Spectrum is limited, but innovative communications formats using ultranarrowband (UNB) modulation may provide relief for added wireless services. UNB modulation formats have their limitations, of course, and it is important to understand the performance capabilities of UNB modulation and how systems can differ by their use of abrupt versus gradual changes in phase. In order to better understand UNB modulation, the relationship between phase modulation and frequency modulation will be examined, and simulated and measured results compared. In addition, the dependency between Fourier sinx/x components and the bit error rate (BER) will be explored, along with analysis of a zero-group-delay filter as instrumental to the implementation of practical UNB formats.

The basic concept of UNB modulation involves achieving almost infinite change or modulation in frequency (f) for a short duration, such as a single cycle at RF. Practical implementations of UNB systems have incorporated minimim-sideband (MSB) modulation and very-minimum-shiftkeying (VMSK) modulation, where the modulated information ideally appears as a single spectral line.¹

In spite of its ideal appearance, any UNB format must comply with Nyquist theory. For example, Fig. 1(a) offers a modified view of a popular presentation on UNB modulation, with the left-hand side showing abrupt phase changes while the right-hand side shows gradual phase changes. Figure 1(b) shows the frequency modulation (FM) that results from the phase modulation of Figure 1(a)). An abrupt change in phase produces instantaneous changes in frequency while gradual changes in phase yield constant changes in frequency. In MSB modulation, a coded baseband is used with abrupt edges, with rise/fall times as close to zero as possible. Of course, some finite rise time is inevitable, due to slew rates in integrated circuits (ICs) and resistive-capacitive (RC) delays. Typically, a UNB system makes use of almost infinite FM during one cycle of the intermediate frequency (IF).¹ For efficient UNB systems, with the phase modulation of Fig. 1(c) yielding the frequency modulation of Fig. 1(d), the rise/fall time is assumed to be one-half an IF cycle. The cycle of the carrier is T_carrier = 1/F_carrier.

The frequency (F) resulting from a modulating input signal is F = F_carrier + Δf, where the modulation frequency, Δf, can be calculated from the basic relationship ωt = φ  = 2Πft. The modulation frequency can also be rewritten in derivative form as Δf = Δφ/2Πt. The rise/fall time, Δt, is equal to one-half a carrier cycle and is fixed by the circuit parameters. During the rise/fall times for modulation formats with abrupt phase modulation, there is a Δφ/Δt value that causes large modulation frequency, f, for a very short duration (about one-half cycle at RF), so that F = F_carrier + Δf = F_carrier + Δφ/2ΠΔt = F_carrier + Π/2(T_carrier) = 2F_carrier.

At other times, when the changes in phase are gradual or constant,  Δφ = 0 and the frequency is constant at F = F_carrier. In this case, the modulated waveform is shown in Fig. 2, with the asterisk denoting a modulated waveform and the dot marking an unmodulated waveform. The modulated waveform employs continuous phase modulation with two kinds of frequencies, F_carrier and 2F_carrier, which includes Bessel products; the phasor vector sum is shown in Fig. 3. A phase detector using F_carrier as a phase reference will detect phase changes as positive and negative voltages, but will ignore large f. In this case, Δφ is considered to be zero for most of the bit (information) period.

For modulation with an abrupt phase change as shown in the lefthand side of Fig. 1(c), when Δt →0,, then Δf → ∞ , so the instantaneous frequency is F = Fcarrier + Δf → ∞ shown in the left-hand side of Fig. 1(b), at time Δt →0 or t = 0. The modulated wave is indicated by the dot in Fig. 4. Although an instantaneous near-infinite frequency modulation is produced, the duration, t, is zero, and the modulated result only includes one kind of frequency component, Fcarrier. This is the familiar form of digital phase modulation, not including Bessel products. The phasor vector sum is shown in the right-hand side of Fig. 4. The phase modulation in this case is not related to the Bessel products, with no Bessel products other than J₀.

The power spectral density (PSD) for 2PSK modulation (MSB modulation with an abrupt change in angle of 90 deg.) is shown in Fig. 5(a). The highest point of continuous spectrum is about –30 dB. The data rate is 270 kb/s with non-return-to-zero (NRZ) baseband coding. The continuous spectrum has been called the sum of all Fourier sinx/x products related to random NRZ data, which do not cause any phase shift and can be filtered with a zerogroup- delay filter that does not modify the demodulation performance.¹ The PSD of all sinx/x products filtered in Fig. 5(a) is plotted in Fig. 5(b). The continuous spectrum is lowered to about -35 dB and the detected result remains constant as shown in Fig. 5(c). When Bessel products are present, the relationship between the phase angle and the Bessel sidebands is 2J₁ = sin and if the relationship is kept for Fourier products, then the detected result will become degraded, with the Fourier sideband lowering, although the actual result is not degraded [Fig. 5(c)]. So the Fourier sideband line spectrum does not have a relationship to the detected phase angle. Figure 5(d) is the PSD of all the Fourier products filtered out, with the continuous spectrum is lowered to -60 dB; the detected performance is not affected.

The bandwidth of a zero-groupdelay filter is only about 2 to 3 kHz as presented in earlier work.¹ In such a narrowband filter, only J₀ is passed after the UNB signal is filtered, with no added group delay. The signal can be demodulated with single carrier J₀ extracted.^2-4 However, digital modulation schemes also contain a discrete spectrum and a continuous spectrum.^5,6 The discrete products can be removed without affecting the detected information, but the modulation information resides in the continuous spectrum. A modulated waveform can only be demodulated when adhering to the principles of Nyquist theory. In reality, the signal bandwidth cannot be compressed to a single spectral line, J₀. The PSD for VMSK modulation can be shown as⁶

SEE EQ. 1

and the energy of the continuous spectrum can be represented by

SEE EQ. 2

with the energy given by

SEE EQ. 3

The continuous spectrum represents the difference of two waveforms. The VMSK continuous spectrum energy is only 1/(2N + 1) that of BPSK, but the VMSK bandwidth is 2N + 1 times the bandwidth of BPSK (Fig. 6). A modulated waveform cannot be demodulated only by the carrier frequency, J₀.

Figure 7 show the equivalent circuit for a zero-group-delay filter.¹ The transfer function for the filter is

The output of the filter is

and using the phasor vector representation method, v₁ = V?

where θ is the original phase,

where

H(ω) = ωC₁R/(1 + ξ²)^0.5

v₂ has a leading phase, φ(ω) =Π /2 - arctanξ,then V₁. It is possible to write K(ω) = VH(ω), then the output signal is

on the resonance point, K(ω₀) = K, which is constant, and φ(ω₀) =Π /2, the output is

and the signal is passed without any loss at an expected phase of 90 deg. This verifies the zero-group-delay characteristics at the resonant point.⁴ When the signal is fully passed, the filter shows high-pass characteristics.

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