New digital demodulator with matched filters and curve segmentation techniques for BFSK demodulation: Analytical description Nuevo demodulador digital con filtros macheados y técnicas de segmentación de curvas para la demodulación de señales BFSK: Descripción analítica

The present article relates, in general, to digital demodulation of Binary Frequency Shift Keying (BFSK). The objective of the present research is to obtain a new processing method for demodulating BFSK-signals in order to reduce hardware complexity in comparison with other methods reported. The solution proposed here makes use of the matched filter theory and curve segmentation algorithms. This paper describes the integration and configuration of a Sampler Correlator and curve segmentation blocks in order to obtain a digital receiver for a proper demodulation of the received signal. The proposed solution is shown to strongly reduce hardware complexity. In this part, a presentation of the proposed solution regarding the analytical expressions is addressed. The paper covers in detail the elements needed for properly configuring the system. In a second article the implementation of the system for FPGA technology and the simulation results in order to validate the overall performance are presented.

These types of detectors are implemented in a two-part structure The first part is an analogue correlator (Egau, 1984); the second part includes procedures for derivation, addition and multiplication, in order to extract the instantaneous frequency.Figure 1 depicts the block diagram of the Balanced Quadricorrelator.However, due to the large number of adders and multipliers needed for the lowpass filter (LPF) implementation, such digital implementations demand a high cost.
Although the low pass filter and the multiplication procedure could be obtained through a flip-flop element, as in the digital logic quadricorrelator scheme of (Ahn et al., 2005), and the differentiator can be accomplished by flip-flop blocks followed by AND-gates, this type of solution is prone to instabilities under conditions of low signal to noise ratio.This causes total interruption of the counting process at the system output, despite its low hardware consumption.The method discussed in this paper outperforms all previous correlator-based schemes, in terms of hardware complexity.A new BFSK detection correlator-based scheme, directed to the slope recognition instead of the instantaneous frequency at the correlator output, is devised.This article addresses in detail the analytical expressions used to describe the proposed solution, which exhibit a strong reduction with regard to hardware complexity.
The rest of the paper is organized as follows: Section 2 summarizes the proposed solution in a descriptive way, for ease of exposition.Section 3 addresses the analytical expressions for configuring the proposed solution.Section 4 addresses a discussion with regard to hardware complexity.Finally, the Conclusions are presented in Section 5.

Receiver conception. Cualitative description.
In the scheme proposed in this paper a Discrete Correlator (Lee et al., 1950) is employed in order to reduce hardware complexity (the scheme is depicted in Figure 2).The correlator is configured with a local tone of frequency w 0 and phase φ L .The received BFSK waveform arrives with a frequency w i , i either 0 or 1, with phase φ 0 and amplitude given by A c .
At the output of the sampler correlator, linear segments are obtained in the temporal series of y p [n] and y q [n] for the in phase and quadrature branches, respectively.The slope of these segments depends on the symbol received.For instance, considering the result shown in Figure 3, it can be noted that with the received lower frequency w 1 the output of the correlator is comprised by segments of zero slope (e.g.AB and CD ).On the other hand, with the received higher frequency w 0 , the output is comprised by segments of non-zero slope (e.g.BC ).The main idea of the proposed receiver is to identify the approximate linear signal ramp observed at the output of the correlator for a sine-wave input (Sklar, 2001), and, by means of this procedure, to identify the symbols received.The change of slope in the curve showed in Figure 3b is directly related to the symbol transmitted.Therefore, methods for recognizing such changes can be employed for demodulation purposes.
The curve in Figure 3b is comprised by the linear segments AB, BC and CD, and the split of a curve into linear segments is a task intensively studied in the field of curve segmentation by means of two categories (Iñesta et al., 1998): feature point approach and error tolerance approach.This paper addresses those methods based on the feature point approach.
Methods of feature point approach identify dominant points, such as B and C, by means of a curvature measurement.Consequently, this measure can be directly related to the slope of each segment.The techniques of curve segmentation can be employed for extracting the tanφ of each segment on Figure 3b.
In average, the slope of segment AB is approximately zero, whereas the slope of segment BC can be described, for the in phase branch, as in (Sklar, 2001): For the quadrature branch a similar results is obtained: Considering the values of the slope in Equations ( 1) and (2), a receiver can be implemented by squaring the tangent functions in Equations ( 1) and ( 2), which yields to: Summarizing, the proposed solution uses the sampler correlator of Figure 2 in order to obtain the linear curves, the slope that can be estimated by curve segmentation techniques.After estimating the slope values in Equations ( 1) and ( 2), the operation in Equation ( 3) is performed for obtaining high and low levels and then recovering the transmitted bits.The general scheme is depicted in Figure 4 and the output is depicted in Figure 5.  From Figure 5 it can be noted that the pulses in b) can be used to detect the frequency received in a); this detection can be accomplished by means of threshold (dotted line), the value that is determined in Section 3.

Curvature measurement methods
Once the signal of the sampler correlation output is obtained, the estimation of the slope of each segment follows.To that end, the use of curvature measurement methods is analyzed.
Methods such as k-cosine (Wang et al., 2010), k-angular bending (Iñesta et al., 1998), Distance Accumulator (Han, Poston, 1993) and Scatter Matrix (Anderson, Bezdek, 1984) are usually employed to measure the curvature from images.Among this methods, the k-angular bending is the only one that does not implement a division procedure for estimating the slope.This is an important issue concerning a practical implementation in hardware, for instance in FPGA, since a division procedure can produce undesirable glitches at the output.Therefore, the implementation of k-angular bending is considered here for implementing the block curvature measurement depicted in Figure 3.

k-angular bending foundations
This method estimates the slope for each segment calculating the ratio between the incoming values and the time index in a sliding window of length k.At the same time, an average procedure is also implemented for reducing the effect of noise.The value for tanφ [n], for the quadrature and the in phase branches, is obtained through the following relation: where:  4) can also be simplified as indicated by: where: Considering the numerator in Equation ( 4), the expression can be simplified as: Considering the denominator in Equation ( 4), the expression can be simplified by means of: Then, when applying this procedure in the curvature measurement block of Figure 3, the output signal will be high or low, corresponding to the higher or lower frequency of the input signal, respectively (see Figure 4).Thus, changes in those levels can be employed for detecting the symbol transmitted.For this purpose, a proper threshold must be derived, which is addressed in the next Section.The expression given in Equation ( 6) is useful in the digital implementation the system, an issue that will considered in the second part of this paper.

Receiver conception. Analytical description
in order to recover the bits transmitted, it is necessary to establish a threshold for identifying the high from the low levels at the output of the receiver.This is indicated by the dotted line in Figure 5b).Besides, the amplitude of these oscillations depends on the levels at the output of the system, and this can be verified on Figure 4b.In this case, the high level exhibits an oscillating behavior with higher amplitude than the low level.
In order to establish a threshold, the oscillations produced by the operation given in Equation ( 3) must be considered.
Taking into account that the amplitude of these oscillations depends on the binary levels, then the threshold is not easily established as the half of the maximum amplitude; it also depends on the oscillating terms.In order to obtain a proper value for the threshold, it is necessary to analytically describe the proposed system, and this is addressed by the following steps: 1.The obtaining of the analytical expressions for y p [n] and y q [n] of Figure 2.
2. The obtaining of the variance around the binary levels at the output of the system of Figure 4.

Analytical expression for y p [n] and y q [n]
To obtain the analytical expressions, there are two cases to be described.The case in which the lower frequency is received, w 1 , and the case of the higher frequency, w 0 .For the latter, the output of the correlator is as follows (details are given in the Appendix A): where: From Equation (8) it can be observed that the first term is directly related to the ramp signal, and this is described by the segment BC on Figure 3b).This segment has a maximum slope given by A c 2 2 , which means that the maximum voltage at the output of the correlator will be given by A c 2 4 N +1 ( ) , where N represents the total number of samples between transitions.Thus, the system used to implement the receiver in Figure 4 should be capable of storing this value.Additionally, once the detection block identifies a transition, then a reset on the accumulator block can be settled.By this mechanism, the values of y p [n] and y q [n] will be set to zero in order to avoid larger values The slope of is the segments in the phase and quadrature branches are the average value to be estimated by the k-bending method.On the other hand, there is also an oscillating term, denoted by y p00 * n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ in Equation ( 9), which produces the oscillating behavior at the output of the system.For the quadrature branch a similar description to Equation ( 8) is obtained, except that φ L must be replaced by ϕ L - On the other hand, the analytical description when the received frequency is w 1 is given by the relation (A3) in the Appendix ,when w i = w 0 , and denoted by y p10 * n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ .In this case, the slope is zero in average; this is graphically described by the segments AB and CD in Figure 3.
Variance at the output of the system.
The variance at the output of the system is given by the oscillating term y p00 * n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ for the high levels and for the low levels.The method of k-bending performs the Taking into account the use of trigonometric identities for the subtraction of sine functions, the term y p00 * can be simplified as: 4 sin w 0 ( ) An upper bound for Equation ( 11) can be obtained by: 2 sin w 0 ( ) According to the upper bound in Equation ( 12), which is independent of the time index n, the sum in Equation ( 10) can be upper bounded as follows: This sum formula is obtained by straightforward methods, described in Appendix B. The relation in Equation ( 13) is also valid for the quadrature branch, since the phase terms φ 0 and φ L have been neglected; this is denoted by ∆ q00 .
Finally, the variance of the output y out [n], for w 1 = w 0 , can be upper bounded by: The quantities given by ∆ p00 and ∆ q00 are equal in value, and they represent the maximum amplitude of the oscillations at the output of the square procedure for both branches in Figure 4.However, these oscillations are orthogonal, thus the sum in Equation ( 14) can be upper bounded as: 15)   given that the maximum value of the sum of a sine and a cosine function is 2 .
Secondly, when w 1 ≠ w 0 the variance of the output y out [n] can be obtained by means of the procedure implemented for obtaining Equation ( 15).In this case, y p10 * n ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ comprises two tones, one of them at the frequency w 0 + w 1 2 and the other tone at w 0 − w 1 2 , as described in Equation (A3), in Appendix A. Similar to Equation ( 15), an upper bound can be obtained as: For the quadrature branch, a similar upper bound is obtained; this is denoted by ∆ q01 .Finally, after squaring and multiplying by 2 c 2 , an upper bound is obtained by: Once the variances are obtained around the high and the low levels, a threshold can be established for detecting the transitions.In case of the high level, the amplitude of the oscillation is given by ∆ 1 in Equation ( 15), with a direct component equal to A c 2 4 , as given in Equation (3).
For the low levels, the amplitude of the oscillation is given by ∆ 0 .As such, the threshold must be established in some and ∆ 0 .Here the threshold is placed on the middle for increasing the margin noise as: sum indicated in Equation ( 6), which can be arranged in a different way for the ease of calculation as follows: The threshold given in Equation ( 18) is employed to distinguish the low and high levels of the output signal; Figure 5 b) shows the result by means of dotted line with ∆ 1 = 0,0637, ∆ 0 = 0,0108, and y th = 0,1250.The terms ∆ 1 and ∆ 0 represent the effect of the oscillating terms.The use of this relation is affordable if the amplitude of the oscillations around the high and low levels are not comparable, that is, if . Otherwise, a lowpass filter can be employed at the output of the system to attenuate the amplitude of such oscillations.The precision in the determination of transitions, given by the threshold and the risetime of the pulses, is directly related to the total number of bits to be demodulated without errors, an issue that will be analyzed in Section 2.4.
Besides, the length of window k can be established for attenuating ∆ 1 or ∆ 0 in Equations ( 15) and ( 17).Depending on which oscillating term is the highest, k may be chosen in order to make zero the sine function.
In case that the value of A C is unknown, the expression given in Equation ( 18) requires to estimate this value.This is due to the fact that the output of the system is comprised by high and low levels.However, these levels can be transformed into bipolar pulses if the direct component (DC level) of the output sequence y p [n] is supressed.This can be achieved through the use of a Notch filter, the transfer function given by (Proakis, Manolakis, 2006): 19)   where r and w c represent the radius of the pole and the center frequency of the filter, respectively.In order to suppress the DC level, the value of w c must be settled to zero.In this case, the expected value will be zero and the threshold to y th = 0.

Discussion
With regard to hardware complexity, the proposed scheme, with the description given in Figures 4, needs 7 adders and multipliers.Three adders comes from the accumulator blocks and the output of the system, two multipliers are from the squaring devices and the others two from the multiplier at the beginning of branch.In addition, the curvature measurement blocks can be implemented with 2 FIR filters by means of a moving average structure (Oppenheim et al., 2010) without multipliers, since all the coefficients are equals to 1 as indicated in 5. Finally, through the use of the notch filter in Equation ( 21) a total of 7 adders and multipliers are also needed if the value of A c is unknown.
On the other hand, considering the system for the Balanced Quadricorrelator in Figure 1, a total of 7 adders and multipliers with another 4 FIR filters are required.Comparing the elements needed for each receiver, the proposed solution may need 7 additional adders and multipliers, but a reduction of FIR filters is achieved.Furthermore, the FIR filters employed do not use multipliers and the order of the filters employed in the Balanced Quadricorrelator could be superior to that of the proposed system, taking into account the difference between w 0 and w 1 .

Conclusions
The

Appendix A
This Section summarizes the obtaining of the analytical expressions for y p [n] and y q [n] in Figure 2. Considering the multiplication of the received signal by the local tone and after applying some trigonometric identities, the following expression is obtained: Where: x ( ) The operation of the accumulator block from the sampler correlator is to perform the following sum: After considering the Euler's theorem and the sum formula of the geometric progression (Gradshteyn 1980), it is possible to obtain closed form expressions for each term After some straightforward manipulations, the description for yp [n] yields: The description given in Equation (A3) considers the case when the frequency received is different to that of the local tone, that is w 1 ≠ w 0 .The case in which w 1 = w 0 can be obtained by taking the limit lim

Appendix B
In this section the sum formula for relation Equation ( 13) is obtained as follows: In this development, the Euler's identity and the formula of the geometric progression have been used (Gradshteyn 1980).

Figure 1 .
Figure 1.Block Scheme of the Balanced Quadricorrelator.

Figure 2 .
Figure 2. Receiver for the BFSK waveform with segmentation curve techniques.

Figure 4 .
Figure 4. Receiver for the BFSK waveform with segmentation curve techniques.
When the operation in Equation (3) is performed, additional oscillation is introduced around the expected values A c 2 4 and 0, in the absence of noise.These oscillations are easily verified in the high level of Figure 4b they are produced by the multiplication of the received signal with the local tone.
branch similar expressions are obtained upon substituting φ L by ϕ L − π solution presented in this paper proposes a new detection scheme for demodulating BFSK signals.Highorder filters are avoided because of one important matter: the symbol detection is accomplished through the constant slope recognition carried out at the output of a Sampler Correlator.This element represents the bridge between digital demodulation and curve segmentation techniques.From this point of view, new digital demodulators for BFSK waveforms can be developed, which advantageously save receiver complexity, in comparison to others reported.The system is only implemented by FIR and IIR filters, devices suitable to be implemented with low complexity.