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 W  MB
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4
• If the data modulation is some form of angle modulation  ( x)
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 < 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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15
• 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, k=1 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 informationsymbol.
– 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 > 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 directsequence 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 directsequence 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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34
•
•
•
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.
1. To provide alignment of the hopping patterns.
2. 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 jammers—the continuous wave
(CW) jammers.
• Suppose the spread spectrum signal is given by
• It is jammed by a sinusoidal signal with frequency
power PJ .
• The received signal is given by
and
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 ([-2π/Tc, 2π/Tc]).
• The despreader spreads the jammer power into a wide
frequency band ([-2π/Tc, 2π/Tc]) while concentrates the
signal power into a much narrower band ([-2π/T, 2π/T]).
• The integrator acts like a low-pass filter to collect power
of the despread signal over the frequency band ([-2π/T,
2π/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 pathdelay 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. 795–799, 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. 900–
904, Jul. 1994.
[5] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA
systems,” IEEE Trans. Commun., vol. 44, no. 2, pp. 238–246, 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. 313–
321, May 1991.
[7] D. Torrieri, “Principles of spread spectrum communications theory,” Springer
2005.
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