Patent Publication Number: US-8115717-B2

Title: Method and system for line by line addressing of RMS responding display matrix with wavelets

Description:
FIELD OF INVENTION 
     The field of invention is related to a line-by-line addressing technique and a technique that is based on wavelets is proposed to obtain gray shades in RMS (root-mean-square) responding matrix displays. A large number of gray shades can be displayed with less number of voltages in the addressing waveforms and hence have simple drivers as compared to amplitude modulation and less number of time intervals as compared to the frame modulation. The technique is demonstrated by displaying 128 gray shades in a liquid crystal display (LCD). 
     BACKGROUND OF INVENTION AND PRIOR ART 
     Amplitude modulation [1] was the first ever technique that attempted to display a large number of gray shades. Hardware complexity of the data drivers is high for this technique because 2(G−1) voltages are necessary in the data waveforms to display G gray shades. Hardware complexity will be less if analog sample and hold type drivers are used instead of the digital drivers but the power consumption will be high. The pulse height modulation [2] for displaying gray shades is similar to the amplitude modulation and hence has the same merits and demerits as the amplitude modulation. The successive approximation techniques [3] can display a large number of gray shades with the simple drivers as that of bi-level displays (with pixels driven to either ON or OFF states). Hardware complexity of the modified voltage level generator and a few analog multiplexers that are necessary to achieve gray shades in a bi-level display is not significant as compared to the increase hardware complexity of the drivers if amplitude modulation or pulse height modulation is used for displaying a large number of gray shades. Reduction in the hardware complexity of the drivers has a high impact on the overall hardware complexity and the cost because the number of drivers is large (sum of the rows and columns in the matrix display). However, supply voltage of the driver circuit of the successive approximation technique increases with the number of gray shades. Wavelets can be used to display gray shades (for driving or matrix addressing) in liquid crystal displays (LCD) and it was demonstrated [4] recently. Wavelets can be used to reduce the supply voltage and hardware complexity of the drivers [5], [6] in RMS responding displays, especially when the number of gray shades is large. All these addressing techniques are based on selecting several address lines simultaneously while scanning the matrix displays. Our objective is to show that the simple line-by-line addressing can be used to display a large number of gray shades without much increase in the hardware complexity of the drivers and without much increase in supply voltage. A line-by-line addressing that is based on using wavelets is proposed in this paper. Effective utilization of wavelets is illustrated with an example: displaying 128 gray shades in a passive matrix LCD using the Haar wavelets. However, the wavelet matrices for 32 and 64 gray shades are also given in this paper. 
    
    
     
       BRIEF DESCRIPTION OF ACCOMPANYING DRAWINGS 
         FIG. 1 : shows typical waveforms when the matrix display is scanned with waveforms derived from wavelets. Scanning (row) waveforms have seven voltages and the column (data) waveforms have twenty-one voltages. 
         FIG. 2 : shows typical waveforms when the row (scanning) and column (data) waveforms are modified to reduce the supply voltage. Both the row and column waveforms are shifted by the same amount. Hence, the waveform across the pixels is same as that in  FIG. 1  although the row and column waveforms differ from that of  FIG. 1 . 
         FIG. 3 : shows block diagram of the prototype of a 32×32 matrix liquid crystal display that is scanned using a line-by-line addressing technique that is based on wavelets. 
         FIG. 4 : are illustrations of a 32×32 matrix LCD driven by the wavelet based line-by-line addressing technique. It is capable of displaying 128 gray shades. 
         FIG. 5 : shows typical scanning (upper trace) and data (lower trace) waveforms of the wavelets based line-by-line addressing technique. 
         FIG. 6 : shows waveform across a pixel in the display (part of a cycle is shown here for the sake of clarity. 
     
    
    
     OBJECT OF THE INVENTION 
     The primary objective of the invention is to develop a method for line-by-line addressing of RMS responding display matrix with wavelets. 
     Another objective of the invention is selecting the wavelets such that energy of them is proportional to an integer power of two to construct a wavelet matrix; 
     Still another objective of the invention is obtaining select waveform profile by summing elements of column in the wavelet matrix; 
     Still another objective of the invention is obtaining column waveforms by dot product of some of the data bits with column of the wavelet matrix; 
     Still another objective of the invention is applying the select waveforms and the corresponding column waveforms by selecting one row of the matrix display at a time. 
     Yet another main objective of the present invention is to develop a system for line-by-line addressing of RMS responding display matrix with wavelets. 
     Still another main objective of the present invention is to develop a matrix liquid crystal display (LCD) to display image and a row driver to drive the rows of the matrix LCD with row waveform. 
     Still another main objective of the present invention is to develop a column driver to drive the columns of the matrix LCD with column waveform. 
     Still another main objective of the present invention is to develop a voltage level generator to facilitate the row driver and the column to generate the desired waveforms. 
     Still another main objective of the present invention is to develop a controller to control the system for line-by-line addressing. 
     STATEMENT OF INVENTION 
     The present invention is related to a method for line-by-line addressing of RMS responding display matrix with wavelets, said method comprises steps of: selecting the wavelets such that energy of them is proportional to an integer power of two to form wavelet matrix; obtaining select waveform profile by summing elements of columns in the wavelet matrix; obtaining column waveforms by dot product of data with column of the wavelet matrix; and applying the select waveform and the corresponding column waveforms by selecting one row of the display matrix at a time; and a system for line-by-line addressing of RMS responding display matrix with wavelets comprising: a matrix liquid crystal display (LCD) to display image; a row driver to drive the rows of the matrix LCD with row waveform; a column driver to drive the columns of the matrix LCD with column waveform; a voltage level generator to facilitate the row driver and the column to generate desired waveforms, and a controller to control the system for line-by-line addressing. 
     DETAILED DESCRIPTION OF INVENTION 
     The primary embodiment of the invention is a method for line-by-line addressing of RMS responding display matrix with wavelets, said method comprises steps of: selecting the wavelets such that energy of them is proportional to an integer power of two to form a wavelet matrix; obtaining select waveform profile by summing elements of column in the wavelet matrix; obtaining column waveforms by dot product of data with column of the wavelet matrix; and applying the select waveform and the corresponding column waveforms by selecting one row of the display matrix at a time. 
     In yet another embodiment of the present invention row of the display matrix is selected with waveform that is proportional to the waveform that is obtained by adding ‘g’ wavelets wherein ‘g’ is number of data bits. 
     In still another embodiment of the present invention, a constant of proportionality is added to the column waveform and the select waveform. 
     In still another embodiment of the present invention, the column waveform is obtained by summing products of value assigned to the data bit and the corresponding wavelets. 
     In still another embodiment of the present invention both the select and the column waveforms are shifted by predetermined time interval in order to reduce power supply voltage of driver circuit. 
     In still another embodiment of the present invention order of voltage in the select waveform is changed as long as one to one correspondence among the wavelets and the data bits is maintained while computing the column waveform to reduce power consumption. 
     In still another embodiment of the present invention the select waveform voltage is distributed into several frames. 
     In still another embodiment of the present invention the data bits is assigned with values ‘+1’ and ‘−1’ when the bit is logical ‘0’ and ‘1’ respectively. 
     In still another embodiment of the present invention rows other then the selected row are maintained at ground. 
     In still another embodiment of the present invention number of voltages in the select waveform and the column waveform determines amount of gray shades of the display matrix. 
     In still another embodiment of the present invention the amount of gray shades increases with increase in the number of voltages in the row waveform and the column waveform. 
     In still another embodiment of the present invention the wavelet is Haar wavelet. 
     In still another embodiment of the present invention the wavelets are DC free. 
     Another main embodiment of the present invention a system for line-by-line addressing of RMS responding display matrix with wavelets comprising: a matrix liquid crystal display (LCD) to display image; a row driver to drive the rows of the matrix LCD with row waveform; a column driver to drive the columns of the matrix LCD with column waveform; a voltage level generator to facilitate the row driver and the column to generate desired waveforms, and a controller to control the system for line-by-line addressing. 
     In yet another embodiment of the present invention the row drivers that are capable applying just one of the two voltages are used; wherein a pair of multiplexers that are common to all the row drivers selects these two voltages. Similarly, the a set of multiplexers that are common to all the data (column) drivers is used to reduce the hardware complexity of the data drivers. 
     LCD is a slow responding device and its electro-optic response does not follow the abrupt changes in the electric field across the pixels. State of a pixel is determined by the energy delivered to it as long as the period of the addressing waveforms is less as compared to the response times of the display (RMS response). Hence, gray shades can be displayed by modulating the energy delivered to the pixels by using a set of orthogonal wavelets. For example, the seven Haar wavelets that are used in our prototype (for displaying 128 gray shades) are shown in the following equations.
 
ψ 6 ={+4,+4,+4,+4,−4,−4,−4,−4}  (1)
 
ψ 5 ={+4,+4,−4,−4,0,0,0,0}  (2)
 
ψ 4 4={0,0,0,0,0,0,+4,−4}  (3)
 
ψ 3 ={0,0,0,0,+2,+2,−2,−2}  (4)
 
ψ 2 ={0,0,0,0,+2,−2,0,0}  (5)
 
ψ 1 ={0,0,+√{square root over (2)},−√{square root over (2)},0,0,0,0}  (6)
 
ψ 0 ={+1,−1,0,0,0,0,0,0}  (7)
 
     These wavelets satisfy the following conditions: 
     1. Energy of the wavelet is proportional to an integer power of two; so that its energy corresponds to one of the gray shade bit. Energy of the wavelet means summation of the square of the amplitudes of the wavelets in all time intervals. For example, if a i  is the amplitude of the wavelet in the time interval i, then the energy of wavelet is proportional to 
               ∑     i   =   1     n     ⁢           ⁢     a   i   2           
wherein n is the number of discrete time intervals in the wavelet. For example, Energy of wavelet ψ 6  is equation 1 is proportional to:
 
(+4) 2 +(+4) 2 +(+4) 2 +(+4) 2 +(−4) 2 +(−4) 2 +(−4) 2 +(−4) 2 =8(4) 2  =2 7 =128.
 
     Similarly, Energy of wavelet ψ 5  is equation 2 is proportional to:
 
(+4) 2 +(+4) 2 +(−4) 2 +(−4) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 =4(4) 2 =2 6 =64,
 
     Energy of wavelet ψ 4  is equation 3 is proportional to:
 
(0) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 +(+4) 2 +(−4) 2 =2(4) 2 =2 5 =32,
 
     Energy of wavelet ψ 3  is equation 4 is proportional to:
 
(0) 2 +(0) 2 +(0) 2 +(0) 2 +(+2) 2 +(+2) 2 +(−2) 2 +(−2) 2 =4(2) 2 =2 4 =16,
 
     Energy of wavelet ψ 2  is equation 5 is proportional to:
 
(0) 2 +(0) 2 +(0) 2 +(0) 2 +(+2) 2 +(−2) 2 +(0) 2 +(0) 2 =2(2) 2 =2 3 =8,
 
     Energy of wavelet ψ 1  is equation 6 is proportional to:
 
(0) 2 +(0) 2 +(√{square root over (2)}) 2 +(√{square root over (2)}) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 =2(√{square root over (2)}) 2 =2 2 =4, and
 
     Energy of wavelet ψ 0  is equation 7 is proportional to:
 
(+1) 2 +(−1) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 +(0) 2 =2 (1) 2 =2 1 =2.
 
     2. DC free wavelets are used to ensure long life of the display; life of the LCD is reduced if DC voltages are present across the pixels. 
     Each one of these wavelets may be used to select an address line and deliver a quantum of energy that is proportional to one bit of the gray shade and this process could be repeated sequentially to exhaust all the bits. Then, the number of time intervals is large (56.N to display 128 gray shades) wherein N is number of address lines, even though it is smaller than (127.N for displaying 128 shades) that of the frame modulation [7] and the pulse width modulation [8]. Suppressing the zeros in the wavelets can reduce the number of time intervals to(24.N). Even then, it is more than that of the wavelet based addressing techniques that are based on selecting several address lines simultaneously [4]-[6]. The number of time intervals to complete a cycle is (8.N) if these orthogonal wavelets are used for scanning the matrix display by selecting several (3 or 4) rows simultaneously for displaying 128 gray shades. Our aim is to display a large number of gray shades with line-by-line addressing and yet ensure that the number of time intervals does not exceed that of the multi-line addressing. This goal has been achieved by selecting each row with a waveform profile that is obtained by adding all the wavelets as proposed in the following section. Although the technique is illustrated with the Haar wavelets, other orthogonal wavelets can also be used. 
     Consider a matrix display with N address lines and M data lines. Let the gray shade to be displayed in a pixel at the intersection of row ‘i’ and column ‘j’ be: 
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       k 
                       = 
                       0 
                     
                     
                       g 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     d 
                     
                       i 
                       , 
                       j 
                       , 
                       k 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Here, the index k corresponds to the gray shade bit (most significant to the least significant) of ‘g’ number of data bits. Let ψ k  be the wavelet that has energy proportional to 2 k , corresponding to the bit-k of the gray shade data. Let the row ‘i’ of the matrix display be selected with the waveform that is proportional to the waveform that is obtained by adding the ‘g’ number of wavelets. The number of wavelets is the same as the number of data bits. The expression for the select waveform is given in the following equation. 
     
       
         
           
             
               
                 
                   
                     Select 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       waveform 
                       i 
                     
                   
                   = 
                   
                     
                       V 
                       r 
                     
                     · 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           g 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         ψ 
                         k 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Wherein, V r  is the proportionality constant. Hence, the select voltages during the eight time intervals in our example are as follows:
 
+9V r ,+7V r ,+√{square root over (2)}V r ,−√{square root over (2)}V r ,0,−4V r ,−2V r  and −10V r   (10)
 
     The other (N−1) address lines (non-selected) in the matrix display are held at ground potential. The column waveform that is to be applied to the column ‘j’ is obtained by summing the products of the value assigned to the data bit and the corresponding wavelets as shown in (11). The value assigned to the bit-k (d i,j,k ) is ‘+1’ if the bit is ‘logic-0’ and a value of ‘−1’ is assigned to the bit otherwise. 
     
       
         
           
             
               
                 
                   
                     Column 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       waveform 
                       j 
                     
                   
                   = 
                   
                     
                       V 
                       c 
                     
                     · 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           g 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ψ 
                           k 
                         
                         · 
                         
                           d 
                           
                             i 
                             , 
                             j 
                             , 
                             k 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Wherein, V c  is amplitude of the unit voltage that serves as the constant of proportionality in the column waveforms. Multiplication of the wavelets with +1 or ‘−1’ reduces to just assigning the appropriate sign to the wavelets ψ k  and summing them. The column waveform can also be obtained by the matrix multiplication by arranging the seven wavelets of equations (1) to (7) in to a 7×8 matrix and multiplying its transpose with the data vector of the pixel as shown in equation (12). 
     
       
         
           
             
               
                 
                   
                     
                       C 
                       j 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                           
                           
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 + 
                                 4 
                               
                             
                             
                               
                                 - 
                                 4 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 + 
                                 2 
                               
                             
                             
                               
                                 + 
                                 2 
                               
                             
                             
                               
                                 - 
                                 2 
                               
                             
                             
                               
                                 - 
                                 2 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 + 
                                 2 
                               
                             
                             
                               
                                 - 
                                 2 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 + 
                                 
                                   2 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   2 
                                 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 + 
                                 1 
                               
                             
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       T 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 6 
                               
                             
                           
                         
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               d 
                               
                                 i 
                                 , 
                                 j 
                                 , 
                                 0 
                               
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       V 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     At a given instant of time; just one voltage is applied to the selected row and it is obtained summing the elements of a column in the matrix of wavelets in (12). The column signal that is proportional to the column voltage is obtained as the dot product of the data with the column of the matrix of orthogonal wavelets. A cycle is complete when all the address lines in the matrix display are selected once. This cycle is repeated fast enough to avoid flicker. Waveforms across the pixels will be DC free as long as all the wavelets are DC free. In case the orthogonal wavelets are not DC free, then the DC free condition can be achieved by reversing the polarity of both the scanning and data waveforms in a periodic manner. Select pulses may be clustered as illustrated in  FIG. 1  or it could be distributed in several frames of a cycle. The order of the voltages in the select waveform profile may be changed as long as the one to one correspondence among the wavelets and data bits is maintained while computing the column (data) voltage. RMS voltage across the pixels will be same as long as the period of a cycle is less than the response time; but the frequency spectrum across the pixels will vary depending on the distribution of select pulses in a cycle. Hence, it can be used to enhance the performance of the display like brightness uniformity of pixels, reducing power dissipation etc. The select voltages may also be distributed in to several frames to achieve such improvements. Typical waveform profiles for selecting the address lines when the number of gray shades ranges from 32 to 128 are shown in Table I. The corresponding matrices of orthogonal wavelets and the number of voltages in the scanning (row) and the data (column) waveforms are also shown in this table. An analysis of the technique is presented in the next section. 
     I. Analysis 
     Energy delivered to a pixel that is located at the intersection of ‘row-i’ and ‘column-j’ during a frame when select and data waveforms are applied to the matrix display is given by the following expression. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               V 
                               r 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   g 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 ψ 
                                 k 
                               
                             
                           
                           - 
                           
                             
                               V 
                               c 
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   g 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   ψ 
                                   k 
                                 
                                 · 
                                 
                                   d 
                                   
                                     i 
                                     , 
                                     j 
                                     , 
                                     k 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     · 
                     T 
                   
                   + 
                   
                     
                       ( 
                       
                         N 
                         - 
                         1 
                       
                       ) 
                     
                     · 
                     
                       
                         ( 
                         
                           
                             V 
                             c 
                           
                           · 
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 g 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 ψ 
                                 k 
                               
                               · 
                               
                                 d 
                                 
                                   i 
                                   , 
                                   j 
                                   , 
                                   k 
                                 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     · 
                     T 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The first term in this equation corresponds to the voltage across the pixel when the corresponding address line (row) is selected while the second term correspond to the data voltages appearing across the pixel when other (N−1) address lines are selected. Orthogonal condition, factors like energy of the wavelets and that of data assigned to the pixels viz. equations (14) to (16) are incorporated in to the equation (13). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∫ 
                             0 
                             T 
                           
                           ⁢ 
                           
                             
                               ψ 
                               i 
                             
                             ⁢ 
                             
                               ψ 
                               j 
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                         = 
                         0 
                       
                     
                     
                       
                         
                           
                             ∀ 
                           
                           
                             
                               i 
                               ≠ 
                               j 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∫ 
                       0 
                       T 
                     
                     ⁢ 
                     
                       
                         ψ 
                         k 
                         2 
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   = 
                   
                     c 
                     · 
                     
                       2 
                       k 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     d 
                     
                       i 
                       , 
                       j 
                       , 
                       k 
                     
                     2 
                   
                   = 
                   
                     
                       
                         + 
                         1 
                       
                       ∵ 
                       
                         d 
                         
                           i 
                           , 
                           j 
                           , 
                           k 
                         
                       
                     
                     = 
                     
                       { 
                       
                         
                           
                             
                               + 
                               1 
                             
                           
                           
                             ∀ 
                           
                           
                             
                               
                                 “ 
                                 
                                   logic 
                                   - 
                                   0 
                                 
                                 ” 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               and 
                             
                           
                         
                         
                           
                             
                               - 
                               1 
                             
                           
                           
                             ∀ 
                           
                           
                             
                               
                                 “ 
                                 
                                   logic 
                                   - 
                                   1 
                                 
                                 ” 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Hence, the energy delivered to a pixel simplifies to the following expression: 
     
       
         
           
             
               
                 
                   
                     E 
                     pixel 
                   
                   = 
                   
                     
                       
                         V 
                         r 
                         2 
                       
                       · 
                       c 
                       · 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             g 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           ψ 
                           k 
                           2 
                         
                       
                     
                     - 
                     
                       2 
                       · 
                       
                         V 
                         r 
                       
                       · 
                       
                         V 
                         c 
                       
                       · 
                       c 
                       · 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             g 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             ψ 
                             k 
                             2 
                           
                           · 
                           
                             d 
                             
                               i 
                               , 
                               j 
                               , 
                               k 
                             
                           
                         
                       
                     
                     + 
                     
                       N 
                       · 
                       c 
                       · 
                       
                         V 
                         c 
                         2 
                       
                       · 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             g 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           ψ 
                           k 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     The RMS voltage across a pixel is shown in (18). 
     
       
         
           
             
               
                 
                   
                     V 
                     RMS 
                   
                   = 
                   
                     
                       
                         c 
                         · 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               g 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               k 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   V 
                                   r 
                                   2 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                     
                                       d 
                                       
                                         i 
                                         , 
                                         j 
                                         , 
                                         k 
                                       
                                     
                                     · 
                                     
                                       V 
                                       r 
                                     
                                     · 
                                     
                                       V 
                                       c 
                                     
                                   
                                 
                                 + 
                                 
                                   NV 
                                   c 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           n 
                           g 
                         
                         · 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Here, n g  is the number of columns in the matrix of wavelets. Sign of the mid term (2·d i,j,k ·V r ·V c ) in (18) is determined by the ‘bit-k’ of the data. RMS voltage across an ON (all the bits are ‘logic-1’ and they are assigned ‘−1’) and OFF (all the g bits are assigned +1 for ‘logic-0’) pixels is given in equation (19) and (20) respectively. 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ON 
                     
                     ⁡ 
                     
                       ( 
                       RMS 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         c 
                         · 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               g 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               k 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   V 
                                   r 
                                   2 
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     V 
                                     r 
                                   
                                   ⁢ 
                                   
                                     V 
                                     c 
                                   
                                 
                                 + 
                                 
                                   NV 
                                   c 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           n 
                           g 
                         
                         · 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                       OFF 
                     
                     ⁡ 
                     
                       ( 
                       RMS 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         c 
                         · 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               g 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               ψ 
                               k 
                               2 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   V 
                                   k 
                                   2 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                     V 
                                     r 
                                   
                                   ⁢ 
                                   
                                     V 
                                     c 
                                   
                                 
                                 + 
                                 
                                   NV 
                                   c 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           n 
                           g 
                         
                         · 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Selection ratio defined as the ratio of RMS voltage across ON pixels to that across OFF pixels is a maximum when the condition in (21) is satisfied. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           V 
                           ON 
                         
                         
                           V 
                           OFF 
                         
                       
                       ⁢ 
                       
                         ❘ 
                         
                           max 
                           . 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             N 
                           
                           + 
                           1 
                         
                         
                           
                             N 
                           
                           - 
                           1 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   when 
                   ⁢ 
                   
                     
 
                   
                   ⁣ 
                   
                     
                       
                         V 
                         r 
                       
                       
                         V 
                         c 
                       
                     
                     = 
                     
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     The expressions for the RMS voltage across the ON and OFF pixels get simplified to: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ON 
                     
                     ⁡ 
                     
                       ( 
                       RMS 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           c 
                           · 
                           
                             ( 
                             
                               
                                 2 
                                 g 
                               
                               - 
                               1 
                             
                             ) 
                           
                           · 
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               N 
                               + 
                               
                                 N 
                               
                             
                             ) 
                           
                           · 
                           
                             V 
                             c 
                             2 
                           
                         
                       
                       
                         
                           n 
                           g 
                         
                         · 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                       OFF 
                     
                     ⁡ 
                     
                       ( 
                       RMS 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           c 
                           · 
                           
                             ( 
                             
                               
                                 2 
                                 g 
                               
                               - 
                               1 
                             
                             ) 
                           
                           · 
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               N 
                               - 
                               
                                 N 
                               
                             
                             ) 
                           
                           · 
                           
                             V 
                             c 
                             2 
                           
                         
                       
                       
                         
                           n 
                           g 
                         
                         · 
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Contrast in the display will be a maximum when OFF pixels are applied a voltage that is near the threshold voltage of the display. Hence, the unit voltage in the column waveform is determined by equating V OFF (RMS) to the V th , threshold voltage of the electro-optic characteristics of LCD. 
     
       
         
           
             
               
                 
                   
                     V 
                     c 
                   
                   = 
                   
                     
                       
                         
                           
                             n 
                             g 
                           
                           · 
                           N 
                         
                         
                           2 
                           · 
                           c 
                           · 
                           
                             ( 
                             
                               
                                 2 
                                 g 
                               
                               - 
                               1 
                             
                             ) 
                           
                           · 
                           
                             ( 
                             
                               N 
                               - 
                               
                                 N 
                               
                             
                             ) 
                           
                         
                       
                     
                     · 
                     
                       V 
                       threshold 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Supply voltage of the drive electronics is determined by the maximum swing in the addressing waveforms. Let the P·V r  be the peak to peak amplitude of the row waveforms then, the supply voltage of the addressing technique (normalized to the threshold voltage of the electro-optic characteristics of the LCD) is: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       W_APT 
                       ) 
                     
                   
                   = 
                   
                     
                       P 
                       · 
                       
                         V 
                         r 
                       
                     
                     = 
                     
                       P 
                       · 
                       
                         N 
                       
                       · 
                       
                         V 
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       V 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       W_APT 
                       ) 
                     
                   
                   = 
                   
                     
                       P 
                       2 
                     
                     · 
                     
                       
                         
                           
                             n 
                             g 
                           
                           
                             c 
                             · 
                             
                               ( 
                               
                                 
                                   2 
                                   g 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                       . 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             
                               N 
                             
                           
                           
                             
                               2 
                               · 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     ( 
                                     
                                       1 
                                       / 
                                       
                                         N 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Supply voltage of the successive approximation technique [3] that is also based on the line-by-line addressing (SA-APT) is as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       SA_APT 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           g 
                           · 
                           
                             2 
                             
                               ( 
                               
                                 g 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         
                           
                             2 
                             g 
                           
                           - 
                           1 
                         
                       
                     
                     · 
                     
                       ( 
                       
                         
                           2 
                           ⁢ 
                           
                             N 
                           
                         
                         
                           
                             2 
                             · 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   ( 
                                   
                                     1 
                                     / 
                                     
                                       N 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Supply voltages of these two techniques are compared using the following ratio: 
     
       
         
           
             
               
                 
                   
                     
                       
                         V 
                         s 
                       
                       ⁡ 
                       
                         ( 
                         W_APT 
                         ) 
                       
                     
                     
                       
                         V 
                         s 
                       
                       ⁡ 
                       
                         ( 
                         SA_APT 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       P 
                       2 
                     
                     · 
                     
                       
                         
                           n 
                           g 
                         
                         
                           c 
                           · 
                           g 
                           · 
                           
                             2 
                             
                               ( 
                               
                                 g 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     
       
         
           
               
             
               
                 TABLE I 
               
             
            
               
                   
               
               
                 SELECT WAVEFORM PROFILES FOR GRAY SHADES. 
               
            
           
           
               
               
               
               
               
            
               
                 N g   
                 Wavelets 
                 Select waveform 
                 N r   
                 N c   
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 32 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                     
                         
                         
                     
                   
                 
                 5 
                 11 
               
               
                   
               
               
                 64 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                     
                         
                         
                     
                   
                 
                 6 
                 15 
               
               
                   
               
               
                 128 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   4 
                                 
                               
                               
                                 
                                   - 
                                   4 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   2 
                                 
                               
                               
                                 
                                   - 
                                   2 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 
                                   + 
                                   
                                     2 
                                   
                                 
                               
                               
                                 
                                   - 
                                   
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   + 
                                   1 
                                 
                               
                               
                                 
                                   - 
                                   1 
                                 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                           
                       
                     
                   
                 
                 
                   
                     
                     
                         
                         
                     
                   
                 
                 7 
                 21 
               
               
                   
               
               
                 N g :—Number of Gray Shades, 
               
               
                 N r :—Number of voltages in row waveforms and 
               
               
                 N c :—Number of voltages in the column waveforms. 
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE II. 
               
             
            
               
                   
               
               
                 SUPPLY VOLATGE COMPARISON BETWEEN WAVELET AND 
               
               
                 SUCCESSIVE APPROXIMATION TECHNIQUES 
               
            
           
           
               
               
               
               
               
            
               
                 N g   
                 Peak to peak voltage in the scanning waveforms (V p     —     APT ) 
                 Reduced peak to peak voltage in the addressing waveforms (V p     —     IAPT ) 
                 
                   
                     
                       
                         
                           
                             
                               V 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               W_APT 
                               ) 
                             
                           
                           
                             
                               V 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               SA_APT 
                               ) 
                             
                           
                         
                         × 
                         100 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         % 
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             
                               V 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               W_IAPT 
                               ) 
                             
                           
                           
                             
                               V 
                               s 
                             
                             ⁡ 
                             
                               ( 
                               SA_IAPT 
                               ) 
                             
                           
                         
                         × 
                         100 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         % 
                       
                     
                   
                 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 32 
                 7V r   
                 4(V r  + V c ) 
                 78.26 
                 89.44 
               
               
                 64 
                 (14 + √2)V r   
                 8(V r  + V c ) 
                 78.66 
                 81.65 
               
               
                 128 
                 (15 + √2)V r   
                 9(V r  + V c ) 
                 77.55 
                 85.04 
               
               
                   
               
            
           
         
       
     
     II. Reducing the Supply Voltage 
     A technique, similar to that proposed by Kawakami et al., [9] to reduce the supply voltage of the conventional line-by-line addressing can be adopted to reduce the supply voltage of the drive electronics. Let us consider the example of the waveforms for displaying 128 gray shades. The voltages in the select waveform starting from the first to the eighth time intervals in order (a profile that is slightly different from that of (10) is chosen here) are:
 
+9V r ,+7V r ,+4V r ,−4V r ,0,−4V r ,−(6−√{square root over (2)})V r  and −(6+√{square root over (2)})V r   (29)
 
     Data voltages in the 1 st  and 2 nd  time intervals are:
 
±9V c ,±7V c , and ±V c .  (30)
 
     Data voltages in the third and fourth time intervals are:
 
±12V c  and ±4V c   (31)
 
     Similarly the data voltages in the fifth and sixth time intervals are:
 
±8V c ,±4V c  and 0  (32)
 
     Finally, the data voltages in the seventh and eighth time intervals are as follows:
 
±(6+√{square root over (2)})V c ,±(6−√{square root over (2)})V c ,±(2+√{square root over (2)})V c  and ±(2−√{square root over (2)})V c   (33)
 
     The maximum amplitude of instantaneous voltages across any pixel is 9(V r +V c ).Hence, the supply voltage can be reduced by: 
     1. Shifting both the row and column waveforms by (9V c )during the first two intervals; 
     2. Shifting the row and column waveforms by (4V r +12V c ) for the third to sixth time intervals; and 
     3. Shifting by 8V r  during the last two time intervals. 
     Then, the voltages in the select waveform are modified as follows:
 
+9(V r +V c ),+(7V r +9V c ),+(8V r +12V c ),+(12V c ),+(4V r +12V c ),+(12V c ),+(2+√{square root over (2)})V r  and +(2−√{square root over (2)})V r   (34)
 
     The modified data voltages are as follows: 
     In the 1 st  and the 2 nd  time intervals:
 
+18V c ,0,+16V c ,2V c ,+10V c  and +8V c   (35)
 
     In the 3 rd  and 4 th  time intervals:
 
+(4V r +24V c ),+(4V r ),+(4V r +16V c ) and +(4V r +8V c )  (36)
 
     In the 5 th  and 6 th  intervals:
 
+(4V r +20V c ),+(4V r +4V c )+(4V r +16V c ),+(4V r +8V c ) and +(4V r +12V c )  (37)
 
     The data voltages in the last two intervals are:
 
(8V r ±(6+√{square root over (2)})V c ),(8V r ±(6−√{square root over (2)})V c ),(8V r ±(2+√{square root over (2)})V c ) and (8V r ±(2−√{square root over (2)})V c )  (38)
 
     The voltage transformation is not unique, the transformation described in the previous paragraphs is well suited for LCD drivers with the lowest voltage as the common rail and all other voltages in the addressing waveforms are positive with reference to this common voltage. An example of the voltage transformation that is better suited for the LCD drivers with most positive voltage as the common rail is outlined next. Both the row and column waveforms are shifted as follows: 
     1. Down by 9V r  during the first two time intervals; 
     2. Down by 4V r  during the third to sixth time intervals; and 
     3. Down by 1.5V r  during the last two time intervals. 
     Then, the voltages in the select waveform get modified as follows:
 
0,−2V r ,0,−8V r ,−4V r ,−8V r ,−(7.5−√{square root over (2)})V r  and −(7.5+√{square root over (2)})V r   (39)
 
     The modified data voltages are as follows: 
     In the 1 st  and the 2 nd  time intervals:
 
−(9V r ±9V c ),−(9V r ±7V c ) and −(9V r ±V c )  (40)
 
     In the 3 rd  and 5 th  time intervals:
 
−(4V r ±12V c ) and −(4V r ±4V c )  (41)
 
     In the 4th and 6th intervals:
 
−(4V r ±8V c ),−(4V r ±4V c ) and −4V r   (42)
 
     The data voltages in the last two intervals are:
 
−(1.5V r ±(6+√{square root over (2)})V c ),−(1.5V r ±(6−√{square root over (2)})V c ),−(1.5V r ±(2+√{square root over (2)})V c )and −(1.5V r ±(2−√{square root over (2)})V c )  (43)
 
     Number of voltages in the scanning waveform increases from 7 to 9 (or 10) as compared to 3 (or 4) in the conventional line-by-line addressing. The non-select voltages when the common rail is the most negative voltage are +9V c , (4V r +12V c ) and 8V r . Similarly, the non-select voltages when the most positive voltage is the common rail are −9V r , −4V r  and −1.5V r . It is important to note that the number of voltages in the data waveforms does not change where as it doubles (from 2 to 4 voltages) in the conventional addressing technique. Supply voltage will be proportional to 9(V r +V c ) as compared to (15+√{square root over (2)})V r . The percentage reduction in supply voltage due to modification in the addressing waveforms is: 
     
       
         
           
             
               
                 
                   
                     
                       Vs 
                       ⁡ 
                       
                         ( 
                         W_IAPT 
                         ) 
                       
                     
                     
                       Vs 
                       ⁡ 
                       
                         ( 
                         W_APT 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               9 
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     N 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             
                               
                                 ( 
                                 
                                   15 
                                   + 
                                   
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 N 
                               
                             
                           
                         
                         ) 
                       
                       · 
                       100 
                     
                     ≈ 
                     
                       45 
                       ⁢ 
                       % 
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     The modification of the addressing waveforms is not unique; the shift in the addressing waveforms has to be such that the difference among these voltages is equal to the maximum voltage across the pixels. For example, it is sufficient to ensure that all the voltages in the scanning and data waveforms lie with in the voltage range of 9(V r +V c ), the maximum voltage across the pixel during a cycle. Supply voltages of the wavelet based line-by-line addressing with modified waveforms (V s (W_IAPT)) and the successive approximation technique based on the line-line-addressing with modified waveforms (V s (SA_IAPT)) are also compared in Table II. 
     III. Drive Electronics 
     A block diagram of the prototype is shown  FIG. 3 . Voltages in the addressing waveforms are generated using a resistor network. The number of voltages in the addressing waveform decides the hardware complexity of the drivers. Number of voltages in the scanning waveform ranges from 6 to 11 as the number of gray shades is increased from 8 to 128. However, at a given instant of time just two voltages viz., select and non-select voltages are necessary for scanning the matrix display. Hence, the hardware complexity is the same as that of the conventional line-by-line addressing for displaying bi-level images. LCD drivers like the HD-44100 can be used along with a few multiplexers to select and route the appropriate voltages from the voltage level generator. Number of voltages in the data waveforms is at least 21 for displaying 128 gray shades as shown in Table I. It demands a display driver having 5-bit shift register, 5-bit latch and an analog multiplexer (21:1) for each output. The number of voltages is limited to just eight at any instant of time and a good reduction in hardware complexity of the driver can be achieved when this observation is incorporated in the design. LCD drivers with 3-bit shift register, 3-bit latch and analog multiplexers (8:1) for each output are adequate as data drivers (for example: TMS57206). Hence, a 40% in the shift register &amp; latches and 58% reduction in the analog switches in the multiplexers are achieved. Number of stages in the column driver is equal to the number of columns and hence this reduction is substantial. The reduction is about 62% for the shift registers &amp; latches 96% for the analog switches of the output stage of the drivers if the technique is compared with amplitude modulation for the same number of gray shades. Although, the column voltages are computed as the dot product of the select pattern and the corresponding data vectors, it is not necessary to compute them on the fly; they can be computed once and they can be incorporated in to the voltage level generator. Analog multiplexers in the driver circuit are used to apply these voltages depending on the select voltage and the data. The controller is designed to select one row at a time with all the eight row select voltages. During each select voltage, the voltages corresponding to the data of the pixels are applied on the column side. The pixel values (data) are stored in an EPROM in continuous memory locations as a 1-dimensional array. The addresses corresponding to the data of the pixels in the selected row are generated repeatedly for all the select voltages (once per each select voltage). Similarly, all the other rows are selected one at a time and the display is refreshed continuously. The controller was implemented using a CPLD (complex programmable logic device). Illustrations of a useful LCD decice are shown in  FIGS. 4A and 4B . Typical row and column (data) waveforms are shown in  FIG. 5 . Typical waveform across a pixel is shown in  FIG. 6 . 
     IV. Comparison 
     Amplitude modulation [1] and the very similar Pulse height modulation (PHM) [2] can display 128 gray shades in (4.N) time intervals with 254 voltages in the column (data) waveforms and 3 voltages in the scanning waveforms. It may be more appropriate to compare the wavelet-based technique with the successive approximation technique [3] since both the techniques are based on delivering energies that are proportional to the bit-weight of the gray shade data in several time intervals. A comparison of the wavelet-based technique with amplitude modulation and successive approximation based on line-by-line addressing is given in Table III. 
     V. Conclusion 
     Number of time intervals to complete a cycle is less in the wavelet-based technique as compared to the successive approximation technique when all other parameters are equal. Slow scanning is helpful to reduce the power consumption [10]. The brightness non-uniformity of pixels due to distortion in the addressing waveforms will also be less because the select time is larger when the number of time intervals is small. A lower supply voltage of the wavelet-based technique is advantageous in portable devices. It has been achieved with out increasing the hardware complexity of the row drivers and a moderate increase in hardware complexity of the column drivers. Increasing the number of voltages in the column waveform by a factor of 4 has paved way to an increase in number of gray shades by factor 64; when wavelet based addressing (WAT) is used for scanning the matrix LCD. 
     
       
         
           
               
             
               
                 TABLE III 
               
             
            
               
                   
               
               
                 Comparison of the Gray Shade Techniques (128 ray 
               
               
                 shades) 
               
            
           
           
               
               
               
               
            
               
                   
                 Amplitude 
                   
                   
               
               
                   
                 modulation 
                 Successive 
                   
               
               
                   
                 and PHM (line- 
                 approximation 
                 Wavelet based 
               
               
                 Parameter 
                 by-line) technique 
                 technique 
                 technique 
               
               
                   
               
               
                 Number of time 
                 4N 
                 14N 
                 8N 
               
               
                 intervals for 128 
               
               
                 gray 
               
               
                 Supply voltage 
                 Low 
                 High 
                 Intermediate 
               
               
                 Number of voltages 
                  3 
                 15 
                 7 
               
               
                 in scanning 
                   
                 (22 for IAPT) 
                 (10 for IAPT) 
               
               
                 Number of voltages 
                 254 
                 14 
                 21 (21 for 
               
               
                 in the data 
                   
                 (22 for IAPT) 
                 IAPT also) 
               
               
                 waveforms 
               
               
                   
               
               
                 * IAPT: Improved Alt and Pleshko Technique