FrequencyHopping

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Chapter 3

• Frequency-Hopping
• Systems

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3.1 Concepts and Characteristics

• Frequency hopping
• The periodic changing of the carrier frequency of
  a transmitted signal.
• Hopset
• The set of M possible carrier frequencies
• Frequency hopping pattern
• The sequence of carrier frequencies.
• Hopping band
• Hopping occurs over a frequency band that
  includes M frequency channels.

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• hop duration
• The time interval between hops
• The hopping band has bandwidth

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• If the data modulation is some form of angle
  modulation then the received signal for
  the ith hop is
• dehopping
• The mixing operation removes the
  frequency-hopping pattern from the received
  signal.
• Frequency hopping enables signals to hop out of
  frequency channels with interference or slow
  frequency-selective fading.
• spectral notching
• Some spectral regions with steady interference or
  a susceptibility to fading may be omitted from
  the hopset.

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• Transmission security
• The specific algorithm for generating the control
  bits is determined by the key and the time-of-day
  (TOD).
• The key is a set of bits that are changed
  infrequently and must be kept secret.
• The TOD is a set of bits that are derived from
  the stages of the TOD counter and change with
  every transition of the TOD clock.
• The purpose of the TOD is to vary the generator
  algorithm without constantly changing the key.
• The generator algorithm is controlled by a
  time-varying key.
• The code clock, which regulates the changes of
  state in the code generator and thereby controls
  the hop rate, operates at a much higher rate than
  the TOD clock.

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• Dwell interval
• A frequency-hopping pulse with a fixed carrier
  frequency occurs during a portion of the hop
  interval.
• dwell time
• The duration of the dwell interval during which
  the channel symbols are transmitted.

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• The hop duration Th is equal to the sum of the
  dwell time Td and the switching time Tsw.
• The switching time is equal to the dead time plus
  the rise and fall times of a pulse.
• dead time is the duration of the interval when no
  signal is present
• The nonzero switching time decreases the
  transmitted symbol duration Ts .
• If Tso is the symbol duration in the absence of
  frequency hopping, then
• The reduction in symbol duration expands the
  transmitted spectrum and thereby reduces the
  number of frequency channels within a fixed
  hopping band.

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• Fast frequency hopping
• If there is more than one hop for each
  information symbol.
• Slow frequency hopping
• If one or more information symbols are
  transmitted in the time interval between
  frequency hops.
• Let M denote the hopset size, B denote the
  bandwidth of frequency channels, and Fs denote
  the minimum separation between adjacent carriers
  in a hopset.
• For full protection against stationary narrowband
  interference and jamming, it is desirable that
  so that the frequency channels are
  nearly spectrally disjoint.
• A hop then enables the transmitted signal to
  escape the interference in a frequency channel.

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• Symbol errors are independent if the fading is
  independent in each frequency channel and each
  symbol is transmitted in a different frequency
  channel.
• If each of the interleaved code symbols is
  transmitted at the same location in each hop
  dwell interval, then adjacent symbols are
  separated by Th after the interleaving.
• The sufficient condition for nearly independent
  symbol errors is
• where Tcoh is the coherence time of the fading
  channel.
• Bcoh is the coherence bandwidth of the
  fading channel.

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• For a hopping band with bandwidth W, and a hopset
  with a uniform carrier separation,
• If nearly independent symbol errors are to be
  ensured, the number of frequency channels is
  constrained by
• If B lt Bcoh equalization will not be necessary
  because the channel transfer function is nearly
  flat over each frequency channel.
• If B ? Bcoh equalization may be used to prevent
  intersymbol interference.

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• In military applications, the ability of
  frequency-hopping systems to avoid interference
  is potentially neutralized by a repeater jammer
  (also known as a follower jammer), which is a
  device that intercepts a signal, processes it,
  and then transmits jamming at the same center
  frequency.
• To be effective against a frequency-hopping
  system, the jamming energy must reach the victim
  receiver before it hops to a new set of frequency
  channels.
• Thus, the hop rate is the critical factor in
  protecting a system against a repeater jammer.

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3.2 Modulations

• FH/MFSK system
• Uses MFSK as its data modulation.
• One of q frequencies is selected as the carrier
  or center frequency for each transmitted symbol,
  and the set of q possible frequencies changes
  with each hop.
• An FH/MFSK signal has the form
• is the average signal power during
  a dwell interval.
• is a unit-amplitude rectangular pulse of
  duration Ts.
• Nh is the number of symbols per dwell interval.

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• The effective number of frequency channels is
• M the hopset size.
• q frequencies or tones in an MFSK set
• For noncoherent orthogonal signals, the MFSK
  tones must be separated enough that a received
  signal produces negligible responses in the
  incorrect subchannels.
• The frequency separation must be
  
• k is a nonzero integer.
• Ts denotes the symbol duration.
• To maximize the hopset size when the MFSK
  subchannels are contiguous, k1 is selected.

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• The bandwidth of a frequency channel for slow
  frequency hopping with many symbols per dwell
  interval is
• Tb is the duration of a bit.
• The hopset size is

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Soft-Decision Decoding

• We consider an FH/MFSK system that uses a
  repetition code and the receiver of Figure
  3.5(b).
• Each information symbol, which is transmitted as
  L code symbols, may be regarded as a codeword or
  as an uncoded symbol that uses diversity
  combining.
• The interference is modeled as wideband Gaussian
  noise uniformly distributed over part of the
  hopping band.
• Slow frequency hopping with a fixed hop rate and
  ideal interleaving.
• The optimal metric for the Rayleigh-fading
  channel and a good metric for the
  additive-white-Gaussian-noise (AWGN) channel
  without fading is the Rayleigh metric which is

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• Linear square-law combining
• Rli is the sample value of the envelope-detector
  output that is associated with code symbol i of
  candidate information-symbol.
• L is the number of repetitions or code symbols.
• This metric has the advantage that no side
  information, which is specific information about
  the reliability of symbols, is required for its
  implementation.
• A performance analysis of a frequency-hopping
  system with binary FSK and soft-decision decoding
  with the Rayleigh metric indicates that the
  system performs poorly against worst case
  partial-band jamming 6 primarily because a
  single jammed frequency can corrupt the metrics.

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• Nonlinear square-law combining
• Noi is the two-sided power spectral density of
  the interference and noise over all the MFSK
  subchannels during code symbol i.
• Variable-gain metric

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• Suppose that the interference is partial-band
  jamming.
• N1/2 the two-sided power-spectral density
• µ the fraction of the hopping band with
  interference .
• It0 the spectral density that would exist if
  the interference power were uniformly spread over
  the entire hopping band.
• Upper bound on the information-bit error
  probability
• where

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• Suppose that the interference is worst-case
  partial-band jamming.
• An upper bound on Pb is obtained by maximizing
  the right-hand side of (3-27) with respect to µ.
• Calculus yields the maximizing value of
• Substituting (3-28) into (3-27), we obtain

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• let L0 denote the minimizing value of L.

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• The upper bound on Pb for worst-case partial-band
  jamming when L L0 is given by
• This upper bound indicates that Pb decreases
  exponentially as
  increases if the appropriate number of
  repetitions is chosen.
• Thus, the nonlinear diversity combining with the
  variable-gain metric sharply limits the
  performance degradation caused by worst-case
  partial-band jamming relative to full-band
  jamming.

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• Substituting (3-30) into (3-28), we obtain
• This result shows that the appropriate choice of
  L implies that worst-case jamming must cover
  three-fourths or more of the hopping band.
• The task may not be a practical possibility for a
  jammer.

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Narrowband Jamming Signals

• Although (3-31) indicates that it is advantageous
  to use nonbinary signaling (m gt 1) when
  .
• This advantage is completely undermined when
  distributed, narrowband jamming signals are a
  threat.
• A sophisticated jammer with knowledge of the
  spectral locations of the MFSK sets can cause
  increased system degradation by placing one
  jamming tone or narrowband jamming signal in
  every MFSK set.

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• To assess the impact of this sophisticated
  multitone jamming on hard decision decoding in
  the receiver of Figure 3.5(b),
• It is assumed that thermal noise is absent and
  that each jamming tone coincides with one MFSK
  tone in a frequency channel encompassing q MFSK
  tones.
• Whether a jamming tone coincides with the
  transmitted MFSK tone or an incorrect one, there
  will be no symbol error if the desired-signal
  power S exceeds the jamming power.
• If It is the total available jamming power, then
  the jammer can maximize symbol errors by placing
  tones with power levels slightly above S whenever
  possible in approximately J frequency channels
  such that

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• If a transmitted tone enters a jammed frequency
  channel and then with probability
  , the jamming tone will not coincide with the
  transmitted tone and will cause a symbol error
  after hard-decision decoding.
• Since J/M is the probability that a frequency
  channel is jammed, the symbol error probability
  is
• Substitution of (3-8), (3-9), and (3-34) into
  (3-35) yields

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• denotes the energy per bit.
• denotes the spectral density of
  the interference power that would exist if it
  were uniformly spread over the hopping band.
• This equation exhibits an inverse linear
  dependence of Ps on
• It is observed that Ps increases with q which is
  the opposite of what is observed over the AWGN
  channel.
• Thus, binary FSK is advantageous against this
  sophisticated multitone jamming.

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3.3 Hybrid Systems

• Frequency-hopping systems reject interference by
  avoiding it, whereas direct-sequence systems
  reject interference by spreading it.
• Channel codes are more essential for
  frequency-hopping systems than for
  direct-sequence systems.
• Because partial-band interference is a more
  pervasive threat than high-power pulsed
  interference.
• When frequency-hopping and direct-sequence
  systems are constrained to use the same fixed
  bandwidth, then direct-sequence systems have an
  inherent advantage.
• They can use coherent PSK rather than a
  noncoherent modulation.
• Coherent PSK has an approximately 4 dB advantage
  relative to noncoherent MSK over the AWGN channel
  and an even larger advantage over fading
  channels.

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• A major advantage of frequency-hopping systems
• It is possible to hop into noncontiguous
  frequency channels over a much wider band than
  can be occupied by a direct-sequence signal.
• This advantage more than compensates for the
  relatively inefficient noncoherent demodulation
  that is usually required for frequency-hopping
  systems.
• Excluding frequency channels with steady or
  frequent interference.
• The reduced susceptibility to the near-far
  problem and the relatively rapid acquisition.

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• A hybrid frequency-hopping direct-sequence system
• A frequency-hopping system that uses
  direct-sequence spreading during each dwell
  interval
• Or, equivalently, a direct-sequence system in
  which the carrier frequency changes periodically.

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• Hops occur periodically after a fixed number of
  sequence chips.
• Because of the phase changes due to the frequency
  hopping, noncoherent modulation, such as DPSK, is
  usually required unless the hop rate is very low.
  
• Serial-search acquisition occurs in two stages.
• To provide alignment of the hopping patterns.
• The phase of the pseudonoise sequence finishes
  acquisition rapidly.
• Because the timing uncertainty has been reduced
  by the first stage to less than a hop duration.

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• A hybrid system curtails partial-band
  interference in two ways.
• The hopping allows the avoidance of the
  interference spectrum part of the time.
• When the system hops into the interference, the
  interference is spread and filtered as in a
  direct-sequence system.
• Large bandwidth limits the number of available
  frequency channels, which increases the
  susceptibility to narrowband interference and the
  near-far problem.
• Hybrid systems are seldom used except perhaps in
  specialized military applications because the
  additional direct-sequence spreading weakens the
  major strengths of frequency hopping.

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3.4 Applications

• Anti-jamming is an important application for
  spread spectrum modulations.
• In addition to anti-jamming, we will briefly
  introduce several other spread spectrum
  applications in this section.
• In describing these applications, we focus on
  DS-SS systems.
• One should note that other spread spectrum
  techniques also have similar applications since
  the main idea behind these applications is the
  spreading of the spectrum.

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3.4.1 Anti-jamming

• We know that we can combat a wide-band Gaussian
  jammer by spreading the spectrum of the data
  signal.
• Here we consider another kind of jammersthe
  continuous wave (CW) jammers.
• Suppose the spread spectrum signal is given by
• It is jammed by a sinusoidal signal with
  frequency and power PJ .
• The received signal is given by
• where n(t) represents the AWGN.

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• We can easily see that the power spectrum of the
  received signal r(t) is given by
• We consider the matched filter receiver in the
  equivalent correlator form in Figure 3.11.
• Figure 3.11 Matched filter receiver (correlator
  form) for DS-SS

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• At the output of the despreader, the signal z(t)
  can be expressed as
• It can be shown that the power spectrum of the
  despread signal z(t) is
• Now the anti-jamming property of the spread
  spectrum modulation can be explained by comparing
  the spectra of the signals before and after
  despreading in Figure 3.12.

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• Figure 3.12 Spectra of signals before and after
  despreading

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• Before despreading, the jammer power is
  concentrated at frequency and the signal
  power is spread across a wide frequency band
  (-2p/Tc, 2p/Tc).
• The despreader spreads the jammer power into a
  wide frequency band (-2p/Tc, 2p/Tc) while
  concentrates the signal power into a much
  narrower band (-2p/T, 2p/T).
• The integrator acts like a low-pass filter to
  collect power of the despread signal over the
  frequency band (-2p/T, 2p/T).

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• As a result, almost all of the signal power is
  collected, but only 1/Nth of the jammer power is
  collected.
• The effective power of the jammer is reduced by a
  factor of N.
• This is the reason why N is called the spreading
  gain.

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3.4.2 Low probability of detection

• Another military-oriented application for spread
  spectrum is low probability of detection (LPD),
  which means that it is hard for an unintentional
  receiver to detect the presence of the signal.
• The idea behind this can be readily seen from
  Figure 3.12.
• When the processing gain is large enough, the
  spread spectrum signal hides below the white
  noise level.
• Without knowledge of the signature sequences, an
  unintentional receiver cannot despread the
  received signal.

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• Therefore, it is hard for the unintentional
  receiver to detect the presence of the spread
  spectrum.
• We are not going to treat the subject of LPD any
  further than the intuition just given.
• A more detailed treatment can be found in 1, Ch.
  10.

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2.4.3 Multipath combining

• Another advantage of spreading the spectrum is
  frequency diversity, which is a desirable
  property when the channel is fading.
• Fading is caused by destructive interference
  between time-delayed replica of the transmitted
  signal arise from different transmission paths
  (multipaths).
• The wider the transmitted spectrum, the finer are
  we able to resolve multipaths at the receiver.

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• Loosely speaking, we can resolve multipaths with
  path-delay differences larger than 1/W seconds
  when the transmission bandwidth is W Hz.
• Therefore, spreading the spectrum helps to
  resolve multipaths and, hence, combats fading.
• The best way to explain multipath fading is to go
  through the following simple example.
• Suppose the transmitter sends a bit with the
  value 1 in the BPSK format, i.e., the
  transmitted signal envelope is pT (t), where T is
  the symbol duration.

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• Assume that there are two transmission paths
  leading from the transmitter to the receiver.
• The first path is the direct line-of-sight path
  which arrives at a delay of 0 seconds and has a
  unity gain.
• The second path is a reflected path which arrives
  at a delay of 2Tc seconds and has a gain of -0.8,
  where
• Tc T/10 is the chip duration of the DS-SS
  system we are going to introduce in a moment.

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• The overall received signal can be written as
• where n(t) is AWGN.
• To demodulate the received signal, we employ the
  matched filter receiver, which is matched to the
  direct line-of-sight signal, i.e., h(t) pT (T -
  t).
• The output of the matched filter is plotted in
  Figure 3.13.
• We can see from the figure that the contribution
  from the second path partially cancels that from
  the first path.

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• Figure 3.13 Matched filter output for the
  two-path channel without spreading

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• We sample the matched filter output at time t
  T.
• The signal contribution in the sample is 0.36T
  and the noise contribution is a zero-mean
  Gaussian random variable with variance N0T.
• Compared to the case where only the direct
  line-of-sight path is present, the signal energy
  is reduced by 87, while the noise energy is the
  same.
• Therefore, the bit error probability is greatly
  increased.

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• Now, let us spread the spectrum by the spreading
  signal
• where
• DS-SS system is
• Again, we consider using the matched filter
  receiver, which is matched to a(t).
• The output of the matched filter is shown in
  Figure 3.14.
• We can clearly see from the figure that the
  contributions from the two paths are separated
  since the resolution of the spread system is ten
  times finer than that of the unspread system.

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• Figure 3.14 Matched filter output for the
  two-path channel with spreading

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• If we sample at t T, we get a signal
  contribution of T, which is the same as what we
  would get if there was only a single path.
• Hence, unlike what we saw in the unspread system,
  multipath fading does not have a detrimental
  effect on the error probability.
• In fact, we will show in Chapter 4 that we can do
  better by taking one more sample at t T 2Tc
  to collect the energy of the second path.
• If we know the channel gain of the second path,
  we can combine the paths coherently.
• Otherwise we can perform equal-gain noncoherent
  combining. This ability of the spread spectrum
  modulation to collect energies from different
  paths is called multipath combining.

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3.5 References

• 1 R. L. Peterson, R. E. Ziemer, and D. E.
  Borth, Introduction to Spread Spectrum
  Communications, Prentice Hall, Inc., 1995.
• 2 M. B. Pursley, Performance evaluation for
  phase-coded spread-spectrum multiple-access
  communication Part I System analysis, IEEE
  Trans. Commun., vol. 25, no. 8, pp. 795799, Aug.
  1977.
• 3 R. A. Scholtz, Multiple access with
  time-hopping impulse modulation, Proc. MILCOM
  93, pp. 11-14, Boston, MA, Oct. 1993.
• 4 N. Yee, J. M. G. Linnartz, and G. Fettweis,
  Multi-carrier CDMA in indoor wireless radio
  networks, IEICE Trans. Commun., vol. E77-B, no.
  7, pp. 900904, Jul. 1994.
• 5 S. Kondo and L. B. Milstein, Performance of
  multicarrier DS CDMA systems, IEEE Trans.
  Commun., vol. 44, no. 2, pp. 238246, Feb. 1996.
• 6 R. L. Pickholtz, L. B. Milstein, and D. L.
  Schilling, Spread spectrum for mobile
  communications, IEEE Trans. Veh. Technol., vol.
  40, no. 2, pp. 313321, May 1991.
• 7 D. Torrieri, Principles of spread spectrum
  communications theory, Springer 2005.