Patent Publication Number: US-2019179446-A1

Title: Hover sensing with multi-phase self-capacitance method

Description:
RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application No. 62/598,347, filed on Dec. 13, 2017, which is incorporated by reference herein in its entirety. 
    
    
     TECHNICAL FIELD 
     This disclosure relates to the field of capacitance sensing and, in particular, to the sensing of hover inputs at capacitive touch sensing surfaces. 
     BACKGROUND 
     Computing devices, such as notebook computers, personal data assistants (PDAs), kiosks, and mobile handsets, have user interface devices, which are also known as human interface devices (HID). One type of user interface device is a touch-sensor pad (also commonly referred to as a touchpad), which can be used to emulate the function of a personal computer (PC) mouse. A touch-sensor pad replicates mouse X/Y movement by using two defined axes which contain a collection of sensor electrodes that detect the position of one or more objects, such as a finger or stylus. The touch-sensor pad provides a user interface device for performing such functions as positioning a pointer, or selecting an item on a display. Another type of user interface device is a touch screen. Touch screens, also known as touchscreens, touch windows, touch panels, or touchscreen panels, are transparent display overlays that allow a display to be used as an input device, removing the keyboard and/or the mouse as the primary input device for interacting with the display&#39;s content. Other user interface devices include buttons, sliders, etc., which can be used to detect touches, taps, drags, and other gestures. 
     Capacitance sensing systems are increasingly used for implementing these and other types of user interface devices, and function by sensing electrical signals generated on electrodes that reflect changes in capacitance. Such changes in capacitance can indicate a touch event or the presence of a conductive object, such as a finger, near the electrodes. The capacitance changes of the sensing electrodes can then be measured by an electrical circuit that converts the capacitances measured from the capacitive sense elements into digital values to be interpreted by a host device. However, the accuracy of existing capacitance measurement circuits can be degraded by noise and fluctuations affecting the drive voltages, current source outputs, switching frequencies, and other signals within the measurement circuit. Such measurement inaccuracy can result in inaccurate positioning or touch detection in a capacitance-based user interface device. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure is illustrated by way of example, and not by way of limitation, in the figures of the accompanying drawings. 
         FIG. 1  is a block diagram illustrating a capacitance sensing system, according to an embodiment. 
         FIG. 2  is a block diagram illustrating components in a processing device of a capacitance sensing system, according to an embodiment. 
         FIG. 3  illustrates circuit diagrams for stages of a multiphase sensing process, according to an embodiment. 
         FIGS. 4A and 4B  illustrate sequences of excitation signal patterns for a hover sensing process, according to an embodiment. 
         FIGS. 5A and 5B  illustrate hover inputs at different locations of a sensor array and the measured signals corresponding to the hover inputs, according to an embodiment. 
         FIGS. 6A-6C  illustrate least squares approximations of measured signal sequences by sine functions, according to an embodiment. 
         FIG. 7  illustrates displacement of a sine function, according to an embodiment. 
         FIG. 8  illustrates division of a sine function period into quarters for referencing a lookup table, according to an embodiment. 
         FIGS. 9A and 9B  illustrate approximation of a subset of signals surrounding a maximum measured value with a parabola, according to an embodiment. 
         FIGS. 10A-11  illustrate approximations of different measured signal sequences by reciprocal quadratic curves, according to an embodiment. 
         FIG. 12  illustrates the calculation of a hover input position based on differentials calculated from a measured signal sequence, according to an embodiment. 
         FIGS. 13A-13C  illustrate sequences of excitation signal patterns for which one sensor electrode is measured for each pattern, according to an embodiment. 
         FIGS. 14A-14C  illustrate sequences of excitation signal patterns that are applied to both row and column sensor electrodes, according to an embodiment. 
         FIG. 15  illustrates measured signal sequences resulting from excitation of both row and column sensor electrodes using different excitation patterns, according to an embodiment. 
         FIG. 16  illustrates a process for detecting the presence and location of a hover input, according to an embodiment. 
         FIG. 17  illustrates a process for detecting the presence and location of a hover input, according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     The following description sets forth numerous specific details such as examples of specific systems, components, methods, and so forth, in order to provide a good understanding of several embodiments of the claimed subject matter. It will be apparent to one skilled in the art, however, that at least some embodiments may be practiced without these specific details. In other instances, well-known components or methods are not described in detail or are presented in a simple block diagram format in order to avoid unnecessarily obscuring the claimed subject matter. Thus, the specific details set forth are merely exemplary. Particular implementations may vary from these exemplary details and still be contemplated to be within the spirit and scope of the claimed subject matter. 
     For a computing device that is capable of accepting input via a capacitive sensing surface, such as a touchscreen or trackpad, a hover input is caused by an object (such as a user&#39;s finger or a stylus) that is near but not touching the sensing surface. For example, a hover input can be detected when a user&#39;s finger is held above the touch sensing surface. The computing device determines the location on the sensing surface that is nearest to the finger as the input location. A computing device that is capable of hover sensing can detect inputs from a user when the user is unable or unwilling to physically touch the device, such as when the user is wearing gloves, the user&#39;s fingers are wet or dirty, the device is inside a protective case, etc. A hover input at a particular location on the sensing surface can be distinguished from a touch input at the same location; accordingly, hover sensing can be used to perform additional types of actions in the computing device (e.g., previewing documents, opening submenus, etc.) that are not initiated by touch inputs. 
     In one embodiment, a computing device performing hover sensing is capable of detecting a finger at a range of at least 35 millimeters from the sensing surface. As an example, a typical capacitance between a finger and an indium tin oxide (ITO) sensor electrode can be approximately 15-30 femtofarads (fF) at this distance, depending on the finger and electrode characteristics and relative positions. As a result, hover sensing is effectively performed using low-noise and high-resolution charge-to-digital conversion circuits that are able to distinguish tiny capacitance changes in the presence of relatively large parasitic capacitances. Temperature drift in the large parasitic capacitances in combination with internal and external noise sources (e.g. liquid crystal display (LCD) noise) substantially complicates the detection of hover inputs, and can result in false inputs, failure to detect real hover inputs, position jitter, and other operational artefacts. In addition, emission limits imposed in some industries (e.g., in automotive or aeronautics applications) constrain the sizes of sensing surfaces and the magnitudes of drive signals that are applied to sensor electrodes for performing hover sensing operations. 
     In one embodiment, a computing device detects hover inputs at a sensing surface by applying a sequence of zero-sum patterns of excitation signals to an array of sensor electrodes in the sensing surface. The device applies complementary transmit (TX) excitation signals (i.e., a positive phase and a negative phase of an excitation signal) to the sensor electrodes, with positive and negative phase signals applied to equal numbers of electrodes so that emissions induced by the excitation signals are cancelled at a sufficient distance from the sensor electrodes. 
     In one embodiment, an initial TX signal pattern includes positive phase excitation signals applied to a subset of contiguous sensor electrodes while negative phase excitation signals are applied to a complementary subset of the sensor electrodes. Through application of the subsequent TX signal patterns in the sequence, the initial excitation signal pattern is rotated in circular fashion through the sensor electrodes. The application of the sequence of TX excitation signal patterns causes a sequence of receive (RX) signals to be generated at the sensor electrodes, which is affected by hover inputs near the sensor electrodes. 
     A least squares approximation is performed to determine parameters for a predetermined function for approximating the measured sequence of RX signals. In one embodiment, a hover input near the sensor electrodes results in a sequence of RX signals that resembles a sine function; accordingly, the least squares approximation determines the parameters of the sine function for approximating the sequence of RX signals. The position of the hover input relative to the sensor electrodes is determined by calculating a displacement parameter for the sine function that approximates the measured RX signal sequence. 
     A correlation coefficient is calculated that describes the degree of correlation between the measured RX signal sequence and the sine function approximating the RX signal sequence. Actual hover inputs caused by a finger or other conductive object near the sensor electrodes cause an RX signal sequence to be measured that resembles a sine function, while signals caused by temperature changes or other unintentional inputs do not. Thus, the correlation coefficient is compared to a correlation threshold, and a hover input is detected when the correlation coefficient exceeds the threshold (i.e., indicating a sufficiently high degree of similarity between the measured RX signal sequence and the approximating sine function). 
     The above approach for detecting a hover input results in a substantial increase in the measurable capacitance signal resulting from the hover input. Signal to noise ratio is also improved. Accordingly, hover inputs are detectable at greater distances from the sensing surface, and with greater immunity to noise (e.g., from a nearby LCD panel). In addition, measurement inaccuracies due to changes in parasitic capacitances of the sensing surface (e.g., due to temperature drift) are substantially reduced, as these changes are compensated by the zero-sum TX excitation signal patterns. Lower electromagnetic emissions are also achieved by the zero-sum TX excitation signal patterns. 
       FIG. 1  illustrates a block diagram of a capacitance sensing system  100 , according to an embodiment. The sensing system  100  includes a host device  101 , a processing device  102 , and a sensor array  103 . In one embodiment, the sensor array  103  is constructed from a transparent conductive material such as indium tin oxide (ITO) and overlies a display  104 . The host device  101  controls the display  104  and updates the display  104  in response to the objects detected based on self capacitances and mutual capacitances measured from the sensor array  103  so that the display  104  and sensor array  103  function together as a touch screen. In one embodiment, the functions of the processing device  102  are integrated into a single integrated circuit with the host device  101 . 
       FIG. 2  illustrates a functional block diagram of components in the capacitance sensing system  100  that perform sensing of hover inputs, according to an embodiment. The processing device  102  is capable of measuring signals (e.g., voltage and/or current) representing self capacitances and mutual capacitances of electrodes in the capacitive sensor array  103 . The sensor array  103  includes a set of one or more TX sensor electrodes and a set of one or more RX sensor electrodes. Each of the TX sensor electrodes is connected to the processing device  102  via one of the TX lines  211 , while each of the RX sensor electrodes is connected to the processing device  102  via one of the RX lines  212 . For mutual capacitance sensing, the TX lines  211  are used to apply excitation signals to the TX sensor electrodes, while the RX lines  212  are used to receive the induced signals from the RX sensor electrodes. In other operational modes (e.g., multiphase self capacitance sensing), either or both of the TX lines  211  and RX lines  212  can also be configured to transmit signals to and/or receive signals from the sensor electrodes when measuring capacitance values. The processing device  102  performs processing on the measured capacitance values to distinguish between intentional touches and non-intentional touches (e.g., liquid on the sensor surface) and to determine locations of the intentional touches. The processing device  102  also detects hover inputs based on RX signal sequences generated from the sensor array  103 . The host device  101  responds to a detected hover input by executing a subroutine associated with the hover input. In one embodiment, the subroutine executed is additionally determined based on the location of the hover input. In one embodiment, the host device  101  executes different subroutines for a hover input and an intentional touch input at the same location. 
     The processing device  102  reports the locations of intentional touches and hover inputs to the host device  101 . The host device  101  executes one or more functions based on the reported input locations. In one embodiment, the processing device  102  reports the measured signals representing changes in self capacitances and/or mutual capacitances to the host device  101 , and further processing of the measured values is performed in the host device  101 . The processing device  102  may report the raw measurement values to the host  101 , or perform preliminary processing on the raw measurement values (e.g., baseline correction, or filtering) prior to reporting them to the host  101 . 
     The processing device  102  includes a number of components for supplying excitation signals to the sensor array  103 , measuring the resulting signals (e.g., current or charge) from the sensor array, and calculating measures of the self capacitances and mutual capacitances (i.e., values representing the self capacitances and mutual capacitances) based on the measurements. The multiplexer  213  includes switching circuitry that selectively connects the different sensor electrodes to excitation signals or measurement channels. In one embodiment, the multiplexer  213  is implemented using a number of multiplexers. For example, separate multiplexers can be used for the TX lines  211  and for the RX lines  212 , or for column and row electrodes. In one embodiment, several multiplexers can be used for each axis, depending on the number of inputs, the drive and receive configurations, and the size of the multiplexers. The TX generator  215  controls the multiplexer  213  to generate a TX signal as an excitation signal that is selectively applied to the TX sensor electrodes in the array  103  via the TX lines  211 . Vtx generator  214  generates a voltage Vtx that can be selectively applied to the sensor electrodes when generating the TX excitation signal. In one embodiment, Vtx generator  214  includes an inverter so that both positive phase (+1) and negative phase (−1) signals can be generated. Multiplexer  213  can also selectively apply a ground voltage to the sensor electrodes. The specific sequences of excitation signals that are applied to the sensor array  103  over time are generated by the sequencer circuit  223 . 
     Multiplexer  213  can also selectively connect the electrodes in sensor array  103  to the charge-to-code converters  216  so that the amounts of charge generated by excitation of the electrodes can be measured. In one embodiment, the charge-to-code converters  216  integrate current over a set time period and convert the resulting measured charge to a digital code that can be used for further processing. The charge to code converters  216  include at least one RX channel  224  for receiving an RX signal from the sensor array  103 . The baseline compensation circuit  217  supplies a baseline compensation signal to the capacitance-to-code converters  216  that reduces the effect of a baseline level of the sensor array. Alternatively, the baseline compensation circuit  217  may apply the compensation signal to a shield electrode under the sensor array  103 . The compensation level can be set for the entire panel, or can be set at different levels for different excitation signal patterns depending on the parasitic capacitance variations of the panel sense electrodes. 
     The channel engine  218  receives the digital codes representing the charge measured from each electrode. For some sensing methods (e.g., multiphase self-capacitance sensing with nonzero-sum TX excitation patterns), the channel engine  218  supplies the raw digital code values to the deconvolutor module  219 . The deconvolutor  219  performs deconvolution operations on the values to generate a mutual capacitance map  220  and a self capacitance vector  221 . The mutual capacitance map  220  is represented as a matrix of values having dimensions corresponding to the number of row electrodes and column electrodes in the sensor array, so that a mutual capacitance for each intersection between one of the row electrodes and one of the column electrodes is represented by an element (i.e., a measure of the mutual capacitance) in the matrix. The self capacitance vector includes an element for each TX electrode (e.g., row electrode), representing the self capacitance of the TX electrode (i.e., a measure of the self capacitance). 
     The mutual capacitances  220  and self capacitances  221  are transmitted to the post processing and communication block  222 . The post processing block  222  performs additional calculations to detect the presence of any intentional touches and to determine the locations of any such touches based on the capacitances  220 - 221 . The touch locations are transmitted from block  222  to the host device  101 . 
     In one embodiment, when the processing device  102  performs hover sensing, the generated sequence of RX signals is received by the channel engine  218  and stored in the self-capacitance vector  221 , bypassing the deconvolutor  219  which is used in other modes of operation. The post processing block  222  determines parameters of a sine function that approximates the RX signal sequence based on a least squares approximation. In one embodiment, values for the sine function are calculated using the lookup table  225 . The post processing block  222  calculates a measure of correlation (e.g., a correlation coefficient) indicating the degree of correlation between the RX signal sequence and the approximating sine function, then detects the presence of an object (e.g., the user&#39;s finger) proximate to the sensor array  103  based on the correlation coefficient. In particular, if the correlation coefficient exceeds a predetermined correlation threshold (i.e., the RX signal sequence is sufficiently similar to the approximating sine function), the presence of the object is detected. 
       FIG. 3  illustrates two stages in the operation of a capacitance sensing circuit  300  that performs multiphase self capacitance sensing, according to an embodiment. The capacitance sensing circuit  300  measures self capacitances for two electrodes RX- 1  and RX-N, representing the first RX electrode and the Nth RX electrode in the sensor array  103 . The capacitances Cs 1  and CsN represent the self capacitances of electrodes RX- 1  and RX-N, respectively. 
     During the precharging stage, the sensing electrodes RX- 1  and RX-N are isolated from the sensing channel  301  by opening switches SW 3 - 1  and SW 3 -N. The sensing electrodes RX- 1  and RX-N are accordingly isolated from each other and can be precharged to different voltages. In the sensor array  103  as a whole, some electrodes can be precharged to Vtx while others are precharged to ground. As illustrated, SW 2 - 1  is closed while SW 1 - 1  is open so that electrode RX- 1  is connected to ground and RX-N is connected to Vtx. The self capacitances Cs 1  and CsN are thus precharged to ground and Vtx, respectively. 
     During the sensing stage, the switches SW 2 - 1  and SW 2 -N are opened to disconnect the sensor electrodes RX- 1  and RX-N from their respective precharging voltages. The sensor electrodes RX- 1  and RX-N are connected to the sensing channel  301  by closing the switches SW 3 - 1  and SW 3 -N. The voltage Vref is maintained at each of the electrodes RX- 1  and RX-N. Charge Q 1  flows into the self capacitance Cs 1  of electrode RX- 1 , since RX- 1  was precharged to a lower voltage than Vref. Charge QN flows out of the self capacitance CsN since RX-N was precharged to a higher voltage Vtx than Vref. When this process is performed for all of the RX electrodes (RX- 1 , RX- 2  . . . RX-N) in the sensor array  103 , the sensing channel  301  receives charge Qin according to Equation 1 below: 
         in= 1+ 2+ . . . +   N;   (Equation 1)
 
     In Equation 1, the values (Q 1 , Q 2 , . . . QN) represent the charge that is stored in the self capacitances (Cs 1 , Cs 2 , . . . CsN), respectively, after the precharging stage. Equation 1 can be rewritten as shown in Equation 2 below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Qin 
                         = 
                         
                           
                             
                               S 
                                
                               
                                   
                               
                                
                               
                                 1 
                                 · 
                                 Utx 
                                 · 
                                 Cs 
                               
                                
                               
                                   
                               
                                
                               1 
                             
                             + 
                             
                               S 
                                
                               
                                   
                               
                                
                               
                                 2 
                                 · 
                                 Utx 
                                 · 
                                 Cs 
                               
                                
                               
                                   
                               
                                
                               2 
                             
                             + 
                             … 
                             + 
                             
                               Sn 
                               · 
                               Utx 
                               · 
                               CsN 
                             
                           
                           = 
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           
                             
                               Utx 
                               · 
                               
                                 [ 
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                    
                                   
                                       
                                   
                                    
                                   S 
                                    
                                   
                                       
                                   
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                                    
                                   … 
                                    
                                   
                                       
                                   
                                    
                                   Sn 
                                 
                                 ] 
                               
                               · 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         C 
                                          
                                         
                                             
                                         
                                          
                                         s 
                                          
                                         
                                             
                                         
                                          
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       
                                         Cs 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                             
                                         
                                          
                                         M 
                                       
                                     
                                   
                                   
                                     
                                       CsN 
                                     
                                   
                                 
                                 ] 
                               
                             
                             = 
                             
                               Utx 
                               · 
                               S 
                               · 
                               Csx 
                             
                           
                         
                         ; 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     2 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 2, (S 1 -Sn) represents the excitation sequence for a measurement cycle represented by elements of 1, −1, and 0. A value of 1 indicates that excitation in a positive direction, a value of −1 indicates excitation in a negative direction, and a value of 0 indicates that no excitation voltage is applied to the sensor electrode. Accordingly, Utx represents the change in voltage applied to the electrode from the precharge stage to the sensing stage. As indicated in Equation 2, Utx is the same for all electrodes; in alternative embodiments, Utx may differ between electrodes. 
     If the sensor is excited with N different excitation sequences serially, the excitation procedure can be represented as an excitation matrix S with values S 11 -SNN, as indicated in Equation 3 below. 
     
       
         
           
             
               
                 
                   
                     Qin 
                     = 
                     
                       
                         Utx 
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   11 
                                 
                               
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   21 
                                 
                               
                               
                                 L 
                               
                               
                                 
                                   SN 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   12 
                                 
                               
                               
                                 
                                   S 
                                    
                                   
                                       
                                   
                                    
                                   22 
                                 
                               
                               
                                 L 
                               
                               
                                 
                                   SN 
                                    
                                   
                                       
                                   
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                                   2 
                                 
                               
                             
                             
                               
                                 
                                   
                                       
                                   
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                                    
                                   
                                       
                                   
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                                   2 
                                    
                                   N 
                                 
                               
                               
                                 L 
                               
                               
                                 SNN 
                               
                             
                           
                           ] 
                         
                         · 
                         
                           [ 
                           
                             
                               
                                 
                                   Cs 
                                    
                                   
                                       
                                   
                                    
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   Cs 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   
                                       
                                   
                                    
                                   M 
                                 
                               
                             
                             
                               
                                 CsN 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         Utx 
                         · 
                         S 
                         · 
                         Csx 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     In the excitation matrix S, elements in the same row (e.g., S 11 , S 21 , . . . SN 1 ) are applied to different electrodes at the same time, while elements in the same column (e.g., S 11 , S 12 , . . . S 1 N) are applied to the same electrode at different times. If the excitation matrix S has an inverse form S −1 , the sensed self capacitances can be determined by performing a deconvolution of the measured charge values Qin as follows in Equation 4 below, where D is the deconvolution matrix: 
     
       
         
           
             
               
                 
                   
                     Csx 
                     = 
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     D 
                                      
                                     
                                         
                                     
                                      
                                     11 
                                   
                                 
                                 
                                   
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                                      
                                     
                                         
                                     
                                      
                                     21 
                                   
                                 
                                 
                                   L 
                                 
                                 
                                   
                                     DN 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     D 
                                      
                                     
                                         
                                     
                                      
                                     12 
                                   
                                 
                                 
                                   
                                     D 
                                      
                                     
                                         
                                     
                                      
                                     22 
                                   
                                 
                                 
                                   L 
                                 
                                 
                                   
                                     DN 
                                      
                                     
                                         
                                     
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                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                         
                                     
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                                      
                                     M 
                                   
                                 
                                 
                                   O 
                                 
                                 
                                   
                                     
                                         
                                     
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                                     1 
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                                     N 
                                   
                                 
                                 
                                   
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                                      
                                     
                                         
                                     
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                                     2 
                                      
                                     N 
                                   
                                 
                                 
                                   L 
                                 
                                 
                                   DNN 
                                 
                               
                             
                             ] 
                           
                           · 
                           
                             [ 
                             
                               
                                 
                                   
                                     Qin 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     Qin 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                         
                                     
                                      
                                     M 
                                   
                                 
                               
                               
                                 
                                   QinN 
                                 
                               
                             
                             ] 
                           
                         
                         Utx 
                       
                       = 
                       
                         
                           D 
                           · 
                           Qin 
                         
                         Utx 
                       
                     
                   
                   ; 
                   
                     D 
                     = 
                     
                       S 
                       
                         - 
                         1 
                       
                     
                   
                   ; 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     Excitation of the sensor electrodes with a combination of opposite-phase signals reduces emissions of the sensor as compared to the excitation of all of the row or column sensor electrodes in-phase. The emission depends on the sum of the excitation sequence elements (e.g., S 11 -SNN). For example, if the sum of elements is equal to 1, the emission observable at a distance is similar to the emission generated by excitation of a single electrode. In addition, charge measurements for multiple sensor electrodes are included in the deconvolution calculation, which results in an averaging effect after the deconvolution that, in turn, renders the sensing result less sensitive to noise injected into the sensor. In one embodiment, the approaches for sensing hover inputs as described herein can be used with multiple charge-to-code sensing circuits (e.g. that use different sensing channels based on active integrators, current conveyors, etc.) that support multiphase self or mutual capacitance sensing approaches. 
       FIG. 4A  illustrates a sequence of TX signal patterns that are applied to sensor electrodes in a sensor array during hover sensing, according to an embodiment. As illustrated in  FIG. 4A , each of the sensor electrodes is referenced by its index  410 . In one embodiment, the electrodes  1 - 24  are row electrodes (or alternatively, column electrodes) in the sensor array  103 , with the electrode indexes  410  corresponding to the spatial order of the electrodes in the array  103 . The TX generator  215  applies each of the TX signal patterns  420  in sequential order to the sensor electrodes in the sensor array  103 . 
     When each signal pattern  420 ( 1 )- 420 ( 24 ) is applied, the individual TX excitation signals in the pattern are applied by the TX generator  215  to respective sensor electrodes  1 - 24  at the same time, with ‘+1’ and ‘−1’ indicating positive and negative phases of the excitation signal, respectively. For example, the TX generator  215  applies pattern  420 ( 1 ) to the sensor electrodes  1 - 24  by applying a positive phase excitation signal to electrodes  1 - 12  while applying a negative phase excitation signal to electrodes  13 - 24 . In one embodiment, the positive and negative phase signals are complementary signals, where the negative phase excitation signal is the inverse of the positive phase excitation signal. 
     Each of the signal patterns  420  is a zero-sum excitation signal pattern; that is, the magnitude of the sum of the excitation signals in the pattern ideally approaches zero. For example, excitation signal pattern  420 ( 1 ) includes twelve positive phase excitation signals (applied to electrodes  1 - 12 ) and twelve negative phase excitation signals (applied to electrodes  13 - 24 ). Thus, each of the positive phase excitation signals is summed with a corresponding complementary negative phase excitation signal so that the sum of all the signals in the pattern is nominally zero. In practice, the sum of the excitation signals in the pattern can deviate from zero due to manufacturing tolerances, environmental conditions, etc. 
     In one embodiment, the magnitude of the sum of the excitation signals in a pattern is less than the magnitude of any one of the excitation signals in the pattern. Accordingly, the entire zero-sum excitation signal pattern generates less electromagnetic emission than singly applying one of the excitation signals to its corresponding sensor electrode. In one embodiment, the sensor array includes an even number of sensor electrodes, and the positive and negative phase excitation signals are each applied to half of the sensor electrodes so that an equal number of electrodes are excited by the positive phase excitation signal as with the negative phase excitation signal, thus achieving a balanced excitation scheme. 
     For each of the excitation signal patterns  420 , the TX generator  215  applies all of the positive phase excitation signals to a contiguous subset of sensor electrodes, which are contiguous in a circular sequence of the set of sensor electrodes. The circular sequence of the electrodes proceeds in order from index  1  to index  24 , then returns to index  1  after index  24  and repeats. Accordingly, in signal pattern  420 ( 24 ), the electrodes  24  and  1 - 11  to which the positive phase excitation signal are applied are contiguous in the circular sequence. All of the negative phase excitation signals are also applied to a contiguous subset of electrodes that are contiguous in the circular sequence. 
     In one embodiment, the same excitation signal is applied to a subset of sensor electrodes by physically connecting the electrodes together by a conductive path (e.g., via switching circuitry) and applying the excitation signal to the connected electrodes. In an alternative embodiment, each of the sense electrodes is separately driven with the same excitation signal. In one embodiment, a majority of the electrodes (i.e., at least half of the electrodes) in the set of sensor electrodes are included in subsets each including at least three or more contiguous sensor electrodes to which the same excitation signal (e.g., the positive phase excitation signal or the negative phase excitation signal) is applied for each signal pattern. 
     Circular rotation is used to generate successive signal patterns in the sequence  400 . In sequence  400 , twelve positive phase excitation signals and twelve negative phase excitation signals are circularly rotated through the sensor electrodes to form 24 different excitation signal patterns  420 . For each signal pattern  420  in the sequence  400 , the sequencer circuit  223  generates the next subsequent excitation signal pattern in the sequence by circularly rotating the pattern by an increment of one; that is, each sensor electrode in the next subsequent pattern receives the excitation signal that was applied to a preceding (i.e., lower indexed) sensor electrode in the circular sequence. For example, in pattern  420 ( 2 ), a negative phase excitation signal is applied to electrode  1  since the negative phase excitation signal was previously applied to electrode  24  (which precedes electrode  1  in the circular sequence) in the prior excitation signal pattern  420 ( 1 ). In pattern  420 ( 2 ), a positive phase excitation signal is applied to electrode  13  because a positive phase excitation signal was applied to electrode  12  in the preceding pattern  420 ( 1 ). 
       FIG. 4B  illustrates an alternative sequence  430  of circularly rotated TX signal patterns that can be applied to sensor electrodes in a sensor array during hover sensing, according to an embodiment. The TX generator  215  applies each of the TX signal patterns  440  in sequential order to the sensor electrodes in the sensor array  103 . In the signal patterns  440 , the sensor electrodes to which the positive (or negative) phase excitation signal is applied are divided into two groups. In pattern  440 ( 1 ), the positive phase excitation signal is applied to electrodes  1 - 6  and  13 - 18 , while the negative phase excitation signal is applied to electrodes  7 - 12  and  19 - 24 . 
     When other nonzero-sum TX excitation patterns sequences are used in multiphase capacitive sensing, deconvolution can be used to determine the measured capacitance values from the raw data. However, the convolution matrixes for zero-sum signal patterns such as patterns  420  and  430  are singular. Therefore, instead of deconvolution, a least squares approximation calculation to match a predetermined function (e.g., a sine function) with the measured sequence of RX signals is performed to determine the presence and location of a hover input relative to the sensor electrodes. 
     In one embodiment, according to the multiphase sensing process, after the precharging phase in which the positive or negative phase excitation signal is applied to the sensor electrode, an RX self-capacitance sensing channel  224  in the charge to code converter  216  measures an RX signal for each sensor electrode during a sensing phase. In one embodiment, an RX signal is measured for each excitation signal pattern. The measured RX signals make up a sequence of RX signals that is correlated with a predetermined function, such as a sine function, to determine whether a hover input is present. 
       FIGS. 5A and 5B  illustrate RX signal sequences resulting from hover inputs near different locations on a sensor array  103 , according to an embodiment. As illustrated in  FIG. 5A , the sensor array  103  has 47 sensor electrodes. Thus, a circularly rotated sequence of excitation signal patterns includes 47 different patterns that are applied to the sensor electrodes in sequence. In  FIG. 5B , the vertical axis represents a signal level of the RX signal as sensed by the RX sensing channel  224 , and the horizontal axis represents the pattern number of the excitation signal pattern that induced the RX signal. 
       FIG. 5B  illustrates RX signal sequences  511 ,  512 , and  513  resulting from respective hover inputs  501 ,  502 , and  503  at different locations of the sensor array  103 . Hover input  501  is located at an edge of the sensor array  103 , while hover inputs  502  and  503  are progressively nearer to the center of the array  103 . Each of the measured RX signal sequences  511 - 513  resembles a sine function; thus, a sine function is used to approximate each of the RX signal sequences  511 - 513  by least squares approximation. 
     A sine function is expressed generally in terms of parameters a, b, c, and d in Equation 5: 
         y=a+b ·sin( c·i+d )  (Equation 5)
 
     In Equation 5, y is the magnitude of the sine function, which corresponds to the magnitude of the RX signal, and i is the sensor electrode index. The parameter d reflects the phase shift of the sine function, and is directly related to the hover position at the sensor array  103 .  FIGS. 6A, 6B, and 6C  illustrate approximations of RX signal sequences  513 ,  512 , and  511 , respectively, by sine functions having different parameters, according to an embodiment. In  FIG. 6A , the RX signal sequence  513  is approximated by a sine function  601  (illustrated as a continuous line). In  FIG. 6B , the RX signal sequence  512  is approximated by a sine function  602 . In  FIG. 6C , the RX signal sequence  511  is approximated by a sine function  603 . Each of the approximating sine functions  601 - 603  is defined using a different set of parameters a, b, c, and d, and approximates the RX signal sequence with a high correlation coefficient. 
     Thus, the parameters a, b, c, and d are calculated in order to determine a sine function that best approximates a measured RX signal sequence. One method for calculating the parameters c and d is based on minimizing Equation 6 below, according to the least squares method: 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       
                         [ 
                         
                           a 
                           + 
                           
                             b 
                             · 
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     c 
                                     · 
                                     i 
                                   
                                   + 
                                   d 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             S 
                             i 
                           
                         
                         ] 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     However, using trigonometric theory, c can be more easily determined as follows, in Equation 7: 
     
       
         
           
             
               
                 
                   c 
                   = 
                   
                     
                       2 
                       · 
                       π 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 7, N represents the number of sensor electrodes, and 2×π is the period of the sine function. In one embodiment, the parameter d defines an amount by which the sine function is shifted along the x axis, as illustrated in  FIG. 7 . In  FIG. 7 , sine function  701  has a value of d that is equal to zero. Sine function  702  has a non-zero value of d; accordingly, features of the sine function  702  are shifted by the amount d relative to the original sine function  701 . The parameter d is calculated according to Equation 8: 
     
       
         
           
             
               
                 
                   d 
                   = 
                   
                     - 
                     
                       
                         2 
                         · 
                         π 
                         · 
                         
                           ( 
                           
                             idxMax 
                             - 
                             
                               
                                 N 
                                 - 
                                 1 
                               
                               4 
                             
                           
                           ) 
                         
                       
                       
                         N 
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     8 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 8, idxMax represents the index of the sensor electrode from which the highest magnitude signal is measured, and N represents the number of sensor electrodes. The parameter d represents the displacement of the hover input (i.e., the position of the object relative to the sensor electrodes in the array  103 ). 
     Based on Equations 7 and 8, the argument arg i  of the sine function is calculated as follows in Equations 9: 
     
       
         
           
             
               
                 
                   
                     
                       arg 
                       i 
                     
                     = 
                     
                       
                         
                           c 
                           · 
                           i 
                         
                         + 
                         d 
                       
                       = 
                       
                         
                           
                             
                               2 
                               · 
                               π 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           · 
                           i 
                         
                         - 
                         
                           
                             2 
                             · 
                             π 
                             · 
                             
                               ( 
                               
                                 idxMax 
                                 - 
                                 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   4 
                                 
                               
                               ) 
                             
                           
                           
                             N 
                             - 
                             1 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     
                       Or 
                        
                       
                         : 
                       
                        
                       
                           
                       
                        
                       
                         arg 
                         i 
                       
                     
                     = 
                     
                       
                         π 
                         · 
                         
                           [ 
                           
                             
                               4 
                               · 
                               
                                 ( 
                                 
                                   i 
                                   - 
                                   idxMax 
                                 
                                 ) 
                               
                             
                             + 
                             N 
                             - 
                             1 
                           
                           ] 
                         
                       
                       
                         2 
                         · 
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     9 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation 10 below follows from Equations 5 and 9: 
     
       
         
           
             
               
                 
                   y 
                   = 
                   
                     
                       a 
                       + 
                       
                         b 
                         · 
                         
                           sin 
                            
                           
                             ( 
                             
                               arg 
                               i 
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       a 
                       + 
                       
                         b 
                         · 
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 π 
                                 · 
                                 
                                   [ 
                                   
                                     
                                       4 
                                       · 
                                       
                                         ( 
                                         
                                           i 
                                           - 
                                           idxMax 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     N 
                                     - 
                                     1 
                                   
                                   ] 
                                 
                               
                               
                                 2 
                                 · 
                                 
                                   ( 
                                   
                                     N 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     Parameters a and b are calculated by minimizing Q in Equation 11 as follows: 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             a 
                             + 
                             
                               b 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     arg 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               S 
                               i 
                             
                           
                           ] 
                         
                         2 
                       
                     
                     ⇒ 
                     min 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 11, S i  is the measured response signal caused by the hover input from the sensor electrode having index i. Partial differentiation yields Equations 12 below: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ∂ 
                               
                               
                                 ∂ 
                                 a 
                               
                             
                             = 
                             
                               
                                 2 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     [ 
                                     
                                       a 
                                       + 
                                       
                                         b 
                                         · 
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               arg 
                                               i 
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                       
                                         S 
                                         i 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               = 
                               0 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∂ 
                             
                             
                               ∂ 
                               b 
                             
                           
                           = 
                           
                             
                               2 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     [ 
                                     
                                       a 
                                       + 
                                       
                                         b 
                                         · 
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               arg 
                                               i 
                                             
                                             ) 
                                           
                                         
                                       
                                       - 
                                       
                                         S 
                                         i 
                                       
                                     
                                     ] 
                                   
                                   · 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         arg 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     12 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations 12 can be expressed as shown in Equations 13: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 N 
                                 · 
                                 a 
                               
                               + 
                               
                                 b 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       0 
                                     
                                     
                                       N 
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         arg 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             = 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 S 
                                 i 
                               
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               a 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       arg 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               b 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     [ 
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           arg 
                                           i 
                                         
                                         ) 
                                       
                                     
                                     ] 
                                   
                                   2 
                                 
                               
                             
                           
                           = 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 S 
                                 i 
                               
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     arg 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     13 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations 13 can be simplified by denoting variables sumSin, sumS, sumSinSin, and sumSsin as shown below in Equations 14: 
     
       
         
           
             
               
                 
                   
                     sumSin 
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             arg 
                             i 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     sumS 
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         S 
                         i 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     sumSinSin 
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             sin 
                              
                             
                               ( 
                               
                                 arg 
                                 i 
                               
                               ) 
                             
                           
                           ] 
                         
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     sumSin 
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           S 
                           i 
                         
                         · 
                         
                           sin 
                            
                           
                             ( 
                             
                               arg 
                               i 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations 13 can thus be expressed as shown in Equations 15: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 N 
                                 · 
                                 a 
                               
                               + 
                               
                                 b 
                                 · 
                                 sumSin 
                               
                             
                             = 
                             sumS 
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               a 
                               · 
                               sumSin 
                             
                             + 
                             
                               b 
                               · 
                               sumSinSin 
                             
                           
                           = 
                           
                             sumS 
                              
                             
                                 
                             
                              
                             sin 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     15 
                   
                   ) 
                 
               
             
           
         
       
     
     Solving the first equation of Equations 15 for parameter a yields Equation 16: 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       sumS 
                       - 
                       
                         b 
                         · 
                         sumSin 
                       
                     
                     N 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     16 
                   
                   ) 
                 
               
             
           
         
       
     
     Substituting Equation 16 into the second equation of Equations 15 yields Equation 17 below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           sumS 
                           - 
                           
                             b 
                             · 
                             sumSin 
                           
                         
                         N 
                       
                       · 
                       sumSin 
                     
                     + 
                     
                       b 
                       · 
                       sumSInSin 
                     
                   
                   = 
                   
                     sumS 
                      
                     
                         
                     
                      
                     sin 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     17 
                   
                   ) 
                 
               
             
           
         
       
     
     Solving Equation 17 for the parameter b yields Equation 18: 
     
       
         
           
             
               
                 
                   b 
                   = 
                   
                     
                       
                         
                           N 
                            
                           
                               
                           
                           · 
                           sumS 
                         
                          
                         
                             
                         
                          
                         sin 
                       
                       - 
                       
                         sumS 
                         · 
                         sumSin 
                       
                     
                     
                       
                         N 
                         · 
                         sumSinSin 
                       
                       - 
                       
                         sumSin 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     18 
                   
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, the equations above can be used to calculate the parameters a, b, c, and d of a sine function that approximates the measured RX signal sequence. Notably, the correlation coefficient between the RX signal sequence and the approximating sine function can be calculated without calculating parameter a. 
     The sine function is calculated in the post processing block  222 . Different approaches can be used to calculate the sine function in devices with limited computing resources. For example, the sine function can be computed using a Coordinate Rotation Digital Computer (CORDIC) calculation or alternatively, the Taylor series for the sine function, as expressed in Equation 19 below, in which arg is the input argument to the sine function: 
     
       
         
           
             
               
                 
                   
                     sin 
                      
                     
                       ( 
                       arg 
                       ) 
                     
                   
                   = 
                   
                     arg 
                     - 
                     
                       
                         arg 
                         3 
                       
                       
                         3 
                         ! 
                       
                     
                     + 
                     
                       
                         arg 
                         5 
                       
                       
                         5 
                         ! 
                       
                     
                     - 
                     
                       
                         arg 
                         7 
                       
                       
                         7 
                         ! 
                       
                     
                     + 
                     … 
                     + 
                     
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             n 
                             - 
                             1 
                           
                         
                         
                           
                             ( 
                             
                               
                                 2 
                                  
                                 n 
                               
                               - 
                               1 
                             
                             ) 
                           
                           ! 
                         
                       
                        
                       
                         arg 
                         
                           
                             2 
                              
                             n 
                           
                           - 
                           1 
                         
                       
                     
                     + 
                     … 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     19 
                   
                   ) 
                 
               
             
           
         
       
     
     In one embodiment, the sine function is tabulated in a lookup table (LUT)  225 , which stores a result of the sine function for each of a range of possible input arguments. Thus, calculation of the correlation coefficient in the post processing block  222  involves performing lookups in the LUT  225 . Performing a table lookup is not an iterative calculation and thus can be performed with limited computing time and resources. Furthermore, the tabulation need not consume a large amount of memory. For a sine function, data stored in the LUT  225  for a first quarter of the period can be used to construct the remainder of the period. In one embodiment, the LUT  225  occupies no more than 65 bytes of memory. 
       FIG. 8  illustrates the division of a period of a sine function into four quarters to reduce the size of the LUT  225 , according to an embodiment. In  FIG. 8 , the sine function y=sin(2 πx/256) has a period T that is equal to 256. Input arguments x that fall in the first quarter  1001  are looked up directly from the LUT  225  to determine the result of the sine function for the argument x. For input arguments x in the second quarter  1002 , x is subtracted from 128 and the result is looked up in the LUT  225 . For input arguments x in the third quarter  1003 , 128 is subtracted from x and the result is looked up in the LUT  225  and negated. For input arguments x in the fourth quarter  1004 , x is subtracted from 256 and the result is looked up in the LUT  225  and negated. 
     The input argument x is also limited to the range between 0 and 256, since the period of the sine function is T=256. In one embodiment, if x is greater than 256, a modulo operation (e.g., x mod 256) is performed to calculate an input argument within the acceptable range for lookup. 
     The input argument arg i  for electrode index i is expressed in terms of i and idxMax in Equation 9 above. Equations 20 show the conversion of Equation 9 from a sine function with period 2π to the sine function with period T=256, by multiplying by the factor 256/2π. 
     
       
         
           
             
               
                 
                   
                     
                       arg 
                       i 
                     
                     = 
                     
                       
                         256 
                         · 
                         π 
                         · 
                         
                           [ 
                           
                             
                               4 
                               · 
                               
                                 ( 
                                 
                                   i 
                                   - 
                                   idxMax 
                                 
                                 ) 
                               
                             
                             + 
                             N 
                             - 
                             1 
                           
                           ] 
                         
                       
                       
                         2 
                         · 
                         π 
                         · 
                         2 
                         · 
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       Or 
                        
                       
                         : 
                       
                        
                       
                           
                       
                        
                       
                         arg 
                         i 
                       
                     
                     = 
                     
                       
                         
                           256 
                           · 
                           
                             ( 
                             
                               i 
                               - 
                               idxMax 
                             
                             ) 
                           
                         
                         
                           N 
                           - 
                           1 
                         
                       
                       + 
                       64 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     20 
                   
                   ) 
                 
               
             
           
         
       
     
     Once the correction of the period has been performed as shown in Equations 20, the resulting input argument arg i  can be looked up in the LUT  225 . 
     The post processing block  222  performs the sine function calculation when calculating the correlation coefficient between the measured sequence of RX signals and its approximating sine function, as defined by the parameters a, b, c, and d. To calculate the correlation coefficient, the average values avSigOriginal and avSigAppr for the measured RX signal sequence and the approximating sine function, respectively, are calculated as shown in Equations 21 and 22 below. 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     avSigOriginal 
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           S 
                           i 
                         
                       
                       N 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     21 
                   
                   ) 
                 
               
             
             
               
                 
                   avSigAppr 
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           [ 
                           
                             a 
                             + 
                             
                               b 
                               · 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     arg 
                                     i 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                       N 
                     
                     = 
                     
                       a 
                       + 
                       
                         
                           b 
                           · 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   arg 
                                   i 
                                 
                                 ) 
                               
                             
                           
                         
                         N 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     22 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation 23 is true for the entire period of the sine function. 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                         
                     
                      
                     
                       sin 
                        
                       
                         ( 
                         
                           arg 
                           i 
                         
                         ) 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     23 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, Equation 22 can be reduced to show that avSigAppr is equal to a. 
     The squared correlation coefficient is calculated as shown in Equation 24. 
     
       
         
           
             
               
                 
                   corrCoeff 
                   = 
                   
                     
                       
                         
                           
                             256 
                             · 
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         S 
                                         i 
                                       
                                       - 
                                       avSigOriginal 
                                     
                                     ) 
                                   
                                   · 
                                   
                                     ( 
                                     
                                       
                                         Sappr 
                                         i 
                                       
                                       - 
                                       avSigAppr 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     
                                       S 
                                       i 
                                     
                                     - 
                                     avSigOriginal 
                                   
                                   ) 
                                 
                                 2 
                               
                               · 
                             
                           
                         
                       
                       
                         
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 ( 
                                 
                                   
                                     Sappr 
                                     i 
                                   
                                   - 
                                   avSigAppr 
                                 
                                 ) 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     24 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation 24 is expressed more simply by denoting deltaAppr i  and deltaOriginal i  as follows in Equation 25: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             deltaAppr 
                             i 
                           
                           = 
                           
                             
                               
                                 Sappr 
                                 i 
                               
                               - 
                               avSigAppr 
                             
                             = 
                             
                               
                                 a 
                                 + 
                                 
                                   b 
                                   · 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       
                                         arg 
                                         i 
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 a 
                               
                               = 
                               
                                 b 
                                 · 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     
                                       arg 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               deltaOriginal 
                               i 
                             
                             = 
                             
                               
                                 S 
                                 i 
                               
                               - 
                               avSigOriginal 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     25 
                   
                   ) 
                 
               
             
           
         
       
     
     As shown in Equations 25, the correlation coefficient can be calculated without calculating the parameter a. Combining Equation 24 with Equations 25 yields an expression for calculating the correlation coefficient, as shown in Equation 26. 
     
       
         
           
             
               
                 
                   corrCoeff 
                   = 
                   
                     
                       256 
                       · 
                       
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 deltaOriginal 
                                 i 
                               
                               · 
                               
                                 deltaAppr 
                                 i 
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           deltaOriginal 
                           i 
                           2 
                         
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             deltaAppr 
                             i 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     26 
                   
                   ) 
                 
               
             
           
         
       
     
     The approximating sine curve is adjusted by moving the coordinate of the largest signal to the left and to the right from the index of the sensor with the largest signal (i.e., idxMax) until the correlation coefficient value corrCoeff, as calculated according to Equation 26, is maximized. 
     In one embodiment, the post processing block  222  detects a hover input due to the presence of an object near the sensor electrodes based on comparing the maximized correlation coefficient with a predetermined correlation threshold. If the correlation coefficient exceeds the correlation threshold, then the measured RX signal sequence is sufficiently close to the approximating sine function and the post processing block  222  detects a hover input. The correlation threshold is a tunable parameter in the range between 0 and 255. In one embodiment, the correlation threshold is 75, so that a hover input is detected if the correlation coefficient is greater than or equal to 75. 
     In one embodiment, a TX excitation signal pattern that does not have a zero sum can be similarly analyzed using a least squares approximation instead of deconvolution. In one embodiment, a parabola is used to approximate the measured sequence of RX signals. According to this approach, a vertex is calculated for the approximating parabola function based on a subset of the measured RX signals surrounding a maximum RX signal value.  FIG. 9  illustrates a portion of an RX signal sequence  900  in which the maximum signal  901  has a value of S i , according to an embodiment. The subset of measured RX signals surrounding the maximum 901 include signals S i−3  to S i+3 . With reference to Equation 27 below, the approximating parabola function is defined by a quadratic curve, with Sappr i−j  representing the value of the parabola function corresponding to a sensor electrode that is j electrodes away from the electrode i at which the maximum signal was measured. 
       Sappr i−j   =a+b·j+c·j   2   (Equation 27)
 
     The extremum of the parabola is calculated by setting the derivative of the quadratic function to zero, as shown in Equations 28 below. 
     
       
         
           
             
               
                 
                   
                     y 
                     = 
                     
                       a 
                       + 
                       
                         b 
                         · 
                         
                           x 
                           0 
                         
                       
                       + 
                       
                         c 
                         · 
                         
                           x 
                           0 
                           2 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       dy 
                       
                         dx 
                         0 
                       
                     
                     = 
                     
                       
                         b 
                         + 
                         
                           2 
                           · 
                           c 
                           · 
                           
                             x 
                             0 
                           
                         
                       
                       = 
                       0 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       x 
                       0 
                     
                     = 
                     
                       - 
                       
                         b 
                         
                           2 
                           · 
                           c 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     28 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation 29 shows the function to be minimized according to the least squares approximation. 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           
                             - 
                             3 
                           
                         
                         3 
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             a 
                             + 
                             
                               b 
                               · 
                               j 
                             
                             + 
                             
                               c 
                               · 
                               
                                 j 
                                 2 
                               
                             
                             - 
                             
                               S 
                               
                                 i 
                                 - 
                                 j 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                     ⇒ 
                     min 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     29 
                   
                   ) 
                 
               
             
           
         
       
     
     Partial differentiation of Equation 29 yields Equations 30. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ∂ 
                               
                               
                                 ∂ 
                                 a 
                               
                             
                             = 
                             
                               
                                 2 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       
                                         - 
                                         3 
                                       
                                     
                                     3 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       [ 
                                       
                                         a 
                                         + 
                                         
                                           b 
                                           · 
                                           j 
                                         
                                         + 
                                         
                                           c 
                                           · 
                                           
                                             j 
                                             2 
                                           
                                         
                                         - 
                                         
                                           S 
                                           
                                             i 
                                             - 
                                             j 
                                           
                                         
                                       
                                       ] 
                                     
                                     2 
                                   
                                 
                               
                               = 
                               0 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ∂ 
                               
                               
                                 ∂ 
                                 b 
                               
                             
                             = 
                             
                               
                                 2 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       
                                         - 
                                         3 
                                       
                                     
                                     3 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       [ 
                                       
                                         a 
                                         + 
                                         
                                           b 
                                           · 
                                           j 
                                         
                                         + 
                                         
                                           c 
                                           · 
                                           
                                             j 
                                             2 
                                           
                                         
                                         - 
                                         
                                           S 
                                           
                                             i 
                                             - 
                                             j 
                                           
                                         
                                       
                                       ] 
                                     
                                     · 
                                     j 
                                   
                                 
                               
                               = 
                               0 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∂ 
                             
                             
                               ∂ 
                               c 
                             
                           
                           = 
                           
                             
                               2 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     
                                       - 
                                       3 
                                     
                                   
                                   3 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     [ 
                                     
                                       a 
                                       + 
                                       
                                         b 
                                         · 
                                         j 
                                       
                                       + 
                                       
                                         c 
                                         · 
                                         
                                           j 
                                           2 
                                         
                                       
                                       - 
                                       
                                         S 
                                         
                                           i 
                                           - 
                                           j 
                                         
                                       
                                     
                                     ] 
                                   
                                   · 
                                   
                                     j 
                                     2 
                                   
                                 
                               
                             
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     30 
                   
                   ) 
                 
               
             
           
         
       
     
     Combining Equations 28 and Equations 30 yields Equation 31 below. 
     
       
         
           
             
               
                 
                   
                     x 
                     0 
                   
                   = 
                   
                     - 
                     
                       
                         3 
                         · 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               
                                 - 
                                 3 
                               
                             
                             3 
                           
                            
                           
                               
                           
                            
                           
                             
                               S 
                               
                                 i 
                                 - 
                                 j 
                               
                             
                             · 
                             j 
                           
                         
                       
                       
                         2 
                         · 
                         
                           ( 
                           
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   
                                     - 
                                     3 
                                   
                                 
                                 3 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   S 
                                   
                                     i 
                                     - 
                                     j 
                                   
                                 
                                 · 
                                 
                                   j 
                                   2 
                                 
                               
                             
                             - 
                             
                               4 
                               · 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     
                                       - 
                                       3 
                                     
                                   
                                   3 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   S 
                                   
                                     i 
                                     - 
                                     j 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     31 
                   
                   ) 
                 
               
             
           
         
       
     
       FIG. 9B  illustrates a graph of the resulting approximation curve  911  along with the measured RX signal  910 . The vertical axis represents the magnitude of the RX signal  910  and the approximation curve  911 , while the bottom axis of the graph represents the indexes of the corresponding sensor electrodes. As illustrated, seven electrodes around the highest measured value are used; however, greater noise immunity for detecting hover inputs is achieved by involving a greater number of sensor electrode measurements in the detection process. 
     In one embodiment, a reciprocal quadratic curve is used as a predetermined function for approximating a measured sequence of RX signals. The approximation using the reciprocal quadratic curve utilizes a larger number of the available RX measurements, thus increasing noise immunity in the hover detection result, since error caused by noise is averaged over a larger number of sensor electrodes. The reciprocal quadratic curve is defined in Equation 32 below. 
     
       
         
           
             
               
                 
                   
                     Sappr 
                     i 
                   
                   = 
                   
                     1 
                     
                       a 
                       + 
                       
                         b 
                         · 
                         i 
                       
                       + 
                       
                         c 
                         · 
                         
                           i 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     32 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equation 32, Sappr i  is the magnitude of the function result, i is the index of the sensor electrode for which the function result is calculated. Parameters a, b, and c correspond to a version of the curve that best approximates the measured sequence of RX signals. A correlation coefficient between the RX signal sequence and its approximating reciprocal quadratic curve is calculated according to Equation 38, derived as shown in Equations 33-38 as follows. For the reciprocal quadratic curve, a vertex x 0  is equal to −b/2c. Accordingly, Equation 32 can be expressed in terms of the vertex x 0  as shown in Equations 33. 
     
       
         
           
             
               
                 
                   
                     
                       Sappr 
                       i 
                     
                     = 
                     
                       1 
                       
                         a 
                         + 
                         
                           b 
                           · 
                           i 
                         
                         + 
                         
                           c 
                           · 
                           
                             i 
                             2 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       Sappr 
                       i 
                     
                     = 
                     
                       1 
                       
                         a 
                         + 
                         
                           c 
                           · 
                           i 
                           · 
                           
                             ( 
                             
                               i 
                               - 
                               
                                 2 
                                 · 
                                 
                                   x 
                                   0 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     33 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equations 33, Sappr i  is the magnitude of the approximating function corresponding to sensor electrode index i. The magnitude of the signal at the vertex x 0  is set to a variable S p  in Equations 34. 
     
       
         
           
             
               
                 
                   
                     
                       S 
                       p 
                     
                     = 
                     
                       1 
                       
                         a 
                         + 
                         
                           c 
                           · 
                           p 
                           · 
                           
                             ( 
                             
                               p 
                               - 
                               
                                 2 
                                 · 
                                 
                                   x 
                                   0 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     a 
                     = 
                     
                       
                         1 
                         
                           S 
                           p 
                         
                       
                       - 
                       
                         c 
                         · 
                         p 
                         · 
                         
                           ( 
                           
                             p 
                             - 
                             
                               2 
                               · 
                               
                                 x 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     34 
                   
                   ) 
                 
               
             
           
         
       
     
     In Equations 34, S p  is the value of the peak magnitude, while  p  is the index corresponding to the electrode where the peak magnitude is located. The function to be minimized, according to the least squares approximation process is expressed below in Equation 35. 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             
                               c 
                               · 
                               
                                 ( 
                                 
                                   
                                     i 
                                     · 
                                     
                                       ( 
                                       
                                         i 
                                         - 
                                         
                                           2 
                                           · 
                                           
                                             x 
                                             0 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     p 
                                     · 
                                     
                                       ( 
                                       
                                         p 
                                         - 
                                         
                                           2 
                                           · 
                                           
                                             x 
                                             0 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               1 
                               
                                 S 
                                 p 
                               
                             
                             - 
                             
                               1 
                               
                                 S 
                                 i 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                     ⇒ 
                     min 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     35 
                   
                   ) 
                 
               
             
           
         
       
     
     Partial differentiation of Equation 35 yields Equations 36 below. 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                       
                       
                         ∂ 
                         c 
                       
                     
                     = 
                     
                       
                         2 
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               [ 
                               
                                 
                                   c 
                                   · 
                                   
                                     ( 
                                     
                                       
                                         i 
                                         · 
                                         
                                           ( 
                                           
                                             i 
                                             - 
                                             
                                               2 
                                               · 
                                               
                                                 x 
                                                 0 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                       - 
                                       
                                         p 
                                         · 
                                         
                                           ( 
                                           
                                             p 
                                             - 
                                             
                                               2 
                                               · 
                                               
                                                 x 
                                                 0 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     S 
                                     p 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     S 
                                     i 
                                   
                                 
                               
                               ] 
                             
                             · 
                             
                               ( 
                               
                                 
                                   i 
                                   · 
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       
                                         2 
                                         · 
                                         
                                           x 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   p 
                                   · 
                                   
                                     ( 
                                     
                                       p 
                                       - 
                                       
                                         2 
                                         · 
                                         
                                           x 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       0 
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     c 
                     = 
                     
                         
                     
                      
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         S 
                                         p 
                                       
                                       - 
                                       
                                         S 
                                         i 
                                       
                                     
                                     ) 
                                   
                                   · 
                                 
                               
                             
                             
                               
                                 
                                   ( 
                                   
                                     
                                       i 
                                       · 
                                       
                                         ( 
                                         
                                           i 
                                           - 
                                           
                                             2 
                                             · 
                                             
                                               x 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       p 
                                       · 
                                       
                                         ( 
                                         
                                           p 
                                           - 
                                           
                                             2 
                                             · 
                                             
                                               x 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             S 
                             i 
                           
                         
                       
                       
                         
                           S 
                           p 
                         
                         · 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               ( 
                               
                                 
                                   i 
                                   · 
                                   
                                     ( 
                                     
                                       i 
                                       - 
                                       
                                         2 
                                         · 
                                         
                                           x 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   p 
                                   · 
                                   
                                     ( 
                                     
                                       p 
                                       - 
                                       
                                         2 
                                         · 
                                         
                                           x 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     36 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations 36 are simplified by denoting variables A and B as shown below in Equations 37. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           A 
                           = 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                                 
                             
                              
                             
                               
                                 
                                   ( 
                                   
                                     
                                       S 
                                       p 
                                     
                                     · 
                                     
                                       S 
                                       i 
                                     
                                   
                                   ) 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       i 
                                       · 
                                       
                                         ( 
                                         
                                           i 
                                           - 
                                           
                                             2 
                                             · 
                                             
                                               x 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       p 
                                       · 
                                       
                                         ( 
                                         
                                           p 
                                           - 
                                           
                                             2 
                                             · 
                                             
                                               x 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               
                                 S 
                                 i 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             B 
                             = 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       i 
                                       · 
                                       
                                         ( 
                                         
                                           i 
                                           - 
                                           
                                             2 
                                             · 
                                             
                                               x 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       p 
                                       · 
                                       
                                         ( 
                                         
                                           p 
                                           - 
                                           
                                             2 
                                             · 
                                             
                                               x 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                            
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     37 
                   
                   ) 
                 
               
             
           
         
       
     
     Equations 38 result from substituting A and B into Equations 36. 
     
       
         
           
             
               
                 
                   
                     Sappr 
                     i 
                   
                   = 
                   
                     
                       1 
                       
                         a 
                         - 
                         
                           2 
                           · 
                           c 
                           · 
                           
                             x 
                             0 
                           
                           · 
                           i 
                         
                         + 
                         
                           c 
                           · 
                           
                             i 
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           S 
                           p 
                         
                         · 
                         B 
                       
                       
                         B 
                         + 
                         
                           A 
                           · 
                           
                             ( 
                             
                               
                                 i 
                                 · 
                                 
                                   ( 
                                   
                                     i 
                                     - 
                                     
                                       2 
                                       · 
                                       
                                         x 
                                         0 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 p 
                                 · 
                                 
                                   ( 
                                   
                                     p 
                                     - 
                                     
                                       2 
                                       · 
                                       
                                         x 
                                         0 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     38 
                   
                   ) 
                 
               
             
           
         
       
     
     The RX signals are normalized, and a squared correlation coefficient is calculated between the normalized RX signals and their approximation by the reciprocal quadratic curve according to Equations 39 below. 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       avSigNorm 
                       = 
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             Snorm 
                             i 
                           
                         
                         N 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       avSigAppr 
                       = 
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             Sappr 
                             i 
                           
                         
                         N 
                       
                     
                      
                     
                       
 
                     
                      
                     
                       corrCoeff 
                       = 
                       
                         
                           256 
                           · 
                           
                             
                               ( 
                               
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           i 
                                           = 
                                           0 
                                         
                                         
                                           N 
                                           - 
                                           1 
                                         
                                       
                                        
                                       
                                           
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             
                                               Snorm 
                                               i 
                                             
                                             - 
                                             avSigNorm 
                                           
                                           ) 
                                         
                                         · 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           Sappr 
                                           i 
                                         
                                         - 
                                         avSigApp 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                         
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     
                                       ( 
                                       
                                         
                                           Snorm 
                                           i 
                                         
                                         - 
                                         avSigNorm 
                                       
                                       ) 
                                     
                                     2 
                                   
                                   · 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     ( 
                                     
                                       
                                         Sappr 
                                         i 
                                       
                                       - 
                                       avSigAppr 
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     39 
                   
                   ) 
                 
               
             
           
         
       
     
     A hover input is detected if the correlation coefficient corrCoeff is greater than a correlation threshold. The correlation threshold is a tunable parameter in the range between 0 and 255. In one embodiment, the correlation threshold has a value of 125. 
       FIG. 10A  illustrates a reciprocal quadratic curve  1003  approximating a sequence of RX signals  1001  resulting from a hover input at sensor electrode  18 , according to an embodiment. The raw RX signal sequence  1001  as measured from the sensor electrodes  0 - 44  is normalized to generate a normalized sequence of RX signals  1002 . The normalized signal sequence  1002  is approximated by curve  1003 . The vertex of the curve  1003  corresponds to the location of the hover input at sensor electrode  18 . In  FIG. 10A , the correlation coefficient is 246; thus, the presence of the hover input is detected because the correlation coefficient is greater than the correlation threshold of 175. 
       FIG. 10B  similarly illustrates a reciprocal quadratic curve  1013  approximating a sequence of RX signals  1011  resulting from a hover input at sensor electrode  12 , according to an embodiment. The raw RX signal sequence  1011  as measured from the sensor electrodes  0 - 44  is normalized to generate a normalized sequence of RX signals  1012 . The normalized signal sequence  1012  is approximated by curve  1013 . The vertex of the curve  1013  corresponds to the location of the hover input at sensor electrode  12 . In  FIG. 10B , the correlation coefficient is 234; thus, the presence of the hover input is detected because the correlation coefficient is greater than the correlation threshold of 175. 
       FIG. 11  illustrates a reciprocal quadratic curve  1103  approximating a sequence of RX signals  1101  resulting from heating the sensor array, rather than from a hover input. The raw RX signal sequence  1101  as measured from the sensor electrodes  0 - 44  is normalized to generate a normalized sequence of RX signals  1102 . The normalized signal sequence  1102  is approximated by curve  1103 . The vertex of the curve  1103  corresponds to the location of the highest signal magnitude at sensor electrode  14 . In  FIG. 11 , the correlation coefficient is 60; thus, the presence of the hover input is not detected because the correlation coefficient is less than the correlation threshold of 175. 
     In one embodiment, a subset of nine signals surrounding the local maximum signal in the RX signal sequence are approximated by a cubic curve in order to determine the location of the hover input. According to this approach, values of the differences Diff i−1 , Diff i , Diff i+1 , and Diff i+2  of the RX signal sequence (corresponding to sensor electrodes i−1, i, i+1, and i+2, respectively) are linearly related with the indexes of their corresponding sensor electrodes.  FIG. 12  illustrates this linear relationship by graphing the differential values Diff i−1 , Diff i , Diff i+1 , and Diff i+2  relative to the sensor electrode indexes, according to an embodiment. The top portion of  FIG. 12  shows the signal magnitudes S i−4  to S i+4  for the sensor electrodes. The intercept of this line with the x axis indicates the hover position. 
     The intercept x is calculated according to Equation 40, where N is the number of sensor electrodes and ResX is the resolution of the sensor array along the length of the X (horizontal) axis in pixels. 
     
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       
                         Re 
                          
                         
                             
                         
                          
                         sX 
                       
                       Nx 
                     
                     · 
                     
                       ( 
                       
                         i 
                         - 
                         
                           
                             Diff 
                             
                               i 
                               - 
                               1 
                             
                           
                           
                             
                               Diff 
                               i 
                             
                             - 
                             
                               Diff 
                               
                                 i 
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     40 
                   
                   ) 
                 
               
             
           
         
       
     
     The coefficients of the approximating cubic curve are determined according to Equation 41. 
         S   i−j   =a−b·j−c·j   2   (Equation 41)
 
     Accordingly, the function to be minimized for the least squares approximation is shown in Equation 42. 
     
       
         
           
             
               
                 
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           
                             - 
                             4 
                           
                         
                         4 
                       
                        
                       
                           
                       
                        
                       
                         
                           [ 
                           
                             
                               S 
                               
                                 i 
                                 - 
                                 j 
                               
                             
                             - 
                             a 
                             - 
                             
                               b 
                               · 
                               j 
                             
                             - 
                             
                               c 
                               · 
                               
                                 j 
                                 2 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                     ⇒ 
                     min 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     42 
                   
                   ) 
                 
               
             
           
         
       
     
     Partial differentiation of Equation 42 yields Equations 43. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 ∂ 
                               
                               
                                 ∂ 
                                 a 
                               
                             
                             = 
                             
                               
                                 
                                   - 
                                   2 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       
                                         - 
                                         4 
                                       
                                     
                                     4 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         S 
                                         
                                           i 
                                           - 
                                           j 
                                         
                                       
                                       - 
                                       a 
                                       - 
                                       
                                         b 
                                         · 
                                         j 
                                       
                                       - 
                                       
                                         c 
                                         · 
                                         
                                           j 
                                           2 
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               = 
                               0 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ∂ 
                               
                               
                                 ∂ 
                                 b 
                               
                             
                             = 
                             
                               
                                 
                                   - 
                                   2 
                                 
                                 · 
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       = 
                                       
                                         - 
                                         4 
                                       
                                     
                                     4 
                                   
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       [ 
                                       
                                         
                                           S 
                                           
                                             i 
                                             - 
                                             j 
                                           
                                         
                                         - 
                                         a 
                                         - 
                                         
                                           b 
                                           · 
                                           j 
                                         
                                         - 
                                         
                                           c 
                                           · 
                                           
                                             j 
                                             2 
                                           
                                         
                                       
                                       ] 
                                     
                                     · 
                                     j 
                                   
                                 
                               
                               = 
                               0 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ∂ 
                             
                             
                               ∂ 
                               c 
                             
                           
                           = 
                           
                             
                               
                                 - 
                                 2 
                               
                               · 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     
                                       - 
                                       4 
                                     
                                   
                                   4 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     [ 
                                     
                                       
                                         S 
                                         
                                           i 
                                           - 
                                           j 
                                         
                                       
                                       - 
                                       a 
                                       - 
                                       
                                         b 
                                         · 
                                         j 
                                       
                                       - 
                                       
                                         c 
                                         · 
                                         
                                           j 
                                           2 
                                         
                                       
                                     
                                     ] 
                                   
                                   · 
                                   
                                     j 
                                     2 
                                   
                                 
                               
                             
                             = 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equations 
                      
                     
                         
                     
                      
                     43 
                   
                   ) 
                 
               
             
           
         
       
     
     Equation 40 can thus be expressed as shown in Equation 44. 
     
       
         
           
             
               
                 
                   x 
                   = 
                   
                     
                       
                         Re 
                          
                         
                             
                         
                          
                         sX 
                       
                       
                         256 
                         · 
                         Nx 
                       
                     
                     · 
                     
                       ( 
                       
                         
                           256 
                           · 
                           i 
                         
                         + 
                         128 
                         - 
                         
                           
                             9856 
                             · 
                             
                               
                                 ∑ 
                                 
                                   j 
                                   = 
                                   
                                     - 
                                     4 
                                   
                                 
                                 4 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   S 
                                   
                                     i 
                                     - 
                                     j 
                                   
                                 
                                 · 
                                 j 
                               
                             
                           
                           
                             5 
                             · 
                             
                               ( 
                               
                                 
                                   3 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         j 
                                         = 
                                         
                                           - 
                                           4 
                                         
                                       
                                       4 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         S 
                                         
                                           i 
                                           - 
                                           j 
                                         
                                       
                                       · 
                                       
                                         j 
                                         2 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   20 
                                   · 
                                   
                                     
                                       ∑ 
                                       
                                         j 
                                         = 
                                         
                                           - 
                                           4 
                                         
                                       
                                       4 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       S 
                                       
                                         i 
                                         - 
                                         j 
                                       
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                      
                     
                         
                     
                      
                     44 
                   
                   ) 
                 
               
             
           
         
       
     
     The value calculated for x according to the above equations provides a coordinate for the detected hover input, in terms of the sensor electrode indexes. 
     In one embodiment, noise immunity from nearby sources (e.g., LCD common mode noise) is improved by reducing the number of sensor electrodes from which RX signals are being measured. This happens because the charge injected into the sensing channel from the LCD panel is proportional to the number of electrodes being sensed by the channel at same time, or in other words, proportional to the coupling capacitance between the equivalent RX sensor electrode and LCD panel.  FIGS. 13A-13C  illustrate sequences of excitation signal patterns according to an embodiment, in which TX excitation signals are applied to the sensor electrodes in a non-circular manner and the RX signal is measured for each pattern from a single sensor electrode via a single RX sensing channel. 
     With reference to  FIG. 13A , each of the excitation signal patterns  1320  (including signal patterns  1320 ( 1 )- 1320 ( 10 )) in the sequence  1300  applies a positive phase excitation signal to sensor electrodes surrounding and including the sensor electrode from which the RX signal is measured (indicated with bold outlines). For example, in pattern  1320 ( 5 ), the positive phase excitation signals are applied to electrodes  3 - 7 . Each of the signal patterns in the sequence  1300  is a zero-sum pattern; thus, the complementary negative phase excitation signals are applied to the remaining electrodes  1 - 2  and  8 - 10  in signal pattern  1320 ( 5 ). A decrease in the magnitude of the measured RX signal due to fewer sensor electrodes being measured is compensated by an associated decrease in external noise (e.g., from an LCD display). 
       FIG. 13B  illustrates a sequence  1301  of signal patterns  1330 , including patterns  1330 ( 1 )- 1330  ( 10 ). For each of the excitation signal patterns  1330 , no excitation signal (as indicated by ‘0’) is applied to some of the sensor electrodes, including the sensor electrode from which the RX signal is measured. In each pattern, the positive phase excitation signals are applied to the sensor electrodes surrounding the sensed electrode. Each of the patterns  1330  is also a zero-sum signal pattern. 
       FIG. 13C  illustrates a signal pattern sequence  1302  in which, for each of the excitation signal patterns  1340 , no excitation signal is applied to a larger number of sensor electrodes, including the sensor electrode from which the RX signal is measured. Each of the patterns  1340  is also a zero-sum signal pattern. 
       FIGS. 14A-14C  illustrate an approach for increasing the measurable RX signal during sensing of a hover input, according to an embodiment.  FIG. 14A  illustrates a sequence of excitation patterns  1402  for detecting a hover input  1401  at the sensor array  103  as previously described, in which the column sensor electrodes are excited by a sequence of column excitation patterns  1403  that are circularly rotated zero-sum patterns. For each of the column patterns  1403 , no excitation signal is applied to the row sensor electrodes. Accordingly, the row excitation patterns  1404  are all zero for each of the sensor patterns  1402 . 
     If the location of the hover input  1401  along the vertical axis is known (e.g., if the row sensor electrodes were previously sensed), the measurable RX signal can be increased by applying positive phase excitation signals to a subset of the row sensor electrodes nearest to the hover input  1401  while applying negative phase excitation signals to the remaining row sensor electrodes.  FIG. 14B  illustrates this approach, according to an embodiment. A sequence of six sensor patterns  1412  is applied to the sensor array  103 . For each of the column excitation signal patterns  1403 , one of the row excitation signal patterns  1414  is applied to the row sensor electrodes based on a previously determined location of the hover input  1401 . Each of the row excitation signal patterns  1414  is a zero sum pattern and applies a positive phase excitation signal to the four row sensor electrodes nearest to the hover input  1401 , while a complementary negative phase excitation signal is applied to the remaining four row sensor electrodes. 
     This measurement procedure increases the baseline of the measured RX signals, but does not increase the dynamic range of the RX signals. In one embodiment, the magnitude of the measured RX signal for each of the sensor patterns  1412  is increased by approximately 175 fF, while the dynamic range of the RX signals (i.e., the difference between the lowest and highest RX signals) remains the same (approximately 150 fF). A similar effect occurs in the opposite direction (i.e., the baseline is decreased) when the phase polarity of the TX excitation signals are switched (i.e., positive phase excitation signals are switched with negative phase excitation signals). 
       FIG. 14C  illustrates an embodiment in which positive and negative phase signals are alternated in the sequence of signal patterns  1424  applied to the row sensor electrodes. The phase polarities are alternated between successive patterns in the sequence  1412 . In alternative embodiments, the phase polarities are alternated after every second pattern, third pattern, etc. The alternation of phase polarities for signal patterns applied the row sensor electrodes increases the dynamic range of the measured RX signal sequence without increasing the baseline. In one embodiment, the dynamic range is increased to approximately 500 fF. 
       FIG. 15  illustrates a graph of RX signal sequences for the different measurement approaches, according to an embodiment. RX signal sequence  1501  results from measurement of a hover input  1401  when no excitation signal is applied to the row sensor electrodes. RX signal sequences  1502  and  1503  represent the cases where positive phase excitation signals and negative phase excitation signals, respectively, are applied to the row sensor electrodes nearest the hover input  1401 . RX signal sequence  1504  is generated when alternating positive and negative phase excitation signals are applied to the row sensor electrodes nearest the hover input  1401 , with the alternation occurring every second pattern. This type of drive method can be referred to as “correlated double sampling”, and reduces drift in the receive channels. 
       FIG. 16  is a flow diagram illustrating a process  1600  for sensing a hover input at a capacitive sensor array  103 , according to an embodiment. The process  1600  is performed by components in the processing device  102 . 
     At block  1601 , the sequencer  223  determines an initial TX signal pattern in a sequence of TX signal patterns for applying the sensor array  103 . At block  1603 , the TX generator  215  applies the initial TX signal pattern the set of TX electrodes in the sensor array  103  by, for each sensor electrode in a set of sensor electrodes (e.g., the column sensor electrodes in the sensor array  103 ), applying either a positive phase excitation signal or a negative phase excitation signal according to the TX signal pattern. The negative phase excitation signal is complementary to the positive phase excitation signal, and the positive and negative phase excitation signals are applied to sensor electrodes that are contiguous in a circular sequence of the sensor electrodes. 
     In one embodiment where zero-sum excitation patterns are used, the positive phase excitation signal is applied to half of the sensor electrodes in the set and the negative phase excitation signal is applied to the remaining half. Accordingly, the magnitude of a sum of the excitation signals applied to the sensor electrodes is reduced to less than the magnitude of any of the excitation signals applied to an individual one of the sensor electrodes, and ideally approaches zero. In alternative embodiments where a nonzero sum excitation pattern is used, the excitation of the sensor electrodes generates at least as much emission as a single sensor electrode to which an excitation signal is applied. At block  1605 , the resulting RX signal induced from the application of the initial TX signal pattern is measured by the RX channel  224  in the charge to code converter  216 . 
     At block  1607 , if there are any TX signal patterns in the sequence that have not yet been applied to the sensor electrodes, then the process  1600  returns to block  1601 . At block  1601 , the sequencer circuit  223  determines the next TX signal pattern in the sequence of TX signal patterns based on a circular rotation of the initial TX signal pattern. In one embodiment, the current pattern of positive and negative phase excitation signals is circularly shifted by one sensor electrode to generate the next TX signal pattern in the sequence. Thus, for each TX signal pattern, the excitation signal applied to each sensor electrode corresponds to the excitation signal that was applied to a preceding (i.e., lower-indexed) sensor electrode in a circular sequence of the set of sensor electrodes during application of the preceding TX excitation signal pattern, where the preceding TX excitation signal pattern is the pattern in the sequence that was most recently applied prior to the current signal pattern. In alternative embodiments, the current pattern is shifted by more than one electrode. 
     Blocks  1601 - 1607  are repeated in a loop so that each of the TX signal patterns in the sequence is applied in order to the set of sensor electrodes and an RX signal response is measured for each TX signal pattern. The measured sequence of RX signals is stored in memory as a vector  211 . At block  1607 , if all of the TX signal patterns have been applied, the post processing block  222  calculates the correlation coefficient between the measured sequence of RX signals and a least squares approximation of the RX signal sequence by a predetermined function, such as a sine function, as provided at blocks  1609 - 1613 . 
     At block  1609 , some parameters of the approximating sine function are calculated; for example, parameters c and d are calculated using Equations 7 and 8. Parameter d represents a displacement that indicates a position of the hovering object relative to the sensor electrodes, and is calculated based on a highest magnitude signal in the measured sequence of RX signals. 
     At block  1611 , the post processing block  222  determines remaining parameters for the sine function using a least squares calculation for approximating the sequence of RX signals. In one embodiment, the LUT  225  stores a result of the sine function for each of a number of possible input arguments. The post processing block  222  accesses the LUT  225  when calculating the remaining parameters (e.g., parameter b according to Equation 18). 
     At block  1613 , the post processing block  222  calculates the correlation coefficient between the measured RX signal sequence and its approximation by the sine function having the calculated parameters. In one embodiment, the approximating sine curve is adjusted by shifting the sine curve relative to the sensor electrode indexes until the correlation coefficient, as calculated according to Equation 26, is maximized. 
     At block  1615 , the post processing block  222  compares the correlation coefficient with a predetermined correlation threshold. At block  1617 , if the correlation coefficient does not exceed the correlation threshold, then no hover input is detected, and the post processing block  222  reports at block  1621  that no hover input is present. From block  1621 , the process  1600  returns to block  1601  to restart the detection process. At block  1617 , if the correlation coefficient exceeds the correlation threshold, the process  1600  continues at block  1619 . At block  1619 , the presence of a valid hover input is detected. From block  1619 , the presence of the detected hover input and its location (as indicated by the displacement parameter d) are transmitted to the host device  101 , which performs some action (e.g., updating a cursor on the display  104 , highlighting or selecting a menu item, etc.). The process  1600  returns to block  1601  to resume applying the TX excitation signal patterns to the sensor electrodes for the next measurement cycle, starting again from the initial TX excitation signal pattern in the sequence. The process  1600  thus repeats blocks  1601 - 1619  continuously to detect the presence of hover inputs and their locations over time. 
       FIG. 17  is a flow diagram illustrating a process  1700  for detecting a hover input based on a measured sequence of RX signals, according to an embodiment, which uses a non-zero sum scan sequence and a deconvolution procedure. In process  1700 , a parabola or reciprocal quadratic function is used for the least squares approximation instead of a sine function. Process  1700  is performed by components in the processing device  102 , including the post processing block  222 . From block  1607  in process  1600 , the process  1700  begins at block  1701 . At this time, an RX signal has been measured for each of the TX excitation signal patterns in the sequence of TX excitation patterns. 
     At block  1701 , the post processing block  222  determines the vertex of the approximating function (i.e., the parabola or reciprocal quadratic curve). The vertex of the approximating function is set equal to the measured RX signal having the highest magnitude. From block  1701 , the process  1601  continues at block  1703 . 
     At block  1703 , the post processing block  222  normalizes the measured RX signal sequence and approximates a portion of the normalized RX signal sequence (e.g., seven RX signals nearest the highest magnitude RX signal) using the parabola with its vertex set as provided at block  1701 . Alternatively, all of the available RX signals in the normalized sequence are approximated using the reciprocal quadratic curve with its vertex set as provided at block  1701 . 
     At block  1705 , a correlation coefficient is calculated that indicates a degree of correlation between the normalized RX signal (scaled down to a global peak value) sequence and the simulated hover profile approximating function, having its peak at a vertex point calculated by according to the parabola function. At block  1707 , the correlation coefficient is compared with a predetermined correlation threshold. At block  1709 , if the correlation coefficient does not exceed the threshold, then no hover input is detected. The absence of any hover input is reported at block  1717 . The process  1700  returns to block  1601  to resume applying the sequence of TX excitation patterns to the sensor electrodes. 
     At block  1709 , if the correlation coefficient exceeds the predetermined correlation threshold, the post processing block  222  detects a valid hover input, as provided at block  1711 . In one embodiment, the location of the hover input relative to the sensor electrodes is determined according to blocks  1713  and  1715 . At block  1713 , a set of differentials is calculated for a subset of the RX signals nearest to the highest magnitude RX signal. At block  1715 , the hover position is calculated from the x-axis intercept of a best fit line calculated for the differentials. From block  1715 , the presence of the hover input and its location are communicated with the host device  101 , which performs some action based on the presence of the hover input and/or its location. The process  1700  returns to block  1601  and resumes applying TX signal patterns to the sensor electrodes for the next measurement cycle. 
     In the foregoing embodiments, various modifications can be made; for example, signals described as being asserted with a high voltage may instead be asserted with a low voltage, or specified components can be replaced with other components having similar functionality. As described herein, conductive electrodes that are “electrically connected” or “electrically coupled” may be coupled such that a relatively low resistance conductive path exists between the conductive electrodes. Quantities, dimensions, or other values described as “substantially” equal may be nominally equal but need not be exactly equal (with variations due to manufacturing tolerances, environmental conditions, quantization or rounding error, and/or other factors), or may be sufficiently close to equal for achieving an intended effect or benefit. 
     Embodiments described herein include various operations. These operations may be performed by hardware components, software, firmware, or a combination thereof. As used herein, the term “coupled to” may mean coupled directly or indirectly through one or more intervening components. Any of the signals provided over various buses described herein may be time multiplexed with other signals and provided over one or more common buses. Additionally, the interconnection between circuit components or blocks may be shown as buses or as single signal lines. Each of the buses may alternatively be one or more single signal lines and each of the single signal lines may alternatively be buses. 
     Certain embodiments may be implemented as a computer program product that may include instructions stored on a computer-readable medium. These instructions may be used to program a general-purpose or special-purpose processor to perform the described operations. A computer-readable medium includes any mechanism for storing or transmitting information in a form (e.g., software, processing application) readable by a machine (e.g., a computer). The computer-readable storage medium may include, but is not limited to, magnetic storage medium (e.g., floppy diskette); optical storage medium (e.g., CD-ROM); magneto-optical storage medium; read-only memory (ROM); random-access memory (RAM); erasable programmable memory (e.g., EPROM and EEPROM); flash memory, or another type of medium suitable for storing electronic instructions. 
     Additionally, some embodiments may be practiced in distributed computing environments where the computer-readable medium is stored on and/or executed by more than one computer system. In addition, the information transferred between computer systems may either be pulled or pushed across the transmission medium connecting the computer systems. 
     Although the operations of the method(s) herein are shown and described in a particular order, the order of the operations of each method may be altered so that certain operations may be performed in an inverse order or so that certain operation may be performed, at least in part, concurrently with other operations. In another embodiment, instructions or sub-operations of distinct operations may be in an intermittent and/or alternating manner. 
     In the foregoing specification, the claimed subject matter has been described with reference to specific exemplary embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the invention as set forth in the appended claims. The specification and drawings are, accordingly, to be regarded in an illustrative sense rather than a restrictive sense.