Patent Application
20090185643
The present invention relates generally to equalization, detection and decoding using maximum-a-posteriori (UP) detection techniques.
Disk-drives contain a signal-processing chip, often referred to as a “read channel chip,” that converts the noisy analog signal from the magnetic recording head into the binary data sequence that is used by the host computer. A major component in a read channel chip is the channel detector. Extremely high-speed read channel chips are required to operate at data rates on the order of 3 GHz. Read channel chips for disk-drives that are used in portable lap-top computers and other battery-operated consumer electronics must be low-power. Since power usage increases with chip area and clock-speed, a good way to achieve low-power and high-throughput is to reduce the clock-rate and process more than one bit at a time with low-complexity signal processing that does not require significant area. Unfortunately, very complex area intensive signal processing is typically required to achieve the bit-error rate requirements for disk-drives.
The analog signal is filtered with an anti-aliasing filter and sampled so that there is one sample per bit. Because the bits are packed very tightly to achieve high storage capacity on the magnetic disk, the received samples have a low signal-to-noise ratio and are typically corrupted by inter-symbol-interference. The channel detector unravels the inter-symbol-interference by representing each possible combination of binary interfering symbols as states in a trellis. Typically, a 16-state trellis with four inter-symbol-interference symbols is required to provide sufficient performance in the channel detector. The area and power of the channel detector increase with the number of states in the trellis.
The binary sequences stored on the disk-drive awe typically encoded with parity codes or error-correction codes that require soft-decisions from the channel detector to decode the sequence with sufficiently low error-rates. There are two main types of soft-output channel detectors. The most common type is a SOVA detector that uses a soft-output Viterbi algorithm (SOVA) to find the path through the trellis with the maximum probability given the received samples, and to calculate a soft-output for each bit along the path. A SOVA detector typically uses a forward stare processor, a path memory, and a soft-output processor. Improved performance can be obtained by a maximum-a-posteriori (MAP) detector that uses a forward state processor, a backward stare processor, and a soft-output processor. Since the backward state processor makes the MAP detector more complex than a SOVA detector, MAP detectors are not typically used. MAP detectors only provide small performance improvements over SOVA detectors and typically require larger chip area and more power. A MAP detector determines which binary value at each bit position has the maximum probability given the received samples, y. For each bit, a MAP detector calculates a log-likelihood ratio, which is the conditional probability that ck is equal to binary zero (0) divided by the conditional probability that ck is equal to binary one (1), given the received samples, as follows:
Pr(ck=0|y)/Pr(ck=1|y).
In a Log-MAP detector, the soft-decision is the natural log of the likelihood ratio, and the hard-decision is the sign of the soft-decision. In a Max-Log-Map detector, the soft-decision is the natural log of the probability associated with the maximum probability path with ck=0 minus the natural log of the probability associated with the maximum probability path with ck=1. The Max-Log Map algorithm is a simplification of the Log-Map algorithm, and the difference in performance is typically small.
A MAP detector comprises a forward detector, a current branch metric detector, and a backward detectors. A full-rate forward detector calculates forward state metrics leading to each of the states at time k−1. In most conventional designs, the backward detector calculates backward state metrics starting from a single state with state metric 0 at time k+Q (where Q is the backward detection period) and leading to each of the states at time k, in the backward direction. Another approach is to let all the states at time k+Q have state metrics equal to 0. In a full-rate MAP detector, the current branch step represents the state transitions from time k−1 to time k. In conventional designs, the number of states at time k and k−1 are typically the same. The current bit label ck is the bit for which the detector is currently calculating a soft-output. The soft-output at time k is generated from combined metrics that are calculated by adding together a forward state metric, a backward state metric and a current branch metric.
In order to keep up with the throughput of emerging magnetic disk drives, a need exists for a high-speed, low-power, high-performance soft-output channel detector. A MAP detector that is smaller than a SOVA detector when implemented at very high-speed, and has better performance is the ideal solution.
Generally, methods and apparatus are provided for high-speed, low-power, high-performance channel detection. According to one aspect of the invention, a soft output channel detector is provided for processing a received signal and generating a log-likelihood ratio for one or more detected bits. The channel detector operates at a rate of 1/N and detects N bits per 1/N-rate clock cycle. The channel detector comprises a plurality, D, of MAP detectors operating in parallel, wherein each of the MAP detectors generates N/D log-likelihood ratio values per 1/N-rate clock cycle and wherein at least one of the plurality of MAP detectors constrains each of the bits. The log-likelihood ratio values can be merged to form an output sequence.
In one exemplary embodiment, N is four and D is two, and each MAP detector generates two log-likelihood ratio values per quarter-rate clock cycle. One MAP detector constrains two out of four bits and the second MAP detector constrains an additional two out of the four bits. Furthermore, the first MAP detector determines a bit value and log-likelihood ratio value for two out of the four bits and the second MAP detector determines a bit value and log-likelihood ratio value for the remaining two out of four bits. The first and second MAP detectors are offset by two bits and a branch metric input to the second MAP detector is delayed by two bits from the first MAP detectors.
According to another aspect of the invention, a single MAP detector is provided for processing a received signal and generating a log-likelihood ratio for one or more detected bits. The MAP detector comprises a forward detector for calculating forward state metrics; a backward detector for calculating backward state metrics; and a current branch detector for calculating a current branch metric, wherein at least two of the forward detector, the backward detector and the current branch detector employ different trellis structures. The forward and backward detectors can use different trellis structures with some bits not constrained, and wherein each bit used to calculate a label on the current branch is constrained. In one exemplary embodiment, the forward detector uses a four state trellis, the backward detector uses a 4 or 8 state trellis, and substantially all possible 4-bit current branches are constrained in a combined trellis structure.
According to a further aspect of the invention, a complexity of the channel detector can optionally be reduced by removing a dependence on one or more bits. For example, the dependence on one or more bits can be removed by selecting a branch metric with a minimum value for a binary value in a respective position.
A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings.
The present invention provides a high-speed, low-power, high-performance channel detectors. According to one aspect of the invention, a channel detector is provided that comprises one or more MAP detectors operating in parallel. In the exemplary MAP detectors, the number of bits on a branch is larger than the number of bits in the state. The bits that are not constrained by a state are either selected based on local feedback or selected to minimize the path metric. When a bit is not constrained, the detector trellis structure does not guarantee that there will be both a surviving path with a 1 and a surviving path with a 0 at that time. Therefore, the detector may not be able to calculate a soft-output for that bit. In addition, assuming that 0 is the value on the minimum metric path, if there is a surviving path with a 1 in that position, there is no guarantee that it will be the minimum metric path with a 1. Therefore, the magnitude of the soft-output generated for a bit that is not constrained may be too large. Furthermore, a NP FIR filter may not be able to adapt that bit properly.
In the exemplary parallel MAP detectors, every bit is constrained in at least one of the MAP detectors. A bit is “detected” by a detector that has that bit constrained. The multiple instantiations of the MAP detector are shifted in time so that all the bits are constrained in at least one of the MAP detectors. Each MAP detector calculates soft-outputs fox a subset of the bits that are constrained in the detector trellis. Then, the output from all the detectors is merged to form the output sequence.
In one exemplary embodiment, two parallel 4-state MAP detectors are employed at quarter-rate Each of the parallel detectors generates two soft-outputs per quarter-rate clock cycle, so there are four soft-outputs generated per quarter-rate clock cycle. Each branch represents four bits, and each state represents two bits. An even MAP detector ties down two out of four possible bits, such as bits 0 and 1. An odd detector ties down the remaining two bits, such as bits 2 and 3. The even and odd notation applies when looking at the quarter-rate clock cycle, which is high for the first two bits and low for the last two bits in a 4-bit clock cycle. The even and odd MAP detectors determine the bit values and soft-outputs for the bits that are constrained by the respective detectors.
In another aspect of the present invention, a single MAP detector is employed, where the forward and backward detectors within the MAP detector may use different trellis structures with some bits not constrained, but each bit used to calculate the label on the current branch is constrained. The current bit label ck is the bit for which the detector is currently calculating a soft-output, and the current branch is the branch that contains that bit label. In one exemplary embodiment, a single MAP detector operates at quarter-rate with four bits represented on each branch, and the forward detector uses a four state trellis, the backward detector uses a 4 or 8 state trellis, and all possible 4-bit current branches are constrained in the combined trellis structure. Four soft-outputs are generated for the 4 bit labels on the current branch.
Channel detectors are often implemented at a half rate using a radix-2 trellis, i e., there are two bits per branch and four incoming branches for each state. For each state, the state metric is added to four branch metrics to obtain four outgoing path metrics. The four incoming path metrics are compared using six comparators in parallel. A half-rate SOVA is typically smaller than a half-rate MAP detector However, at quarter-rate, all the known prior implementations of SOVA and MAP detectors with good performance are typically too large to consider for a high-speed, low-cost channel detector. The present invention significantly reduces the complexity of a quarter-rate MAP detector, and increases the speed so that it is close to double that for a half-rate implementation.
The branch label (c0,c−1,c−2,c−3) for the quarter-rate transition from state [b−4b−5] to state [b0b−1] is usually given by (c0,c−1,c−2,c−3)=(b0,b−1b−2,b−3), but may be different if there is precoding. In a four state trellis, the bits b0 and b−1 are constrained by the state and the bits b−2 and b−3 are not.
Four-Stare Max-Log MAP Algorithm
As previously indicated, a Max-Log-MAP detector combines forward and backward state metrics to calculate a soft-decision for each bit label.
In one exemplary embodiment of the Max-Log MAP algorithm, the branch metrics that are used to compute forward state metrics are delayed to compute metrics associated with backward processing in the forward direction. Such a Max-Log MAP detector computes combined metrics at time k+Q that correspond to the maximum probability path with bit label 0 at time k and the maximum probability path with bit label 1 at time k. In a Max-Log-MAP detector, the maximum probability path is the same as the minimum metric path. The difference between the two combined metrics is the soft-decision for the bit label at time k. The combined state metrics 200 represent the path metrics at time k+Q for the best paths going through each state at time k. For many applications, Q=8 is sufficient for good error rate performance. This look-ahead method eliminates the warm-up period typically associated with priori backward detectors. All the metrics that are calculated in the backward detector are accurate.
In this particular embodiment, the backward detector is approximately twice the size of the forward detector. The backward detector is implemented in the forward direction as if there are four independent forward detectors associated with each of the states at time k. The forward detector associated with state S=[bkbk−1] with local feedback F=bk−2 starts with state metrics Mk(bk bk−1 bk−2) equal to 0 for [bk bk−1 bk−2]=[S, F] and equal to infinity for all the other states. The forward detector for state S finds the best path at time k+Q leaving state S at time k. There are 8 forward detectors in the backward detector, one for state S and local feedback F combination. If the number of bits on a branch is large, i.e., 4, the look-ahead length Q is small, i.e., 8, and the number of state and local feedback bits is small, i.e., 4 or 8, then the forward detectors associated with each state can be unraveled so that there are no state metrics that are fedback to generate new state metrics. Since there is no state metric feedback, the backward detector can be pipelined to achieve high-speeds and it does not have to use a single trellis structure to represent each trellis step. Each step can have a different number of states or local feedback.
With conventional designs, the number of bits on a branch is typically small, i.e., 1 or 2, and the number of states at time k is large, i.e., 16. When there are only 1 or 2 bits on each branch, the accurate backward detection period, Q, divided by the number of bits on a branch is large. For a half-rate 16-state MAP detector, there are 4 trellis branch steps associated with the look-ahead period Q=8. The size of an unraveled backward detector would be about 4 times the size of the forward detector. In contrast, the prior-art backward detector that warms-up from a single state is about the same size as the forward detector. When the backward detector uses the same branch metrics as the forward detector, the backward detector cannot use local feedback in the same way that the forward detector does. The local feedback needs to be treated as a state bit. Therefore, one local feedback bit doubles the number of states and two quadruples it. The easiest thing to do is to drop the local feedback and use a backward detector that has worse performance than the forward detector, but has the same number of states. In most conventional detectors, good performance requires 16 states with no local feedback, in which case the backward detector would be approximately the same size as the forward detector. Since conventional forward and backward detectors have the same speed and approximately the same size, there is no incentive to unravel the backward detector to make the backward detector faster and larger. However, the problem with conventional MAP and SOVA detectors is that at half-rate they are unable to achieve high data rates, and a much larger quarter-rate implementation is required.
The disclosed MAP detector uses only a 4-state trellis in the forward detector, but achieves performance similar to a 16-state detector by using two local feedback taps for the computation of branch metrics. However, in one exemplary implementation, the MAP detector minimizes one local feedback tap out so that only one local feedback tap is required. It has been found that this branch metric computation technique achieves nearly the same performance as the one with two local feedback taps.
As discussed hereinafter, an exemplary MAP detector implements two 4-state modified Max-Log Map detectors that are offset by two bits to generate 4 log-likelihood ratios (LLRs) per quarter-rate cycle. The branch metric input to the odd phase detector is delayed by two from the even phase detector. The even phase detector computes LLR values (and preliminary decisions) for bits b0b1 b4b5 b8b9. The odd phase detector calculates LLR values for bits b2b3 b6b7 b10b11.
In a 16-state trellis, each state has four bits. When using a four state trellis to approximate the performance of a 16-state trellis, two bits are obtained from the state label, and two bits are obtained from local feedback to select the branch metrics. Each state has 16 incoming branches, which reduce to four branches after comparing and selecting the best of the parallel branch metrics. The exemplary feedback loop contains an adder to add the state metric to the current branch metric, a 4-way compare and a 4-way select. The exemplary embodiment calculates two sets of 256 (for a total of 512) quarter-rate branch metrics. The branch metrics in the even phase detector can be expressed as BM0(b0b−1b−2b−3[b−4b−5b−6b−7]) and the branch metrics in the odd phase detector can be expressed as BM−2(b−2b−3b−4b−5[b−6b−7b−8b−9]). In one variation, only 256 quarter-rate branch metrics are calculated, reducing the complexity and increasing the speed, by removing the dependence on the last bit (b−7 and b−9, respectively). As discussed hereinafter, the dependence on the last bit is removed by selecting the branch metric with the minimum value for either binary 0 or 1 in the respective position.
Generally, the output of stage 520 is the four sets of 32 full rate branch metrics when operating at quarter rate, B0(b0[b−1b−2b−3b−4]), B−1(b−1[b−2b−3b−4b−5]), B−2(b−2[b−2b−3b−4b−5]), and B−3(b−3[b−4b−5b−63b−7]). The output of the 32 square difference operators associated with time t0 are squared error values, sqn0(b0b−1b−2b−3b−4). For the correct path, the squared error represents the squared noise associated with equalized sample y0. For incorrect paths, the squared error includes both a noise and a signal component. The square difference operator 520 computes the squared noise value sqn0(b0b−1b−2b−3b−4) as follows:
sqn
0(b0[b−1b−2b−3b−4])=(z0(b0b−1b−2b−3b−4)−mean(b0b−1b−2b−3b−4))2
where mean (b0b−1b−2b−3b−4) represents the average value of z0(b0b−1b−2b−3b−4) given that b0b−1b−2b−3b−4 is the correct path. The value of mean(b0b−1b−2b−4) is computed by adding the ideal (noise-less) output for the effective equalization target (combination of the FIR target and the SDNP FIR filter), ideal(b0b−1b−2b−3b−4), and a bias value, bias(b0b−1b−2b−3b−4), that corresponds to the average error between z0(b0b−1b−2b−3b−4) and ideal(b0b−1b−2b−3b−4) when b0b−1b−2b−3b−4 is the correct path,
mean(b0b−1b−2b−3b−4)=ideal(b0b−1b−2b−3b4)+bias(b0b−1b−2b−3).
In a soft-input/soft-output detector, the soft-input LI0 is added to the squared error to form the full-rate branch metric, according to:
B
0(b0b−1b−2b−3b−4)=sqn0(b0b−1b−2b−3b−4) if c0=b0=0
sqn
0(b0b−1b−2b−3b−4)+LI0 if c0=b0=1
The 128 full rate branch metrics, B, are applied to a branch metric unit 530 that calculates 512 quarter rate branch metrics, BM, from the 128 full-rate branch metrics and 64 delayed full-rate branch metrics, according to the equation shown within element 530. As used herein, in the notation BM0(0000[0000]), the underlined leading zeroes indicate the sum of four fall rate branch metrics for times t0, t1, t2, t3, respectively. As shown in
As shown in
The four bit labels leaving each state, such as 00b−2b−3, comprise two bits indicating the new state (first two bits) and then two bits that are not constrained by the state (therefore, there are four possible state values). In addition, the branch metric representing the minimum branch metric of the four parallel branches, is denoted by
BM
0(00 . . . [00s0−6s0−7]=min{BM0(0000[00s0−6s0−7], BM0(0001[00s0−6s0−7], BM0(0010[00s0−6s0−7], BM0(0011[00s0−6s0−7]}
The best parallel path (minimum value) is selected to remove the dependency on bits b−2b−3. The four bits inside the square brackets represent the historical information of the prior state and local feedback.
Comparators 760-1 through 760-4 select the minimum metric. Comparator 760-1 selects the best path having a value of 0 at time k−1. Comparator 760-2 selects the best path having a value of 1 at time k−1. Comparator 760-3 selects the best path having a value of 0 at time k. Comparator 760-4 selects the best path having a value of 1 at time k. Each comparator 760-1 through 760-4 selects, Ci(x), equal to −ln(Pr{bi=x}). The soft-output of the M detector 750 represents the log-likelihood ratio Li equal to ln(Pr{bi=0}/Pr{bi=1}).
According to one aspect of the invention, the backward processing is unfolded and combined with the forward processing so that both share the same branch metric computations. Backward processing is implemented as a look-ahead to find the best path leaving the state. While the exemplary MAP detector 750 is illustrated as a max-log MAP detector, any type of MAP algorithm can be implemented, as would be apparent to a person of ordinary skill in the art.
Reduced Complexity Approach
In a variation of the present invention, complexity is reduced and speed is increased by feeding back only one of the two local feedback bits, and minimizing out the dependence on the other bit. As used herein, “reduced complexity” means the dependence on the oldest bit (h) is removed by selecting the value of h that minimizes the oldest full-rate branch metric, as follows:
BM0(b0b−1b−2b−3[b−4b−5b−6 . . . ])
where the dependence on bit b−7 has been minimized out. The minimum value is selected, with a trade off in reduced complexity at the expense of some lost performance. The complexity of the branch metric calculation block is reduced in half. Thus, only 256 quarter-rate branch metrics are pre-calculated at stage 530 of
BM0(b0b−1b−2b−3[b−4b−5b−6 . . . ]) and BM−2(b−2b−3b−4b−5[b−6b−7b−8 . . . ])
Now, when the parallel branch metrics for each feedback condition are compared and the minimum parallel branch metric is selected that matches the local feedback, there are only 64 compare/selections to get:
BM0(b0b−1 . . . [b−4b−5b−6]) and BM−2(b−2b−3 . . . [b−5b−7b−8])
Finally, the 32 quarter-rate branch metrics (16 to each of the even and odd MAP detectors 750) are selected to feed to the channel detector, as follows:
BM
0(b0b1 . . . [b−4b−5 . . . ])=BM0(b0b−1 . . . [b−4b−51 . . . ]) if sb−4 b−5−6=1 BM0(b0b1 . . . [b−4b−50 . . . ]) if sb−4 b−5−6=0
and
BM
−2(b−2b−3 . . . [b−6b−7 . . . ])=BM2(b−2b−3 . . . [b−6b−71 . . . ]) if sb−6 b−7−8=1 BM−2(b−2b−3 . . . [b−6b−70 . . . ]) if sb−6 b−7−8=0
BM0(b0b−1 . . . [b−4b−5 . . . ]) indicates the branch metric associated with a transition from state at time −4 to the state at time 0 that is added to the state metric M(b−4b−5). The selection signal, s, controls a corresponding multiplexer as discussed further below in conjunction with
As previously indicated, the four lower multiplexers at stage 810 are controlled by selection signal, s0k−6, to select one of the branch metrics having the appropriate value in the k−6 position. The outputs of the four lower multiplexers at stage 810 are the four branch metrics leaving state 0. The adders at stage 820 add the four branch metrics leaving state 0 with Mk−4(00), the best path leading up to state 0 at time k−4. The sums are the path metrics Mk(00 . . . 00). The additional path metrics 825 are calculated in other state blocks for branch metrics leaving states 1, 2, 3.
The path metrics Mk(00 . . . 00), Mk(01 . . . 00), Mk(10 . . . 00) and Mk(11 . . . 00) generated by the adders 820 are delayed at stage 830 by twice the 4T quarter rate to obtain the Mk−8 values for backwards processing. In this manner, the block 800 allows circuitry to be shared and reused by the forward and backward processing. In addition, a multiplexer at stage 830 compares the four branch metrics leaving state 0 and selects the best one (minimum value) to and then delay it by 4T to obtain BMk−4( . . . [00]), the best branch metric leaving state 0. It is noted that in the notation BMk−4( . . . [00]), the values inside the square brackets are tied down at times k−8 and k−9.
The adders at stage 850 add the values at times k−8 and k−9 which are tied down. In addition, the multiplexer 840 at stage 850 compares the four path metrics Mk(00 . . . 00), Mk(01 . . . 00), Mk(10 . . . 00) and Mk(11 . . . 00) entering state 0 (and generated by the adders 820) and selects the minimum value which is then delayed by 4T at delay 842 to generate the state metric Mk−4(00). The selection signal s0k−6 is selected by multiplexer 846 under control of the selected state metric and delayed by 4T at delay 848. The minimum combined metric Mk−8( . . . 00) for state 0 through state 0 eight time periods ago (k−8) is selected by a multiplexer at stage 860 and then delayed by 4T. A selection signal for time k−4, selk−4, is generated by a delay and multiplexer stage 870.
Single Detector Approach
Another embodiment of the present invention involves a single MAP detector with three parts: a forward detector, a current branch metric, and a backward detector; where all three parts may have different trellis structures (i.e, different number of states and/or different local feedback) In this embodiment, the forward trellis and backward trellis structures do not constrain all the bits, and the current branch constrains all the current branch bits. In order to guarantee that the soft-output for the current branch labels is calculated accurately, the combination of the three parts constrains all the bits that are needed to calculate the labels on the current branch.
M
k(bkbk−1 . . . bk−4bk−5)=BMk(bkb−1 . . . [bk−4bk−5bk−6bk−7])+Mk−4(bk−4bk−5),
where the local feedback is selected by the survivor paths in the forward detector, bk−6=Selk−6[bk−4bk−5] and bk−7=Selk−7[bk−4bk−5]. There are four bits associated with the current branch and future state fox the current branch (bkbk−1bk−2bk−3) and there are four bits associated with the initial state [bk−4bk−5] and local feedback [bk−6bk−7]. The current branch detector 1120 adds the branch metric for the current branch, BMk(bkbk−1bk−2bk−4[bk−4bk−5bk−6bk−7]) with selected local feedback bk−6=Selk−6[bk−4bk−5] and bk−7=Selk−7[bk−4bk−5], to the forward state metric Mk−4(bk−4bk−5) to generate a current metric Mk(bkbk−1bk−2bk−3) 1130. Ideally, there would be 8 or 16 states in the backward detector, due to the local feedback. However, in this particular embodiment the backward detector has reduced performance and complexity. The backward detector has 4 states with no local feedback everywhere except on the state connected to the current branch, which has 4 state bits [bkbk−1bk−2bk−3]. The dependence on local feedback is removed by selecting the values that minimize the branch metrics. The backward detector 1140 calculates a look-ahead branch metric Bk( . . . [bkbk−1bk−2bk−3]) 1150 that represents the best 8-bit branch leaving state [bkbk−1bk−2bk−3]. The look-ahead branch metric is calculated by selecting the minimum 4-bit look-ahead branch metric leaving state [bk+4bk+3], adding it to the minimum branch metric associated with the transition from state [bkbk−1bk−2bk−3] to state [bk+4bk+3], and minimizing over the dependence on the connecting state bits:
B
k( . . . [bkbk−1bk−2bk−3])=min over bk+4bk+3{Bk+8( . . . [bk+4bk+3])+Bk+4(bk+4bk+3 . . . [bkbk−1bk−2bk−3])}
Then, the look-ahead branch metric is added to a delayed version of the current metric to obtain the combined metric Mk( . . . bkbk−1bk−2bk−3) 1160
L
i=ln(Pr{bi=1})−ln(Pr{bi=0})
The single MAP detector embodiment may be a more flexible design than the parallel MAP detector embodiment for achieving higher performance with higher complexity, beyond the examples described in detail. However, storing all the quarter-rate branch metrics and the more complex combined metric make the single detector described by
Conclusion
While exemplary embodiments of the present invention have been described with respect to digital logic blocks, as would be apparent to one skilled in the art, various functions may be implemented in the digital domain as processing steps in a software program, in hardware by circuit elements of state machines, or in combination of both software and hardware. Such software may be employed in, for example, a digital signal processor, micro-controller, or general-purpose computer. Such hardware and software may be embodied within circuits implemented within an integrated circuit.
Thus, the functions of the present invention can be embodied in the form of methods and apparatuses for practicing those methods. One or more aspects of the present invention can be embodied in the form of program code, for example, whether stored in a storage medium, loaded into and/or executed by a machine, or transmitted over some transmission medium, wherein, when the program code is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the invention. When implemented on a general-purpose processor, the program code segments combine with the processor to provide a device that operates analogously to specific logic circuits. The invention can also be implemented in one or more of an integrated circuit, a digital signal processor, a microprocessor, and a micro-controller.
It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention.
