# Patent

## Title
Method and arrangement for sample-rate conversion

## Abstract

Method and arrangement for converting the sample rate of a higher sample rate discrete time signal to a lower sample rate discrete time signal or vice versa. A recursive signal processing algorithm with low pass filtering function is used, which entirely takes place at the lower sample rate. The impulse response of the low pass filtering function is a power series expansion

## Background
The invention relates to a method of converting the sample rate of a discrete-time input signal with sample rate q.fs to a discrete-time output signal with a sample rate fs which is a sub multiple of the sample rate of the input signal (q=integer larger than 1), the method comprising the steps of series to parallel converting the input signal to generate non-overlapping words of q samples of the input signal and of processing said words at the sub multiple rate fs in accordance with a recursive algorithm comprising a low-pass filtering function. Such method is known from the article “Recursive Bitstream Conversion” by E. Roza in IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal-Processing, Vol. 40, NO 2, February 1993. The invention further relates to the reverse method of converting the sample rate of a discrete-time input signal with sample rate fs to a discrete-time output signal with a sample rate q.fs which is a multiple of the sample rate of the input signal (q=integer larger than 1), the method comprising the steps of processing said input signal at the sample rate fs in accordance with a recursive algorithm comprising a low-pass filtering function to obtain non-overlapping words of q samples of the output signal and of parallel to series converting said words to generate the discrete-time q.fs sample rate output signal.
The above-described methods have become known under the name: “Rebic” (recursive bit-stream conversion) and in this application the expression “forward Rebic” shall be used for the conversion of a low-precision high sample rate signal to a high precision low sample rate signal, while the expression “reverse Rebic” shall be used for the conversion of a high precision low sample rate signal to a low precision high sample rate signal. The low precision high rate signal may often be a “single bit” bitstream, where each sample comprises only one bit, although this signal occasionally may comprise more than one bit per sample.
Usually the generation of a “single bit” bitstream, either from an analog input signal or from a high precision low sample rate discrete signal, is done by a conventional ΣΔ modulator, which basically consists of a low pass filter and a clock-synchronized quantizer in feedback arrangement. This is a low cost and robust arrangement for performing the conversion. However, when signals with large base-bandwidth, such as e.g. video signals, have to be converted, the sample rate has to be extremely high in order to obtain sufficient noise shaping and then several drawbacks of the conventional ΣΔ modulator become apparent. The basic advantage of the above-described Rebic configuration is that it has a loop-configuration that operates at the lower clock rate, which is a fraction (1/q) of the clock rate that is required in conventional ΣΔ modulators. An out-of-loop parallel-to series converter is necessary and sufficient to produce a serial bitstream with similar properties as obtained from conventional configurations. The advantages of Rebic over conventional ΣΔ modulators are: reduction of high-speed circuitry, lower power consumption, less interference tones and increased stability in high-order configurations.
In the above referenced article the low-pass filter operation is achieved by an expansion of its impulse response into a series of exponential functions. In contradistinction, the present invention is based on the recognition that such exponential expansion is not obligatory and the method of sample rate conversion according to the present invention is therefore characterized in that the algorithm is based on a power-series expansion of the impulse-response of the low-pass filtering function. Such impulse responses may be obtained from discrete-time integrators and can therefore be obtained from actual digital circuitry more easily than conventional impulse responses. Preferably, the method according to the invention, both for the forward Rebic conversion and for the reverse Rebic conversion, is characterized in that the said processing is in accordance with the formulas (1) to (3e) of the accompanying formula sheet, wherein an represents the sequence of samples of the discrete-time signal with sample rate q.fs, wherein bj represents the sequence of samples of the discrete time signal with sample rate fs, wherein αm represent the coefficients of the power series of the impulse response of the low-pass filtering function and αm′ the said coefficients divided by the sum of all said coefficients. In this algorithm it is assumed that the impulse response of the low pass filtering function




  
    
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      ⁡
      
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          (
          t
          )
        
      
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            m
            =
            0
          
          M
        
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  ,




which implies that the impulse response is zero at time t=0. Hereafter it will be shown that also other power series expansions of the impulse response may be used.

The invention also relates to an arrangement for sample rate conversion, which arrangement then usually comprises a number of integrators (I2-I3, I4-I5-I6) in cascade i.e. that a former integrator in the cascade sends its output signal, together with one or more other signals, to the next integrator in the cascade. The Rebic structure may be of certain order, which is the order of the low pass filter function used in the Rebic algorithm. Normally the number of cascaded integrators in a low pass filter determines the order of the filter. This also applies to the Rebic algorithm. However it appears that, in case of a forward Rebic structure, one of the integrators is redundant so that the arrangement may preferably be characterized in that the number of integrators in the cascade of integrators is one less than the order of the algorithm.
It appears that in a reverse Rebic structure according to the invention it is not possible to calculate the samples an of the high sample rate signal directly from the samples of the low sample rate signal. From equation (3c) of the enclosed formulas sheet it follows that the sequence an is hidden in the sum Σαm′Amj, which is herein called the “bunch value” at the time j. In order to derive said sequence the reverse basic structure according to the invention is further characterized by comprising a mapper receiving a mapper input signal from said integrators in cascade and generating said non-overlapping words of q samples of the output signal. The task of the mapper is to establish a q-length binary sequence an such that the bunch value Σαm′Amj is close to the mapper input sequence. Because there are 2q possible binary sequences of length q, there are also 2q values of the bunch value to which the mapper input sequence will be mapped. One possible strategy could be to compare the most recent value of the mapper input sequence with a predefined table of all possible bunch values on the basis of a best fit. However, this can be quite cumbersome for large values of q. Therefore a preferred reverse Rebic structure according to the invention may preferably be characterized in that said mapper comprises a cascade of quantizer-subtracter combinations, each of said combinations comprising a 1-bit quantizer for generating one of the bits of said non-overlapping words of q bits and a subtracter for subtracting the input- and output-signals of the 1-bit quantizer, that the first of said combinations in the cascade receives the mapper input signal, that each of the other of said combinations in the cascade receives a weighed sum of the output signals of the subtracters of the previous combinations in the cascade and that a final quantizer receives a weighed sum of the output signals of all subtracters in the cascade and generates the final bit of said non overlapping words of q bits.
Further, in order to prevent the Rebic structure from becoming unstable, the arrangement according to the invention may further be characterized in that at least one of said integrators comprises a clipper.



The invention will be described with reference to the accompanying formula sheet and to the accompanying figures. Herein shows:
FIG. 1 an embodiment of a forward Rebic sample rate converter according to the invention,
FIG. 2 an embodiment of a reverse Rebic sample rate converter according to the invention,
FIG. 3 an embodiment of a mapper for use in the reverse Rebic sample rate converter of FIG. 2,
FIG. 4 a modification of part of the reverse Rebic sample rate converter of FIG. 2 and
FIG. 5 a modification of an integrator for use in a Rebic sample rate converter according to the invention.



The algorithm used for the sample rate conversion is based on equating the response of the low pass filter operation to the high sample rate signal an with the response of the said low pass filter operation to the low sample rate signal bj. The impulse response of the low pass filter operation is an expansion in terms of a power series and may be expressed by the following formula:




  
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        M
      
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              α
              m
            
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                /
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        .
      
    
  




Herein is U(t) Heaviside's step function, M defines the order of the filter and αm are the filter coefficients that can freely be chosen. When the two mentioned responses are equated the formula (1) of the formula sheet is obtained. In formula (2) this equation is split into four terms respectively representing from left to right: the most recent sample of the signal bj, the filtered contribution from the q most recent samples of the signal an, the filtered history of the signal an and finally the filtered history of the signal bj.

In formula (2a) the coefficients αm are replaced by the normalized coefficients αm′ that are obtained by dividing each coefficient αm by the sum of all coefficients, with the result that




  
    
      ∑
      
        m
        =
        0
      
      M
    
    ⁢
    
      α
      m
      ′
    
  
  =
  1.




When Aij and rij are defined as shown in formulas (3a) and (3b), the formula (2a) for bj becomes





  
    b
    j
  
  =
  
    
      ∑
      
        m
        =
        0
      
      M
    
    ⁢
    
      
        α
        m
        ′
      
      ⁡
      
        (
        
          
            A
            mj
          
          +
          
            r
            mj
          
        
        )
      
    
  




as is shown in formula (3c). The value rmj may be recursively expanded so as to obtain the equation (3d) wherein the signal Tij is defined by equation (3e). The symbol (im) in equation (3d) stands as usual for the binomial coefficient





  
    
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          (
          
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  .




The set of equations (3a) . . . (3e) easily allows to determine a structure for implementing the algorithm, as will be shown afterwards with reference to FIGS. 1, 2 and 3.

The value of M determines the order of the Rebic. A 3rd order Rebic is obtained when M=2, a 4th order is obtained when M=3 and so on. Usually a filter of certain order requires an equal number of cascaded integrators. However, it appears that, in a forward Rebic, one of such integrators is redundant so that the number of cascaded integrators can be one less than the order of the Rebic. This follows from the formulas (4a) and (4b) of the formulas sheet. Formula (3c) represents the basic equation from which the new sample bj is calculated. One sample period earlier the sample bj-1 has been calculated by the corresponding formula (4a). Because




  
    
      ∑
      
        m
        =
        0
      
      M
    
    ⁢
    
      α
      m
      ′
    
  
  =
  1




the term bj-1 in this formula can be brought between the brackets, so that, using the Tij-definition of formula (3e), the equation (4b) is obtained. This equation allows to calculate the last term TMj without an integrator, when all other terms T0j . . . T(M-1)j are obtained through integration.

FIG. 1 shows a structure for the implementation of the algorithm for the case that M=2, {αm}={0, α1, α2}and q=4. Structures for other combinations of M, αm and q can be readily established. In practice larger values of q will usually be applied. With these values for M and αm the formula (3c) for bj becomes as is shown in formula (5a). Because from equation (3d) it follows that r0j=T0j, r1j=T0j+T1j and r2j=T0j+2T1j+T2j the equation (5a) can be rewritten as shown in equation (5b). A structure for calculating bj in accordance with this formula would require (at least) three integrators. However, with equation (4b) it is follows that α1′T1j+α2′T2j=0 so that equation (5b) simplifies to equation (5c).
The left-hand part of FIG. 1 shows a series to parallel converter SP with 1-bit latches L, with which non-overlapping (“isolated”) words of q 1-bit symbols of an are down-sampled with a factor q. The words of q bits are applied to an adder D0 for generating the signal A0j, to a set of multipliers M1 and an adder D1 for generating the signal A1j and to a set of multipliers M2 and an adder D2 for generating the signal A2j. The multipliers M1 multiply the bits with the weighing factors 1/q, 2/q, 3/q . . . q/q respectively and the multipliers M2 multiply these bits with the weighing factors (1/q)2, (2/q)2, (3/q)2 . . . (q/q)2 respectively, so that the generation of the signals A0j, A1j and A2j is in accordance with formula (3a).
A subtracter S1 generates the signal A0j-bj that is applied to an integrator I1. This is a “delaying” integrator, which has a Z-transform 1/(z-1) i.e. which has a 1-sample delay in its forward path and no delay in its feedback path. This is in contradistinction with a “non delaying” integrator, which has a Z-transform z/(z-1) i.e. which has a 1-sample delay in its feedback path and no delay in its forward path. The delaying integrator I1 outputs the previous sample of the input signal plus the previous sample of the output signal. It easily follows from equations (3d) and (3e) that the output of integrator I1 is then equal to A0(j-1)−b(j-1)+r0(j-1)=T0j=r0j. This signal is added to A1j in an adder D3 and the result is multiplied in a multiplier M3 with the weighing factor α1′ to obtain the signal α1′(A1j+T0j).
A second subtracter S2 and a second delaying integrator I2 receive the same input signals as the subtracter S1 and the integrator I1 and consequently they generate the same output signal T0j. A subtracter S3 subtracts the signals A1j and bj and in an adder D4 the result is added to the output T0j of integrator I2 to generate the signal A1j−bj+T0j. This is applied to a third delaying integrator I3. From equations (3d) and (3e) it can be derived that integrator I3 generates the signal T1j=A1(j-1)−bj-1+T0(j-1)+T1(j-1).
A multiplier M4 multiplies the signal T1j with the factor 2 and an adder D5 adds the signal A2j from the adder D2, the signal T0j from the integrator I2 and the signal 2T1j from the multiplier M4 together to generate the signal A2j+T0j+2T1j, which is subsequently multiplied in a multiplier M5 by the weighing factor α2′ to generate the signal α2′(A2j+T0j+2T1j). The signals from the multipliers M3 and M5 are finally added in an adder D6 to obtain the output signal bj in accordance with equation (5c) of the formulas sheet.
It has to be noted, that the structure of FIG. 1 may be modified in several ways with the same end-result. For instance, the subtracter S1 and the integrator I1 can be deleted while the signal T0j for the adder D3 can be obtained from the output of integrator I2. Furthermore, the adders D3, D5, D6 and the multipliers M3, M4 and M5 can be replaced by a multiplier for multiplying the signal A1j from the adder D1 by α1′, a multiplier for multiplying the signal A2j from the adder D2 by α2′, a multiplier for multiplying the signal T1j from integrator I3 by 2α2′ and a single adder for adding the signals from these three multipliers and the signal T0j from integrator I2 together to obtain the signal bj=α1′A1j+α2′A2j+T0j+2α2′T1j. This expression is equal to that of formula (5c) because α1′+α2′=1.
The reverse Rebic process implies the determination of a bitstream an from a given high precision multibit sequence bj. The formulas (1) to (3e) of the formulas sheet equally apply to the reverse Rebic of this invention, although formula (3c) is preferably written as:




  
    
      
        ∑
        
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          =
          0
        
        M
      
      ⁢
      
        
          α
          m
          ′
        
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    =
    
      
        b
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      -
      
        
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            0
          
          M
        
        ⁢
        
          
            α
            m
            ′
          
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            r
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  ,




thereby indicating that the “bunch-value”





  
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  ⁢
  
    
      α
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is calculated from the input sequence bj and the calculated sequence





  
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    M
  
  ⁢
  
    
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      .
    
  




It is clear that this formula does not allow to calculate an directly, because the sequence an is hidden in the bunch value





  
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  ⁢
  
    
      α
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Therefore a mapping process is needed to assign a particular word of q bits for each bunch value





  
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    M
  
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Several strategies for this mapping process are possible, such as “best fit, binary search”, see the article “Recursive Bitstream Conversion, the reverse mode.” By E. Roza in ”IEEE Transactions on Circuits and Systems, vol. 41, no.5, pp. 329–336, May 1994 or ”recursive mapping”, see the article “Reduced-sample-rate sigma-delta modulation using recursive deconvolution.” by D. Birru in “Int. Journal of Circuit Theory and Applications”, vol. 25, pp. 419–437, 1997. So, once the value of





  
    ∑
    
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    M
  
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has been determined and mapping has taken place, a parallel to series conversion of bunches of q bits is necessary and sufficient to obtain q consecutive bits of the bitstream an.

FIG. 2 shows a possible structure for the implementation of the reverse Rebic process. To facilitate the comparison with the forward Rebic of FIG. 1, the same parameters M=2, {αm}={0, α1, α2} and q=4 have been chosen for the reverse Rebic of FIG. 2. A subtracter S4 receives the signals bj and A0j and generate there from the difference bj−A0j. This signal is integrated to the signal −T0j in the delaying integrator I4. A subtracter S5 and an adder D7 generate the signal bj−A1j−T0j and a delaying integrator I5 integrates this signal to the output signal −T1j. A multiplier M6 doubles this signal to −2T1j. A subtracter S6 and a one-sample delay F generate from the two signals A2j and bj the output signal b(j-1)−A2(j-1) and this signal is, together with the signal −2T1j from the multiplier M6 and the signal −T0j from the integrator I4, added in an adder D8 and subsequently supplied to a non-delaying integrator I6 to generate the signal −r2j at the output thereof. Moreover, the signals −T0j from the integrator I4 and the signal −T1j from the integrator I5 are added in an adder D9 to generate the signal −r1j.
The signals bj and −r1j are added in an adder D10 and multiplied by the coefficient α1′ in a multiplier M7 to obtain the signal α1′{bj−r1j}. Equally, the signals bj and −r2j are added in an adder D11 and multiplied by the coefficient α2′ in a multiplier M8 to obtain the signal α2′{bj−r2j}. α2′{bj−r2j}. Finally the two output signals of the multipliers M7 and M8 are added in an adder D12 to obtain the mapper input signal bj−α1′r1j−α2r2j, which, according to equation (5a) of the formulas sheet, should correspond with the bunch value




  
    
      ∑
      
        m
        =
        0
      
      2
    
    ⁢
    
      
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        ′
      
      ⁢
      
        A
        mj
      
    
  
  =
  
    
      
        
          α
          1
          ′
        
        ⁢
        
          A
          
            1
            ⁢
            j
          
        
      
      +
      
        
          α
          2
          ′
        
        ⁢
        
          A
          
            2
            ⁢
            j
          
        
      
    
    ..
  




The mapper P generates during each sample period at its output a parallel word of q bits which, when put in series, represents q bits of the output signal an. The weighted sum of the mapper output signals is the bunch value





  
    
      
        ∑
        
          m
          =
          0
        
        2
      
      ⁢
      
        
          α
          m
          ′
        
        ⁢
        
          A
          mj
        
      
    
    =
    
      
        
          α
          1
          ′
        
        ⁢
        
          A
          
            1
            ⁢
            j
          
        
      
      +
      
        
          α
          2
          ′
        
        ⁢
        
          A
          
            2
            ⁢
            j
          
        
      
    
  
  ,




which should be as close as possible to the mapper input signal bj. This will be guaranteed by a correctly designed mapper.

The mapper P generates during each sample period at its output a parallel word of q bits which, when put in series, represents q bits of the output signal an. This parallel word of q bits is applied to a parallel to series converter PS for conversion to the high sample rate signal an, to an adder D13 for generating the signal A0j, to a set of multipliers M9 with adder D14 for generating the signal A1j and to a set of multipliers M10 with adder D15 for generating the signal A2j. The arrangement of adders D13, D14, D15 and multipliers M9, M10 is identical to the arrangement of adders D0, D1, D2 and multipliers M1, M2 of FIG. 1.
The mapper of FIG. 3, which is designed for q=4, comprises three quantizer-subtracter combinations Q1−S7, Q2−S8, Q3−S9 and a final quantizer Q4. Each of the quantizers is a 1-bit quantizer, which outputs one bit of the q-bit output sequence. In each of the combinations the subtracter calculates the difference between the input and the output of the quantizer. The difference signal of the first combination Q1−S7 is weighed in a multiplier M11 and the so weighed difference signal is applied as input signal to the second combination Q2−S8. The difference signal of the first combination and the difference signal of the second combination are each weighed in a multiplier M12 and M13 respectively and added in an adder D16. The output of this adder constitutes the input signal of the third combination. The difference signal of the first, second and third combinations are each weighed in a multiplier M14, M15 and M16 respectively and added in an adder D17 to constitute the input signal of the final quantizer Q4. Preferably, the multipliers M11, M13 and M16 have the same weighing factor and also the multipliers M12 and M15 have the same weighing factor. The values of the weighing factors may be calculated with the algorithm shown in the above-mentioned article in “Int. Journal of Circuit Theory and Applications”, vol. 25, pp. 419–437, 1997.
In the reverse Rebic algorithm the mapping process is not free of errors with the consequence that the signals A0j, A1j and A2j, which are fed back, are also not free of errors. This is the basic reason that formulas (4a) and (4b) cannot be applied in a reverse Rebic algorithm and therefore it is not possible to make a reverse Rebic structure with one integrator less than the order of the Rebic is.
As with the structure of the forward Rebic, also the reverse Rebic structure of FIG. 2 can be modified in several ways. An example thereof is given in FIG. 3, which comes in place of the elements S6, F, M6, D8, D9 and I6 of FIG. 2. The arrangement comprises an integrator I7 with two input signals bj−A2j and −r1j−T1j. For the input signal bj−A2j the integrator operates as a delaying integrator with z-transform 1/(z-1). For the input signal −r1j−T1j the integrator operates as a non-delaying integrator with z-transform z/(z-1). The output signal of the integrator I7 is: b(j-1)−A2(j-1)−r1j−T1j−r2(j-1)=−r2j.
As indicated above, the algorithm as defined in the formulas sheet is based on a low pass filter function with impulse response




  
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      ⁢
      
        
          
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                α
                m
              
              ⁢
              
                (
                
                  t
                  /
                  T
                
                )
              
            
            )
          
          m
        
        .
      
    
  




However, impulse responses with another power series expansion may also be used, so that





  
    
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          t
          )
        
      
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                α
                m
              
              ⁢
              
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                    (
                    
                      t
                      +
                      CT
                    
                    )
                  
                  /
                  T
                
                )
              
            
            )
          
          m
        
      
    
  
  ,




wherein C is any real constant. For instance, when C=1 the impulse response is





  
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    ⁢
    
      (
      t
      )
    
  
  =
  
    
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        )
      
    
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        ∑
        
          m
          =
          0
        
        M
      
      ⁢
      
        
          (
          
            
              α
              m
            
            ⁡
            
              (
              
                
                  (
                  
                    t
                    +
                    T
                  
                  )
                
                /
                T
              
              )
            
          
          )
        
        m
      
    
  




and the formulas of the formulas sheet equally apply, except in that the summation-borders for the counter n have to be changed as follows:





  
    
      
        
          ∑
          
            n
            =
            0
          
          
            jq
            +
            q
            -
            1
          
        
        ⁢
        
          
            a
            n
          
          ⁢
          
            
          
          ⁢
          ...
        
      
    
    
      
        
      
    
  




in equation (1) should read





  
    
      ∑
      
        n
        =
        0
      
      jq
    
    ⁢
    
      
        a
        n
      
      ⁢
      
        
      
      ⁢
      ...
    
  
  ,
  
    
      ∑
      
        n
        =
        jq
      
      
        jq
        +
        q
        -
        1
      
    
    ⁢
    
      
        a
        n
      
      ⁢
      
        
      
      ⁢
      ...
    
  




in equations (2), (2a) and (3a) should read





  
    ∑
    
      n
      =
      
        jq
        -
        q
        +
        1
      
    
    jq
  
  ⁢
  
    
      a
      n
    
    ⁢
    
      
    
    ⁢
    ...
  




and finally





  
    ∑
    
      n
      =
      0
    
    
      jq
      -
      1
    
  
  ⁢
  
    
      a
      
        n
        ⁢
        
          
        
      
    
    ⁢
    ...
  




in equations (2), (2a) and (3b) should read





  
    ∑
    
      n
      =
      0
    
    
      jq
      -
      q
    
  
  ⁢
  
    
      a
      
        n
        ⁢
        
          
        
      
    
    ⁢
    ...
  




The implication in the structures of FIGS. 1 and 2 is that the weighing factors of the multiplying sets M1 and M9 should be 1, 1+1/q, 1+2/q . . . 2−1/q in stead of 1/q, 2/q, 3/q . . . q/q and that the weighing factors of the multiplier sets M2 and M10 should be 1, (1+1/q)2, (1+2/q)2. . . (2−1/q)2 in stead of (1/q)2, (2/q)2, (3/q)2 . . . (q/q)2 

As any higher order feedback system the Rebic system suffers from instability and this risk is higher the higher the order of the system is. Because the third order system of FIG. 3 is noise free, this system is stable and operation is possible without further measures. If noise is introduced, like it is the case in the reverse mode as a consequence of the mapping process, the system may be unstable. In fact, the larger the noise, the higher the instability. This is due to the fact that internal noise overloads the loop-quantizer, thereby violating the amplitude and phase margins of the feedback loop. There are two independent mechanisms that can be used to control the stability for higher order systems: a linear one and a non-linear one. In the third-order example of FIG. 2 the linear one consists in increasing the parameter α1′. The more increase is made, the more stability is obtained, because the system tends to move from third order towards second order behaviour. The non-linear mechanism consists in the introduction of a clipper G in an integrator, as is shown in FIG. 4. This measure changes the spectral behaviour of the integrator for large signals into a more direct connection. For effective stability control both mechanisms can be applied simultaneously.
Formulas:




  
    
      
        
          
            ∑
            
              m
              =
              0
            
            M
          
          ⁢
          
            
              α
              m
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                J
              
              ⁢
              
                
                  
                    b
                    k
                  
                  ⁡
                  
                    (
                    
                      j
                      +
                      1
                      -
                      k
                    
                    )
                  
                
                m
              
            
          
        
        =
        
          
            ∑
            
              m
              =
              0
            
            M
          
          ⁢
          
            
              α
              m
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  0
                
                
                  jq
                  +
                  q
                  -
                  1
                
              
              ⁢
              
                
                  
                    a
                    n
                  
                  ⁡
                  
                    (
                    
                      j
                      +
                      1
                      -
                      
                        n
                        /
                        q
                      
                    
                    )
                  
                
                m
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  
  
    
      
        
          
            b
            j
          
          ⁢
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              α
              m
            
          
        
        =
        
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              
                α
                m
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    jq
                  
                  
                    jq
                    +
                    q
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      a
                      n
                    
                    ⁡
                    
                      (
                      
                        j
                        +
                        1
                        -
                        
                          n
                          /
                          q
                        
                      
                      )
                    
                  
                  m
                
              
            
          
          +
          
            

          
          ⁢
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              
                α
                m
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    0
                  
                  
                    jq
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      a
                      n
                    
                    ⁡
                    
                      (
                      
                        j
                        +
                        1
                        -
                        
                          n
                          /
                          q
                        
                      
                      )
                    
                  
                  m
                
              
            
          
          -
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              
                α
                m
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  
                    j
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      b
                      k
                    
                    ⁡
                    
                      (
                      
                        j
                        +
                        1
                        -
                        k
                      
                      )
                    
                  
                  m
                
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
          b
          j
        
        =
        
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              
                α
                m
                ′
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    jq
                  
                  
                    jq
                    +
                    q
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      a
                      n
                    
                    ⁡
                    
                      (
                      
                        j
                        +
                        1
                        -
                        
                          n
                          /
                          q
                        
                      
                      )
                    
                  
                  m
                
              
            
          
          +
          
            

          
          ⁢
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              
                α
                m
                ′
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    0
                  
                  
                    jq
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      a
                      n
                    
                    ⁡
                    
                      (
                      
                        j
                        +
                        1
                        -
                        
                          n
                          /
                          q
                        
                      
                      )
                    
                  
                  m
                
              
            
          
          -
          
            
              ∑
              
                m
                =
                0
              
              M
            
            ⁢
            
              
                α
                m
                ′
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  
                    j
                    -
                    1
                  
                
                ⁢
                
                  
                    
                      b
                      k
                    
                    ⁡
                    
                      (
                      
                        j
                        +
                        1
                        -
                        k
                      
                      )
                    
                  
                  m
                
              
            
          
        
      
    
    
      
        (2a)
      
    
  
  
    
      
        
          A
          ij
        
        =
        
          
            ∑
            
              n
              =
              jq
            
            
              jq
              +
              q
              -
              1
            
          
          ⁢
          
            
              
                a
                n
              
              ⁡
              
                (
                
                  j
                  +
                  1
                  -
                  
                    n
                    /
                    q
                  
                
                )
              
            
            i
          
        
      
    
    
      
        (3a)
      
    
  
  
    
      
        
          r
          ij
        
        =
        
          
            
              ∑
              
                n
                =
                0
              
              
                jq
                -
                1
              
            
            ⁢
            
              
                
                  a
                  n
                
                ⁡
                
                  (
                  
                    j
                    +
                    1
                    -
                    
                      n
                      /
                      q
                    
                  
                  )
                
              
              i
            
          
          -
          
            
              ∑
              
                n
                =
                0
              
              
                j
                -
                1
              
            
            ⁢
            
              
                
                  b
                  k
                
                ⁡
                
                  (
                  
                    j
                    +
                    1
                    -
                    k
                  
                  )
                
              
              i
            
          
        
      
    
    
      
        (3b)
      
    
  
  
    
      
        
          b
          j
        
        =
        
          
            ∑
            
              m
              =
              0
            
            M
          
          ⁢
          
            
              α
              m
              ′
            
            ⁡
            
              (
              
                
                  A
                  mj
                
                +
                
                  r
                  mj
                
              
              )
            
          
        
      
    
    
      
        (3c)
      
    
  
  
    
      
        
          r
          mj
        
        =
        
          
            ∑
            
              i
              =
              0
            
            m
          
          ⁢
          
            
              (
              
                
                  
                    m
                  
                
                
                  
                    i
                  
                
              
              )
            
            ⁢
            
              T
              ij
            
          
        
      
    
    
      
        (3d)
      
    
  
  
    
      
        
          T
          ij
        
        =
        
          
            A
            
              i
              ⁡
              
                (
                
                  j
                  -
                  1
                
                )
              
            
          
          -
          
            b
            
              j
              -
              1
            
          
          +
          
            r
            
              i
              ⁡
              
                (
                
                  j
                  -
                  1
                
                )
              
            
          
        
      
    
    
      
        (3e)
      
    
  
  
    
      
        
          b
          
            j
            -
            1
          
        
        =
        
          
            ∑
            
              m
              =
              0
            
            M
          
          ⁢
          
            
              α
              m
              ′
            
            ⁡
            
              (
              
                
                  A
                  
                    m
                    ⁡
                    
                      (
                      
                        j
                        -
                        1
                      
                      )
                    
                  
                
                +
                
                  r
                  
                    m
                    ⁡
                    
                      (
                      
                        j
                        -
                        1
                      
                      )
                    
                  
                
              
              )
            
          
        
      
    
    
      
        (4a)
      
    
  
  
    
      
        
          
            ∑
            
              m
              =
              0
            
            M
          
          ⁢
          
            
              α
              m
              ′
            
            ⁢
            
              T
              mj
            
          
        
        =
        0
      
    
    
      
        (4b)
      
    
  




bj=α1′(A1j+r1j)+α2′(a2j+r2j)  (5a)

bj=α1′(A1j+T0j+T1j)+α2′(A2j+T0j+2T1j+T2j))  (5b)

bj=α1′(A1j+T0j)+α2′(A2j+T0j+2T1j)  (5c)

## Claims

1. A method of converting the sample rate of a discrete-time input signal with sample rate q.fs to a discrete-time output signal with a sample rate fs which is a sub multiple of the sample rate of the input signal (q=integer larger than 1), the method comprising the steps of series to parallel converting the input signal to generate non-overlapping words of q samples of the input signal and of processing said words at the sub multiple rate fs in accordance with a recursive algorithm comprising a low-pass filtering function, characterized in that the algorithm is based on a power-series expansion of the impulse-response of the low-pass filtering function.

2. A method of converting the sample rate of a discrete-time input signal with sample rate fs to a discrete-time output signal with a sample rate q.fs which is a multiple of the sample rate of the input signal (q=integer larger than 1), the method comprising the steps of processing said input signal at the sample rate fs in accordance with a recursive algorithm comprising a low-pass filtering function to obtain non-overlapping words of q samples of the output signal and of parallel to series converting said words to generate the discrete-time q.fs sample rate output signal, characterized in that the algorithm is based on a power-series expansion of the impulse-response of the low-pass filtering function.

3. An apparatus for sample rate conversion for converting the sample rate of a discrete-time input signal with sample rate q.fs to a discrete-time output signal with a sample rate fs which is a sub multiple of the sample rate of the input signal (q=integer larger than 1), comprising:
means for series to parallel converting the input signal to generate non-overlapping words of q samples of the input signal;
means for processing said words at the sub multiple rate fs in accordance with a recursive algorithm comprising a low-pass filtering function, characterized in that the algorithm is based on a power-series expansion of the impulse-response of the low-pass filtering function, said means for processing comprising a number of integrators in cascade.

4. The apparatus of claim 3, characterized in that the low pass filtering function is of predetermined order and that the number of integrators in said cascade of integrators is one less than the said predetermined order.

5. The apparatus as claimed in claim 3 characterized in that at least one of said integrators comprises a clipper.

6. An apparatus for converting the sample rate of a discrete-time input signal with sample rate fs to a discrete-time output signal with a sample rate q.fs which is a multiple of the sample rate of the input signal (q=integer larger than 1), comprising:
means comprising a number of integrators in cascade for processing said input signal at the sample rate fs in accordance with a recursive algorithm comprising a low-pass filtering function to obtain non-overlapping words of q samples of the output signal, wherein the algorithm is based on a power-series expansion of the impulse-response of the low-pass filtering function; and
means for parallel to series converting said words to generate the discrete-time q.fs sample rate output signal; and
receiving a mapper input signal from said integrators in cascade and generating said non-overlapping words of q samples of the output signal.

7. The apparatus as claimed in claim 6, characterized in that said mapper comprises a cascade of quantizer-subtracter combinations, each of said combinations comprising a 1-bit quantizer for generating one of the bits of said non-overlapping words of q bits and a subtracter for subtracting the input- and output-signals of the 1-bit quantizer, that the first of said combinations in the cascade receives the mapper input signal, that each of the other of said combinations in the cascade receives a weighed sum of the output signals of the subtracters of the previous combinations in the cascade and that a final quantizer receives a weighed sum of the output signals of all subtracters in the cascade and generates the final bit of said non overlapping words of q bits.