     x   ,   2     )                          x           +       ∫         L                 Δ                  2     ∞              (     x   -         (     L   -   1     )        Δ     2       )     2            p   x          (     x   ,   2     )                          x                        Equation  4.0                                
     Thus given log 2 L bits available in the DAC, an optimal quantizer may be found by minimizing the mean-squared error given in Equation 4.0. This can be accomplished by scaling the input signal x to adjust the signal variance with respect to the maximum output signal level of DAC  104 ,  106 . 
     The scaling is necessary to reduce the loss of signal quality resulting from D/A conversion while fixing the granularity of the D/A converter. The optimal scaling factor is determined empirically through simulation of D/A conversion of typical test data under different scaling factors. After scaling, the input signal variance is adjusted as a function of the square of the scaling other words, given that the input signal x with variance σ x   2  is scaled by a factor K, the resulting signal variance is K 2  σ x   2 . 
     The scaling factor, which results from the minimum mean-squared error for x, may also be found analytically. FIG. 3A illustrates the signal-to-quantization noise ratio simulated for different scaling factors optimized over an enter pdf distribution. Sample data are passed through the baseband pulse-shaping filter, scaled, and quantized, using 8 bits of quantization were available. The results are shown for different scaling factors in FIG.  3 A. An adaptive quantizer, which changes with the changing pdf in time is impractical with a typical DAC. Adapting the scaling prior to the DAC  112 ,  114  can result in spectral distortion. This is due to the fact that the scaling adaptation rate is at 2 times the chipping rate; although the change in scaling may be slight, this could result in significant spectral growth. Therefore, it is desirable to optimize the quantizer over the time-varying pdf. FIG. 3B illustrates the signal-to-quantization noise ration achieved for different scaling factors optimized for an instantaneous pdf. FIG. 3B is shows optimal scaling for the two pdf&#39;s present in a cdma 2000  reverse link channel in FIG.  3 B. In this case, the signal-to-quantization noise ratios are shown for different scaling factors based on all instantaneous pdf&#39;s appearing in the input signal, while in FIG. 3A the signal-to-quantization noise ratios were determined with respect to the input signal regardless of instantaneous pdf. Comparing FIG.  3 A and FIG. 3B, it is observed that although the peak SNR&#39;s of the distribution depicted in FIG. 3B fall above and below the peak SNR of the quantizer optimized over the entire distribution by nearly the same distance. 
     The cdma 2000  signal filter output of FIG. 1 has two positive scaling factors, denoted as k 1  and k 2 . If the scaling factors are represented as a square wave defined as:                k        (   l   )       =     {             k   1     ,     l   =   …                ,     -   8     ,     -   7     ,     -   4     ,     -   3     ,   0   ,   1   ,   4   ,   5   ,   8   ,   9   ,   12   ,   13   ,   …                 k   2     ,     l   =   …                ,     -   6     ,     -   5     ,     -   2     ,     -   1     ,   2   ,   3   ,   6   ,   7   ,   10   ,   11   ,   14   ,   15   ,   …                     Equation  5.0                                
     and a square function t(l) is defined as: 
       t ( l )= sgn (cos( l ))  Equation 6.0 
     Then the following may be derived:                k        (   l   )       =         (       t        (   l   )       +   1     )                   k   2     -     k   1       2            +     (     min        (       k   1     ,     k   2       )       )               Equation  7.0                                
     Since a square function may be written in terms of its Fourier series as a sum of sinusoids, we can redefine t(l) as                t        (   l   )       =       ∑     i   =   0     ∞                           (     -   1     )     i         2      i     +   1            cos        (       π   2          (       2      i     +   1     )        l     )                   Equation  8.0                                
     The discrete-time Fourier transform of t(l) is given by:                T        (        jω     )       =       ∑     i   =   0     ∞                           π        (     -   1     )       i         2      i     +   1              ∑     k   =     -   ∞       ∞                     [                  δ        (     ω   -       π   2          (       2      i     +   1     )       +     2                 π                 k       )       +     δ        (     ω   +       π   2          (       2      i     +   1     )       +     2      π                 k       )         ]                   Equation  9.0                                
     The discrete-time Fourier transform of k(l) may now be found in terms of the discrete-time Fourier transform of t(l) as                K        (        jω     )       =                  k   2     -     k   1       2               T        (        jω     )         +       (                k   2     -     k   1       2          +     min        (       k   1     ,     k   2       )         )                       ∑     k   =     -   ∞       ∞                     2        πδ        (     ω   +     2      π                 k       )                       Equation  10.0                                
     In Equation 9.0, the second term in the sum is essentially a DC-offset term. The leftover square wave has a spectral response with the fundamental frequency centered at the chipping frequency. Multiplying this scaling signal by x(l) will result in a convolution of the spectral responses of the two signals. Therefore, the resultant spectrum of x(l) is                  K        (        jω     )       *       Y   chan          (        jω     )         =                  k   2     -     k   1       2                 ∑     i   =   0     ∞                π        (     -   1     )       i         2      i     +   1              ∑     k   =     -   ∞       ∞          [         Y   chan          (          j        (     ω   -       π   2          (       2      i     +   1     )       +     2      π                 k       )         )       +       Y   chan          (          j        (     ω   +       π   2          (       2      i     +   1     )       +     2      π                 k       )         )         ]             +       (                k   2     -     k   1       2          +     min        (       k   1     ,     k   2       )         )            ∑     k   =     -   ∞       ∞          2      π                     Y   chan          (          j        (     ω   +     2      π                 k       )         )                       Equation  11.0                                
     Since x(l) is band limited to half of the chipping frequency, then the spectrum after adaptive scaling could reach as high as 1½ times the chipping frequency, due to the first term in Equation 11.0. In addition, higher-order spectral products resulting from the odd harmonics of k(l) also result in spectral distortion. The second term in Equation 11.0 results in a DC-offset. However, as k 1  and k 2  move closer together in value, the more k(l) resembles a constant, which results in no spectral modification of Y chan (l). This can be seen by examining Equation 10.0, which leads to                  lim       k   1     →     k   2              k        (   l   )         =       k   1     =     k   2               Equation  12.0                                
     As a result, the following spectrum results                  lim       k   1     →     k   2                K        (        jω     )       *       Y   chan          (        jω     )           =         k   1            ∑     k   =     -   ∞       ∞                     2      π                     Y   chan          (          j        (     ω   +     2      π                 k       )         )             =       k   2            ∑     k   =     -   ∞       ∞                     2      π                     Y   chan          (          j        (     ω   +     2      π                 k       )         )                       Equation  13.0                                
     If the optimal scaling factor is different for the different distributions, an adaptive quantization algorithm may result in significant spectral growth. There exists a tradeoff between spectral growth and quantization noise. 
     The spectral distortion after D/A conversion may be undone by resealing the signal before transmission. This involves multiplying the converted signal by 1/k(l). Additional amplification may be needed if the resealing operation results in a deviation from the desired output power. 
     This may be accomplished by adjusting an amplifier which follows the D/A converter at the same rate at which the scaling constant prior to D/A conversion is being adjusted. This may be implemented directly in the lowpass filter which normally follows a D/A converter, as this filter is usually an active filter. 
     Referring now to FIG. 2, therein is a functional block diagram of a TDMA transmitter according to an embodiment of the invention. Transmitter  200  comprises slot data input  202 , serial-to-parallel converters  204 , differential phase encoder  206 , root-raised cosine filter  208  and  210 , adaptive scaler  214  and  216 , D/A converter  218  and  220 , and lowpass filter  222  and  224 . 
     For the embodiment of FIG. 2, root-raised cosine filters  208  and  210  have a length of N=25 taps and an oversampling ration, L, of 4. Since N is not an integer multiple of L in this case, the number of pdf&#39;s is equal to L−|R−Ceil(L/2)|, where 0&lt;R≦L−1, and ceil (L/2) is the smaller integer greater than or equal to L/2, or the number of pdf&#39;s=4−|3−2|=3. Scaling constants and adaptive scaling are found for each pdf as described previously for FIG.  1 . 
     Although the method an apparatus of the present invention has been illustrated and described with regard to particular embodiments thereof, it will be understood that numerous modifications and substitutions may be made to the embodiments described and that numerous other embodiments of the invention may be implemented without departing from the spirit and scope of the invention as defined in the following claims.