Frequency-Hopping Systems
Abstract
Frequency hopping is the periodic changing of the carrier frequency of a transmitted signal. This time-varying characteristic potentially endows a communication system with great strength against interference. Whereas a direct-sequence system relies on spectral spreading, spectral despreading, and filtering to suppress interference, the basic mechanism of interference suppression in a frequency-hopping system is avoidance. When avoidance fails, it is only temporary because of the periodic changing of the carrier frequency. The impact of the interference is further mitigated by the pervasive use of channel codes, which are more essential for frequency-hopping systems than for direct-sequence systems. The basic concepts, spectral and performance aspects, and coding and modulation issues of frequency-hopping systems are presented in this chapter. The effects of partial-band interference and multitone jamming are examined, and the most important issues in the design of frequency synthesizers are described.
Frequency hopping is the periodic changing of the carrier frequency of a transmitted signal. This time-varying characteristic potentially endows a communication system with great strength against interference. Whereas a direct-sequence system relies on spectral spreading, spectral despreading, and filtering to suppress interference, the basic mechanism of interference suppression in a frequency-hopping system is avoidance. When avoidance fails, it is only temporary because of the periodic changing of the carrier frequency. The impact of the interference is further mitigated by the pervasive use of channel codes, which are more essential for frequency-hopping systems than for direct-sequence systems. The basic concepts, spectral and performance aspects, and coding and modulation issues of frequency-hopping systems are presented in this chapter. The effects of partial-band interference and multitone jamming are examined, and the most important issues in the design of frequency synthesizers are described.
3.1 Concepts and Characteristics
The frequency-hopping pattern produced by the receiver synthesizer of Figure 3.2 (b) is synchronized with the pattern produced by the transmitter, but is offset by a fixed intermediate frequency, which may be zero. The mixing operation removes the frequency-hopping pattern from the received signal and is hence called dehopping. The mixer output is applied to a bandpass filter that excludes double-frequency components and power that originated outside the appropriate frequency channel and produces the data-modulated dehopped signal, which has the form of (3.1) with f_{ ci } for all hops replaced by the common intermediate frequency.
Although it provides no advantage against white noise, frequency hopping enables signals to hop out of frequency channels with interference or slow frequency-selective fading. To fully escape from the effects of narrowband interference signals, disjoint frequency channels are necessary. The disjoint channels may be contiguous or have unused spectral regions between them. Some spectral regions with steady interference or a susceptibility to fading may be omitted from the hopset, a process called spectral notching.
To ensure that a frequency-hopping pattern is difficult to reproduce or dehop by an opponent, the pattern should be pseudorandom with a large period and an approximately uniform distribution over the frequency channels. The pattern generator is a nonlinear sequence generator that maps each generator state to the pattern-control bits that specify a frequency. The linear span or linear complexity of a nonlinear sequence is the number of stages of the shortest linear feedback shift register that can generate the sequence or the successive generator states. A large linear span inhibits the reconstruction of a frequency-hopping pattern from a short segment of it. More is required of frequency-hopping patterns to alleviate multiple-access interference when similar frequency-hopping systems are part of a network (Section 7.4 ).
Frequency hopping may be classified as fast or slow. Fast frequency hopping occurs if there is more than one hop for each information symbol. Slow frequency hopping ensues if one or more information symbols are transmitted in the time interval between frequency hops. Although these definitions do not refer to the absolute hop rate, fast frequency hopping is an option only if a hop rate that exceeds the information-symbol rate can be implemented. Slow frequency hopping is preferable because the transmitted waveform is much more spectrally compact (see Section 3.4) and the overhead cost of the switching time is greatly reduced.
To obtain the full advantage of block or convolutional channel codes in a slow frequency-hopping system, the code symbols should be interleaved in such a way that the symbols of a block codeword or the symbols within a few free distances in a convolutional code fade independently. In frequency-hopping systems operating over a frequency-selective fading channel, the realization of this independence requires certain constraints among the system parameter values (Section 6.7 ).
Frequency-selective fading and Doppler shifts make it difficult to maintain phase coherence from hop to hop between frequency synthesizers in the transmitter and the receiver. Furthermore, the time-varying delay between the frequency changes of the received signal and those of the synthesizer output in the receiver causes the phase shift in the dehopped signal to differ for each hop interval. Thus, frequency-hopping systems use noncoherent or differentially coherent demodulators unless a pilot signal is available, the hop duration is very long, or elaborate iterative phase estimation (perhaps as part of turbo decoding) is used.
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 carrier frequency. Thus, the hop rate is the critical factor in protecting a system against a repeater jammer. Required hop rates and the limitations of repeater jamming are analyzed in [98].
3.2 Frequency Hopping with Orthogonal CPFSK
An orthogonal FH/CPFSK system adds frequency hopping to orthogonal CPFSK signals (Section 1.2 ). Figure 3.5 (b) depicts the main elements of a noncoherent orthogonal FH/CPFSK receiver. The frequency-hopping carrier frequency is removed, and the resulting orthogonal CPFSK signal is applied to the demodulator and decoder. Each matched filter in the bank of matched filters corresponds to a CPFSK subchannel.
Illustrative System
To illustrate some basic issues of frequency-hopping communications, we consider an orthogonal FH/CPFSK system that uses a repetition code as its channel code, suboptimal metrics, and the receiver of Figure 3.5 (b). The system has significant deficiencies, such as the very weak repetition code, but is amenable to an approximate mathematical analysis that provides insight into the design issues of a frequency-hopping system. Each information symbol is transmitted as a codeword of n identical code symbols. The interference is modeled as wideband Gaussian noise uniformly distributed over part of the hopping band. Along with perfect dehopping and negligible switching times, either slow frequency hopping with ideal interleaving over many hop intervals or fast frequency hopping is assumed. Both ensure the independence of code-symbol errors.
The difficulty of implementing the maximum-likelihood metric ( 1.80 ) leads to consideration of the suboptimal metrics. The square-law metric defined by ( 1.84 ), which performs linear square-law combining, has the advantage that no channel-state information or side information providing information about the reliability of code symbols is required for its computation. However, this metric is unsuitable against strong partial-band interference because a single large symbol metric can corrupt the codeword metric.
Further analysis requires some facts about convex functions. A real-valued function f over an interval is a convex function if \(f\left [ \alpha x+\left ( 1-\alpha \right ) y\right ] \leq \alpha f\left ( x\right ) +\left ( 1-\alpha \right ) f\left ( y\right ) \) for all x and y in the interval and \(\alpha \in \left [ 0,1\right ]\). A sufficient condition for f to be a convex function is for it to have a nonnegative second derivative (see Section 7.3 ). A local minimum of a convex function over an interval is a global minimum, and similarly a local maximum is a global maximum [19].
For frequency hopping with binary CPFSK and the nonlinear square-law metric, a more precise derivation [44] that does not use the Chernoff bound confirms that (3.23) provides an approximate upper bound on the information-bit error rate caused by worst-case partial-band interference when the thermal noise is negligible, although the optimal number of repetitions is much smaller than is indicated by (3.22). Thus, the appropriate weighting of terms in the nonlinear square-law metric prevents the domination by a single corrupted symbol metric and limits the inherent noncoherent combining loss resulting from the fragmentation of the symbol energy.
Multitone Jamming
When the CPFSK subchannels are contiguous, it is not advantageous to a jammer to transmit the jamming in all the subchannels of an CPFSK set because only a single subchannel needs to be jammed to cause a symbol error. A sophisticated jammer with knowledge of the spectral locations of the CPFSK sets can cause increased system degradation by placing one jamming tone or narrowband jamming signal in every CPFSK set, which is called multitone jamming.
3.3 Frequency Hopping with DPSK and CPM
In a network of frequency-hopping systems and a fixed hopping bandwidth, it is highly desirable to choose a spectrally compact data modulation so that the hopset is large and hence the number of collisions among the frequency-hopping signals is kept low. A spectrally compact modulation also helps to ensure that the bandwidth of a frequency channel is less than the coherence bandwidth (Section 6.3 ) so that equalization in the receiver is not necessary.
The limiting of spectral splatter is another desirable characteristic of data modulation for frequency hopping. Spectral splatter is the interference produced in frequency channels other than that being used by a frequency-hopping pulse. It is caused by the time-limited nature of transmitted pulses. The degree to which spectral splatter causes errors depends primarily on the separation F_{ s } between carrier frequencies and the percentage of the signal power included in a frequency channel. In practice, this percentage must be at least 90% to avoid signal distortion and is often more than 95%. Usually, only pulses in adjacent channels produce a significant amount of spectral splatter in a frequency channel.
The adjacent splatter ratio K_{ s } is the ratio of the power due to spectral splatter from an adjacent channel to the corresponding power that arrives at the receiver in that channel. For example, if B is the bandwidth of a frequency channel that includes 97% of the signal power and F_{ s } ≥ B, then no more than 1.5% of the power from a transmitted pulse can enter an adjacent channel on one side of the frequency channel used by the pulse; therefore, K_{ s } ≤ 0.015/0.97 = 0.0155. A given maximum value of K_{ s } can be reduced by an increase in F_{ s }, but eventually the number of frequency channels M must be reduced if the hopping bandwidth is fixed. As a result, the rate at which users hop into the same channel increases. This increase may cancel any improvement due to the reduction of the spectral splatter. The opposite procedure (reducing F_{ s } and B so that more frequency channels become available) increases not only the spectral splatter but also signal distortion and intersymbol interference, so the amount of useful reduction is limited.
To avoid spectral spreading due to amplifier nonlinearity, it is desirable for the data modulation to have a constant amplitude, as it is often impossible to implement a filter with the appropriate bandwidth and center frequency for spectral shaping of a signal after it emerges from the final power amplifier. Since they produce constant amplitudes and do not require coherent demodulation, a good modulation candidate is differential phase-shift keying (DPSK) or some form of spectrally compact continuous-phase modulation.
FH/DPSK
A DPSK demodulator compares the phases of two successive received symbols. If the magnitude of the phase difference is does not exceed π/2, then the demodulator decides that a 1 was transmitted; otherwise, it decides that a 0 was transmitted.
Consider multitone jamming of an FH/DPSK system with negligible thermal noise. Each tone is assumed to have a frequency identical to the center frequency of one of the frequency channels. The composite signal, consisting of the transmitted signal and the jamming tone, has a constant phase over two successive received symbols in the same hop dwell interval if a 1 was transmitted and the thermal noise is absent; thus, the demodulator correctly detects the 1.
FH/CPM
The main difference between CPFSK and FSK is that h can have any positive value for CPFSK but is relegated to integer values for FSK so that the tones are orthogonal to each other. Both modulations may be detected with matched filters, envelope detectors, and frequency discriminators. Although CPFSK explicitly requires phase continuity and FSK does not, FSK is almost always implemented with phase continuity to avoid the generation of spectral splatter, and hence is equivalent to CPFSK with h = 1. Minimum-shift keying (MSK) is defined as binary CPFSK with h = 1/2, and hence the two frequencies are separated by f_{ d } = 1/2T_{ s }.
With multisymbol noncoherent detection [110], CPFSK systems can provide a better symbol error probability than coherent BPSK systems without multisymbol detection. For r-symbol detection, the optimal receiver correlates the received waveform over all possible r-symbol patterns before making a decision. The drawback is the considerable implementation complexity of multisymbol detection, even for three-symbol detection.
As implied by Figure 3.7, the bandwidth requirement of DPSK with K_{ s } > 0.9, which is the same as that of BPSK or QPSK and less than that of orthogonal FSK, exceeds that of MSK. Thus, if the hopping bandwidth W is fixed, the number of frequency channels available for FH/DPSK is smaller than it is for noncoherent FH/MSK. This increase in B and reduction in frequency channels offsets the intrinsic performance advantage of FH/DPSK and implies that noncoherent FH/MSK gives a lower P_{ s } than FH/DPSK in the presence of worst-case multitone jamming, as indicated in (3.37). Alternatively, if the bandwidth of a frequency channel is fixed, an FH/DPSK signal experiences more distortion and spectral splatter than an FH/MSK signal. Any pulse shaping of the DPSK symbols alters their constant amplitude. An FH/DPSK system is more sensitive to Doppler shifts and frequency instabilities than an FH/MSK system. Another disadvantage of FH/DPSK is the usual lack of phase coherence from hop to hop, which necessitates an extra phase-reference symbol at the start of every dwell interval. This extra symbol reduces \(\mathbb {E}_{s}\) by a factor (N − 1)/N, where N is the number of symbols per hop or dwell interval and N ≥ 2. Thus, DPSK is not as suitable a means of modulation as noncoherent MSK for most applications of frequency-hopping communications, and the main competition for MSK comes from other forms of CPM.
3.4 Power Spectral Density of FH/CPM
Normalized bandwidth (99%) for FH/CPFSK
Deviation ratio | ||
---|---|---|
Symbols/dwell | h = 0.5 | h = 0.7 |
1 | 18.844 | 18.688 |
2 | 9.9375 | 9.9688 |
4 | 5.1875 | 5.2656 |
16 | 1.8906 | 2.1250 |
64 | 1.2813 | 1.8750 |
256 | 1.2031 | 1.8125 |
1024 | 1.1875 | 1.7969 |
No hopping | 1.1875 | 1.7813 |
Fast frequency hopping, which corresponds to N = 1, entails a very large 99-% bandwidth. This fact and the long switching times are the main reasons why slow frequency hopping is preferable to the fast form and is the predominant form of frequency hopping. Consequently, frequency hopping is always assumed to be the slow form unless explicitly stated otherwise.
An advantage of FH/CPFSK with h < 1 or FH/GMSK is that it requires less bandwidth than orthogonal CPFSK (h = 1). The increased number of frequency channels due to the decreased bandwidth does not improve performance over the additive white Gaussian noise (AWGN) channel. However, the increase is advantageous against a fixed number of interference tones, optimized jamming, and multiple-access interference in a network of frequency-hopping systems (Section 9.4 ).
3.5 Digital Demodulation of FH/CPFSK
In principle, the frequency hopping/continuous-phase frequency-shift keying (FH/CPFSK) receiver does symbol-rate sampling of matched-filter outputs. However, the details of the actual implementation are more involved, primarily because aliasing should be avoided. A practical digital demodulation is described in this section.
A critical choice in the design of the digital demodulator is the sampling rate of the ADCs. This rate must be large enough to prevent aliasing and to accommodate the IF offset. To simplify the demodulator implementation, it is highly desirable for the sampling rate to be an integer multiple of the symbol rate 1/T_{ s }. Thus, we assume a sampling rate f_{ s } = L/T_{ s }, where L is a positive integer.
3.6 Partial-Band Interference and Channel Codes
If a large amount of interference power is received over a small portion of the hopping band, then unless accurate channel-state information is available, soft-decision decoding metrics for the AWGN channel may be ineffective because of the possible dominance of a path or code metric by a single symbol metric (see Section 2.6 on pulsed interference). This dominance is reduced by hard decisions or the use of a practical two- or three-bit quantization of symbol metrics instead of unquantized symbol metrics.
Reed-Solomon Codes
Much better performance against partial-band interference can be obtained by inserting erasures (Section 1.1 ) among the demodulator output symbols before the symbol deinterleaving and hard-decision decoding. The decision to erase is made independently for each code symbol. It is based on channel-state information, which indicates the codeword symbols that have a high probability of being incorrectly demodulated. The channel-state information must be reliable so that only degraded symbols are erased.
Channel-state information may be obtained from N_{ t } known pilot symbols that are transmitted along with the data symbols in each dwell interval of a frequency-hopping signal. A hit is said to occur in a dwell interval if the signal encounters partial-band interference during the interval. If δ or more of the N_{ t } pilot symbols are incorrectly demodulated, then the receiver decides that a hit has occurred, and all N symbols in the same dwell interval are erased. Only one symbol of a codeword is erased if the interleaving ensures that only a single symbol of the codeword is in any particular dwell interval. Pilot symbols decrease the information rate, but this loss is negligible if N_{ t } << N, which is assumed henceforth.
There are other options for generating channel-state information in addition to demodulating pilot symbols. A radiometer (Section 10.2 ) may be used to measure the energy in the current frequency channel, a future channel, or an adjacent channel. Erasures are inserted if the energy is inordinately large. This method does not have the overhead cost in information rate that is associated with the use of pilot symbols. Other methods include attaching a parity-check bit to each code symbol representing multiple bits to check whether the symbol was correctly received, or using the soft information provided by the inner decoder of a concatenated code.
Consider the receiver for noncoherent detection of orthogonal CPFSK signals shown in Figure 3.5 (b). The envelope-detector outputs provide the symbol metrics used in several low-complexity schemes for erasure insertion [4]. The output threshold test (OTT) compares the largest symbol metric with a threshold to determine whether the corresponding demodulated symbol should be erased. The ratio threshold test (RTT) computes the ratio of the largest symbol metric to the second largest one. This ratio is then compared with a threshold to determine an erasure. If the values of both \(\mathbb {E}_{b}/N_{0}\) and \(\mathbb {E}_{b}/I_{t0}\) are known, then optimal thresholds for the OTT, the RTT, or a hybrid method can be calculated. It is found that the OTT is resilient against fading and tends to outperform the RTT when \(\mathbb {E}_{b}/I_{t0}\) is sufficiently low, but the opposite is true when \(\mathbb {E}_{b}/I_{t0}\) is sufficiently high. The main disadvantage of the OTT and the RTT relative to the pilot-symbol method is the need to estimate \(\mathbb {E}_{b}/N_{0}\) and either \(\mathbb {E}_{b}/I_{t0}\) or \(\mathbb {E}_{b}/(N_{0}+I_{t0})\). The joint maximum-output ratio threshold test (MO-RTT) uses both the maximum and the second largest of the symbol metrics. It is robust against both fading and partial-band interference.
Proposed erasure methods are based on the use of orthogonal CPFSK symbols, and their performances against partial-band interference improve as the alphabet size q increases. For a fixed hopping band, the number of frequency channels decreases as q increases, thereby making an FH/CPFSK system more vulnerable to multitone jamming or multiple-access interference (Chapter 7 ).
Figure 3.11 depicts P_{ b } for an FH/CPFSK system over the AWGN channel with q = 8, an extended Reed-Solomon (8,3) code, N_{ t } = 4, and δ = 1. A comparison of Figures 3.11 and 3.10 indicates that reducing the alphabet size while preserving the code rate has increased the system sensitivity to \(\mathbb {E}_{b}/N_{0}\), increased the susceptibility to interference concentrated in a small fraction of the hopping band, and raised the required \(\mathbb {E}_{b}/I_{t0}\) for a specified P_{ b } by 5 to 9 dB.
Trellis-Coded Modulation
Trellis-coded modulation (Section 1.3 ) is a combined coding and modulation method that is usually applied to coherent digital communications over bandlimited channels. Multilevel and multiphase modulations are used to enlarge the signal constellation while not expanding the bandwidth beyond what is required for the uncoded signals. Since the signal constellation is more compact, there is some modulation loss that detracts from the coding gain, but the overall gain can be substantial. Since a noncoherent demodulator is usually required for frequency-hopping communications; thus, the usual coherent trellis-coded modulations are not suitable. Instead, the trellis coding may be implemented by expanding the signal set for q/2-ary CPFSK to q-ary CPFSK. Although the frequency tones are uniformly spaced, they can be nonorthogonal to limit or avoid bandwidth expansion.
There is a tradeoff in the choice of Δ because a small Δ allows more frequency channels and thereby limits the effect of multiple-access interference or multitone jamming, whereas a large Δ tends to improve the system performance against partial-band interference. If a trellis code uses four orthogonal tones with spacing Δ = 1/T_{ b }, where T_{ b } is the bit duration, then B ≈ 4/T_{ b }. The same bandwidth results when an FH/CPFSK system uses two orthogonal tones, a rate-1/2 code, and binary channel symbols because B ≈ 2/T_{ s } = 4/T_{ b }. The same bandwidth also results when a rate-1/2 binary convolutional code is used and each pair of code symbols is mapped into a 4-ary channel symbol. The performance of the four-state, trellis-coded, rate-1/2, 4-ary CPFSK frequency-hopping system [118] indicates that it is not as strong against worst-case partial-band interference as an FH/CPFSK system with a rate-1/2 convolutional code and 4-ary channel symbols or an FH/CPFSK system with a Reed-Solomon (32,16) code and errors-and-erasures decoding. Thus, trellis-coded modulation is relatively weak against partial-band interference. The advantage of trellis-coded modulation in a frequency-hopping system is its relatively low implementation complexity.
Turbo and LDPC Codes
Turbo and LDPC codes (Chapter 1 ) are potentially the most effective codes for suppressing partial-band interference if the system latency and computational complexity of these codes are acceptable. A turbo-coded frequency-hopping system that uses spectrally compact channel symbols also resists multiple-access interference. Accurate estimates of channel parameters, such as the variance of the interference and noise and the fading amplitude, are needed in the iterative decoding algorithms. When the channel dynamics are slower than the hop rate, all the received symbols of a dwell interval may be used in estimating the channel parameters associated with that dwell interval. After each iteration by a component decoder, its log-likelihood ratios are updated and the extrinsic information is transferred to the other component decoder. A channel estimator can convert a log-likelihood ratio transferred after a decoder iteration into a posteriori probabilities that can be used to improve the estimates of the fading attenuation and the noise variance for each dwell interval (Section 9.4 ). The operation of a receiver with iterative LDPC decoding and channel estimation is similar.
Known symbols may be inserted into the transmitted code symbols to facilitate the estimation, but the energy per information bit is reduced. Increasing the number of symbols per hop improves the potential estimation accuracy. However, since the reduction in the number of independent hops per information block of fixed size decreases the diversity, and hence the independence of errors, there is an upper limit on the number of symbols per hop beyond which a performance degradation occurs.
A turbo code can still provide a fairly good performance against partial-band interference, even if only the presence or absence of strong interference during each dwell interval is detected. Channel-state information about the occurrence of interference during a dwell interval can be obtained by hard-decision decoding of the output of one of the component decoders of a turbo code with parallel concatenated codes or the inner decoder of a serially concatenated turbo code. The metric for determining the occurrence of interference is the Hamming distance between the binary sequence resulting from the hard decisions and the codewords obtained by bounded-distance decoding.
3.7 Hybrid Systems
Frequency-hopping systems reject interference by avoiding it, whereas direct-sequence systems reject interference by spreading and then filtering 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 because they can use coherent BPSK rather than a noncoherent modulation. Coherent BPSK has an approximately 4 dB advantage relative to noncoherent MSK over the AWGN channel and an even larger advantage over fading channels. However, the potential performance advantage of direct-sequence systems is often illusory for practical reasons. A major advantage of frequency-hopping systems relative to direct-sequence systems is that 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. Other major advantages of frequency hopping are the possibility of excluding frequency channels with steady or frequent interference, the reduced susceptibility to the near-far problem (Section 7.7 ), and the relatively rapid acquisition of the frequency-hopping pattern (Section 4.7 ). A disadvantage of frequency hopping is that it is not amenable to transform-domain or nonlinear adaptive filtering (Section 5.3 ) to reject narrowband interference within a frequency channel. In practical systems, the dwell time is too short for adaptive filtering to have a significant effect.
Serial-search acquisition occurs in two stages. The first stage provides alignment of the hopping patterns, whereas the second stage over the unknown timing of the spreading sequence finishes acquisition rapidly because the timing uncertainty has been reduced by the first stage to a fraction of a hop duration.
In principle, the receiver of a hybrid FH/DS system curtails partial-band interference by both dehopping and despreading, but diminishing returns are encountered. The hopping of the FH/DS signal allows the avoidance of the interference spectrum part of the time. When the system hops into a frequency channel with interference, the interference is spread and filtered by the hybrid receiver. However, during a hop interval, interference that would be avoided by an ordinary frequency-hopping receiver is passed by the bandpass filter of a hybrid receiver because the bandwidth must be large enough to accommodate the direct-sequence signal that remains after the dehopping. This large bandwidth also limits the number of available frequency channels, which increases the susceptibility to narrowband interference and the near-far problem. Thus, hybrid FH/DS systems are seldom used, except perhaps in specialized military applications because the additional direct-sequence spreading weakens the major strengths of frequency hopping.
3.8 Frequency Synthesizers
Direct Frequency Synthesizer
Example 3.1
It is desired to produce a 1.79 MHz tone. Let f_{ r } = 1 MHz and f_{ b } = 5 MHz. The ten tones provided to the decade switches are 5, 6, 7, …, 14 MHz so that f_{1} and f_{2} can range from 0 to 9 MHz. Equation (3.133) yields f_{ a } = 4 MHz. If f_{1} = 7 MHz and f_{2} = 9 MHz, then the output frequency is 1.79 MHz. The frequencies f_{ a } and f_{ b } are such that the designs of the bandpass filters inside the modules are reasonably simple. \(\Box \)
Direct Digital Synthesizer
The sine table stores 2^{ n } words, each comprising m bits, and hence has a memory requirement of 2^{ n }m bits. The memory requirements of a sine table can be reduced by using trigonometric identities and hardware multipliers. Each stored word represents one possible value of sin θ in the first quadrant or, equivalently, one possible magnitude of sin θ. The input to the sine table comprises n + 2 parallel bits. The two most significant bits are the sign bit and the quadrant bit. The sign bit specifies the polarity of sin θ. The quadrant bit specifies whether sin θ is in the first or second quadrants or in the third or fourth quadrants. The n least significant bits of the input determine the address in which the magnitude of sin θ is stored. The address specified by the n least significant bits is appropriately modified by the quadrant bit when θ is in the second or fourth quadrants. The sign bit along with the m output bits of the sine table are applied to the DAC. The maximum number of table addresses that the phase accumulator can specify is 2^{ ν }, but if ν input lines were applied to the sine table, the memory requirement would generally be impractically large. Since n + 2 bits are needed to address the sine table, the ν − n − 2 least significant bits in the accumulator contents are not used to address the table.
Example 3.2
A direct digital synthesizer is to be designed to cover 1 kHz to 1 MHz with E_{ q } ≤ − 45 dB and E_{ s } ≤−60 dB. According to (3.140), the use of 8-bit words in the sine table is adequate for the required quantization noise level. With m = 8, the table contains 2^{8} = 256 distinct words. According to (3.139), n = 8 is adequate for the required E_{ s }, and hence the sine table has n + 2 = 10 input bits. If 2.5 ≤ q ≤ 4, then because \(f_{\max }/f_{\min }=10^{3}\), (3.137) yields ν = 12. Thus, a 12-bit phase accumulator is needed. Since 2^{12} = 4096, we may choose N = 4000. If the frequency resolution and smallest frequency is to be \(f_{\min }=\) 1 kHz, then (3.134) indicates that f_{ r } = 4 MHz is required. When the frequency \(f_{\min }\) is desired, the phase increments are so small that 2^{ν−n−2} = 4 increments occur before a new address is specified and a new value of sin θ is produced. Thus, the 4 least-significant bits in the accumulator are not used for addressing the sine table. \(\Box \)
The direct digital synthesizer can be easily modified to produce a modulated output when high-speed digital data is available. For amplitude modulation, the table output is applied to a multiplier. Phase modulation may be implemented by adding the appropriate bits to the phase accumulator output. Frequency modulation entails modification of the accumulator input bits. For a quaternary modulation, the quadrature signals may be generated by separate sine and cosine tables.
A direct digital synthesizer can produce nearly instantaneous, phase-continuous frequency changes and a very fine frequency resolution despite its relatively small size, weight, and power requirements. A disadvantage is the limited maximum frequency, which restricts the bandwidth of the covered frequencies following a frequency translation of the synthesizer output. For this reason, direct digital synthesizers are sometimes used as components in hybrid synthesizers. Another disadvantage is the stringent requirement for the lowpass filter to suppress frequency spurs generated during changes in the synthesized frequency.
Indirect Frequency Synthesizers
Phase detectors in frequency-hopping synthesizers are usually digital devices that measure zero-crossing times rather than the phase differences measured when mixers are used. Digital phase detectors have an extended linear range, are less sensitive to input-level variations, and simplify the interface with a digital divider.
To decrease the switching time while maintaining the frequency resolution of a single loop, a coarse steering signal can be stored in a read only memory (ROM), converted into analog form by a DAC, and applied to the VCO (as shown in Figure 3.19) immediately after a frequency change. The steering signal reduces the frequency step that must be acquired by the loop when a hop occurs. An alternative approach is to place a fixed divider with modulus M after the loop so that the divider output frequency is f_{0} = Nf_{ r }/M + f_{1}/M. By this means, f_{ r } can be increased without sacrificing resolution, provided that the VCO output frequency, which equals Mf_{0}, is not too large for the divider in the feedback loop. To limit the transmission of spurious frequencies, it may be desirable to inhibit the transmitter output during frequency transitions.
The switching time can be dramatically reduced by using two synthesizers that alternately produce the output frequency. One synthesizer produces the output frequency while the second is being tuned to the next frequency following a command from the pattern generator. If the hop duration exceeds the switching time of each synthesizer, then the second synthesizer begins producing the next frequency before a control switch routes its output to a modulator or a mixer.
A divider , which is a binary counter that produces a square-wave output, counts down by one unit every time its input crosses zero. If the modulus or divisor is the positive integer N, then after N zero crossings, the divider output crosses zero and changes state. The divider then resumes counting down from N. As a result, the output frequency is equal to the input frequency divided by N. Programmable dividers have limited operating speeds that impair their ability to accommodate a high-frequency VCO output. A problem is avoided by the down-conversion of the VCO output by the mixer shown in Figure 3.19, but spurious components are introduced. Since fixed dividers can operate at much higher speeds than programmable dividers, placing a fixed divider before the programmable divider in the feedback loop could be considered. However, if the fixed divider has a modulus N_{1}, then the loop resolution becomes N_{1}f_{ r }; thus, this solution is usually unsatisfactory.
Example 3.3
The Bluetooth communication system is used to establish wireless communications among portable electronic devices. The system has a hopset of 79 carrier frequencies, its hop rate is 1600 hops per second, its hop band is between 2400 and 2483.5 MHz, and the bandwidth of each frequency channel is 1 MHz. Consider a system in which the 79 carrier frequencies are spaced 1 MHz apart from 2402 to 2480 MHz. A 10/11 divider with f_{ r } = 1 MHz provides the desired increment, which is equal to the frequency resolution. Equation (3.142) indicates that t_{ s } = 25 μs, which indicates that 25 potential data symbols have to be omitted during each hop interval. Inequality (3.146) indicates that f_{1} = 2300 MHz is a suitable choice. Then (3.147) is satisfied by B_{ max } = 18. Therefore, dividers A and B require 4 and 5 control bits, respectively, to specify their potential values. If the control bits are stored in an ROM, then each ROM location contains 9 bits. The number of ROM addresses is at least 79, the number of frequencies in the hopset. Thus, a ROM input address requires 7 bits. \(\Box \)
Multiple Loops
Example 3.4
Consider the Bluetooth system of Example 3.3 but with the more stringent requirement that t_{ s } = 2.5 μs, which only sacrifices three potential data symbols per hop interval. The single-loop synthesizer of Example 3.3 cannot provide this short switching time. The required switching time is provided by a three-loop synthesizer with f_{ r } = 10 MHz. The resolution of 1 MHz is achieved by taking M = 10. Equation (3.149) indicates that A_{ min } = 11 and A_{ max } = 20. Inequalities (3.150) and (3.151) are satisfied if f_{1} = 2300 MHz, B_{ min } = 9, and B_{ max } = 16. The maximum frequencies that must be accommodated by the dividers in loops A and B are A_{ max }f_{ r } = 200 MHz and B_{ max }f_{ r } = 160 MHz, respectively. Dividers A and B require 5 control bits. \(\Box \)
Fractional-N Synthesizer
A delta-sigma modulator encodes a higher-resolution digital signal into a lower-resolution one. When its input is a multiple-bit representation of A/M, the delta-sigma modulator generates the sequence \(\{b\left ( n\right ) \}\). If this sequence were periodic, then high-level fractional spurs, which are harmonic frequencies that are integer multiples of f_{ r }/M, would appear at the VCO input, thereby frequency-modulating its output signal. Fractional spurs are greatly reduced by modulus randomization, which randomizes \(b\left ( n\right ) \) while maintaining (3.153). Modulus randomization is implemented by dithering or randomly changing the least significant bit of the input to the delta-sigma modulator. Quantization noise \(q\left ( n\right ) =b\left ( n\right ) -\overline {b\left ( n\right ) }\) is introduced into the loop because A/M is approximated at the delta-sigma modulator output by a bit \(b\left ( n\right )\). To limit the effect of the quantization noise, the modulus randomization is designed such that the quantization noise has a high-pass spectrum. A lowpass loop filter can then eliminate most of the noise.
Example 3.5
Consider a fractional-N synthesizer for the Bluetooth system of Example 3.4 in which t_{ s } = 2.5 μs. If the output of the fractional-N synthesizer is frequency-translated by 2300 MHz, then the synthesizer itself needs to cover 102 to 180 MHz. The switching time is achieved by taking f_{ r } = 10 MHz. The resolution is achieved by taking M = 10. Equation (3.152) indicates that the required frequencies are covered by varying B from 10 to 18 and A from 0 to 9. The integers B and A require 5 and 4 control bits, respectively. \(\square \)
3.9 Problems
1
Consider FH/CPFSK with soft-decision decoding of repetition codes and large values of \(\mathbb {E}_{b}/I_{t0}\). Suppose that the number of repetitions n is not chosen to minimize the potential impact of partial-band interference. Then, the first line of (3.20) upper-bounds P_{ b }. Show that a nonbinary modulation with m =log_{2}q bits per symbol gives a better performance than binary modulation in the presence of worst-case partial-band interference if \(n>(m-1)\ln (2)/\ln (m)\).
2
Consider FH/CPFSK with soft-decision decoding of repetition codes. (a) Prove that \(f\left ( n\right ) =\left ( q/4\right ) \left ( 4n/e\gamma \right ) ^{n}\) is a strictly convex function over the compact set \(\left [ 1,\gamma /3\right ]\). (b) Find the stationary point that gives the global minimum of \(f\left ( n\right )\). (c) Prove (3.22).
3
The autocorrelations of the complex envelopes of practical signals have the form \(R_{l}(\tau )=f_{1}\left ( \tau /T_{s}\right )\). (a) Prove that the power spectral density of the complex envelope has the form \(S_{l} (f)=T_{s}f_{2}\left (fT_{s}\right )\). (b) Prove that if the bandwidth B of a frequency channel is determined by setting F_{ ib }(B/2) = c, then the required B is inversely proportional to T_{ s }.
5
6
Derive (3.112) by following the steps described in the text.
7
Approximate μ = J/M in (3.116) by a continuous variable over the interval \(\left [0,1\right ]\). What is the worst-case value of μ for binary orthogonal FH/CPFSK, noncoherent detection, hard decisions, and the AWGN channel in the presence of strong interference? What is the corresponding worst-case symbol error probability? Why does it not depend on the number of frequency channels?
8
Consider binary orthogonal FH/CPFSK, noncoherent detection, hard decisions, and the Rayleigh channel in the presence of strong interference. Show that interference spread uniformly over the entire hopping band hinders communications more than equal-power interference concentrated over part of the band.
9
This problem illustrates the importance of a channel code to a frequency-hopping system in the presence of worst-case partial-band interference. Consider binary orthogonal FH/CPFSK, noncoherent detection, hard decisions, and the AWGN channel. (a) Use the results of Problem 7 to calculate the required \(\mathbb {E}_{b}/I_{t0}\) to obtain a bit error rate P_{ b } = 10^{−5} when no channel code is used. (b) Calculate the required \(\mathbb {E}_{b}/I_{t0}\) for P_{ b } = 10^{−5} when a (23,12) Golay code is used. As a first step, use the first term in ( 1.22 ) to estimate the required symbol error probability. What is the coding gain?
10
It is desired to cover 198-200 MHz in 10-Hz increments using double-mix-divide modules. (a) What is the minimum number of modules required? (b) What is the range of acceptable reference frequencies? (c) Choose a reference frequency and f_{ b }. What are the frequencies of the required tones? (d) If an upconversion by 180 MHz follows the DMD modules, what is the range of acceptable reference frequencies? Is this system more practical?
11
It is desired to cover 100-100.99 MHz in 10-kHz increments with an indirect frequency synthesizer containing a single loop and a dual-modulus divider. Let f_{1} = 0 in Figure 3.19 and Q = 1 in Figure 3.20 (a) What is a suitable range of values for A? (b) What are a suitable value for P and a suitable range of values for B if it is required to minimize the highest frequency applied to the programmable dividers?
12
It is desired to cover 198-200 MHz in 10-Hz increments with a switching time equal to 2.5 ms. An indirect frequency synthesizer with three loops in the form of Figure 3.21 is used. It is desired that \(B_{\max } \leq 10^{4}\) and that f_{1} is minimized. What are suitable values for the parameters f_{ r }, M, \(A_{\min }\), \(A_{\max },B_{\min }\), \(B_{\max }\), and f_{1}?
13
Specify the design parameters of a fractional-N synthesizer that covers 198-200 MHz in 10-Hz increments with a switching time equal to 250 μs.
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