`By Charan Langton
`www.compieX’coreai.::om
`
`Understanding Frequency Modulation (FM), Frequency Shift Keying (FSK), Sunde’s FSK and MSK and
`some more
`
`The process of modulation consists of mapping the information on to an electromagnetic medium (a
`carrier). This mapping can be digital or it can be analog. The modulation takes place by varying the three
`
`parameters of the sinusoid carrier.
`
`1. Map the info into amplitude changes of the carrier
`2. Map the info into changes in the phase of the carrier
`
`3. Map the info into changes in the frequency of the carrier.
`
`The first method is known as amplitude modulation. The second and third are botl1 a from of angle
`modulation, with second known as phase and third as frequency modulation.
`
`Let's start with a sinusoid carrier given by its general equation
`
`c(r) = Ac cos(2Jrfcr + (D 0)
`
`This wave has an amplitude Ac a starting phase of fdo and the carrier frequency, f:. The carrier in Figure
`1 has amplitude of 1 v, with ft of 4 Hz and starting phase of 45 degrees.
`
`Generally when we refer to amplitude, we are talking about the maximum amplitude, but amplitude also
`
`means any instantaneous amplitude at any time t, and so it is really a variable quantity depending on where you
`specify it.
`
`? AM 2
`
`Figure 1 — A sinusoid carrier of frequency 4, starting phase of 45 degrees and amplitude of 1 volt.
`
`The amplitude modulation changes the amplitude (instantaneous and maximum) in response to the
`information. Take the following two signals; one is analog and the other digital
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 1, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 1, IPR2016-00766
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`
`
`
`
`a. Analog message signal, ma)
`
`III
`
`I
`2
`
`I
`-1-
`
`t
`
`I
`6
`
`I
`3
`
`10
`
`b. Binary message signal, mft)
`
`Fig 2 — Two arbitrary message signals, ma)
`
`About Amplitude Modulation (AM)
`
`The amplitude modulated wave is created by multiplying the ampiimde of a sinusoid carrier with the
`message signal.
`
`s(t) = m(t) c(t)
`
`= Acm(t) cos(2Jrf‘_,t +4130)
`
`The modulated signal shown in Figure 3, is of carrier frequency ft but now the amplitude changes in
`response to the information. We can see the analog information signal as the envelope of the modulated signal.
`
`Same is true for the digital signal.
`
`
`
`a. Amplitude modulated carrier shown with analog message signal as its envelope
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`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 2, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 2, IPR2016-00766
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`
`
`
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`c. Amplitude modulated carrier shown with binary message signal as its envelope
`
`Figure3—AmplitudemoduIatedcan'ier;a.anaIog, hdigital
`
`Let’s look at the argument of the carrier. What is it?
`
`T his part is
`phase.
`
`I
`e(.t) = A5 cos(2:*rf‘1 +;;'ru)
`
`This whole
`part is angle.
`
`The argument is an angle in radians. The argument of a cosine function is always an angle we know that
`
`from our first class in trigonometry. The second term is what is generally called the phase. In amplitude
`modulation only the amplitude of the carrier changes as we can see above for both binary and analog messages.
`
`Phase and frequency retain their initial values.
`
`Any modulation method that changes the angle instead of the amplitude is called angle modulation. The
`
`angle consists of two parts, the phase and the frequency part. The modulation that changes the phase part is
`called phase modulation (PM) and one that changes the frequency part is called frequency modulation (FM).
`
`How do you define frequency? Frequency is the number of 2 pt’) revolutions over a certain time period.
`
`Mathematically, we can write the expression for average frequency as
`
`= ¢,.(t + At) —(b‘. (t)
`2JrAt
`
`f“
`
`This equation says; the average frequency is equal to the difference in the phase at time t + D6! and
`
`time t, divided by 2pm? (or 360 degrees if we are dealing in Hz.)
`
`Example: a signal changes phase from 45 to 2700 degrees over 0.1 second. What is its average fre-
`quency?
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 3, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 3, IPR2016-00766
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`
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`_ 2700- 45
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`73-75%
`
`This is the average frequency over time period t = 0.1 secs. Perhaps it will be different over 0.2 secs or
`some other time period or maybe not, we don’t know.
`
`What is the instantaneous frequency of this signal at any particular moment in the 0.] second period? We
`
`don’t really know given this information.
`
`The instantaneous frequency is defined as the limit of the average frequency as Dfit gets smaller and
`smaller and approaches 0. So we take limit of equation 1 to create an expression for instantaneous frequency,
`f}(t).-
`
`The fi(t) is the limit of fm(t) as Dot goes to 0. The phase change over time Dét is changed to a differential
`to indicate change from discrete to continuous.
`
`f.-(r) = }gg10fm(r)
`
`=
`
`¢,'('t + A't)—¢,‘(t)
`235,3;
`
`A:—>o
`
`_ ; dm
`2::
`dt
`
`This last result is very important in developing understanding of both phase and frequency modulation!
`
`The 2p6 factor has been moved up front. The remaining is just the differential of the phase.
`
`Another way we can state this is by recognizing that radial frequency wfi is equal to
`
`(0:21: fi.(t)
`
`It is also equal to the rate of change of phase,
`
`(1)
`
`_d(¢’.-(0)
`_
`dr
`
`so again we get,
`
`Intuitively, it says; the frequency of a signal is equal to its phase change over time. When seen as a
`phasor, the signal phasor rotates in response to phase change. The faster it spins (phase change), the higher its
`
`frequency.
`
`What does it mean, if I say: the phasor rotates for one cycle and then changes directions, goes the oppo-
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 4, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 4, IPR2016-00766
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`
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`site way for one cycle and then changes direction again? This is a representation of a phase modulation.
`Changing directions means the signal has changed its phase by 180 deg.
`
`We can do a simple minded phase modulation this way. Go N spins in clockwise directions in response to
`
`a l and N spins in counterclockwise directions in response to a 0. Here N represents frequency of the phasor.
`
`Figure 4 — The carrier as a pkasor, the faster it spins, the higher the frequency.
`
`lffrequency is the rate of change of phase, then what is phase in terms of frequency? As we know from
`
`definition of frequency that it is number of full 2 p6 rotations in a time period.
`
`Given, a signal has traveled for 0.3 seconds, at a frequency of 10 Hz with a starting phase of 0, what is
`its phase now?
`
`Phase nowqb = 2Jrf;.t =10Hz><2Jr ><.3 = 20 radians
`
`This is an integration of the total number of radians covered by the signal in 0.3 secs. Now we write this
`as an integral,
`
`.3
`nm=mLfimm
`
`and since this is average frequency, it is constant over this time period, we get
`
`Phase nowqb = 2Jrf;.t =10Hz><2Jr><.3= 20 radians
`
`We note the phase and frequency are related by
`
`Phase a Integral of frequency
`
`Frequency a Differential of phase
`
`Phase modulation
`
`Let the phase be variable. Going back to the original equation of the carrier, change the phase (the
`
`underlined term only) from a constant to a function of time.
`
`(‘(0 = Ac cos(27rf[_I w)
`
`3
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`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 5, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 5, IPR2016-00766
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`
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`We can phase modulate this carrier by changing the phase in response to the message signal.
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`<9,-(I) = k,,m(r)
`
`Now we can write the equation for the carrier with a changing phase as
`
`s(t_) = AL, cos('27rf£t + kpmm)
`
`4
`
`The factor kP is called the phase sensitivity factor or the modulation index of the message signal. For
`
`analog modulation, this expression is called the phase modulation. In phasor representation of an analog PM,
`the phasor slows down gradually to full stop and then again picking up speed in the opposite direction gradu-
`
`ally.
`
`In binary case, the phasor does not slow down gradually but stops abruptly. This is easy to imagine. Take
`a look at equation 4. Replace the underlined terms by a 0 or a 1, or better yet, replace it by 180"’ if m(t) is 1
`and -180” if it is a 0. Now, we have a binary PSK signal. The phase changes from -180” to +180” in re-
`sponse to a message bit.
`
`This is the main difference between analog and digital phase modulation in that in digital, the phase
`
`changes are discrete and in analog, they are gradual and not obvious. For binary PSK, phase change is mapped
`very simply as two discrete values of phase.
`
`s(t)=ac cos(2Jrf__,t+«’,lJJ.)
`
`(bi. =0orJr
`
`nu ll um "WU
`
`2
`
`4
`
`t
`
`5
`
`s
`
`in
`
`2 H
`
`-2
`
`n
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`a. Phase modulation is response to a binary message. The phase changes are abrupt.
`
`'2 In
`
`2
`
`4
`
`1
`
`-5
`
`s
`
`10
`
`b. Phase modulation is response to an analog message. The phase changes are smooth.
`
`Figure 5 - Phase modulated carrierfor both binary and analog messages.
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`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 6, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 6, IPR2016-00766
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`
`
`Frequency modulation
`
`FM is a variation of angle modulation where instead of phase, we change the frequency of the carrier in
`
`response to the message signal.
`
`Vary the frequency by adding a time varying component to the carrier frequency.
`
`fi(t) = f; +kJ,m(t)
`
`where fc is the frequency of the unmodulated carrier, and kf a scaling factor, and m(t), the message signal.
`The term kf m(t) can called a deviation from the carrier frequency.
`
`Example, a carrier with fc = 100, k, = 8 and message bit rate = 1. Assume the message signal is polar so
`we have 1 fora 1 and -1 for 0.
`
`For m(t) = 1, we get
`
`f, = 100+ 8(1): 108 Hz
`
`for m(t):-1
`
`f2 = 100- 8(1) = 92 Hz
`
`The phasor rotates at a frequency of 108 Hz, as long as it has a signal of l and at 92 Hz for a signal
`indicating a 0 bit.
`
`Remember the equation relating phase and frequency
`
`1 d9. (I)
`. t = : '
`f'( )
`2::
`dt
`
`which can also be written as
`
`rm) = 2:: j f,-(t)dt
`
`Now look at the carrier equation,
`
`s(t) = A0 cos(2Jrfct +¢l.)
`
`5
`
`5
`
`7
`
`We define the instantaneous frequency of this signal as the sum of a constant part, which is the carrier
`
`frequency, and a changing part as show below.
`
`f,(t)=f,,+k,m(t)
`
`3
`
`The argument of the carrier is an angle. So we need to convert this frequency term to an angle. Do that by
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 7, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 7, IPR2016-00766
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`
`
`taking tl1e integral of this expression (invoke equation 6)
`
`J‘
`
`I
`
`9,.(:) = 2:: I fcdt + 22:19. I m(t)dt
`
`and since fc is a constant, this is just equal to
`
`t‘
`
`9l.(r) = 271'_f;£+ Zzrkf Im(r)dt
`
`9
`
`Now we plug this as the argument of the carrier into equation 7 and we get
`
`50‘) = Ac cos 0:: )3: + Ztrkf m(t)dt)
`
`10
`
`This represent the equation of a PM modulated signal. A decidedly unpleasant looking equation! This is
`about as far as you can get from intuitive. There is no resemblance at all to anything we can imagine.
`
`Analog FM is one of the more complex ideas in communications. It is very difficult to develop an intuitive
`
`feeling for it. Bessel functions rear their multi-heads here and things go from bad to worse. Thankfully this is
`where binary signals come to the rescue. The binary or digital form of Frequency Modulation is very easy to
`understand .
`
`But before we do that, examine the following relationship between FM and PM.
`
`Let’s write out both equations side by side so we can see what’s going on.
`
`PM;
`
`s(.') = A cos(2xfi.t+ kPm(t))
`
`FM
`
`s(t‘) = Ac cos (22: fit +2:rkf I m(t)a't)
`
`11
`
`12
`
`Note that in FM, we integrate the message signal before modulating. Both FM and PM modulated signals
`are conceptually identical, the only difference being in the first case phase is modulated directly by the message
`
`signal and in FM case, the message signal is first integrated and then used in place of the phase.
`
`Using a Phase modulator we can create a PM signal by just integrating the message signal first. Similarly
`with a FM modulator we can create a PM signal by differentiating the message signal before modulation.
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 8, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 8, IPR2016-00766
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`
`
`
`
`I111}
`
`
`
`Figure 6 — Pwe and Frequency modulators are interchangeable by just changing the form of the message signal.
`Dtfiereutiate the message signal and then feed to a PM modulator to get a FM signal and vice-verso for FM.
`
`Frequency Shift Keying — the digital form of FM
`
`The binary version of FM is called me Frequency Shift Keying or FSK. Here the frequency does not keep
`
`changing gradually over symbol time but changes in discrete amounts in response to a message similar to
`binary PSK where phase change is discrete.
`
`The modulated signal can be written very simply as consisting of two dijferent carriers.
`
`.910‘) = All cos(2Jrflt)
`
`s2 (I) = Ac cos(2Jrf2t)
`
`sl(t) in response to a 1 and s2(t) in response to a 0.
`
`Getting sophisticated, this can be written as a deviation from the carrier frequency.
`
`S1(t) = An 00S(2-’f(f.~ —Af)t)
`
`S2 (r) : All cos(2Jr(fl + Af)t)
`
`Here Ddf is called the frequency deviation. This is excursion of the signal above and below the carrier
`
`frequency and indicates the quality of the signal such in stereo FM reception.
`
`Each of these two frequencies fl and fl, are an offset from the carrier frequency, fc. Let’s call the higher of
`these fl and lower fl,. Now we can create a very simple FM modulator as shown the Figure 7. We can use this
`table to modulate the incoming message signal.
`
`m(t)
`
`-1
`
`+1
`
`Amplitude of
`fl]
`fl
`
`0
`
`1
`
`1
`
`0
`
`We send fl in response to a -1 and 0 in response to a +1. When we get a change from a -1 to +1, we
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 9, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 9, IPR2016-00766
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`
`
`switch frequency, so that only one signal, either fh or fl is transmitted.
`
`The combined signal can be written as, keeping mind that two terms are orthogonal in time and never
`occur at the same time.
`
`ssm (t) = A, cos(271'ffit +gD,,) + A, cos(27rf,r +gD,)
`
`13
`
`cc-s(2:'r_r;.E +rfi,¢)
`lg!-
`
`An
`1
`
`cos(2r.rjj£+¢,)
`
`3355:: (5)
`
`\‘'‘n.
`M
`®,,-
`T
`As
`
`Figure 7 — A FSK modulator
`
`In the modulator shown here, we are changing the amplitude A1 and A}. which are either 0 or 1 in response
`to the message signal, but never present at the same time.
`
`The above process looks very much like amplitude modulation. The only difference is that the signal
`polarity goes from 0 to 1(because Ah is either 0 or 1), rather than -1 to +1 as it does in BPSK. This is an
`important difference and we can try to understand FM better by making this clever substiultion (Ref. 1).
`
`Am) =%+%A;.<r)
`
`]
`]
`.
`AK?) =§+§/M‘)
`
`These transformations turn Ah and A1 from 0,1 to -1, +1.
`Now we substitute these into equation 13 and get,
`
`.v3,SK (I) = cos(2Jrf,,t +¢,,) + cos(2Jrf,t +¢,)
`+A',, cos(2zrf,: +¢,)+ A‘, cos(2.+rf,: +¢,)
`
`14
`
`15
`
`The first two terms are just cosines of frequencies fl] and fl, so their spectrum contribution is an impulse at
`frequencies fh and The second two terms are ampliulde modulated carriers. These last two terms give rise to
`a sin xix type of spectrum of the square pulse as shown below.
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 10, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 10, IPR2016-00766
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`
`
`III”
`
`Min.
`
`Figure 8 — Sin x/ x spectrum ofa square signal pulse
`
`The basic idea behind a PM spectrum is that it is a superimposition of impulses and sin xix spectrum of
`
`the pulses. However, this is Inle only if the separation or deviation is quite large and the overlap is small. The
`combinations are not linear and not easily predicted unless of course you use a program like SPW, Matlab or
`Mathcad as I do.
`
`The following figures show the spectrum of an FSK case, where the carrier frequency is 10, the message
`signal frequency is 1, Dot’ is equal 2.
`
`1.0
`
`0,0
`
`-1.0
`
`I
`500
`
`I
`
`10 III
`
`5
`
`I
`
`I
`2000
`
`I
`
`I
`250
`
`I
`
`a. Binary message signal, ma) = 2
`
`19
`
`0
`
`lH
`lH lll H_!_[_I_l_l llllnl l...I.l,I|ll
`In IIIIIII III!
`II
`'““'H“HHl
`
`-1.0
`
`b. Modulated signal, s(t) with ft: = 4
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 11, IPRZO16-00766
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`MTel., Exhibit 2004, ARRIS v. MTel., Page 11, IPR2016-00766
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`
`
`
`
`Lil“
`
`Mill.
`
`Figure 9- Spectrum of the modulated signal, Dfif is equal 2. Note the two impulses atfiequencies 2 and 6 Hz.
`
`Here we see clearly tl1at there are two impulses representing the carrier frequencies, 2 and IS. If you
`ignore, these, the remaining are just two sin x)’ x speclrums centered at frequencies 2 and 6, added together.
`
`We define a term called the modulation index of a FM signal.
`
`m=—
`
`16
`
`where Ddf is what is called the frequency deviation, and fm is the frequency of the message signal.
`
`f2_fi
`Af:2
`
`This factor m, determines the occupied bandwidth and is a measure of the bandwidth of the signal.
`
`The FM broadcasting in the US takes place at carrier frequencies of app. 50 MHz. The message signal
`which is music, has frequency content up to about 15 kl-Iz. The FCC allows deviation of 75 kHz. For this, the
`mis
`
`Af'}"5
`m:—: —=15
`f...
`5
`
`m, is independent of the carrier frequency and depends only on the message signal frequency and the
`
`allowed deviation frequency. This factor is not constant for a particular signal as is the AM modulation index.
`In FM broadcasting m can go very high, since the message signal has some pretty low frequencies, such as 50
`
`Hz. For FM it can vary any where from 1500 to 5 on the low end.
`
`FM signal is classified in two categories,
`
`Signals with m << 1 are called Narrowballd FM
`
`Signals with m >> 1 are called Wideband FM
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 12, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 12, IPR2016-00766
`
`
`
`FM radio is an obviously a wideband signal by this definition.
`
`The smaller the deviation, the closer the main lobes lie and in fact overlap and larger the deviation, the
`
`further out the signal spreads. The following figures show the effect of increasing modulation index on the
`signal spectrum and bandwidth.
`
`
`
`a. Binary message signal, ma) = I
`
`10
`
`EHHIIHHI
`
`nu
`
`'
`-0.5
`
`-1.0
`
`b. M0dulatedsignal_f;=4,D5f=5
`
`I
`.|ii|.
`
`"IN
`
`H
`
`.||
`!!l.
`
`.
`!"!.i'
`
`-20
`
`-10
`
`I)
`
`10
`
`2|]
`
`3|]
`
`...nI.|lliiAM
`
`°‘
`
`M“Ii|il.ilIInt..
`
`c.Spec1:rumskmvstu-0 impuIsesat_f}5+4=9and4-5=-I
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 13, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 13, IPR2016-00766
`
`
`
`1.0 I
`0.5 l
`|.||||.‘_l I Hllli.‘
`0 0 N_‘.|II|.|
`mm HUflW"'|||\|
`
`-0.5
`
`-1.o'
`
`
`
`||||.l
`‘.|HI.|.l
`|"H|||' W
`
`'
`
`I
`M
`
`H
`a'HH||||J'
`
`a‘. Modulated signal, L = 4, Ddf = 10
`
`s...mHlMMM
`
`[Hr
`=1'
`
`MMm“|l1m..
`
`.9. Spectrum shows two impulses atf; 10+4=14and4-I0:-6
`Note that the spectrum does look like a superposition of impulses and sim: fimcrions.
`
`HmJ|,|,UHHlll\JMJHHI.,Ll!l1Ly Wz«[lII.mHlllII
`
`
`
`
`
`
`
`
`
`
`
`
`HIHIH”‘hii|||lIHW||||l W“ WM “WIJHIII
`
`,eModuta:eds:gnai;;=4;oaf=1
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 14, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 14, IPR2016-00766
`
`
`
`
`
`..|ll
`
`Mlln.
`
`g. Specnumskowsmoinapulsesatfil+4=5and4-1:3
`
`,|I\ll_\_|4WllHIH
`a“HmWu
`
`H
`[l ;I:I_r
`1"
`
`ImMn_.n_LL1mhmmnfllu._:_Ll=ml
`“UHHHIw-\'!HIIyHH=»'u'!H|NW'NW'H‘“%
`
`H
`
`HI
`
`-0.5
`
`-1.0
`
`1:. Modu1axeds:gm1,,;_=4,Daf=.4
`
` .
`
`.
`
`.
`
`u. I.1l.lI.l.llLll|
`
`Llillhllllll I. L.
`
`.
`
`.
`
`.
`
`i. Specmun shows two impulses at}? 4 + -1 = 4.1 and 4 - .11 = 3.9
`
`Figure 10 — Relafionship of FM modulation index and its spectrum. As index gets large, the signal bandwidth
`increases. In these figures, we can defne the bandwidth as the space between the main lobes or irnpulses.
`
`In fact as kf gets very small, the general rules of identifying the impulses do not apply and we have to
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 15, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 15, IPR2016-00766
`
`
`
`know about Bessel functions in order to compute the spectrum.
`
`There are two very special cases of FM that bear discussing.
`
`Sundes’ FSK
`
`Sunde’s FSK is a special case of the general form we have described above. Here the frequency spacing
`of the two carriers, f and fl is exactly equal to the symbol rate.
`
`Dfif = fm
`
`or
`
`m = 1
`
`Two tones appear at frequencies which are exactly the symbol rate and these help in the demodulation of
`
`this signal without external timing information. When you can extract timing information from a signal, the
`detection task is called coherent and generally has better BER perfonnance.
`
`
`
`a. Message signal with bit rate = 1.0
`
`1.0
`
`0 5|
`0.0 i
`-0.5 ii
`
`-1.0
`
`IlllllHHILHIHHHILH|HHHlll_HHI|llHI
`HHIWHHIHllH‘il|H|'|lHIHIIHHH"
`
`b. Modulated signal, 1; = 8, deviation = 1.0 kf = 1.0
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 16, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 16, IPR2016-00766
`
`
`
`Figure 11 — Suude’s FSK spectrum with tones at = 9 and 7 Hz.
`
`The spectrum of Sunde’s FSK is given by
`
`so°)=§ 5(f--5Ti)+5(f+-5Ti)+
`
`.9
`
`.9
`
`4?;
`2:2
`
`cos(7rfTs
`4f2:r§—1
`
`2
`
`The first two underlined terms are the two impulses (batman’s ears in above spectrum) and the last term is
`
`the spectrum of the message signal, which is not quite sin xix as I had said earlier.
`
`Minimum Shift Keying - MSK
`
`Minimum Shift Keying, another special from of FSK (also called Continuous phase FSK or CPSFK) and
`
`a very important one. It is used widely in cellular systen1s.. MSK is also a special from of PSK owing to the
`
`equivalence of phase and frequency modulation which we will discuss in more detail again. For now, let it
`suflice to say that MSK is a special case where
`
`Dof = .5 fm
`
`or
`
`m = 0.5
`
`The MSK spectrum has no discrete components unlike all the other FSKs including Sunde’s above. The
`spectrum is just a bit wider than QPSK but its side lobes fall off much more quickly than QPSK In wireless
`
`systems, it often offers better properties than QPSK due to its constant envelope characteristics, as we can see
`in the comparison above.
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 17, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 17, IPR2016-00766
`
`
`
`nmmmnnmHumInnnmunmnntn
`EIIHIHWHHHHWHIIJIHMIHHIIHH
`
`-1.
`
`a. MSK modulated signal, m(t) = 2.0, Dflf = 1
`
`-10
`
`0
`
`1|]
`
`2|]
`
`mmm
`
`Mn
`
`MSK specrnuns does look a lot like a QPSK spectrum but is not the same.
`
`Figum I2 — MSK signal and spectrum. Note that it has no “Barman ears” as in all other FSKS.
`
`The spectrum of MSK is given by
`
`2
`
`16?;
`
`cos(2JrfT5
`
`(We will discuss MSK again from PSK point of view in the next tutorial.)
`
`M-ary FSK
`
`A M-ary FSK is just an extension of BFSK. Instead of two carriers, we have M. These carriers can be
`orthogonal or not, but the orthogonal case would obviously give better BER.
`
`M-ary FSK requires considerably larger bandwidth than M-PSK but as M increases, the BER goes down
`
`unlike M-PSK. Infact if number of frequencies are increased, M-FSK becomes an OFDM like modulation.
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 18, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 18, IPR2016-00766
`
`
`
`Analog FM
`
`Response to a sinusoid — not as simple as we would like
`
`I have side-skirted the issue of analog FM’s complicated spectrum. Binary or digital FSK allows us to
`
`delve into FM and is easy to understand but a majority of FM transmission such for radio is analog. So we will
`touch on that aspect to give you the full understanding.
`
`Now we will look at what happens to a message signal that is a single sinusoid of frequency fm when it is
`FM modulated.
`
`s(t) = A6 cos (22: 1;: + Zzrkf I m(t)dr)
`
`19
`
`Let’s take a message signal as shown below.
`
`
`
`20
`
`40
`
`
`100 .-1 140 160 130 o 220 240 230 230
`
`so
`
`so
`
`-2
`
`Figure 13 - Just a lowly sinusoid that is about to be mangled to high—fldelity heights by a FM’ modulator.
`
`s(t) = A‘, cos(2Jr ff: + 2Jrkf sin f_t)
`
`Now set Ac = 1 and switch to radial frequency for convenience (less typing.)
`
`s(t) = cos((oct + kf sin (amt)
`
`By the well-known trigonometric relationship, this equation is becomes
`
`s(t) = cos((oct + kf sin (amt)
`= cos (Oct cos(kI sin (om!) — sin (Oct sin(kf sin (amt)
`
`20
`
`Now the terms 00387; Sin Wm?) can be expanded. How and why, we leave it to the mathematicians, and
`
`see that we again get a pretty hairy looking result.
`
`cos(kf sinromt) = Jn(kf)+ 2J2(kf )cos(2(omr) + 2J4(kJ, ) cos(44
`
`21
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 19, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 19, IPR2016-00766
`
`
`
`The sin(k, sin mm!) is similarly written as
`
`sin(kf sin (amt) = 2J1(kf ) sin((umt) + 2J3 (kf )sin(3(omt) + 2.15 (R
`
`22
`
`(Note kf = m)
`
`The underlined functions are Bessel] functions and are function of the modulation index m or kf. They are
`seen many places where harmonic signals analysis and despite my bad-mouthing, they are quite harmless and
`
`benign. There are just so many of them. They ones here are of first kind and of order n. The terms of the cosine
`expansion since it is an even function, contain only even harmonics of wm and sin expansion has only odd
`harmonics and this is obvious if you will examine the above equations even casually.
`
`Figure 14 shows the general shape of 4 Bessel functions as the order is increased. The x-axis is modula-
`tion index k‘ and the y-axis is the value of the Bessel function value. For example, for kf= 2, the values of the
`various Bessel functions are
`
`H}: .25
`J] = .65
`
`J2: .25
`J3 = .18
`
`The PM carriers take their amplitude values from these functions as we will see below.
`
`
`
`Figure 14 — Besselfunction of the first kind of order :1, shown for order = 0, 1, 2 and 4
`X-arisisreadasthe modulation indexk, andy-arisistkevalue oftkeatnplitudeoftheassociatedkatvnonie.
`Except for the 0 - orderfunction, all others start at 0.0 and damp down with cycle.
`
`Below we see a plot of what the cos and sin expansions (equations 22, 21) containing the various Bessel
`functions look like for just four terms.
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 20, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 20, IPR2016-00766
`
`
`
`,_0.96.
`
`0.5
`
`,- 1.023,.
`
`15
`
`0.2
`
`0.4
`
`0.6
`
`pig“, 5 _ pm of cos(kf sin (amt) = J0 (kf) + 2J'2(kf )COS(2COmt) + 2J4(kf )cos(4+
`
`0
`
`0.2
`
`0.4
`
`0.6
`
`pggm, I6 _ pgm of sin(kf sin (om!) = 2J1(kf)sin((omr)+ 2J3 (kf ) sin(3(omt) + 2J5 (I;
`
`Now we put it all together (plug equations 21, 22 into equation 20)
`And by trigonometric magic get, an expression for the FM modulated signal.
`
`s(t) = J0 (kf)cos(a)ct)
`
`-1, (kf )[cos((o£ — (om )t — cos((q_, + (om )t]
`
`+J2(kf )[cos(ro__, — 20),“ )r —cos(rq_, + 2(1),“ )1']
`
`-13 (kf )[cos((oc — 360," )t — cos((oc + 360,“ )r]
`+...
`
`Remember this is a response to just one single solitary sinusoid. This signal contains the carrier with
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 21, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 21, IPR2016-00766
`
`
`
`amplitude set by Bessel function of 0 order (the frst underlined term), and sideband on each side of the carrier
`
`at harmonically related separations of mm, 260“, 3(1),“ , 460," , .
`
`This is very different from AM, in that we
`
`know that in AM, a single sinusoid would give rise to just two sidebands (FFT of the sinusoid) on each side of
`
`the carrier. So many sidebands, in fact an infinite number of them, for just one sinusoid!
`
`Each of these components has a Bessel function as its amplitude value. For certain values of kf, we can
`see that J0 function can be 0. In such case there would no carrier at all (MSK) and all power is in sidebands.
`For the case of kf = O, which also means that there is no modulation, all power is in the carrier and all sideband
`Bessel function values are zero.
`
`FM is called a constant envelope modulation. As we know the power of a signal is function of its ampli-
`
`tude only. We note that the total power of the signal is constant and not a function of the frequency. For FM,
`the power gets distributed amongst the sidebands, the total power always remains equal to the square of the
`
`amplitude and is constant..
`
`Here we plot the above signal for kf = 2 and kf = 1.
`
`
`
`‘DJ
`
`III
`
`I
`
`I12
`
`I
`
`0.4
`
`CIJS
`
`‘
`
`0.3 I
`
`1
`
`Figure 17 — Modulated FM signaifor kf = 2 and kf = 1, the red signal is movingfasier indicated higkerfrequency
`components .
`
`Examples of FM spectrums to a sinusoid.
`
`In the following signals, the only thing we are changing is the modulation index kt. This increases the
`frequency separation (or deviation and increases the bandwidth.)
`
`mummnmmmn
`:3 I an H i H“ H H“ in 3!‘ ~
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 22, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 22, IPR2016-00766
`
`
`
`=.mnnmmmmn
`M
`NH?!
`33;: l W V H 5' W W H W 7 "
`
`mnnnnmmnmm
`jjj ' H W H '71 7 W H 7”“ W
`
`ennm AI1_nnH&|% nmn
`:3 “WW VJWHWHJ W
`
`Time domain FM signals for various k values
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 23, IPR2016-00766
`
`
`
`1
`
`|..l-
`
`I
`
`.
`
`-20
`
`-40
`
`".115
`
`
`
`.I'Lt|:lI1I.1|.l;EI.-.
`
`x
`
`\
`
`-+-——+————H—————+————+———
`[I
`10
`20
`
`.‘
`
`-+-——+————a—————+————+———
`[I
`10
`20
`'
`
`n'
`
`L|-
`
`;
`
`,
`
`L,
`
`'2']
`
`-40
`
`-so
`
`
`
`_".1r‘,'l1Iu'.1.-..LI:l
`
`|
`I
`'30
`-+-———+————+—————+————+———
`0
`10
`20
`
`‘
`
`I
`——+————H—————h————+————+———
`-"
`10
`20
`
`J
`
`H
`
`.".1I1'IItI1.L-_'.1:i
`
`|.,.
`
`-fl--+--F--fl--1--+--h--fl--+--+--F--H--+-
`4|]
`
`-20
`0
`
`-1|]
`
`1|)
`
`10
`_
`v'—.'E;',‘:.'L%r|-iL'[,".
`
`2|]
`
`."
`
`0
`
`3|]
`
`100
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 24, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 24, IPR2016-00766
`
`
`
`Figure 19 — Spectrum of FM signals for various kf values.
`
`Bandwidth of a PM signal
`
`The bandwidth of an FM signal is kind of slippery thing. Carson’s rule states that is bandwidth of a FM
`signal is equal to
`
`Bandwidthm = 2(Af + fm)
`
`f,..=—
`
`or is equal to the baseband symbol rate, Rs.
`
`lfthe frequencies f] and f2 chosen satisfy the following equation, the cross-correlation between the
`carriers is zero, then this is an orthogonal set.
`
`p = _fcos(2Jrf,t)cos(2Jrf2t)dt = 0
`
`The output of I is maximum just when output of Q channel is zero. The decision at the receiver is a
`
`simple matter of determining if the voltage is present.
`
`The BER of a BFSK system is given by
`
`-2
`
`E
`1
`1
`.F;=— 1—e;f— S
`
`~5[No]
`
`2
`
`If however, if the cross-correlation is not zero,
`
`To
`p = jcos(2:rf,:)cos(2:rf,:)d: ya 0
`0
`
`then energy is present on both channels at the same time and in presence of noise, symbol decision may be
`
`flawed. Expanding above equation, we get,
`
`.0
`
`= sin(?r(f2 -f1)T,) c0s(?r(f2 - f1)T,)
`?F(f2-f.)1'l
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 25, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 25, IPR2016-00766
`
`
`
`Now set
`
`(f2—f})?1=%=4k,
`
`p = sin(4Jrkf) COS(4J1'kf)
`4:rkf
`
`1
`
`0.5 —
`
`0
`
`“S
`
`1
`
`2
`
`3
`
`4
`
`5
`
`Figure 20 — Cross-correlation between I and Q FSK signals. The zero crossing points are the preferable points of
`operation.
`
`Plotting this equation against the 2 times modulation index kf, we get Figure 20. The y-axis which is
`correlation between I and Q channels is 0 at specific values of kf. These values are 1, 2, 3 and 4 and so on.
`
`The point with the lowest correlation (negative value is considered the most efficient place to operate an
`FSK. At first zero crossing, we note that the symbols are located only half a symbol apart so, coherent detec-
`
`tion is not possible at this point. The next zero crossing, where kf = 1, is the Sunde’s FSK. Coherent detection
`is possible at this point because symbols are at least ((f2-fl )Ts = 1) one symbol apart. The 4"‘ zero crossing is
`MSK.
`
`Bandwidth of Sunde’s FSK is given by
`
`Hz
`
`3 3
`
`'1
`
`2
`B_(f2—f,)+?_
`.9
`
`And for MSK,
`
`B=(f2—f;)+Ti=% Hz
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 26, IPRZO16-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 26, IPR2016-00766
`
`
`
`Sur1cle’s FSK gives us the most spectrally efficient form of FSK that can be detected incoherently,
`
`whereas MSK gives us a spectrum that is a lot like QPSK but rolls-off much faster.
`
`Questions, corrections?
`Please contact me,
`
`Charan Langton
`1:nntcastle@earth]jnk.net
`
`www.oomplextorea1.oom
`
`Copyright Feb 2002, All rights reserved
`
`MTe|., Exhibit 2004, ARRIS V. MTe|., Page 27, |PR2016-00766
`
`MTel., Exhibit 2004, ARRIS v. MTel., Page 27, IPR2016-00766