Modulated Signals in Chemical Reaction Networks
MModulated Signals inChemical Reaction Networks ∗ Titus H. Klinge and James I. Lathrop Drake University, Des Moines, IA 50311 USA [email protected] Iowa State University, Ames, IA 50011 USA [email protected]
Abstract
Electrical engineering and molecular programming share many ofthe same mathematical foundations. In this paper, we show how tosend multiple signals through a single pair of chemical species usingmodulation and demodulation techniques found in electrical engineer-ing. Key to our construction, we provide chemical implementations ofclassical linear band-pass and low-pass filters with induced differentialequations that are identical to their electrical engineering counterparts.We show how to modulate arbitrary independent input signals withdifferent carrier frequencies for transmission through a shared medium.Specific signals in the medium can then be isolated and demodulatedusing band-pass and low-pass filters. Such programmable chemicalband-pass filters also offer a way to monitor chemical systems to verifythat they are operating between a prescribed set of frequencies.
The chemical reaction network model is commonly used to prototype nanode-vices [6, 17, 19]. In particular, a chemical reaction network (CRN) modelsthe molecular interactions of chemical species and is related to distributedmodels of computation such as population protocols [1, 7]. One of the mostcommon variants of the CRN model is deterministic and is equivalent inpower to Shannon’s general-purpose analog computer [11, 15] and is Turing ∗ This research was supported in part by National Science Foundation Grants 1247051,1545028, and 1900716. a r X i v : . [ c s . ET ] S e p omplete [8]. Deterministic CRNs model the evolution of chemical speciesas real-valued concentrations whose rate of change is governed by massaction kinetics [10, 14]. Since distributed biochemical systems communi-cate via molecular concentration signals, investigation of various molecularcommunication techniques is ongoing [3, 9, 12].In this paper we show how to multiplex multiple chemical signals on asingle pair of chemical species using radio communication techniques. Weshow that classical amplitude modulation and demodulation of signals onspecific carrier frequencies can be accomplished with simple, small, and naturalchemical reaction networks. Our work is related to that of Cardelli, Tribastone,and Tschaikowski who recently showed that any linear electric circuit can beconverted to a chemical reaction network that approximates its behavior [4].In contrast, our chemical low-pass and band-pass implementations yieldsolutions that exactly simulate their electronic counterparts. Moreover, theirgeneral approach requires foreknowledge of the input signal ODEs, whereasour implementations are entirely input-agnostic.Modulation is accomplished through two reactions that multiply aninput signal with its corresponding carrier frequency signal, producing adual-rail signal using species M + and M − ; these two species representthe shared communication channel. Our modulation scheme uses s + 2 reactions to combine the carrier signals, where s is the number of signalsbeing modulated onto the medium. However, each modulated signal must alsohave a corresponding carrier frequency encoded as a chemical signal. For eachinput, we use four additional reactions to create a dual-rail sinusoidal signal,tuned to its target carrier frequency using the concentration of a catalystspecies. It is important to note that our construction does not require aperfect sinusoidal wave; almost any waveform of sufficient frequency can beused.Recovering the AM modulated signal is accomplished using a band-pass filter to isolate a particular modulated carrier frequency, followed by arectification and low-pass filter to reconstruct the original signal. Both thechemical low-pass and band-pass filters admit an arbitrary input, allowingthe system to operate even with unknown signals. The band-pass filteralone may be of interest for detecting if a system is oscillating in a range offrequencies. For example, many biological and chemical systems only functionwith oscillations within a frequency range [13, 18]. Since the band-pass filtercan be constructed using catalytic single rail inputs, a chemical reactionsystem can be devised that alarms when a target chemical species is notoperating between prescribed frequencies without affecting the system.The rest of this paper is divided as follows. Section 2 gives some basic2nformation about chemical reaction networks and filters. Section 3 describesthe construction of a programmable band-pass filter as well as a low-passfilter. Section 4 shows how to modulate an arbitrary signal, transmit it via amedium (with other signals), and then demodulate the signal. In this paper, we are concerned with the chemical reaction network ( CRN )model, which is frequently used in molecular programming [10, 14]. CRNsare an abstraction of modern chemistry, Turing complete [8], and deployableat the nanoscale with motifs such as DNA strand displacement [2, 5, 16].Abstract molecule types in CRNs are called species and are denoted withcapital Roman characters such as A , B , and C and other decorations such as X , X , and Y + , Y − . Although there are many variations of the CRN model,here we use CRNs under deterministic mass action semantics since they areintrinsically analog . These deterministic CRNs are similar to other analogdevices such as electrical circuits and Shannon’s general purpose analogcomputer (GPAC) [11, 15] and consist of systems of polynomial differentialequations.Formally, a chemical reaction network ( CRN ) N is a finite collection of reactions of the form ρ = X + X + · · · + X n k −−−→ Y + Y + · · · + Y m . (2.1)Here the species X , . . . , X n are the reactants , the species Y , . . . , Y m are the products , and k is the rate constant of the reaction ρ . Intuitively, a reactionspecifies a relationship between molecular species, and in particular, howreactants combine to form products. It is important to note that reactantsand products may not be unique; for example, X + X + Y −−−→ X + X + X is a valid reaction and is commonly written X + Y −−−→ X for convenience.The net effect of a reaction ρ on a species X , written ∆ ρ ( X ) , is the differenceof the multiplicities of X in ρ ’s products and reactants. For example, the neteffect of X + Y −−−→ Z on X , Y , and Z is -2, -1, and 1, respectively. If areaction has a net effect of zero on a reactant X , then X is called a catalyst of ρ .We now describe the semantics of deterministic chemical reaction networksunder mass action kinetics. Let N be a CRN consisting of a finite set ofreactions R over the species X , . . . , X n . Then N induces a polynomial initialvalue problem (PIVP) x = ( x , x , . . . , x n ) where each variable x i represents3he real-valued concentration of the species X i . Each variable x i in the PIVPobeys the polynomial ordinary differential equation dx i dt = (cid:88) ρ ∈ R ∆ ρ ( X i ) · rate ρ ( t ) (2.2)where rate ρ ( t ) is the rate of reaction ρ at time t , defined to be the product of itsrate constant along with each of its reactants. Providing initial concentrations x (0) = x , the PIVP yields a unique solution x ( t ) .As an example, consider the CRN defined by the reactions X + Y k −−−→ ZX + Z k −−−→ XY + Z k −−−→ Y. According to equation (2.2), these reactions induce the ODEs dxdt = − k xy + k xzdydt = − k xy + k yzdzdt = 2 k xy − k xz − k yz. Electronic filters are used in many electronic devices, including radios,power lines, headphones, radar terminals, and many others. Filters take aninput signal and produce an output signal, and are often characterized by thisinput-output relationship. This relationship is called a transfer function andis simply the output divided by the input. In linear systems, the output isrelated to the input through a linear differential equation and can be realizedin electronic circuits with resistors, capacitors, and inductors. One method ofcharacterizing these filters utilizes the transfer function and Laplace transformto give a Bode plot, the response of the filter in terms of frequency.Four common categories for filters include low-pass, high-pass, band-pass,and notch, characterized by how much input signal is transmitted at differentfrequencies. For example, a low-pass filter transmits the input signal to theoutput at low frequencies but attenuates the input signal at higher frequencies.This is depicted by a graph that shows the ratio of the output voltage to theinput voltage with respect to frequency. This ratio is measured using dB,
20 log outputinput . For example, if the output signal is half that of the input at4 M agn i t ude ( d B ) -2 -1 -180-135-90-450 P ha s e ( deg ) Frequency (rad/s)
Pass Band Transition Band Stop Band
Figure 2.1: Example Bode plot of a first-order low-pass filter, showingpassband, transition band, and stopband at -60 dB.a particular frequency, then it is approximately -6 dB lower. If the outputsignal is double the input at a frequency, then the filter has a gain (at thatfrequency), and in this case, it is a gain of 6 dB. In addition to the magnitudeof the output to the input, it is also necessary to define the phase changefrom the output to the input at each frequency. These two graphs takentogether are commonly referred to as a Bode plot of the response of the filter.Figure 2.1 shows a low-pass filter with transfer function given as Y ( s ) X ( s ) = 1010 s + 10 , where Y ( s ) is the input and X ( s ) is the output in the frequency domain.Besides characterizing a specific filter, Bode plots are also used to specifythe requirements for filters. For a low-pass filter, there are several criticalparameters. The parameters pertinent to our discussion here are describedbelow.1. Passband.
The passband specifies the region of frequencies thattransmit the input signal to the output signal, and is typically specified5 M agn i t ude ( d B ) f c Transition Band Transition BandPass BandStop Band Stop Band
Figure 2.2: Bode plot of a band-pass filter showing the passband, transitionband, and stopband regions. f c labels the center frequency of the passbandregion. For this filter, the stopband is 40 dB below the input signal. Notethat the phase plot is not shown is this figure.by a single frequency f c that defines the highest frequency where theoutput is above -3 dB of the input.2. Stopband.
The stopband specifies the frequency range below anacceptable level of input leakage to the output.3.
Transition band.
The transition band is the range of frequenciesbetween the passband and the stopband.Figure 2.1 depicts a low-pass filter specification where the stopband is definedas 60 dB below the input signal level. A specification for a band-pass filter isshown in Figure 2.2. Here the center frequency of the passband is denoted f c , and the band width of the filter is the range of frequencies that give anoutput signal that is above -3 dB of the input signal. The high-pass filter isanalogous to the low-pass filter except that the stopband is at low frequencies,and the passband is at high frequencies. A notch filter is similarly an invertedband-pass filter that rejects (stop signals) in a specified range of frequencies.Thus, these filters can also easily be specified in terms of Bode plots. In this section, we describe a programmable band-pass filter and a naturalimplementation using chemical reaction networks. By “natural,” we meanthat the reaction network is small, straightforward, and does not approximateinputs, components, or functions of inputs. In effect, the transfer function for6 in R V out C Figure 3.1: Circuit diagram of a low-pass filterthe band-pass filter directly follows from the differential equations derivedfrom a simple CRN. Our construction will proceed by first constructing anatural low-pass filter. This low-pass filter implementation is integral to theband-pass filter construction and demonstrates why high-pass filters cannotbe implemented without approximation.It is well known that a simple electrical low-pass filter can be constructedusing a resistor and capacitor, as depicted in Figure 3.1. We construct thechemical reaction equivalent of this electrical circuit by reverse-engineeringthe CRN from the ordinary differential equations (ODEs) given by Kirchhoffcircuit laws and the low-pass circuit, which are shown below. iR = v in − v out (3.1) i = C dv out dt (3.2)Substituting equation (3.2) into equation (3.1) yields the first-order ordinarydifferential equation: dv out dt = v in − v out RC . (3.3)We can convert this ODE directly into CRN 3.2 with two species V in and V out , representing the two voltages. This construction is commonly called pure pursuit because one signal is “pursuing” the other. In this case, theoutput species V out is “chasing” the input species V in , and the rate constantof RC determines how fast V out chases V in . It is easy to verify that this CRNimplements a low-pass filter described in Section 2 with cutoff frequency f c = 1 /RC and gain factor k = 1 /RC . We decouple the gain factor fromthe cutoff frequency by decoupling the rate constants in the CRN. Thisis achieved by CRN 3.3 where the output concentration chases the inputconcentration times the gain factor k . This CRN’s input/output behavior isexactly the standard transfer function for a first-order low-pass filter H ( s ) = ks + c , (3.4)7 out RC −−−→ ∅ V in RC −−−→ V in + V out CRN 3.2: Simple low-pass filter as pure pursuit V out c −−−→ ∅ V in k −−−→ V in + V out CRN 3.3: Simple low-pass filter with gainwhere c is the cutoff frequency and k is the gain.The standard implementation of a band-pass filter is to compose a low-pass filter with a high-pass filter. Unfortunately, the standard implementationof a high-pass filter using circuits yields a differential equation in which thederivative of the output voltage is dependent on the derivative of the inputvoltage. For example, dv out dt = dv in dt − v out RC , (3.5)is the differential equation derived from the standard first-order high-passfilter. Although this is easily implemented as an electrical circuit using acapacitor, a CRN cannot compute the derivative of an arbitrary input signalexactly, unless the input signal can be anticipated and hard-coded into theCRN construction.Approximating circuit behavior with CRNs has been studied in [4]. In thispaper we are interested in producing a band-pass filter with an arbitrary inputwithout approximation, over all time, in a relatively small CRN. Somewhatserendipitous, a simple approximation of the derivative leads directly tothe band-pass filter that we desire. A simple approximation of a derivativeinvolves computing the difference between the input signal at two differenttimes; unfortunately, creating a perfect time-delayed signal is not possiblewith a CRN. Nevertheless, we can crudely approximate a time-delayed signalby using pure pursuit in the same way as described in the low-pass filter above.8t is then possible to approximate the derivative by subtracting the pursuingsignal from the original, which gives a crude estimate of the rate of changeover a short time interval. Using this approximation, the resulting CRNinduces a differential equation and transfer function that exactly correspondto a second-order band-pass filter without approximation . Motivated by this,we consider the parameterized transfer function for a second-order filter shownbelow. H ( s ) = ass + bs + c (3.6)Letting x represent the input signal and y represent the output signal, wehave the corresponding differential equation dydt = ax − by − cz, where z ( t ) is the function z = (cid:82) y ( t ) dt . We can now realize a CRN for thisdifferential equation. However, the − cz term in this ODE cannot be realizeddirectly by a CRN, so we employ a construction introduced by Fages et al.that implements such terms using the difference of two species [8]. Thisis commonly called the difference construction and requires splitting eachvariable into two parts: x + , x − , y + , y − , z + , and z − . For this technique towork, it is critical to maintain the invariants: x ( t ) = x + ( t ) − x − ( t ) ,y ( t ) = y + ( t ) − y − ( t ) ,z ( t ) = z + ( t ) − z − ( t ) . Using this technique, we transform these ODEs into the following equivalentsystem of ODEs dy + dt − dy − dt = a ( x + − x − ) − b ( y + − y − ) − c ( z + − z − ) dz + dt − dz − dt = z + − z − . We can now generate the corresponding CRN given below.9 + a −−−→ X + + Y + X − a −−−→ X − + Y − Y + b −−−→ ∅ Y − b −−−→ ∅ Z + c −−−→ Z + + Y − Z − c −−−→ Z − + Y + Y + 1 −−−→ Y + + Z + Y − −−−→ Y − + Z − Note that we also add the reactions Y + + Y − −−−→ ∅ Z + + Z − −−−→ ∅ to bound the concentrations of Y + , Y − , Z + , and Z − , but this does not affectthe solution to the differential equations. The rate constants a , b , and c canbe replaced by catalyzed biomolecular reactions with unity rate constants.This gives us the ability to tune the filter using concentrations of speciesrather than using rate constants in the reactions.The Laplace transform of a second-order band-pass filter transfer functionin terms of gain k , quality Q , and center frequency ω is given by H ( s ) = k ( ω /Q ) ss + ( ω /Q ) s + ω . By assigning a , b , and c appropriately, we can realize any second-order band-pass filter of this form in a CRN with no approximation over all time. Thuswe have that a = k ( ω /Q ) b = ω /Qc = ω . + X + 1 −−−→ A + X + + Y + A + X − −−−→ A + X − + Y − B + Y + 1 −−−→ BB + Y − −−−→ BC + Z + 1 −−−→ C + Z + + Y − C + Z − −−−→ C + Z − + Y + Y + 1 −−−→ Y + + Z + Y − −−−→ Y − + Z − Y + + Y − −−−→ ∅ Z + + Z − −−−→ ∅ -30-20-100 M a gn it ud e ( d B ) -3 -2 -1 -90-4504590 P h a s e ( d e g ) Frequency (rad/s)
Figure 3.4: Band-pass filter with center frequency . rads/sec, bandwidth . rads/sec, and gain of , ( a = 0 . , b = 0 . , c = 0 . )For example, Figure 3.4 shows the CRN implementation and Bode plots of aband-pass filter with gain , center frequency . radians per second, andbandwidth . radians per second. Figure 3.5 shows a Matlab simulationof this band-pass filter with three different input signal frequencies: (1) exactlythe center frequency (0.009 rads/sec), (2) twice the center frequency, and(3) half the center frequency. As expected, the output signal of the filterwith input exactly the center frequency outputs the signal at unity gain. Theoutput of the other two frequencies are appropriately reduced. Encoding and transmitting many signals through a single medium is utilizedin a variety of applications, including AM and FM radio, cable TV, ADSL,and others. In this section, we describe how a chemical reaction network maybe designed to encode and decode amplitude modulated (AM) chemical con-centrations signals. We further describe a method where these concentrationsignals can also be encoded using frequency modulation (FM).11
Time -1-0.500.51 << Back Forward >>
Figure 3.5: Matlab simulation of band-pass filter. Input signal is depicted inblack, output signal depicted in red. Top graph input is sin wave at band-passcenter frequency of 0.009 rads/sec. Middle graph is twice the venter frequency,and bottom is half the center frequency.
Amplitude modulation (AM) encodes a signal on a carrier frequency bymodulating the amplitude of the wave proportional to the signal. Due tothe superposition principal, multiple signals can be encoded using differentcarrier frequencies and then transmitted over a single medium. In the case ofchemical concentrations, our goal is to transmit and receive multiple signalconcentrations sent through a single species that encodes all the transmittedsignals.It is relatively straight-forward to encode an input signal with amplitudemodulation. The input signal u ( t ) is multiplied by a carrier, sin( θt ) where θ is the carrier frequency . Thus, implementing an AM modulator in achemical reaction network requires (1) generating the carrier sin( θt ) , and(2) multiplying the input signal u ( t ) by the carrier signal.We begin with the CRN construction for generating the carrier signalwhich consists of the reactions: 12 + C + −−−→ F + C + + S + F + C − −−−→ F + C − + S − F + S − −−−→ F + S − + C + F + S + −−−→ F + S + + C − S + + S − −−−→ ∅ C + + C − −−−→ ∅ The above CRN is designed in a dual rail scheme such that s + ( t ) − s − ( t ) = sin( f t ) c + ( t ) − c − ( t ) = cos( f t ) is satisfied for all t ≥ as long as s + (0) − s − (0) = 1 and c + (0) − c − (0) = 0 is satisfied. For convenience, we define the function s by s ( t ) = s + ( t ) − s − ( t ) for all t ≥ . It is important to observe that the net effect on the species F is zero. By design, F is constant and serves as a means of tuning thecarrier frequency of the sine wave, akin to tuning an AM radio. F can alsobe changed dynamically while the CRN is active.With the carrier signal generated via s = s + − s − , it remains to computethe modulated signal from the input signal by a simple multiplication. Our ap-proach uses the pure pursuit technique described in Section 3 to approximatean exact multiplication of the two signals. The ODE for this approximationis dmdt = us − m where m = m + − m − is also a dual-rail signal. In this case, the value of m ( t ) is “pursuing” the value of the product u ( t ) · s ( t ) . In effect, this is alow-pass filter with the input signal modulating the gain. The approximationcan be improved by uniformly increasing the rate constants of the reactionsbelow. M + −−−→ ∅ M − −−−→ ∅ U + S + −−−→ U + S + + M + U + S − −−−→ U + S − + M − + C + −−−→ F + C + + S + F + C − −−−→ F + C − + S − F + S − −−−→ F + S − + C + F + S + −−−→ F + S + + C − S + + S − −−−→ ∅ C + + C − −−−→ ∅ M + −−−→ ∅ M − −−−→ ∅ U + S + −−−→ U + S + + M + U + S − −−−→ U + S − + M − CRN 4.1: Modulation CRNIt is easy to verify that the reactions above induce the ODEs dm + dt − dm − dt = (cid:0) − m + + us + (cid:1) − (cid:0) − m − + us − (cid:1) = u (cid:0) s + − s − (cid:1) − (cid:0) m + − m − (cid:1) = us − m. Figure 4.1 defines the complete CRN that modulates a signal on a specifiedcarrier frequency and Figure 4.2 is a Matlab simulation of this CRN. In thissimulation, the signal to modulate is a simple sine wave that is one tenth thefrequency of the carrier wave.It is easy to combine the modulation scheme described above for n > signals u ( t ) , . . . , u n ( t ) with corresponding carriers s ( t ) , . . . , s n ( t ) andtransfer all of them simultaneously through the single dual-railed signal m ( t ) .This is accomplished by summing all of the modulated signals m ( t ) via a CRNthat approximate the sum (cid:80) ni =1 u i ( t ) · s i ( t ) . The CRN for this approximationis shown in CRN 4.3. It is easy to verify that the ODE for m ( t ) is dmdt = n (cid:88) i =1 u i · s i − m, which can again be made arbitrarily precise by increasing the rate constants.14 Time -1-0.500.51 << Back Forward >>
Figure 4.2: Single signal modulation with carrier frequency of . rads/secand a simple sine wave signal of . rads/sec. The top graph is the carriersignal and the bottom graph is the signal superimposed over the modulatedsignal. M + −−−→ ∅ M − −−−→ ∅ U + S +1 −−−→ U + S +1 + M + U + S − −−−→ U + S − + M − ... U n + S + n −−−→ U n + S + n + M + U n + S − n −−−→ U n + S − n + M − . CRN 4.3: Superposition of all modulated signals15 mpmmmf1cp1sp1cm1sm1u1f2cp2sp2cm2sm2u2
Time << Back Forward >>
Figure 4.4: Two signals are shown in the top graph. The combined modulatedsignal is shown in the bottom graph.Input m Band-passfilter Low-passfilter Output y Figure 4.5: Block diagram...Note that only one copy of the reactions M + → ∅ and M − → ∅ are nec-essary even though there are n signals being passed through m ( t ) . Figure 4.4shows two signals that are modulated at two different carrier frequencies, 0.1rads/sec and 0.2 rads/sec. The combined superimposed signal is also shown. We now describe a CRN that given a signal encoding many modulated signalsencoded outputs an approximation of the original signal, similar to how anAM radio may be tuned to decode a signal for a specific radio station ata specific frequency. A simple method for doing this utilizes a band-passfilter to select a specific carrier frequency to pass, followed by a low-passfilter on just the positive signal to remove the carrier frequency but leavethe original signal. The equivalent circuit is known as a diode detector,one of the simplest methods to demodulate amplitude modulated signals.Figure 4.5 shows a high-level block diagram. Both the band-pass and low-passfilters were discussed in Section 3, thus it remains to compose them together16nd select the appropriate parameters. In Figure 4.6, we show an exampleof an input signal, modulated by a carrier frequency of 0.1 rads/sec, andthen demodulated to give an output signal. The Modulated Signal and theBand-pass Filtered Signal are very close to the same since there is only asingle modulated signal. It is very simple to rectify the Band-pass filteredsignal so that only the positive portion remains by only utilizing the positivecomponent yp of y shown in Figure 4.6. The bottom two graphs depictthe output of the Rectified Filter signal after passing though a low-pass filterto recover the original signal. Two different low-pass filters are demonstrated.The first is a simple first-order filter, used when radio was first invented. Thesecond shows an improvement in the output when a second-order filters isused with some gain. Finally in Figure 4.7 we show two signals that aremodulated on two different frequencies of 0.1 rads/sec and 0.2 rads/sec, theirresulting modulated signal, and the demodulated output of the two signals.It is worth noting again that this scheme is not limited two signals. This section shows that it is possible to amplitude modulate signals and thendemodulate them using chemical reaction networks. In fact, any number ofsignals can be encoded with carrier waves of different frequencies, summedtogether, and then transmitted through a single dual-railed species. Whilethe examples above utilize only first-order differential equations, higherperformance filters are easily generated by composing first-order filters, ordirectly implementing a higher-order differential equation. These lead to all ofthe standard low-pass and band-pass filters found in literature and textbooks,including Butterworth, elliptical, and Chebyshev filters. However, any linearcircuit may be implemented with chemical reaction networks, provided thatthe differential equation(s) do not require the derivative of an arbitrary inputsignal. These filters operate identical to their circuit counterparts, and theperformance does not degrade over time. Note that it is possible to createa band-limited version of the high-pass filter using a band-pass filter, or anotch filter by composing two band-pass filters in series.Frequency modulation (FM) is also used to transmit data over a carrierfrequency, except that the frequency is modulated instead of the amplitude.This modulation is easily accomplished with the CRNs above by using thesine wave generator to again create a carrier frequency. However, insteadof using a constant-value species F to set the frequency, we instead add inthe signal to transmit to the constant frequency, and then use this signalto dynamically set the frequency of the sin wave generator. This gives17 -0.4-0.200.2 Original Signal(u1) -0.4-0.200.2 Modulated Signal (m = mp - mm) -0.4-0.200.2 Band-pass Filtered Signal (y1 = yp1 - ym1) -0.4-0.200.2 Rectified Filtered Signal (yp1) -0.4-0.200.2 First-order Low-pass Filter (dm1) time (second) -0.4-0.200.2 Second-order Low-pass FIlter(out1)
X X
Y Y -0.03904-0.03904
Figure 4.6: Complete Mod demod...18 Signal 1(u1) Signal 2(u2) -0.500.5 Modulated Signal(m = mp - mm) Demodulated Signal 1(out1) time (second) Demodulated Signal 2(out2)
Figure 4.7: Multi Signals Example...19he desired frequency modulated signal. The demodulation of frequencymodulated signals is accomplished with two band-pass filters and a low-passfilter.
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