Abstract:
A sound system for a locomotive mounts a loudspeaker line array for broadside firing forward from the vehicle. Additional loudspeakers are installed on the locomotive for projecting sound to the sides. A control system applies drive signals to the loudspeakers of the line array with the control system providing phase adjustment of drive signals applied to each loudspeaker of the line array to control side to side directional steering of a projected sound beam. The control system includes an automated, location dependent sub-system for selecting beam width and directional steering of a projected sound beam.

Description:
REFERENCE TO RELATED APPLICATION AND PRIORITY CLAIM 
       [0001]    This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/600,027 filed 17 Feb. 2012. 
     
    
     BACKGROUND 
       [0002]    1. Technical Field 
         [0003]    The field relates to warning systems and more particularly to a locomotive mounted, directional, acoustic warning system. 
         [0004]    2. Description of the Problem 
         [0005]    Train horns can either supplement or provide the only acoustic alarm at railway/road crossings. In the United States train horn use is regulated by the Federal Railroad Administration (FRA). Since 2005 regulations have provided that for trains moving slower than 45 mph the locomotive horn be sounded at least 15, but not more than 20, seconds before a locomotive enters a crossing. For trains moving faster than 45 mph the horn is to be sounded at designated locations. The train horn is to be sounded using is two long tones, a short tone and one additional long tone. This pattern is repeated until the lead locomotive has entered the crossing. 
         [0006]    Despite the effectiveness of horns in giving warning to motorists and others, the use of horns in some areas is unpopular. The State of Florida attempted to ban the sounding of locomotive horns, but such a blanket prohibition ran afoul of federal preemption issues. Provisions have been made to allow local authorities an option of establishing quiet zones provided effective alternative safety measures are in place. 
       SUMMARY 
       [0007]    A sound system for a ground vehicle comprises a line array of loudspeakers installed across the front of the ground vehicle. The line array is operated in a broadside firing mode to project a sound beam generally forward from the vehicle while allowing steering of the sound beam from side to side off center line. First and second loudspeakers are installed on the ground vehicle for projecting sound to the sides of the ground vehicle. A control system applies drive signals to the loudspeakers of the line array and to the first and second side loudspeakers. The control system provides phase adjustment of drive signals applied to each loudspeaker of the line array to control beam width and side to side directional steering of the sound beam relative to the direction of travel of the ground vehicle. The control system includes an automated, geographical location sensitive sub-system for selecting beam width and directional steering of a projected sound beam to be generated by the line array. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]    Understanding of the following description may be enhanced by reference to the accompanying drawings, wherein: 
           [0009]      FIG. 1  is a side elevation of a locomotive on which a loudspeaker system has been installed as a substitute or supplement for an air horn system. 
           [0010]      FIG. 2  is a top plan view of the locomotive of  FIG. 1  illustrating the location of a front directed loudspeaker array and two side directed loudspeakers. 
           [0011]      FIG. 3  depicts an operating scenario for the system. 
           [0012]      FIG. 4  is a perspective view of the loudspeaker line array. 
           [0013]      FIG. 5  is a cross sectional view of the loudspeaker line array. 
           [0014]      FIG. 6  is a simple, theoretical polar plot for a directivity pattern for the front directed loudspeaker line array. 
           [0015]      FIG. 7  is a graph illustrating frequency shading using a two speaker example. 
           [0016]      FIG. 8  is a block diagram of a control arrangement for the loudspeaker system. 
           [0017]      FIG. 9  is a high level flow chart for operation of the loudspeaker system. 
           [0018]      FIG. 10  is a polar plot of acoustic radiation emitted at 350 Hz for the front directed loudspeaker line array. 
           [0019]      FIG. 11  is a polar plot of acoustic radiation emitted at 400 Hz for the front directed loudspeaker line array. 
           [0020]      FIG. 12  is a polar plot of acoustic radiation emitted at 500 Hz for the front directed loudspeaker line array. 
           [0021]      FIG. 13  is a polar plot of acoustic radiation emitted at 630 Hz for the front directed loudspeaker line array. 
           [0022]      FIG. 14  is a polar plot of acoustic radiation emitted at 800 Hz for the front directed loudspeaker line array. 
           [0023]      FIG. 15  is a polar plot of acoustic radiation emitted at 1000 Hz for the front directed loudspeaker line array. 
           [0024]      FIG. 16  is a polar plot of acoustic radiation emitted at 1250 Hz for the front directed loudspeaker line array. 
           [0025]      FIG. 17  is a polar plot of acoustic radiation emitted at 1600 Hz for the front directed loudspeaker line array. 
           [0026]      FIG. 18  is a polar plot of acoustic radiation emitted at 2000 Hz for the front directed loudspeaker line array. 
           [0027]      FIG. 19  is a polar plot of acoustic radiation emitted at 2500 Hz for the front directed loudspeaker line array. 
           [0028]      FIG. 20  is a polar plot of acoustic radiation emitted at 3150 Hz for the front directed loudspeaker line array. 
           [0029]      FIG. 21  is a polar plot of acoustic radiation emitted at 4000 Hz for the front directed loudspeaker line array. 
           [0030]      FIG. 22  is a polar plot of acoustic radiation emitted at 5000 Hz for the front directed loudspeaker line array. 
           [0031]      FIG. 23  is a polar plot of all frequency components for the front directed loudspeaker line array. 
           [0032]      FIG. 24  is a pressure and phase response plot against frequency for the front directed loudspeaker line array. 
           [0033]      FIG. 25  illustrates delay against frequency for the line array. 
           [0034]      FIG. 26  is an energy over time response over all frequencies for the line array. 
           [0035]      FIG. 27  is a response curve for a single impulse input over time for the line array. 
           [0036]      FIG. 28  is a response curve for a double impulse input over time for the line array. 
           [0037]      FIG. 29  is a graph of total harmonic distortion for the line array in terms of percentage for each frequency band. 
           [0038]      FIG. 30  is system frequency response with harmonic tracking overlay of second, third, fourth, fifth and sixth order distortion in decibels against frequency. 
           [0039]      FIGS. 31-34  are directively patterns. 
           [0040]      FIGS. 35-38  are exemplary directivity patterns. 
           [0041]      FIGS. 39-45  are still further exemplary directivity patterns illustrating operation of the train mounted line array. 
       
    
    
     DETAILED DESCRIPTION 
       [0042]    Referring to the figures, and particularly to  FIGS. 1 and 2 , a ground vehicle such as a locomotive  10  is illustrated on which a loudspeaker system has been installed for the purpose of emitting acoustic warning of approach of the ground vehicle to a location, particularly locations adjacent a level grade road crossing with railroad tracks. The loudspeaker system comprises a forward directed loudspeaker line array  12  and first and second side directed loudspeakers  14  and  16 . The line array  12  is positioned horizontally with respect to the tracks supporting the locomotive and has a primary or neutral sound projection axis aligned on the longitudinal axis of the locomotive  10 . The first and second side directed loudspeakers  14  and  16  oriented to project sound away from the right and left sides of locomotive  10 . The front directed loudspeaker line array  12  and the side directed loudspeakers  14 ,  16  are mounted on top of the locomotive  10  to provide unobstructed projection of sound. 
         [0043]    A loudspeaker line array  12  is illustrated in  FIGS. 4 and 5 . Loudspeaker line array  12  comprises at least three loudspeakers  13 A,  13 B and  13 C. Traditionally air horns have been used on locomotives in order to obtain the desired sound level output. The present applicant has submitted an application for an electro/acoustical transducer system utilizing opposed transducers directed into a waveguide assembly entitled RADIAL WAVEGUIDE FOR DOUBLE CONE TRANSDUCER, U.S. patent application Ser. No. 13/346,077 filed 9 Jan. 2012, which is incorporated herein by reference. Transducer and waveguide assemblies such as disclosed in the incorporated application may be utilized for the loudspeakers  13 A-C of line array  12  and for side directed loudspeakers  14  and  16 . This device is sometimes referred to herein as a “Tandem Horn.” 
         [0044]    Loudspeaker line array  12  is constructed with its center loudspeaker  13 B which, when mounted on a locomotive  10 , is intended to be directed straight ahead aligned on the longitudinal axis of the locomotive  10 . Outboard loudspeakers  13 A,  13 C are canted outwardly away from the longitudinal axis of the locomotive  10  and located slightly behind the center loudspeaker  13 B making the line array a gently curved or staggered line array. Gently curved line arrays are known from several sources including U.S. Patent Application Publication No. 2008/0212805 (a symmetric system) and U.S. Pat. No. 6,870,942 (a non-symmetric system). Line array  12  is disposed on the locomotive  10  with its axis of elongation generally parallel to the ground, or more precisely, generally parallel to the local plane of the tracks that the locomotive  10  rides on. The acoustic centers of adjacent loudspeakers are spaced from one another by about 18 inches. 
         [0045]    Referring to  FIG. 3  a locomotive  10  located on a railroad track  18  is equipped with a forward directed loudspeaker array  12 . Locomotive  10  is shown in a position along an approach to a level grade crossing  28  of a road  30  with track  18 . The approach path of the locomotive  10  to the level grade crossing  28  is not along a straight line as a bend  34  in the tracks  18  occurs between the location of the crossing guard triggers  24  and the level grade crossing  28 . The locations of vehicles and pedestrians along their respective lines of approach to the level grade crossing  28 , that is on or along road  30 , may vary from straight ahead of the locomotive  10  to locations well of the longitudinal axis of the locomotive. Approach of a locomotive  10  to the region into which a warning is to be broadcast by the locomotive horn system may not be straight for reasons other than bends in the tracks  18 . For example, the road or path crossing the tracks  18  may incorporate turns and/or cross the tracks at other than a perpendicular angle and the angle may change either away from or closer to the center line of the locomotive  10  as it approaches. In addition, the speed at which the locomotive  10  is traveling affects the timing of sounding of an alert from the locomotive. A car  31  is located on the road  30  approaching level grade crossing  28  short of a flasher/bell warning system  32  to illustrate a possible location where sound energy is to be directed. 
         [0046]    Under normal circumstances, upon locomotive  10  passing crossing guard triggers  24 , signals are sent along a crossing guard trigger cable  26  to an automatic crossing guard controller  20  which controls activation of the flasher/bell warning system  32 . This system can be used to trigger operation of a crossing guard transponder  22  which can transmit data to or be interrogated by a control system on locomotive  10  for reports on operating condition of the flasher/bell warning system  32 . Crossing guard transponder  22  may be equipped to provide local weather conditions, particularly wind direction and speed at the level grade crossing  28 . Crossing guard transponder  22  may be locally programmed to provide special instructions or a beam profile to an approaching locomotive  10  as described below. 
         [0047]    Forward directed loudspeaker line array  12  can serve to supplement the flasher/bell warning system  32  by emulating a train horn targeting the approaches to level grade crossing  28 . Line arrays compress sound emitted from the array into a primary/major and secondary lobes extending radially from the line array  12  in a plane parallel to the ground and aligned on the locomotive, assuming no beam steering. The horizontally disposed array allows the primary/major lobe or beam of sound from the array to be steered in the horizontal plane parallel to the ground using techniques of phase adjustment, amplitude shading and frequency shading among the loudspeakers  13 A-C of the line array  12 . Confining most sound energy to lobes, and controlling the direction and width of the lobes can be used to reduce sound spill over into areas away from the approaches to the level grade crossing  28 , compensating for bends  34  in the track  18 , or non-perpendicular approaches of roads  30  ton the tracks. 
         [0048]      FIG. 6  illustrates a simple directivity pattern  38  at a particular frequency for forward directed loud speaker line array  12 . The line array  12  is set for broadside firing with a minimum of phase adjustment and frequency and amplitude shading. Directivity pattern  38  is in a plane to the ground with the central projection axis/major lobe and back lobe usually aligned on the longitudinal axis of locomotive  10 . The major lobe can be displaced in either direction outwardly (but parallel to the ground) from the locomotive&#39;s longitudinal axis by beam steering. The spread of the major lobe to −10 dB may be varied by adjusting phase differences between the loudspeakers of the line array  12 . 
         [0049]    While the primary beam lobe is normally set for a narrow beam of 42-45 degrees (about 22 and ½ degrees each side of center) at the primary frequency, for an emergency condition such as a vehicle on the track the beam could be actively focused the a minimum beam waist and steered directly at the target to create the maximum available acoustic power to the target. Video or radar could be used to determine precise location bearing to the target and processing applied to deliver maximum energy density to the selected target(s) 
         [0050]    The primary beam lobe emulates the sound pressure of a standard pneumatic train horn, however the substantial decrease in acoustic sound power at all angles of the system other than the primary beam lobe decrease the noise pollution to surrounding areas. Additional settings (enhancements) of the signal processing allow the array system to have additional decreased output to the null areas where sound energy is to be minimized due to the proximity of houses and businesses. In testing average side attenuation of −18 to −24 db from the primary beam lobe was achieved, however alternative DSP settings produced attenuation levels as great as −42 db from the primary beam lobe in portions of the acoustic spectrum. 
         [0051]    The ability to program waveforms allows for high contrast ratio lower duty-cycle alert tones could be mixed with the train 5 tone sounds to create a louder and higher percentage attention getting signal for use in conditions where the standard train horn sounds are ineffective. In addition, selectable “engineered per species” sound tracks could be chosen to directly deter wildlife from the front of the trains path in the case of obstruction of the tracks. A secondary passive noise absorption housing can be applied to the system to lower the side/rear emission levels even beyond the adaptive null created with the array. The system can be operated via remote location via live data links and or operated in an autonomous response mode eliminating the requirements of a live systems operator on-board. 
         [0052]    Focusing sound energy from the front directed loudspeaker line array  12  into lobes avoids spillover into areas adjacent the tracks where the sound is not needed and beam steering allows sound energy to be directed to compensate for level grade crossings which are non-standard. The provision of side directed loudspeakers  14 ,  15 , which are not installed in arrays and are less directional than the array allows sound to be directed more to the sides of the locomotive  12 . Active steering (left-center-right-center-left etc.) of the main primary acoustic lobe would allow the system to produce a sound in motion effect that would increase the attention getting capability of the system for emergency operations. Increasing the number of loudspeakers in array  12  provided greater control over beam steering and lobe spreading. 
         [0053]    In a test arrangement an array  12  was built with a mechanical splay angle was set at 35 degrees so center horn was at 0 degrees with the left side horn set at 35 degrees to center and the right side at 35 degrees to center. A 42 degree beam was formed when the center horn was phase delayed 0.318 ms from the outside horns. A 60 degree beam was formed when the center horn was phase delayed 0.120 ms from the outside horns. A 85 degree beam was formed when the center horn was phase delayed 0.060 ms from the outside horns. A 120 degree beam was formed when the outside horns were phase delayed 0.298 ms from the center horn. A 198 degree beam was formed when the outside horns were phase delayed 0.918 ms from the center horn. A right steered beam of 20 degrees was formed with a left phase delay 0.00 ms, center 0.121 ms, right 0.815 ms setting. 
         [0054]      FIG. 7  illustrates frequency shading for a pair of loudspeaker with the output amplitude for one speaker ramped up to a plateau between 300 Hz and 10 KHz and another loudspeaker having a more gradual ramp up to the plateau from below 1 KHz. 
         [0055]      FIG. 8  is a block diagram for a control system for side directed loudspeakers  14 ,  16  and front directed loudspeaker line array  12 . An on board computer or matrix select controller  40  generates/supplies selected audio input signals to each loudspeaker channels  60 A-E. The audio input signal may be virtually any signal including frequencies in the human range of hearing, but usually includes a frequency mix which emulates a train horn or captures a voice input. Each of channels  60 A-E is physically substantially identical, however, channels  60 A-C, which drive loudspeakers  13 A-C, are operated in a coordinated manner to provide that the loudspeakers operate as a generally broad side firing array with beam steering. Signals applied to loudspeakers  13 A-C are generally identical, but phase shifted with respect to one another to achieve line array operation with beam steering. 
         [0056]    Each of channels  60 A-E comprises a digital signal processor  61 , an amplifier  63  and a loudspeaker, respectively loudspeakers  13 A-C,  14  and  16 . Matrix select controller  40  directs generation of a sound output either automatically or in response to operator interaction with the system using a graphical user interface  44  and, possibly, a local audio input  42  (such as a microphone). Generally only the lead locomotive of a tandem pair of locomotives  10  is allowed to use its loudspeakers, or at least its forward directed loudspeaker line array  12 . Accordingly a slave/master circuit  45  is provided which supplies an enable/disable signal to the matrix select controller  40  depending upon whether a particular locomotive is the lead or a trailing machine. 
         [0057]    Matrix select controller  40  may be programmed to respond to other inputs. Proximity sensor  58  may be a short range radar unit located with respect to the loudspeakers  13 A-C,  14  and  16  which generates a disable/degrade signal in case a person is located in close proximity to the mouth of the waveguide from a loudspeaker unit. Output from a unit can be blocked or limited to prevent hearing damage to an individual standing in proximity to the unit. A telematics receiver unit  46  may be connected to the matrix select controller  40 . Telematics units may be used to allow matrix select controller  40  to access geographic information system databases and maps allowing it to locate and characterize level grade crossings which the locomotive  10  is approaching. It may also be used to provide location information to the matrix select controller  40  as may a global positioning system (GPS) unit  50  installed on the locomotive  10  and connected to provide location data to the matrix select controller  40 . 
         [0058]    Timing of generation of a warning signal using the loudspeaker system of locomotive  10  depends on the speed and route which the locomotive is traveling. A speed signal source  48  may be provided or speed may be determined by GPS unit  50 . The output generated by the system may be adjusted depending upon weather conditions  52 , particularly wind direction, which can affect beam steering. Transponder trigger unit  54  communicates with crossing guard transponder  22  (if available) to determine if local conditions might be otherwise than indicated in the data base/look up table (LUT)  57  stored in memory  56 . 
         [0059]    Memory  56 , and the LUT  57  relating to level grade crossings, is of particular relevance to control over the audio channels  60 A-E. The database/LUT for level grade crossings is indexed by location and can include a topology classification, a risk factor index, a beam form type to use and a direction for aiming the beam/major lobe produced by the array  12  (which may be adjusted for wind). The beam profile to use may be further defined by amplitude to use, modulation and wafting of the signal. Alternatively, local transponders  22  may broadcast a crossing guard classification enabling the database to simply provide a beam profile to use for the general classification. 
         [0060]    Once matrix select controller  40  has identified from a specific level grade crossing entry or type categorization a beam profile and warning alert type to use, and ambient conditions and locomotive  10  speed obtained, a configuration for each DSP  61  is available. The DSP configuration for each of channels  60 A-E determines if a given channel is used at all, the delay for each channel, frequency shading filters to be implemented by each DSP to obtain a selected beam width and a gain for each channel&#39;s amplifier  63 . A compressor limit may be implemented to shape audio waveforms to create higher average sound without exceeding peak to peak limits of the system. Signal strength can also be enhanced through other well known techniques such as passing more low frequency power (at the cost of beam spreading) or altering the harmonic content to affect human perception of the sound. 
         [0061]    The active DSP system  61  for each horn could be replaced by modified “canned” tracks of the signal with the DSP filters applied to the waveform fed each respective horn. This would have the same effect as an active DSP but utilize independent processed and filtered tracks emulating the active DSP function without requiring the control DSP processing onboard. 
         [0062]    The flow chart of  FIG. 9  is a broad, high level depiction of this operation, occurring upon arming of the system by an operator at step  80 . At step  82  the matrix select controller  40  determines if the system is in automatic or manual mode. If manual an audio input signal can be buffered at step  84 . Next the presence of a trip condition, such as provided by proximity sensor  58  is checked for. If a trip condition is detected the operator is alerted (step  90 ) using the GUI  44  and operation returns to step  84  to continue buffering until the trip condition is cleared and the sound can be generated (step  88 ). 
         [0063]    Under automatic mode it may be determined if the locomotive is a leader(master) or follower (slave), step  92 ). As long as the unit is a slave it may be disabled by looping the test. If the unit is a master its operational status is displayed (step  94 ) and approach to a level grade crossing is monitored, as may be indicated by a receiving a response to a transponder signal (step  96 ). Alternatively, GPS unit  50  may be used for this function through use of location signals to continually interrogate the LUT  58  for a match. Once approach to an crossing is indicated ambient conditions are read (step  98 ), speed of approach to the crossing is determined (step  100 ), the LUT  58  is interrogated to fetch the proper alert (step  102 ) and the several DSP units  61  have configurations set (step  104 ) allowing the audible warning signal to be generated (step  106 ) and the process loops back to step  96 . 
         [0064]      FIGS. 10-22  are polar plots relating to directivity data and frequency response of line array  12  taken at a plurality of frequencies, particularly 315 Hz, 400 Hz, 500 Hz, 630 Hz, 800 Hz, 1000 Hz, 1250 Hz, 1600 Hz, 2000 Hz, 2500 Hz, 3150 Hz, 4000 Hz, 5000 Hz.  FIG. 23  is a composite view. A more distinct front directed major lobe appears with increasing frequency. 
         [0065]      FIG. 24  is a pressure and phase response plot against frequency for the front directed loudspeaker line array  12 . The plots were generated from 1024 samples in 5.5 seconds. The frequency resolution was 35.1 Hz and the time resolution was 28.46 ms (32.16 feet). 
         [0066]      FIG. 25  plots group delay against frequency at the same resolutions used in  FIG. 24 . 
         [0067]      FIG. 26  is an energy over time response over all frequencies for the line array. 
         [0068]      FIG. 27  is a response curve for a single impulse input over time for the line array. 
         [0069]      FIG. 28  is a response curve for a double impulse input over time for the line array. 
         [0070]      FIG. 29  is a graph of total harmonic distortion for the line array in terms of percentage for each frequency band. 
         [0071]      FIG. 30  is system frequency response with harmonic tracking overlay of second, third, fourth, fifth and sixth order distortion in decibels against frequency. 
         [0072]      FIGS. 31 through 34  illustrate directivity of a free standing horn loaded loudspeaker such as loudspeakers  14 ,  16 , or of line array  12  where only one loudspeaker is operated, such as loudspeaker  13 B at the frequencies of 250, 500, 1000 and 2000 Hz. 
         [0073]      FIGS. 35 through 38  illustrate directivity patterns over the same set of frequencies, 250 through 2000 Hz for line array  12  with all of loudspeakers  13 A-C receiving the same drive signals. The drive signal applied to the center loudspeaker  13 B is delayed by 1 millisecond in the bottom view for each frequency set. A delay is analogous to a phase delay except that the phase delay increases for each drawing pair due to the increasing drive signal frequency. Results at 500 and 1000 Hz indicate generation of extra lobes with suppression of a straight ahead primary lobe at 500 Hz. 
         [0074]      FIGS. 39 through 45  illustrate the use of delay and amplitude and/or frequency shading to shape sound lobes from a line array  12  and to steer the resulting lobes/beams.  FIG. 39  is a pair of polar graphs with baselines for the top directivity pattern at 1000 Hz and at 500 Hz, respectively.  FIG. 40  includes three directivity patterns. The top pattern is a pattern for a horn emission frequency of 250 Hz with center horn loaded loudspeaker  13 B of the line  12  delayed 1.1 milliseconds. The middle and bottom patterns are for 250 and 500 Hz. At 250 Hz the line array  12  exhibits a pronounced forward lobe, two side directed lobes and a very small rearward lobe. At 500 Hz the rearward directed lobe substantially disappears and the pair of left and right side lobes are folded forward and gain intensity. The forward major lobe is attenuated in comparison. 
         [0075]      FIG. 41  relate to directivity patterns at 1000 Hz (top) and 500 Hz (bottom) with differentiated delays applied to the center and the rightside of the side loudspeakers relative to the remaining side loudspeaker, respectively. The delays are 0.5 &amp; 1 ms. At 500 Hz the lobes are canted to the left and the right most lobe increases in strength relative to the left and forward lobes. 
         [0076]      FIG. 42  is labeled “steered+−6 −3 shading.” The directivity pattern is for an emission frequency of 500 Hz and reflect 0.5 millisecond and 1.0 millisecond delays to the center and right loudspeakers of a three speaker line array  12 . The output of the center and left loudspeakers is attenuated. The center speaker is attenuated −6 db and the left loudspeaker is attenuated −3 db. In  FIG. 43  the attenuation is increased −9 db for the center and −6 db for the left. This illustrates the use of selective amplitude shading to achieve substantial lobe steering and shaping. 
         [0077]      FIG. 44  is labeled “steered+−12 −9 shading.” It is also for 500 Hz. The center loudspeaker is attenuated by −12 db and the left by −9. The phase relationship is set by a center 0.5 msec delay and a right speaker delay of 1 msec. In  FIG. 45  the center loudspeaker is attenuated by −9 db and the left loudspeaker is shut off. There is no delay of the right loudspeaker with respect to the left.