Patent Publication Number: US-7917834-B2

Title: Apparatus and method for computing LLR

Description:
CROSS-REFERENCE TO-RELATED APPLICATION 
     This application claims priority to and the benefit of Korean Patent Application Nos. 2005-119388, filed Dec. 8, 2005, and 2006-87422, filed Sep. 11, 2006, the disclosures of which are incorporated herein by reference in their entirety. 
     BACKGROUND 
     1. Field of the Invention 
     The present invention relates to an apparatus and method for efficiently computing a log likelihood ratio (LLR) using an improved maximum a posteriori (MAP) algorithm known as block combining. 
     2. Discussion of Related Art 
     The ongoing development of communication systems has generated demand for a high-speed channel coding method reliable for information transmission. However, information loss is caused by noise in a communication channel, fading, interference, and so on. In order to minimize such information loss and correct errors, an error correction code is indispensable. Ever since Shannon published the results of his research in 1948, error correction codes have been widely studied. One type of error correction code known as turbo code, suggested by Berrou, Glavieux and Thitimaishima in 1993, has excellent error correction performance approaching Shannon&#39;s limit and thus is being actively researched. Decoding of turbo code restores the original information by repetitive decoding using a MAP decoder or soft-output Viterbi algorithm (SOVA) decoder. In comparison with a SOVA algorithm, a MAP algorithm is more complex but has superior bit error rate (BER) performance and thus is widely used. 
     The MAP algorithm, first suggested by Bahl, et al. in 1974, calculates an a posteriori probability (APP) from a signal mixed with noise. The MAP decoding algorithm is aimed at determining information bits having the highest probability with respect to a received symbol after data is received.  FIG. 1  is a 4-state trellis diagram from time (k−1) to time (k+1). 
     In  FIG. 1 , alpha (α) is called a forward state metric (FSM) and denotes a state metric for transition of information bits from state S′ before time (k−1) to state S after time k. Beta (β) is called a backward state metric (BSM), and a current beta value can be calculated by repeatedly using a beta value of a previous state after all information is received in the same way as alpha. Gamma (γ) is defined as a branch metric (BM). 
     In order to calculate an LLR, a gamma value is calculated using received data, and then an alpha value and a beta value are calculated using the calculated gamma value. In this process, Formulae 1-3 given below are used: 
     
       
         
           
             
               
                 
                   
                     
                       α 
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                   ⁢ 
                   1 
                 
               
             
             
               
                 
                   
                     
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                   ⁢ 
                   
                       
                   
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                   2 
                 
               
             
             
               
                 
                   
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                     ⁢ 
                     
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                   3 
                 
               
             
           
         
       
     
     Here, U k  is a data bit and V k  is a parity bit. X k  is a data bit mixed with noise passed through a channel, and y k  is a parity bit mixed with noise passed through a channel. When alpha, beta and gamma values are calculated, an LLR can be calculated using Formulae 4 and 5: 
     
       
         
           
             
               
                 
                   
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                     ⁡ 
                     
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                   4 
                 
               
             
             
               
                 
                   
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                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
               
             
           
         
       
     
     Formulae 4 and 5 enable calculation of LLRs at time k and time (k+1), respectively. Using these formulae, information bits having the highest probability of transition from a previous state to a current state are determined. 
     However, the MAP algorithm described above requires a large memory size and a significant amount of calculation. Thus, the MAP algorithm imposes heavy restrictions on system design and drives up the cost of system building. 
     In order to solve the above problem of the MAP algorithm requiring a large memory size, a block processing algorithm has been suggested. The block processing algorithm is a MAP algorithm capable of more efficiently using a memory according to a principle described below. 
       FIG. 2  is a 4-state trellis diagram employing a block processing technique from time (k−1) to time (k+1) according to conventional art. The algorithm using the block processing technique calculates alpha values and beta values from time (k−1) to time (k+1), at time (k+1) rather than at time k. Thus, the algorithm can reduce the amount of memory required to store an alpha value and a beta value at time k in the middle of the process. Alpha and beta values are calculated by Formulae 6 to 8 given below: 
     
       
         
           
             
               
                 
                   
                     
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                   6 
                 
               
             
             
               
                 
                   
                     
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                   7 
                 
               
             
             
               
                 
                   
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                   8 
                 
               
             
           
         
       
     
     The algorithm using the block processing technique reduces the number of data access operations by efficiently using a memory according to the conventional MAP algorithm, thereby reducing a required memory size and power. However, the algorithm performs more multiplication operations than the conventional MAP algorithm in LLR calculation, and thus decoding speed is reduced. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to an apparatus and method for computing a log likelihood ratio (LLR) capable of increasing decoding speed while improving memory use efficiency in turbo decoding. 
     The present invention is also directed to an apparatus and method for computing an LLR capable of maintaining decoding efficiency while allowing a simplified hardware configuration. 
     Thus, the present invention aims to provide an apparatus and method for computing an LLR capable of minimizing multiplication operations and comparison operations. 
     A method for computing an LLR according to the present invention is based on the spirit of: using the above-described block processing; in order to calculate an LLR, selecting and using one of a plurality of LLR calculation formulae according to the highest transition probability of respective states instead of using a complicated conventional formula. 
     That is, in order to obtain the highest transition probability of respective states, performing a comparison operation for some of the transition probabilities to determine the highest, selecting one of the other transition probabilities according to the determined high value, and comparing the determined value with the selected value to select the higher value. 
     One aspect of the present invention provides a method for computing an LLR, comprising the steps of: calculating alpha values, beta values and gamma values of at least two time sections; calculating transition probabilities in the at least two time sections using the alpha values, beta values and gamma values; performing a comparison operation between some of the transition probabilities to determine a high value, selecting one of the other transition probabilities according to the determined high value, comparing the determined high value with the selected value to select a higher value, and thereby obtaining the highest of the transition probabilities; and determining an operation to apply according to the highest transition probability and calculating an LLR. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other features and advantages of the present invention will become more apparent to those of ordinary skill in the art by describing in detail preferred embodiments thereof with reference to the attached drawings in which: 
         FIG. 1  is a 4-state trellis diagram from time (k−1) to time (k+1) according to conventional art; 
         FIG. 2  is a 4-state trellis diagram employing a block processing technique from time (k−1) to time (k+1) according to improved conventional art; 
         FIG. 3  is a 4-state combined-block trellis diagram from time (k−1) to time (k+1) according to an exemplary embodiment of the present invention; 
         FIG. 4  is a flowchart showing a process of obtaining the highest transition probability according to an exemplary embodiment of the present invention; 
         FIG. 5  is a block diagram of an apparatus for computing a log likelihood ratio (LLR) according to an exemplary embodiment of the present invention; and 
         FIG. 6  is a circuit diagram of a transition probability calculator shown in  FIG. 5  according to an exemplary embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Hereinafter, exemplary embodiments of the present invention will be described in detail. However, the present invention is not limited to the embodiments disclosed below, but can be implemented in various forms. Therefore, the following embodiments are described in order for this disclosure to be complete and enabling to those of ordinary skill in the art. The embodiments are described with a detailed process of calculating a log likelihood ratio (LLR) for 4-state transition in two time sections. 
     A method for computing an LLR according to an exemplary embodiment first calculates an alpha value (forward state metric (FSM)), a beta value (backward state metric (BSM)) and a gamma value (branch metric (BM)) for each state of two time sections from time (k−1) to time (k+1). 
     In this embodiment, a block combination method is used to reduce multiplication operations in a maximum a posteriori (MAP) algorithm employing a block processing method.  FIG. 3  is a 4-state trellis diagram from time (k−1) to time (k+1).  FIG. 3  illustrates a process of combining two decoding processes into one and performing decoding. Using a gamma value BM defined for every state, an alpha value FSM and a beta value BSM are calculated. The gamma value BM defined for every state is calculated as the product of a gamma value BM from time (k−1) to time k and a gamma value BM from time k to time (k+1). The calculations are made according to Formulae 9 to 11 given below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                           k 
                           
                             
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                             k 
                             
                               
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                         = 
                           
                         ⁢ 
                         
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                           ⁢ 
                           
                             { 
                             
                               
                                 2 
                                 
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                   Formula 
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                   ⁢ 
                   9 
                 
               
             
             
               
                 
                   
                     
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                   10 
                 
               
             
             
               
                 
                   
                     
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                   11 
                 
               
             
           
         
       
     
     γ k   s′,s″  is the combined gamma value obtained by taking the product of the gamma value for transition from state S′ at time (k−1) to state S at time k with the gamma value for transition from state S at time k to next state S″. Using the combined gamma value, an alpha value and beta value at time (k+1) are obtained. Here, the alpha value and beta value are calculated by the same calculation method as a conventional MAP algorithm. More specifically, alpha denotes a state metric to make transition from all the states at time (k−1) to one state at time (k+1). Likewise, beta is obtained by repetitively using a beta value of a previous state. 
     Otherwise, alpha and beta may be obtained by the same method as the block processing algorithm described above as improved conventional art using Formula 6 to Formula 8 given above. 
     When alpha values, beta values and gamma values of respective states are obtained, transition probabilities of the respective states in two time sections are calculated using the obtained values. 
     When data (0, 0) is received, transition is made from all the four states at time (k−1) to state ‘0’ at time (k+1). When data (0, 1) is received, transition is made to state ‘1’ at time (k+1). When data (1, 0) is received, transition is made to state ‘2’ at time (k+1). When data (1, 1) is received, transition is made to state ‘3’ at time (k+1). Probabilities of transition to the respective states are calculated by Formulae 12 to 15 given below: 
     
       
         
           
             
               
                 
                   
                     p 
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                   12 
                 
               
             
             
               
                 
                   
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                   13 
                 
               
             
             
               
                 
                   
                     p 
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                   14 
                 
               
             
             
               
                 
                   
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                           k 
                         
                         = 
                         
                           11 
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                             : 
                           
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                             k 
                           
                         
                       
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                   = 
                   
                     
                       ∑ 
                       
                         1 
                         , 
                         1 
                       
                     
                     ⁢ 
                     
                       
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                           - 
                           1 
                         
                       
                       ⁢ 
                       
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                             s 
                             ′ 
                           
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                             s 
                             ″ 
                           
                         
                       
                       ⁢ 
                       
                         β 
                         
                           k 
                           + 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     p(u k =00:y k ) denotes a probability that when data y k  is received, the received data value is (0, 0), and p(u k =01:y k ) denotes a probability that when data y k  is received, the received data value is (0, 1). Likewise, p(u k =10:y k ) denotes a probability that received data value is (1, 0), and p(u k =11:y k ) denotes a probability that received data value is (1, 1). 
     After the respective probabilities are obtained, the highest among them is determined. The reason why the highest is determined is because when an LLR is calculated by determining a path of transition from time (k−1) to time (k+1), the calculation process varies according to the highest. It is possible to further simplify the process of obtaining the highest using the characteristics of a turbo decoding algorithm. The simplified method of obtaining the highest value is described below. 
       FIG. 4  is a flowchart showing a process of obtaining the highest probability by determining the highest transition probability according to this embodiment. As illustrated in  FIG. 4 , the process of obtaining the highest probability may be divided into the following detailed steps: 
     a first step of comparing a transition probability to (0, 0) with a transition probability to (0, 1) and selecting the higher value; 
     a second step of selecting a transition probability to (1, 0) when the transition probability to (0, 0) is more than the transition probability to (0, 1), and selecting a transition probability to (1, 1) when the transition probability to (0, 1) is more than the transition probability to (0, 0); and 
     a third step of selecting the higher one of the selected values of the first step and the second step. 
     In the first step, A 1  is compared with A 2 , and the higher value is selected and stored as r 1 . When A 1  is selected in the first step, A 3  is stored as r 2 . On the contrary, when A 2  is selected in the first step, A 4  is stored as r 2 . In the third step, r 1  is finally compared with r 2 , and the higher value is determined and output. 
       FIG. 6  illustrates a circuit structure capable of performing the process of obtaining the highest probability. In  FIG. 5 , among four data A 1 (0, 0), A 2 (0, 1), A 3 (1, 0) and A 4 (1, 1), A 1 (0, 0) is compared with A 2 (0, 1) by MUX 1 , and one of them is selected. When MUX 1  selects A 1 (0, 0), a control signal automatically selects A 3 (1, 0). On the contrary, when MUX 1  selects A 2 (0, 1), the control signal automatically selects A 4 (1, 1). MUX 2  selects the higher one of a value selected by MUX 1  and a value selected by the control signal and outputs it. 
     A 1 (0, 0) is the sum of a probability that first decoding data is ‘0’ and a probability that second decoding data is ‘0.’ In other words, A 1 (0, 0) is the sum of a probability that L(u k ) is 0 and a probability that L(u k+1 ) is 0. A 2 (0, 1) is the sum of a probability that first decoding data is ‘0’ and a probability that second decoding data is ‘1.’ A 3 (1, 0) is the sum of a probability that first decoding data is ‘1’ and a probability that second decoding data is ‘0.’ A 4 (1, 1) is the sum of a probability that first decoding data is ‘1’ and a probability that second decoding data is ‘1.’ 
     When MUX 1  selects A 1 (0, 0), the control signal selects A 3 (1, 0) because A 1 (0, 0), which is the sum of the probability that L(u k ) is 0 and the probability that L(u k+1 ) is 0, is higher than the sum of the probability that L(u k ) is 0 and the probability that L(u k+1 ) is 1. In other words, since the probability that L(u k+1 ) is 0 is higher than the probability that L(u k+1 ) is 1 while the same probability that L(u k ) is 0 is added, the comparison of the probability that L(u k+1 ) is 1 is removed from the next step and only the probability that L(u k ) is 0 needs to be compared with the probability that L(u k ) is 1. In the result, when A 1 (0, 0) is selected, the control signal selects A 3 (1, 0) because only the probability that L(u k ) is 1 needs to be compared with the probability that L(u k ) is 0. 
     In the same manner, when MUX 1  selects A 2 (0, 1), the control signal selects A 4 (1, 1). This means that A 2 (0, 1), which is the sum of the probability that L(u k ) is 0 and the probability that L(u k+1 ) is 1, is higher than the sum of the probability that L(u k ) is 0 and the probability that L(u k+1 ) is 0. In other words, since the probability that L(u k+1 ) is 1 is higher than the probability that L(u k+1 ) is 0 while the same probability that L(u k ) is 0 is added, the comparison of the probability that L(u k+1 ) is 0 is removed from the next step and only the probability that L(u k ) is 0 needs to be compared with the probability that L(u k ) is 1. In the result, when A 2 (0, 1) is selected, the control signal selects A 4 (1, 1) because only the probability that L(u k ) is 1 needs to be compared with the probability that L(u k ) is 0. 
     When the highest probability is determined by the above-described process, an LLR is obtained by a method employing different formulae according to the highest probability. An LLR is calculated by Formulae 16 to 19 given below: 
     
       
         
           
             
               
                 
                   
                     case 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                     ⁢ 
                     
                       ( 
                       
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                         ⁡ 
                         
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                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   16 
                 
               
             
             
               
                 
                   
                     case 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     
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                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   17 
                 
               
             
             
               
                 
                   
                     case 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
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                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   18 
                 
               
             
             
               
                 
                   
                     case 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
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                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   19 
                 
               
             
           
         
       
     
     L(u k ) is a value decoded at time k, and L(u k+1 ) is a value decoded at time (k+1). In L(u k ) of casel, p(u k =00: y k ) is obtained by multiplying a probability of receiving ‘0’ and making transition from time (k−1) to time k by a probability of receiving ‘0’ and making transition from time k to time (k+1). Thus, a value decoded at time (k+1) should be counterbalanced. Consequently, p(u k =10: y k ) is divided by p(u k =00: y k ) as shown in Formula 16. 
     In this manner, the decoded value L(u k ) can be obtained. Likewise, L(u k+1 ) can be also obtained by counterbalancing a probability of transition from time (k−1) to time k. In the same way, L(u k ) and L(u k+1 ) are calculated in case 2 , case 3  and case 4  as well. 
     Meanwhile, Formula 12 may be simplified as shown in Formula 20 given below: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           p 
                           ⁡ 
                           
                             ( 
                             
                               
                                 u 
                                 k 
                               
                               = 
                               
                                 00 
                                 ⁢ 
                                 
                                   : 
                                 
                                 ⁢ 
                                 
                                   y 
                                   k 
                                 
                               
                             
                             ) 
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               0 
                               , 
                               0 
                             
                           
                           ⁢ 
                           
                             
                               α 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
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                                   ″ 
                                 
                               
                             
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                         = 
                           
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                                 ( 
                                 0 
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                             ⁢ 
                             
                               γ 
                               
                                 0 
                                 , 
                                 0 
                               
                             
                             ⁢ 
                             
                               
                                 β 
                                 
                                   k 
                                   + 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
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                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
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                                   - 
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                                 ( 
                                 1 
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                             ⁢ 
                             
                               γ 
                               
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                                 , 
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                             ⁢ 
                             
                               
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                                   k 
                                   + 
                                   1 
                                 
                               
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                                 ( 
                                 0 
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                           + 
                           
                             
                               
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                                   k 
                                   - 
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                               ⁡ 
                               
                                 ( 
                                 2 
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                             ⁢ 
                             
                               γ 
                               
                                 2 
                                 , 
                                 0 
                               
                             
                             ⁢ 
                             
                               
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                                   k 
                                   + 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 0 
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                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               α 
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               3 
                               ) 
                             
                           
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                               3 
                               , 
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                           ⁢ 
                           
                             
                               β 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         
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                                             k 
                                             - 
                                             1 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           0 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         γ 
                                         
                                           0 
                                           , 
                                           0 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           α 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           1 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         γ 
                                         
                                           1 
                                           , 
                                           0 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           α 
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         ⁡ 
                                         
                                           ( 
                                           2 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         γ 
                                         
                                           2 
                                           , 
                                           0 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         α 
                                         
                                           k 
                                           - 
                                           1 
                                         
                                       
                                       ⁡ 
                                       
                                         ( 
                                         3 
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       γ 
                                       
                                         3 
                                         , 
                                         0 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           × 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             β 
                             
                               k 
                               + 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             0 
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               α 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                           × 
                           
                             
                               β 
                               
                                 k 
                                 + 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Formula 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   20 
                 
               
             
           
         
       
     
     In Formula 20, since an alpha value at time (k+1) includes alpha values of all previous states, a probability that received data is (0, 0) can be simplified using only alpha and beta of a current state. Likewise, Formulae 13 to 15 can be simplified. 
     An exemplary embodiment of an apparatus for computing LLR to perform the above-described LLR computing method is shown in  FIG. 5 . The illustrated apparatus for computing an LLR comprises an FSM calculator  140 , a BSM calculator  160 , a BM calculator  120 , a transition probability calculator  220 , a highest probability determiner  240 , and an LLR calculator  260 . The FSM calculator  140  calculates alpha values of at least two time sections. The BSM calculator  160  calculates beta values of the at least two time sections. The BM calculator  120  calculates gamma values of the at least two time sections. The transition probability calculator  220  calculates transition probabilities of respective states in the at least two time sections using the alpha, beta and gamma values. The highest probability determiner  240  determines the highest of the transition probabilities. The LLR calculator  260  performs an operation specified according to the highest probability, thereby calculating an LLR. 
     The BM calculator  120  calculates gamma values of the at least two time sections using Formula 9 according to the block processing algorithm. 
     The FSM calculator  140  receives the gamma values calculated by the BM calculator  120  and applies Formula 11 according to the block processing algorithm, thereby calculating alpha values of at least two time sections. 
     The BSM calculator  160  receives the gamma values calculated by the BM calculator  120  and applies Formula 10 according to the block processing algorithm, thereby calculating beta values of the at least two time sections. 
     The transition probability calculator  220  receives the gamma, alpha and beta values calculated by the BM calculator  120 , the FSM calculator  140  and the BSM calculator  160  and applies Formulae 12 to 15, thereby calculating transition probabilities. 
     An exemplary embodiment of the highest probability determiner  240  used in the LLR computing apparatus of  FIG. 5  is shown in  FIG. 6 . The illustrated highest probability determiner  240  includes a first comparator  242 , a switch  244 , and a second comparator  246 . The first comparator  242  compares a transition probability to (0, 0) with a transition probability to (0, 1) and selects the higher value. The switch  244  selects a transition probability to (1, 0) when the value selected by the first comparator  242  is (0, 0) and selects a transition probability to (1, 1) when the value selected by the selected by the first comparator  242  with the value selected by the switch  244  first comparator  242  is (0, 1). The second comparator  246  compares the value and selects the higher value. 
     The LLR calculator  260  selects and applies one of Formulae 16 to 19 according to the determination result of the highest probability determiner  240 , thereby calculating an LLR. 
     Operation of the LLR calculator illustrated in  FIG. 5  can be easily deduced from the description of the present invention and thus will be omitted. 
     According to the LLR computing method and apparatus having the above-described structure, it is possible to efficiently calculate an LLR in an LLR calculation structure. 
     While the invention has been shown and described with reference to certain exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.