Abstract:
In an environment where a block transmission takes place via a frequency selective channel that is characterized by channel parameters that change with time in the course of the transmission of a block, an arrangement employs pilot signals to ascertain some of the parameters, and estimates the remaining parameters through an interpolation process. In some embodiments, channel coefficients estimates are improved by employing estimates from previous blocks. In an OFDM system, the pilot signals are advantageously selected to be in clusters that are equally spaced from each other in the time or the frequency domains. This approach applies to multiple antennae arrangements, as well as to single antenna arrangements, and to arrangements that do, or do not, use space-time coding.

Description:
RELATED APPLICATIONS 
   This is a continuation of U.S. patent application Ser. No. 10/152,609, filed May 21, 2002 now U.S. Pat. No. 7,173,975, which claims priority from Provisional Application No. 60/307,759, filed Jul. 25, 2001. 

   BACKGROUND OF THE INVENTION 
   This invention relates to wireless communication and, more particularly, to wireless communication in an environment where transmission channel characteristics change relatively rapidly. 
   The explosive growth of wireless communications is creating a demand for high-speed, reliable, and spectrally efficient communications. There are several challenges to overcome in attempting to satisfy this growing demand, and one of them relates to the time variations in the transmission of multicarrier-modulated signals. 
   Multicarrier transmission for wireless channels has been well studied. The main advantage of orthogonal frequency division multiplexing (OFDM) transmission stems from the fact that the Fourier basis forms an eigenbasis for time-invariant channels. This simplifies the receiver, which leads to an inexpensive hardware implementations, since the equalizer is just a single-tap filter in the frequency domain—as long as the channel is time invariant within a transmission block. Combined with multiple antennas, OFDM arrangements are attractive for high data rate wireless communication, as shown, for example, in U.S. application Ser. No. 09/213,585, filed 17 Dec. 1998. 
   Time invariability within a transmission block cannot be guaranteed at all times, for example, with the receiving unit moves at a high speed, and that leads to impairments because the Fourier basis at such times no longer forms the eigenbasis, and the loss of orthogonality at the receiver results in inter-carrier interference (ICI). Depending on the Doppler spread in the channel, and the block length, ICI can potentially cause severe deterioration of quality of service 
   SUMMARY 
   In an environment where a block transmission takes place via a frequency selective channel that is characterized by channel parameters that change with time in the course of the transmission of a block, an advance in the art is realized by employing pilot signals to ascertain some of the parameters and by estimating the remaining parameters through an interpolation process. More specifically, in the aforementioned environment, the number of channel parameters that potentially can change in value from one block to the next is such that pilot signals cannot be used to ascertain the values of the channel parameters. In accordance with the principles disclosed herein, a number of pilot signals can be used that number is less than the number of pilot signals that are required for estimation of parameters. The channel coefficients are determined through judicious selection of the pilot signals that are used, coupled with an interpolation process. In some embodiments, channel coefficients estimates are improved by employing estimates from previous blocks. 
   In an OFDM system, the pilot signals are advantageously selected to be in clusters that are equally spaced from each other in the time or the frequency domains. This approach applies to multiple antennae arrangements, as well as to single antenna arrangements, and to arrangements that do, or do not, use space-time coding. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  presents a block diagram of a prior art arrangement with a multi-antenna transmitter and a multi-antenna receiver; 
       FIG. 2  shows a signal arrangement where transmitted symbol blocks are augmented with a prefix code that is used to remove inter-block interference; 
       FIG. 3  presents a block diagram of a prior art OFDM arrangement with a multi-antenna transmitter and a multi-antenna receiver; 
       FIG. 4  presents a block diagram of an OFDM arrangement with a multi-antenna transmitter and a multi-antenna receiver in accord with principles of this invention; and 
       FIG. 5  presents a flow diagram of the method carried out in a receiver that employs interpolation to obtain channel attribute estimates. 
   

   DETAILED DESCRIPTION 
     FIG. 1  presents a general block diagram of a wireless arrangement where a transmitter  10  includes M t  transmitting antennas  12 - 1 ,  12 - 2 , . . .  12 -M t , each of which transmits a signal s i  that is obtained by processing input signal x i  through a processing unit g i , shown by units  11 - 1 ,  11 - 2  and  11 -M t . It is assumed that the channel&#39;s impulse response function is time-varying. A receiver  20  includes M r  receiving antennas  21 - 2 ,  21 - 2 , . . .  21 -M r , and the received signals are applied to processing unit  22 . 
   When considering only a single transmitting antenna and a single receiving antenna, the received signal y, at time t, can be expressed by
 
 y ( t )=∫ b ( t , τ) s ( t−τ ) dτ+z ( t )  (1)
 
where b(t, τ) is the impulse response of the time-varying channel between the transmitting antenna and the receiving antenna, as a function of τ, at time t, and s(t−τ) is the transmitted signal at time t−τ. When sampled at a sufficiently high rate (above 2W t +2W s , where W t  is the input bandwidth and W s  is the bandwidth of the channel&#39;s time variation), equation (1) can be written in discrete form; and when the transmission channel between the transmitting antenna and the receiving antenna is represented by a discrete, finite, impulse response corresponding, for example, to a tapped filter having v samples of memory, then equation (1) can be expressed as
 
                   y   ⁡     (   k   )       =       y   ⁡     (     k   ⁢           ⁢     τ   s       )       =         ∑     l   =   0     v     ⁢       h   ⁡     (     k   ,   l     )       ⁢     x   ⁡     (     k   -   l     )           +     z   ⁡     (   k   )                   (   2   )               
where y(k) and z(k) are the received signal and the received noise at sample time k, h(k,l) represents the sampled (time-varying) channel impulse response that combines the transmit filter g i  with the channel response b i , and x(k−l) is the input signal at sample time k−l.
 
   When there are M r  receiving antennas, and M t  transmitting antennas, equation (2) generalizes to 
                   y   ⁡     (   k   )       =         ∑     l   =   0     v     ⁢       H   ⁡     (     k   ,   l     )       ⁢     x   ⁡     (     k   -   l     )           +     z   ⁡     (   k   )                 (   3   )               
where y(k) is an M r -element vector corresponding to the signals received at the receiving antennas ( 31 ,  32 ,  33 ) at sample time k, H(k,l) is an M r  by M t  matrix of the l th  tap of the transmission medium filter at sample time k (k is a parameter because the transmission medium varies with time), x(k−l) is an M t -element input vector at sample time k−l that corresponds to the signals of transmitting antennas  21 ,  22 ,  23 , and z(k) is the M r -element noise vector at sample time k.
 
   When considering the transmission of information from transmitter  10  to receiver  20  in blocks, one has to realize that the memory/delay of the transmission medium will cause interference between one block and the next, unless a guard-time interval is provided that corresponds to at least the delay introduced by the transmission medium. One can simply send no signals during this interval, but one can also send a symbol sequence, of any length greater than the memory/delay of the transmission medium, i.e., v or more symbols. Thus, one can send a block of N symbols from antenna  12 - 1  with a prefix of v symbols, as shown in  FIG. 2 , for example, and a receiving antenna can ignore the first v receiving signals—because those first v signals suffer from interference by the previous block—and focus on only the remaining N received signals. This can be expressed for one transmitting antenna and one receiving antenna, by the equation
 
 y=Hx+z   (4)
 
where y is a vector with elements y(k), y(k+1), . . . y(k+N−1), x is a vector with elements x(k−v), x(k−v+1), . . . x(k−1), x(k), x(k+1) . . . x(k+N−1), and H is a N by N+v element matrix (channel coefficients between the transmitting antenna and the receiving antenna during the N symbol intervals when the receiving antenna pays attention to the signal). When the prefix is selected so that
 
 x ( −v+i )= x ( N −1 +i ) for i=1, 2, . . . v−1,  (5)
 
then the x vector reduces to an N element vector, and H reduces to a N by N element matrix.
 
   When the full complement of M t  transmitting antennas and M r  receiving antennas are considered, equation (4) holds, but H becomes an N·M r  by N·M t  matrix, y and z become N·M r -element vectors, and x becomes an N·M r -element vector. 
   It may be noted that there is no loss of generality in assuming that M t =M r =M, which makes H a square matrix of size NM. In the treatment below, therefore, this assumption is made, but it should be understood that the principles disclosed herein apply to situations where M t  is not necessarily equal to M r . 
   At a particular receiving antenna, for example  21 - j , the received signal can be expressed by equation (4) where H is an N by NM matrix. H can also be considered as an M element vector, where each element is an N by N matrix of H ij  of transfer coefficients between transmitting antenna i and receiving antenna j, i.e., vector [H 1j , H 2,j , . . . H Mj ]. 
   In an OFDM system like the one shown in  FIG. 3 , each block of N signals in sequence x i  is generated by block  13 - i , which performs the inverse discrete Fourier transform (IDFT) of an information-bearing signal X, i.e.,
 
 x=Q   H   X,   (6)
 
which in the case of an arrangement where there are M transmitting antennas and M receiving antennas, Q H  is the Hermitian of Q, and Q is an NM by NM matrix with elements {tilde over (Q)} on the diagonal, and 0s elsewhere, where {tilde over (Q)} is the N-point DFT transform matrix
 
                       Q   ~     ⁡     (     l   ,   k     )       =         1     N       ⁢     ⅇ       -   j     ⁢           ⁢       2   ⁢   π     N     ⁢   lk       ⁢           ⁢   for   ⁢           ⁢   0     ≤   l       ,     k   ≤     N   -   1.               (   7   )               
In the case of a single transmitting antenna and a single receiving antenna, equation (6) is simply x={tilde over (Q)} H X.
 
   At the receiver, the signal of each antenna  21 - j  is applied to element  24 - j , which performs a N-point DFT, generating an N-element vector Y j . This can be expressed by 
                   Y   j     =         Q   ~     ⁡     (         ∑     i   =   1     M     ⁢       H   ij     ⁢       Q   ~     H     ⁢     X   i         +   z     )       .             (   8   )               
where H ij  is the N by N matrix of coefficients between transmitting antenna i and receiving antenna j, and X i  is an N-element vector applied to inverse FFT element  13 - i  (advantageously, N is a power of 2 integer). The signal at the output of element  24 - i  at clock interval p (within a block) can be written as
 
                     Y   j     ⁡     (   p   )       =         ∑     l   =   1     M     ⁢     (       ∑     q   =   1     N     ⁢         G   lj     ⁡     (     p   ,   q     )       ⁢       X   l     ⁡     (   q   )           )       +     Z   ⁡     (   p   )                 (   9   )               
where Z(p) is the transformed noise at clock interval p, X l (q) is the signal at clock interval q of the block applied to inverse FFT element  13 - l , and G lj (p,q) is the (p,q) th  element of matrix
   G   lj   ={tilde over (Q)}H   lj   {tilde over (Q)}   H .  (10) 
Equation (9) can also be written as
 
                       Y   j     ⁡     (   p   )       =         ∑     l   =   1     M     ⁢     (         G   lj     ⁡     (     p   ,   p     )       +       ∑       q   =   1       q   ≠   p       N     ⁢         G   lj     ⁡     (     p   ,   q     )       ⁢       X   l     ⁡     (   q   )             )       +     Z   ⁡     (   p   )           ,           (   11   )               
and generalizing to the multiple receiving antenna case, equation (11) can be expressed by
 
   
     
       
         
           
             
               
                 
                   
                     
                       
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   When the transmission medium coefficients do not vary with time, matrix H of equation (4) becomes a circulant matrix, and it can be shown that when H is circulant, {tilde over (Q)}H{tilde over (Q)} H  is a diagonal matrix. However, when H does vary with time, {tilde over (Q)}H{tilde over (Q)} H  is no longer a diagonal matrix, and consequently, signal Z ICI (p) of equation (12) is non-zero. Stated in other words, when the transmission coefficients do vary with time, the received signal contains inter-carrier interference. If the Z ICI (p) were to be eliminated, however, then a conventional decision circuit can be used to arrive at the N elements of Y(p) that form the received block of information signals. Obviously, therefore, it is desirable to eliminate—the inter-carrier interference signals, but in order to do that one must know the values of all of the coefficients of matrix H. 
   In the time span of N symbol intervals, i.e., in a block, it is possible for each coefficient to change with each sample interval. The matrix H ij  between any transmitting antenna i and any receiving antenna j (in the time span of a block) in an N by N matrix (as demonstrated by above), but only v terms in each row are non-zero. Consequently, only Nv coefficients need to be ascertained for each H matrix between a transmitting antenna and a receiving antenna (rather than N 2  coefficients). Nevertheless, this number is still much too large to ascertain, because only N signals are transmitted in a block and, therefore, it is not possible to estimate Nv coefficients, even if all N sample intervals in a block were devoted to known (pilot) signals—which, of course, one would not want to do because it would leave no capacity for communicating any information. 
   The disclosure below presents a novel approach for developing the necessary coefficients of H ij , but alas, some of those coefficients are likely to be inexact. Consequently, one can only reduce the inter-carrier interference to some minimum level, rather than completely eliminate it. Still, while recognizing that the channel coefficients that are available in receiver  20  are not all totally accurate, in the treatment below it is assumed that all coefficients of matrix H ij  are known. 
   To reduce this inter-carrier interference in accord with the principles disclosed herein, a filter element  25  with transfer function W is interposed between the receiving antennas and the FFT elements  24 - j  (where index j ranges from 1 to M r ), as shown in the  FIG. 4  embodiment. The signal developed at the outputs of the FFT elements is
 
 Y=QWHQ   H   X+QWz=  G X+  Z ,   (13)
 
where Q, W, and H are NM by NM matrices, and X and z are NM-element vectors. Defining e m  as an NM-element vector with a 1 in the m th  element and zeros elsewhere, then vector q m =Q H e m  represents the m th  column of matrix Q H , which is an NM element vector, comprising M concatenated sets of values
 
             1     N       ⁢     ⅇ     j   ⁢           ⁢       2   ⁢   π     N     ⁢   mk             
for 0≦k≦N−1.
 
   Defining h m =Hq m , w m =W H q m , and R m =HH H −h m h m   H , and further, assuming that w m   H w m =1 for 0≦m≦N−1, it can be shown that the optimum vector at symbol interval m, (i.e., for frequency bin m), w m , is one that results from solving the optimization problem 
   
     
       
         
           
             
               
                 
                   
                     
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   It can be shown that by defining 
             R   yy     =         1   SNR     ⁢   I     +     HH   H             
and computing the inverse matrix R yy   −1 , the optimum vector for frequency bin m is
 
   
     
       
         
           
             
               
                 
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   Repeating the computations leading to equation 15 for all values of m=0,1, . . . NM yields the various vectors that correspond to the columns of matrix W H Q H , by definitions of the relationships w m =W H q m  and q m =Q H e m . Forming the matrix, post-multiplying it by Q and taking the Hermitian of the result yields the matrix W. 
   In the  FIG. 4  arrangement, controller  50  is responsive to all of the signals acquired by antennas  21 - j  (via bus  51 ), computing the values of the channel coefficients (elements of H), and also computing the coefficients of matrix W as disclosed above. The coefficients of W are applied to filter  25 , and the coefficients of H are applied to detection element  23 . Computation of channel coefficients from received pilot signals is well known in the art. An innovative approach for estimating channel coefficients in an OFDM system is disclosed, for example, in a patent application by Ariyavisitakul et al, titled “Channel Estimation for OFDM Systems with Transmitter Diversity” which was field on Dec. 18, 1998 and bears the Ser. No. 09/215,074. This application is incorporated by reference herein. In the work by Ariyavistakul et al the channels do not vary within an OFDM block. 
   As indicated above, computation of the optimum filter that is placed following each antenna requires knowledge of the channel coefficients. The following discloses a method and corresponding apparatus that ascertains a selected number of coefficients of H through the use of pilot signals, and obtains the remaining coefficients of H through interpolation. For sake of simplicity of the mathematical treatment, it is assumed that M=1 because the generality of the treatment is not diminished by this assumption. Also, in accord with the principles disclosed herein, it is assumed that the H matrix coefficients in the course of transmitting a number of adjacent symbols within a block do not vary significantly and that, therefore, if two rows of coefficients of matrix H that are fairly close to each other are known, then the coefficient rows between them can be obtained through linear interpolation of the known rows. 
   A row of coefficients effectively defines the channel at the clock interval corresponding to the row, and in the treatment below it is designated by h(n,l), where index n corresponds to the row with matrix H (i.e., an integer between 1 and N, inclusively) and index l corresponds to the v potentially non-zero coefficients on a row of H. 
   Extending this thought, if the channel coefficients are known at P clock intervals, where P is any selected number, i.e., if P rows of H are known, then the remaining rows of H can be obtained by interpolation of the P known rows. Intuitively it is apparent that the error in estimating the coefficients of H decreases as the value of P is increased (i.e., more rows of H are known). Stating the interpolation mathematically, generally, a row h(n,l) can be obtained from 
                   h   ⁡     (     n   ,   l     )       =       ∑     i   =   1     P     ⁢       a   n   i     ⁢     h   ⁡     (       m   i     ,   l     )                   (   17   )               
where coefficients a n   i  are members of a set of coefficients a n  and h(m i ,l) is the i th  known set of channel coefficients. In vector notation,
   h ( n,l )= a   n               (18) 
where           is a P-element vector          =[h(m 1 ,l)h(m 2 ,l) . . . h(m P ,l)] T , and a n  is a vector with elements a n   i . If H c (i) designates the N by N H matrix if it were not time variant and had the coefficients of the i th  channel that is known, i.e., channel h(m,l), then, the channel estimate, {tilde over (H)}, can be expressed by

                   H   ~     =       ∑     i   =   1     P     ⁢       A     m   ⁡     (   i   )         ⁢       H   c     ⁡     (     m   i     )                   (   19   )               
where A i  is an N by N diagonal matrix with elements [A m(i) ] m,n  that is equal to 1 when n corresponds to the known channel h(m,l), i.e., when n equals m(i), is equal to 0 when n corresponds to the other P−1 known channels, and is equal to a n   i  otherwise.
 
   To illustrate, suppose N=5, and rows 1, 3 and 5 of H are known through detection of pilot signals at that are sent during clock intervals 1, 3 and 5. That is, index i has values 1, 3 and 3, and m(i) has values 1, 3 and 5. From the known rows we can then construct H c (m 1 ) from h(1,l), H c (m 2 ) from h(3,l) and H c (m 3 ) from h(5,l). For the two missing rows, we have vector a 2  that has three elements, a 2   1 , a 2   2 , a 2   3 , for example [0.2, 0.3, 0.5] and vector a 4  that has three elements, for example, [0.1, 0.2, 0.7]. According to the above, 
                     A   1     =       [         1       0       0       0       0           0         a   2   1         0       0       0           0       0       0       0       0           0       0       0         a   4   1         0           0       0       0       0       0         ]     =     [         1       0       0       0       0           0       0.2       0       0       0           0       0       0       0       0           0       0       0       0.1       0           0       0       0       0       0         ]         ,           (   20   )                   A   3     =       [         0       0       0       0       0           0         a   2   2         0       0       0           0       0       1       0       0           0       0       0         a   4   2         0           0       0       0       0       0         ]     =     [         0       0       0       0       0           0       0.3       0       0       0           0       0       1       0       0           0       0       0       0.3       0           0       0       0       0       0         ]         ,   and           (   21   )                   A   5     =       [         0       0       0       0       0           0         a   2   3         0       0       0           0       0       0       0       0           0       0       0         a   4   3         0           0       0       0       0       1         ]     =     [         0       0       0       0       0           0       0.5       0       0       0           0       0       0       0       0           0       0       0       0.7       0           0       0       0       0       1         ]         ,           (   22   )               
and
   {tilde over (H)}=A   1   H   c (1)+ A   3   H   c (3)+ A   5   H   c (5).  (23) 
   Two remaining considerations are the placement of the pilot tones, and the values employed in the a n  vectors, for n=1, 2, . . . ,P. 
   It can be shown that for time-selective channels, pilot tones should be grouped together. On the other hand, in frequency selective time-invariant channels, placing the pilot tones equally spaced on the FFT grid is the optimal scheme. Therefore, for purposes of the  FIG. 4  arrangement, it is advantageous to partition the pilot tones into equally spaced groups on the FFT grid. The pilot tones are generated in element  15  of  FIG. 4 , and are applied to coding unit  13 - 0 , to be applied to elements  13 - j  as disclosed above. Of course, in the frequency bins where pilot tones are placed (or time intervals where pilot tones are placed), no information that is to be communicated can be sent. 
   As for the values employed in the a n  vectors, without imposing any assumptions or the underlying channel variations, linear interpolation appears to be the simplest method for choosing the weight vectors. On the other hand, if a priori knowledge about the underlying channel model is available, more sophisticated channel interpolation schemes can be devised. 
   In the case of the linear interpolation, each a n  vector consists of two non-zero terms that correspond to the two closest known rows of H (one on either side), and the values of the two non-zero terms reflect the relative distance of the row corresponding to n to the two known rows. For example, if H contains 48 rows and rows 1, 2, 3, 16, 17, 18, 31, 32, 33, 46, 47 and 48 are known, the a 4  vector is {0,0, 12/13, 1/13,0,0,0,0,0,0,0,0} 
   To give an example of a situation where a priori knowledge about the channel is available assume, for example that the channels follow the Jakes model (see W. C. Jakes,  Microwave Mobile Communications , John Wiley &amp; Sons, Inc. 1994) where E[(h(m,l)h H (n,l)]=J 0 (2πf d (m−n)T) with f d  denoting the Doppler frequency, and T denoting the symbol period, then the calculation of the interpolation weights is straightforward. For example if we fix rows h 1 , h N/2 , h N , the set of weights a n =[a n (1), a n (N/2), a n (N)] that minimizes E[|h(n,l)−a n   H {tilde over (h)}(l)| 2 ] where {tilde over (h)}(l)=[h(1,l), h(N/2,l), h(N,l)] T  can be obtained using the orthogonality principle
         a n   H =R h     n     {tilde over (h)} R {tilde over (h)}{tilde over (h)}   −1 
 
where
 
 R   h     n     {tilde over (h)}   =[J   0 [1− n], J   0   [N/ 2− n], J   0   [N−n]]   (24)
 
and
       

                   R       h   ^     ⁢           ⁢     h   ^         =     [         1           J   0     ⁡     [       N   /   2     -   1     ]               J   0     ⁡     [   N   ]                   J   0     ⁡     [       N   /   2     -   1     ]           1           J   0     ⁡     [     N   /   2     ]                   J   0     ⁡     [   N   ]               J   0     ⁡     [     N   /   2     ]           1         ]             (   25   )               
with J 0 (n) being equal to J 0 (2πf d nT). Typically, however, for Doppler values of practical importance, there is little to be gained by adopting the Jakes-based estimator in place of the linear interpolator. Hence, from an implementation point of view, the linear estimator appears to be an attractive solution, as it dispenses with the estimation of the Doppler frequency, without sacrificing performance.
 
   An additional enhancement is achieved through channel tracking. In channel tracking, it is assumed that matrix H of one block is related to matrix H of the previous blocks and, therefore, given a matrix Ĥ u−1  that is used during block u−1, and an estimate of the H matrix derived from the pilot signals for block u, {tilde over (H)} u , a matrix to be employed during block 1 is obtained from
 
 Ĥ   u   =α{tilde over (H)}   u +(1+α) Ĥ   u−1   (26)
 
where α is a preselected constant less than 1.
 
   It should be noted that the above-disclosed approach could be used in conjunction with any coding technique in coder  13 - 0  of  FIG. 4 , including space-time coding as described, for example, in U.S. Pat. No. 6,185,258. 
   It should also be realized that the receiver embodiment shown in  FIG. 4  intends to clearly demonstrate the signal flow in the receiver and that the actual, physical, embodiment will likely have a somewhat different block diagram. Specifically, the coefficients of W for block u of the received signals (the block being the M r  received signals during N symbol intervals) need to be developed from the coefficients of the channel transmission matrix, as best estimated by processor  50 . In some applications, as disclosed above, this matrix is Ĥ u , which is developed pursuant to equation (26). Equation (26) needs to have access to Ĥ u−1  and to {tilde over (H)} u . The former implies a memory within controller  50 , and the latter implies access to the received signals of block u and processing time that is necessary to develop {tilde over (H)} u . To obtain this processing time, filter W (or an element between antennas  21 - j  and the filter W might advantageously include memory that can store at least one block&#39;s work of received signals. In an embodiment where the functions of controller  50 , filter  25 , FFT elements  24 - j , and detector &amp; decoder element  23  are implemented with a stored program controlled processor, more than one block&#39;s work of memory is needed, though the precise amount is dependent of the specific code that the artisan will write to implement to method disclosed herein.