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Fractional out-of-band power (FOBP) has for some time been a standard metric of spectral containment for digital data signals.1 For any given modulation scheme the FOBP can be measured experimentally or calculated theoretically. The theoretical calculation often may take the form of numerical integration on an existing tabulation of power spectral density (PSD) obtained from a Monte Carlo simulation or, perhaps, may be evaluated directly from a closed-form expression for PSD. If an analytical expression is available for PSD, then the option is open for the derivation of a closed-form expression for the FOBP as well. Though numerical integration can generally be refined to yield whatever precision is desired, the closed-form expression may be preferable for the insight it provides into asymptotic behavior, or for the validation of similar closed-form results available from other sources.

Fig. 1 Power spectral densities for BPSK, QPSK and MSK normalized for unit transmitter power.

This article offers closed-form expressions for FOBP for classical binary phase-shift keying (BPSK), quadrature phase-shift keying (QPSK) (including offset QPSK (OQPSK)) and minimum-shift keying (MSK) modulation. Of these three cases, the classical MSK presents the greatest challenge. In all cases the FOBP is expressible in terms of well-known, widely tabulated functions. The derivation of closed-form FOBP begins, of course, with the analytic expression for PSD. All of the PSDs mentioned thus far are shown in Figure 1. The FOBP characteristics are shown in Figure 2. The FOBP expressions reported here were derived and subsequently verified against the results of numerical integration on known PSD.



The expressions for the PSD of classical BPSK, QPSK and OQPSK take the same familiar general form.1 For the present purposes it is convenient to normalize all PSDs for unit signal power so that the FOBP is simply equal to the out-of-band power itself. When the data signal is denoted by s(t), the PSD is usually denoted by |S(f)|2, whose normalized expression becomes

Note that the factor T in the denominator is not squared. As usual, fc indicates the RF carrier.

The FOBP is readily determined in closed form in terms of the widely2 tabulated Si(x) function:

The frequency variable for FOBP has been defined by various authors as either the upper band edge frequency or the normalized doublesided bandwidth WT as previously illustrated in the PSD plots. The FOBP (designated here as FOBP1) becomes

for BPSK.

The power spectral density of QPSK (and OQPSK) takes the same form as that of BPSK, except for a factor of 2 compression in the frequency domain.1 That compression is the well-known doubling of spectral efficiency attained by reverting to QPSK or OQPSK. The resulting FOBP characteristic is similarly compressed in frequency, leading to FOBP2:



The normalized PSD of MSK is simply expressed as3

However, the expression for FOBP is somewhat more complicated than for the BPSK case. Here it is convenient to invoke a new function Xi(x) based on the widely2 tabulated Ci(x) function:


Now the required expression is given as FOBP3:



Most contemporary mathematical software packages either include the Si(x) and Ci(x) functions explicitly or allow their inclusion implicitly as user-defined integrals for numerical evaluation. All of these software packages will most likely suffice for the production of plots over the abscissa range shown previously in the FOBP plot. However, for large WT some of the available software may produce unsteady, ragged plots. Therefore, it may be helpful to state the Taylor series expansions of Si(x) and Xi(x):

If the upper limit of • is replaced, for example, by p = 60 in both series, then the series provide accuracy to within a truncation error of less than 2 ¥ 10–14 for values of x up to 36. In addition, the following asymptotic expressions may be substituted for the required functions for extremely large values of x:



The MSK modulation format is a special case of binary frequency-shift keying (FSK) in which the selected modulation index is h = 0.5, which is the value that forces a phase change of ±p/2 on each bit interval. In general, the normalized PSD for FSK over a large range of h is given by4,5


F = (f – fc)T

This family of normalized PSD is shown in Figure 3. To date, no closed-form expression for FOBP has been made widely available. Expressions for the PSD of more complex digital FSK schemes are available from more recent sources.6,7



Closed-form expressions for FOBP for classical BPSK, QPSK and MSK modulation have been derived and compared to PSD calculations to verify their accuracy. These expressions may be used in place of numerical integration to determine asymptotic behavior or to validate other similar closed-form results for FOBP analysis of digital data signals.


1. F. Amoroso, "The Bandwidth of Digital Data Signals," IEEE Communications Magazine, Vol. 18, No. 6, November 1980, pp. 13–24.

2. E. Jahnke and F. Emde, Tables of Functions, Dover, New York, 1945.

3. F. Amoroso, "The Use of Quasi-bandlimited Pulses in MSK Transmission," IEEE Transactions on Communications, Vol. COM-27, No. 10, October 1979, pp. 1616–1624.

4. M.G. Pelchat, "The Autocorrelation Function and Power Spectrum of PCM/FM with Random Binary Modulating Waveforms," IEEE Transactions on Space Electronics and Telemetry, Vol. SET-10, March 1964, pp. 39–44.

5. T.T. Tjhung, "Band Occupancy of Digital FM Signals," IEEE Transactions on Communication Technology, Vol. COM-12, No. 4, December 1964, pp. 211–216.

6. J.G. Proakis, Digital Communications, McGraw-Hill, New York, 1983.

7. J.B. Anderson et al., Digital Phase Modulation, Plenum Press, New York, 1986.

Frank Amoroso received his BS and MS degrees in electrical engineering from the Massachusetts Institute of Technology in 1958, then pursued additional graduate studies at Purdue University of Turin, Italy. His career in the US industry included such organizations as the Hughes Aircraft Co., the RCA David Sarnoff Research Laboratories, the Mitre Corp. and the US Army Signal Corps Laboratories at Fort Monmouth, NJ. In 1989, Amoroso retired from industrial employment to begin a new career as a private consultant to a number of industrial firms, including Lincom Inc., Los Angeles, CA. He holds five US patents, has published 30 papers in various archival-quality journals and has taught seminar courses at George Washington University. Amoroso's accomplishments are recognized in the current edition of Marquis Who's Who in America.