the preferred embodiments in accordance with the present invention will now be described with reference to accompanying drawings . fig3 shows an in - phase averaging method used for carrying out synchronization establishment in the present embodiment . in the in - phase averaging , the output of the correlator is averaged in - phase in complex numbers , followed by powerizing the average by a power detector 207 . generally , the in - phase averaging has higher averaging effect against noise and interference than the power averaging . fig4 and 5 illustrate in - phase averaging utilizing averaging windows . it is usually difficult to estimate the phase of the output of the correlator because of sharp fluctuations in the phase of a signal on a propagation path of the mobile communication . the phase fluctuations , however , are small during an interval short enough as compared with the speed of the phase fluctuations due to fading on the propagation path . accordingly , in - phase averaging in short intervals will bring about averaging effect higher than the conventional power averaging method . in the example as shown in fig4 , the averaging window range n for averaging the correlation values 201 is three slots , and the moving step m of the averaging windows is one slot . within the averaging window range , the output of the correlator is averaged in phase in complex number . after that , to eliminate the effect of the phase fluctuations between the windows , the powerizing is carried out , followed by further averaging between the windows , thereby producing a result with higher reliability . in the example as shown in fig5 , the averaging window range n and the moving step m of the averaging windows are both three slots . placing n = m can reduce the buffer size needed for the in - phase averaging of the mobile station . thus , although the example of fig4 requires the buffer size of three slots for the in - phase averaging , the example of fig5 requires the buffer size of only one slot . fig6 shows an example of carrying out the in - phase averaging with assigning weights within each averaging window in the in - phase averaging utilizing averaging windows as shown in fig4 . here , w 0 , w 1 and w 2 designate weight factors of the averaging . decreasing the weight factors toward the ends of each averaging window can reduce the effect of the phase fluctuations of the correlation values due to fading . fig7 shows an example of exponential weighting in - phase averaging . here , λ is a weight factor taking a value between zero and one . as the λ increases , the averaging period becomes shorter . on the contrary , as the λ decreases , the averaging period is increased because the number of the past correlation values to be considered is increased in this case . in the exponential weighting in - phase averaging , the number of buffers required for the in - phase averaging is only one . in the example of fig7 , the power averaging after the in - phase averaging is carried out at every m = 1 slot interval . as the m is increased , the averaging effect will become smaller . however , since the amount of the calculation can be reduced in this case , the m can be increased when the calculation amount must be reduced because of hardware management . fig8 and 9 each show an example of differentially coherent detection type in - phase averaging . in the differentially coherent detection type in - phase averaging , the phase of the correlation value in a given slot is corrected using the correlation values of previous l slots as a reference signal ( returning the phases to zero ). such phase correction can align the phases of the correlation values . averaging these values in the complex numbers , and powerizing the result ( that is , extracting the real part of the result ) by an operator 803 make it possible to carry out highly reliable correlation detection . in the example of fig8 , the differentially coherent detection type in - phase averaging is performed at l = 1 , and in that of fig9 , it is carried out at l = 2 . fig1 shows a method of assigning weights in the differentially coherent detection type in - phase averaging as shown in fig9 . as the l increases ( that is , the time difference from the reference signal is large ), the effect of the phase fluctuations of the correlation values due to fading can be reduced by decreasing the weight factors . fig1 shows an example of the in - phase averaging utilizing corresponding averaging windows , when tstd is applied to the sch , that is , when tstd - on . the mobile station has buffers for signals sent from respective antennas of a base station , and carries out in - phase averaging of the correlation values of the sch that is assumed to be sent from the same antenna . in other words , the in - phase averaging is carried out every other slot . by preparing the buffers for respective antennas , the mobile station can carry out the in - phase averaging every other slot not only when using the averaging windows , but also when performing the exponential weighting in - phase averaging or differentially coherent detection type in - phase averaging . the mobile station may recognize whether the tstd is applied to the mobile communication system or not . in the mobile communication system in which each base station can select the application of the tstd , the mobile station cannot recognize whether the tstd is applied or not at least when the mobile station is turned on . when tstd - off / on is unknown , the mobile station can carry out one of the following three methods as the synchronization establishment operation . a first method assumes both tstd - off and tstd - on , and combines or selects the averaging results of the two corresponding methods . fig1 shows an example of a system for implementing the synchronization establishing method according to the first method . in the example as shown in fig1 , an averaging circuit 1203 assuming the tstd - off and an averaging circuit 1205 assuming the tstd - on are connected to a correlator 1201 , and the correlation output is produced by combining or selection using a correlation value combiner / selector 1207 . the combining of the correlation values can be achieved by assigning predetermined weights to the outputs of the two averaging circuits , followed by summing them up . on the other hand , the selection of the correlation values can be achieved by assigning predetermined weights to maximum correlation values obtained by the two averaging methods , followed by comparing them , and by selecting the output result of the averaging circuit that will provide a greater correlation value . a second method switches between an averaging method assuming the tstd - off and an averaging method assuming the tstd - on . fig1 shows a system for implementing the second synchronization establishing method . in fig1 , the reference numeral 1303 designates a controller for controlling the selection of one of the averaging circuits by switching the connection between a correlator 1301 and the two averaging circuits . as a switching control method , there is a switching method of the averaging circuits according to the number of selections of the averaging circuits . for example , when the number of trials of the synchronization establishment is odd , the averaging by the averaging circuit 1203 assuming the tstd - off is carried out , whereas when it is even , the averaging by the averaging circuit 1205 assuming the tstd - on is carried out so as to obtain an averaged result . alternatively , in the first to nth trial of the synchronization establishment , the averaging assuming the tstd - off is carried out , and if the synchronization is not established up to the end of the n consecutive trials , a decision is made that tstd - on . thus , the averaging assuming the tstd - on is carried out from the ( n + 1 ) th trial onward , enabling the method to try the tstd - off in - phase averaging with higher effect , first . a third method comprises a tstd decision section 1403 for deciding the tstd - off / on between a correlator 1401 and a controller 1405 as shown in fig1 , makes a tstd - off / on decision before the averaging , and carries out switching of the averaging method in response to the decision result . the controller 1405 decides as to which one of the averaging circuits is to be used according to the decision result of the tstd decision section 1403 . as the tstd - off / on decision method , such a method can be adopted that calculates correlation between correlation values of adjacent slots , makes a decision as the tstd - on when the correlation value is larger than a predetermined value , and as the tstd - off when the correlation value is smaller . alternatively , there is a method in which the visited location cell notifies the tstd decision section 1403 of the tstd - off / on by means of a control signal or the like when the mobile station is in the idle state or traffic state . in the in - phase averaging , as the number of the averaging period increases , that is , as the number of slots used for the averaging increases , the averaging effect of the noise and interference will be improved in general . however , since the phase of the signal fluctuates momentarily because of fading or the like on the propagation path of the mobile communication , too long an averaging period can deteriorate the peak detection characteristic because of the phase fluctuations . thus , there is an optimum averaging period depending on the state of the propagation path , the moving speed of the mobile station , etc . accordingly , it is important to set an appropriate averaging period . the following two methods can be conceived as a method of setting the averaging period in the in - phase averaging . a first method carries out n ways of averaging of the output of the correlator 1501 over n different averaging periods t 0 − t n − 1 using n averaging circuits 1503 - 1 - 1503 - n as illustrated in fig1 , and combines or selects the results by a correlation value combiner / selector 1505 to supply the peak detection circuit 205 with the result . as a combining method , a method can be adopted that assigns weights to the outputs of the averaging circuits 1503 at respective timings , followed by summing them up . as a selection method , there is a method that assigns weights to maximum correlation values of the averaging circuits 1503 - 1 - 1503 - n , followed by comparing them and selecting the output of the averaging circuit that gives the maximum correlation value as the result of the comparison . a second method adaptively varies the averaging period over which the output of the correlator 1601 is averaged . more specifically , it detects the speed of the mobile station by a speed detector 1603 as shown in fig1 . in response to the detection result , it controls the averaging period by an averaging period controller 1605 of the averaging circuit 1203 . the state of the mobile station carrying out the synchronization establishment can be roughly divided into three states : a power up state , an idle state , and a traffic state . these states will vary conditions involved in the synchronization establishing operation such as the intensity of the received signal and the magnitude of the frequency drift at the base station with which the mobile station tries to establish synchronization . therefore , selecting the appropriate synchronization establishing method depending on the state , the mobile station can achieve the synchronization establishment at higher reliability , in a shorter time . thus , it is generally effective to control the averaging period of the in - phase averaging according to the states . in addition , at power up , it is usually difficult for the mobile station to carry out the in - phase averaging because it receives no signal from the base station , and its frequency stability is generally low . in such a case , it is effective for the mobile station to take power average during power up , and to carry out in - phase averaging in the idle state or traffic state . even during the idle state , the frequency stability of the mobile station can be low because of the power saving . in such a case , it is preferable for mobile station to use power average during the power up and idle state , and to carry out the in - phase averaging only in the traffic state . furthermore , since the mobile station tries to establish synchronization with a signal sent from a base station other than the base station in the traffic state in the synchronization establishing operation during the traffic state , the signal from the base station in the traffic state becomes interference , and hence the s / n becomes small of the signal from the base station with which the mobile station tries to establish synchronization . therefore , it is effective for the mobile station to employ the in - phase averaging only during the traffic state , and to use the rather simple power averaging method during the power up and idle state considering to simplify hardware configuration of the mobile station or to promote the power saving . the present invention has been described in detail with respect to preferred embodiments , and it will now be apparent from the foregoing to those skilled in the art that changes and modifications may be made without departing from the invention in its broader aspect , and it is the intention , therefore , in the apparent claims to cover all such changes and modifications as fall within the true spirit of the invention .