WO1995001033A1 - Quadrature minimum-shift keying - Google Patents

Abstract

Transmission-efficient digital signaling methods, provided by modulation and subsequent detection of multi-dimensional, non-orthogonal signal basis sets (V (t), V (t), V (t), V (t)), are disclosed herein. In accordance with the invention, Quadrature Minimum-Shift Keyed (QMSK), M-are Quadrature Minimum-Shifted Keyed (M-QMSK), and constant-envelope QMSK (CE) signaling methods are provided for carrier-based transmission. In additional accordance with the invention, a baseband QMSK (BB) signaling method, suitable for conductive transmission, is provided. Bandwidth efficiency and power efficiency (Fig. 20 and Fig. 21) are dramatically improved by the signaling methods of the present invention. For example, band-limited QMSK signaling yields 100 % improvement in bandwidth efficiency in comparison to band-limited QPSK or MSK while retaining esentially antipodal power efficiency. For identical data rates, M-QMSK signaling provides significant bandwidth efficiency and power efficiency improvements in comparison to M-QAM. In still additional accordance with the invention, power-efficient signal detection for the non-orthogonal signals disclosed herein is provided by a non-conventional, augmented correlating demodulator (Fig. 17).

Description

QUADRATURE MINIMUM-SHIFT KEYING

Field of the Invention

The present invention relates to digital communication systems. More specifically, the present invention relates to methods for generating and demodulating transmission-efficient digital information signals wherein such signals are formed by combining elements of quadrature carrier waveforms, staggered symbol waveforms, and correspondingly-staggered digital data streams.

BACKGROUND OF THE INVENTION

Generally

Due to superior performance characteristics, digital (discrete) communication methods are rapidly supplanting analog (continuous) communication methods. Existing applications for digital communication methods include computer networks, digital telephone service, facsimile service, etc. Emerging applications for such methods include HDTV (high definition television) and cellular telephone service; as technology improves, digital sound broadcasting will emerge. Both existing and emerging communication applications place increasing pressure on available frequency resources.

The demands placed on bandwidth availability by our information-driven environment have made bandwidth an increasingly scarce and valuable resource. As a result, a continuing need for improvements in bandwidth efficiency of digital signaling methods exists. Bandwidth efficiency is defined as the numerical value of the ratio R_b/W, where R_b is the transmitted information bit rate in bits per second and W is the effective transmitted signal bandwidth in Hertz. Larger values indicate higher bandwidth efficiency.

Bandwidth-efficient digital signal transmission is implemented by maximizing the information rate that can be sustained in a given transmission bandwidth. Practical digital signaling is implemented by using time-limited waveforms. The power spectral density of such waveforms is non-zero for all but possibly certain discrete values of frequency. Thus to avoid interference between communicators, it is generally necessary to band-limit (i.e., to filter) both transmitted and received signals. Filtering, however, must be done in such a way that all frequency components: which contribute substantially to total signal power are retained, while those which contribute little to total signal power are effectively removed.

In band-limiting a practical digital signal, that is, in filtering an originally time-limited waveform, two penalties are paid. The first penalty is that some fraction of original signal power is lost. The second penalty is that the filtered signal extends in time beyond its original, prescribed duration. That portion of the filtered signal which extends beyond its allocated time interval distorts the bit decision process in succeeding intervals. Such distortion, called inter-symbol interference (ISI), reduces the effective signal-to-noise power ratio (SNR), and thereby increases the probability of bit error, P_b(e).

ISI increases as signal bandwidth is reduced. Accordingly, the effect of ISI on

P_b(e) plays a significant role in determining an acceptable lower limit on signal bandwidth. The effect of ISI on P_b(e) depends both on signal waveform design and filter design; a precise formulation of the relationship is difficult. It is generally necessary to make simplifying assumptions in order to obtain calculable numerical results. While it is usually necessary, band-limiting a practical digital signal always causes some finite loss in effective SNR and a corresponding increase in P_b(e). Thus, the concept of bandwidth efficiency is intimately related to the concept of power efficiency.

For a time-limited waveform (alternately referred to as a time-limited signal) processed in an "ideal" channel, and in which the only perturbing influence is additive white Gaussian noise (AWGN), the ideal correlating demodulator provides optimum signal detection. Its output SNR is proportional to E_b/N_o, where E_b is the energy per bit of the signal and N_o is the one-sided power density of the AWGN. The

proportionality constant depends on signal geometry. When conditions are less than ideal, the effective output SNR retains a functional dependence on the ratio E_b/N_o, so it is customary in the art to describe power efficiency by the ordered pair [E_b/N_o, P_b(e)]. Shannon's work in communications theory showed that methods of digital signaling in an ideal AWGN channel exist such that error-free signalling can be achieved at any bit rate lower than channel capacity C bits per second. C is related to signal power P_s, one-sided noise power density N_o, and channel bandwidth W by the equation

C/W = log₂(1 +P_s/N₀0) . ^{(1 )}

An ideal method of digital signaling may be defined as one for which bit rate R_b = C. Using this definition, and the identity P_s = E_bR_b, one can manipulate Equation (1) to obtain

Equation (2) defines a curve in the (R_b/W, E_b/N_o) plane. Its implication is that for any positive value of R_b/W, error-free digital signaling is theoretically possible for any value of E_b/N_o greater than that defined by Equation (2).

Practical digital signaling methods are not error-free. However, the concept of plotting, for an i^th signaling method, the ordered pair [R_b/W)_i, (E_b/N_o)_i] parametrically with P_b(e)_j in the (R_b/W, E_b/N_o) plane has great utility. One may intuitively think of the shortest distance D_i between the i^th point and the ideal curve of Equation (2) as representing an inverse measure of transmission efficiency. Because D_i depends on the quantities [(R_b/W)_i, (E_b/N_o)_i, P_b(e)_i], this concept of transmission efficiency elegantly relates the concepts of bandwidth efficiency and power efficiency.

The digital signals of interest herein may generally be defined by the equation

during the m^th symbol interval. The coefficients C_mn contain the digital information transmitted on the n^th signal basis set component v_n(t,m,T_s) during the m^th symbol interval. In general, the C_mn may contain both antipodal information (±1) and discrete amplitude information. The n^th signal basis component is the product of the n^th symbol waveform p_n(t-mT_s) and the n^th carrier waveform c_n(t). The information sequences (C_mn) are in time-coincidence with their associated symbol waveforms. T_s = 2T is the symbol period common to all symbol wave forms; R_s = 1/T_s is the symbol rate. The symbol waveforms are specific to particular signaling methods. Carrier waveforms c_n(t) are normally selected from the quadrature waveforms [cos(ω_ct), sin(ω_ct)], where ω_c = 2 πf_c and f_c is the signal carrier frequency.

When the signal basis set [v_n(t,m,T_s)] is orthogonal, the individual information coefficients C_mn can be recovered (detected), in the absence of noise, by the parallel, ideal correlating demodulator operations

The integrations of Equation (4) are each performed over appropriate symbol intervals (possibly staggered) of length T_s. K is a constant dependent on signal geometry. If all symbol waveforms are time-coincident, orthogonality of the signal basis set over each common symbol interval ensures proper signal detection. If a subset of symbol waveforms is staggered (i.e., offset in time by T = T_s/2 with respect to the remaining symbol waveforms), the more stringent requirement of orthogonality over each joint half-symbol interval is necessary for proper detection. If the signal basis set is orthogonal under either condition, it spans (defines) a vector signal space of dimension N.

Review of the Prior Art

Many different modulation schemes have been proposed or developed in order to meet the need for transmission-efficient digital communication systems. Quadrature Phase Shift Keying (QPSK), for example, is a two-dimensional, constant-envelope digital signaling method. QPSK carrier waveforms are the quadrature (i.e., having 90-degree phase difference) pair [cos(ω_ct), sin(ω_ct)]. Basic QPSK symbol waveforms are time-coincident rectangular pulses as illustrated in Figure 1A.

QPSK C_mn assume values from the set (B_mnA), where antipodal data bits B_mn = (±1) and A is a constant related to signal power. For independent,

identically-distributed (IID) data sequences (B_mn), the power spectral density of the QPSK signal. depends only on the spectral characteristics of the rectangular symbol waveform. Therefore, the main lobe of the power spectral density function has width Δf = 2R_s. However, when the QPSK signal is band-limited to W = 2R_s, only 90.7% of the original signal power is retained. Nevertheless, W = 2R_s is often used as a measure of bandwidth occupancy of the QPSK signal.

The QPSK signal carries one antipodal bit on each of its two signal basis components. Thus, there are M = 2^N = 4 possible signal states during a given symbol interval. Because R_b = [log₂(M)]R_s = 2R_s, one obtains R_b/W = 1.0 as a measure of bandwidth efficiency of the QPSK signal. Ignoring ISI and power loss, the ideal, antipodal power efficiency of QPSK can be characterized by [E_b/N_o, P_b(e)] =

(10.53 dB, 10^-6). The transmission efficiency of band-limited QPSK can then be characterized by [R_b/W, E_b/N_o, P_b(e)] = (1.0, 10.53 dB, 10^-6).

Staggered QPSK (SQPSK) is a variant of QPSK in which the antipodal data sequences (B_m1, B_m2) are staggered. Their correspondingly-staggered basic symbol waveforms are illustrated in Figure 1 B. The transmission efficiencies of identically band-limited SQPSK and QPSK are identical.

On the other hand, Minimum-Shift Keying (MSK) is a two-dimensional, constant-envelope digital signaling method which has structure similar to that of SQPSK in that each uses staggered data sequences and symbol waveforms. However, in contrast to SQPSK, the basic MSK symbol waveforms are the staggered half-cosinusoid and half-sinusoid of frequency f_o = 1/2T_s = 1/4T as illustrated in

Figure 1 C. The MSK carrier waveforms are the quadrature pair [cos(ω_ct), sin(ω_ct)]. Strict orthogonality of the MSK signal basis set over the joint half-symbol interval of length 7 occurs when the carrier frequency f_c is an integer multiple of f_o; in practice, orthogonality is satisfied if f_c is much greater than f_o.

The MSK power spectral density function, which is determined by the spectral characteristics of its half-cosinusoidal symbol waveforms, has main lobe width J = 3R_s. However when filtered to an effective bandwidth W = 2R_s, the band-limited MSK signal retains 96.9% of its original power. MSK signals filtered to W = 2R_s retain more power and suffer less ISI than do identically-filtered QPSK and SQPSK signals. Thus, in comparison to QPSK and SQPSK, band-limited MSK can more reliably be said to have the transmission efficiency characterized by (1.0, 10.53 dB, 10^-6).

M-ary Phase Shift Keying (M-PSK) is a two-dimensional, constant-envelope digital signaling method which, for M greater than four, offers increased bandwidth efficiency at the expense of power efficiency. Its symbol waveforms are identical to those of QPSK, and its carrier waveforms are rotated by 45 degrees from those of QPSK. The 4-PSK signal has transmission efficiency identical to that of QPSK. M is chosen as M = 2ⁱ, where j is a positive integer. Ignoring ISI and power loss, the transmission efficiencies of. band-limited 8-PSK and 16-PSK are characterized by (1.5, 13.95 dB, 10^-6) and (2.0, 18.44 dB, 10^-6), respectively. The power efficiency of M- PSK decreases rapidly with increasing M.

M-ary Quadrature Amplitude Modulation (M-QAM) is a two-dimensional digital signaling method which has non-constant envelope. Its symbol and carrier waveforms are identical to those of QPSK. Its information coefficients are given by C_mn = B_mnA_mn. During the m^th symbol interval, (B_m1, B_m2) are antipodal data bits and (A_m1, A_m2) are discrete amplitude levels, each of which is determined by an additional k data bits. The signal basis set carries N(k+1) = 2(k+1) data bits per symbol interval, so M = 2^(2k+2). The M-QAM bandwidth efficiency is R_b/W = (k+1). For optimum spacing of the discrete amplitude levels, and ignoring ISI and power loss, band-limited 16-QAM and 64-QAM have transmission efficiencies characterized by (2.0, 14.4 dB, 10^-6) and

(3.0, 18.78 dB, 10^-6), respectively. A variant of M-QAM, M-SQAM, has symbol and carrier waveforms in common with SQPSK. Also in analogy to M-QAM, an M-MSK signal can be generated by applying identically-defined (though staggered) information coefficients to the MSK signal basis set. Theoretically, M-MSK has transmission efficiency equivalent to that of M-QAM; however, when each is identically band-limited,

M-MSK performs better because of its higher percentage of retained power and better ISI characteristics.

Quadrature-Quadrature Phase Shift Keying (Q²PSK) is a family of four-dimensional digital signaling methods disclosed by Saha in U.S. patent 4,680,777, dated July 14, 1987. Of the possible family members, Q²PSKH has the best power efficiency. Q²PSKH symbol waveforms are the time-coincident half-cosinusoid and half-sinusoid illustrated in Figure 1 D; its carrier waveforms are-the quadrature pair [cos(ω_ct), sin(ω_ct)]. Both the symbol waveform pair and the carrier waveform pair are orthogonal over a given symbol interval. Thus by forming all distinguishable products of these symbol and carrier wave forms, a set of four orthogonal Q²PSKH signal basis set components is defined. The antipodally-modulated Q²PSKH signal occupies the vertices of a four-dimensional hypercube.

For any signaling method within the Q²PSK family, symbol waveform orthogonality requires that one member of the time-coincident symbol waveform pair has zero average value, and therefore its power density function is equal to zero at the origin. As a result, the modulated Q²PSK signal has significant power density at frequencies well-removed from the carrier frequency. In addition, the truncated half-sinusoid symbol waveform of Q²PSKH has abrupt discontinuities, which cause additional spectrum broadening. Band-limiting the Q²PSKH signal to reasonable bandwidth introduces significant signal power loss and ISI. Using the analysis provided by Saha, the Q²PSKH transmission efficiency can be characterized by (1.67,

12.80 dB, 10^-6). Q²PSKH can thus be considered to have a 67% bandwidth efficiency advantage over MSK while sustaining an effective 2.27 dB loss in E_b/N_o at P_b(e) =

10^-6.

The Q²PSK signal, as previously defined, does not have constant envelope. However, as demonstrated by Saha in U.S. patent 4,730,344, dated March 8, 1988, the Q²PSK signal can be constrained to have constant envelope by using a coding technique of code rate 3/4. Thus, constant-envelope Q²PSKH(CE) signals can be obtained at a transmission efficiency characterized by (1.25, 12.80 dB, 10^-6).

Figure 2 is useful to illustrate the differences in the equivalent baseband power spectral density functions of QPSK, MSK, and Q²PSKH as a function of the normalized frequency variable x = f/R_b. Alternatively, Figure 3 indicates the relative transmission efficiencies of digital signaling methods discussed previously; for comparison, the Shannon channel capacity limit of Equation (2) is also shown.

From the foregoing discussion, and from examination of Figure 3, it is understood that, while reasonably-efficient band-limited digital signaling methods are available, there is much room for improvement in the art. Coding can be used to improve the power efficiency of various signaling methods, but coding inherently reduces bandwidth efficiency. On the other hand, for the two-dimensional signaling methods, bandwidth efficiency can be improved by resorting to simultaneous antipodal and discrete amplitude modulation as in M-MSK and M-QAM, but this inherently reduces power efficiency. A reasonable expectation of improving bandwidth efficiency while simultaneously retaining antipodal power efficiency derives from consideration of using signal spaces of higher dimension. The four-dimensional, band-limited Q²PSK signaling method, as disclosed by Saha, does achieve improved bandwidth efficiency; however, ISI and inefficient power density resulting from its use of certain prescribed orthogonal symbol waveforms prevent it from achieving antipodal power efficiency.

SUMMARY OF THE INVENTION

In accordance with one aspect of the invention, digital signaling methods based on four-dimensional data transmission are provided. The resulting transmitted signals are composed of elements of staggered data streams, staggered symbol waveforms, and quadrature carrier waveforms. The combination of four dimensional signal transmission and staggered symbol waveforms of the present invention provides improved bandwidth efficiency over that of modulation methods described above.

The quadrature carrier waveforms are normally cosinusoidal and sinusoidal, respectively, each having common carrier frequency. The two basic symbol waveforms are staggered (i.e., identical waveshapes offset in time by one-half symbol period). By generating all distinguishable products of said symbol waveforms and said carrier waveforms, a set of four distinguishable signal basis set components is both defined and physically realized.

The set of four distinguishable signal basis set components occupies a four-dimensional vector signal space; however, due to the prescribed staggering of the symbol waveform set, the signal basis set is not orthogonal. Nevertheless, the distinguishable, non-orthogonal components of this signal basis set can each be modulated by independent, appropriately-staggered data streams. The four components of this signal basis set can be shown to consist of two distinct pairs such that each pair is orthogonal to the other pair, and such that pair components are not themselves orthogonal. Thus each distinct, non-orthogonal pair identifies and occupies a distinct hyperplane in the four-dimensional signal space; the distinct hyperplanes are themselves orthogonal.

In accordance with a further aspect of the invention, the two distinct, staggered symbol waveforms are specified to be a half-cosinusoid and a half-sinusoid, respectively, during their appropriate symbol periods. The distinguishable components of the product set of these specific symbol waveforms and the quadrature carrier waveforms define a particularly intriguing signal basis set. The four components of this specific signal basis set form two distinct pairs, each of which can be identified as the signal basis set for an independent MSK signal, provided that one considers the separate MSK modulations to be performed simultaneously on two carriers which are in phase quadrature. Thus the composite signal formed by summing the results of independent, antipodal modulation on each of the four components of this specific signal basis set is identical to, and indistinguishable from, the composite modulation resulting from summing two independent, time-coincident MSK modulations performed on quadrature carriers. Thus, this specific embodiment of the invention is called Quadrature Minimal-Shift Keying (QMSK). Appropriately band-limited QMSK signaling achieves 100% bandwidth efficiency improvement in comparison to band-limited QPSK and MSK, and, when demodulated in accordance with other aspects of the invention, simultaneously achieves essentially antipodal power efficiency.

In accordance with a yet further aspect of the invention, each component of the multi-dimensional, non-orthogonal QMSK signal basis set can be jointly modulated by an appropriately-timed, independent, antipodal data stream and a sequence of discrete amplitude levels determined by other independent data streams which are in time-coincidence with the antipodal data stream. Summing a QMSK signal basis set modulated as described above provides a composite M-ary QMSK (M-QMSK) signal which has improved bandwidth efficiency. The M-QMSK signal is generated by appropriately applying both antipodal and discrete amplitude modulation to all components of the multi-dimensional, non-orthogonal, QMSK signal basis set. For a pre-determined bandwidth efficiency, M-QMSK signaling achieves dramatic power efficiency improvement in comparison to previously-known M-PSK, M-QAM, and M-MSK signaling methods.

QMSK signals do not have constant envelope. Therefore, in accordance with a still further aspect of the invention, when the QMSK signal basis set is modulated by appropriately-timed antipodal data streams, three of which are independent and the fourth of which is dependent on the first three, a constant-envelope QMSK(CE) signaling method is provided. The bandwidth efficiency of the QMSK(CE) signal is only three-fourths that of the QMSK signal, but is nonetheless 50% greater than that of QPSK or MSK. The resulting QMSK(CE) signal is identical to that formed by summing two time-coincident, quadrature MSK signals provided that both are constrained to have identical frequency during each joint half-symbol interval. QMSK(CE) signaling achieves three-fourths the bandwidth efficiency of QMSK while retaining its

near-antipodal power efficiency.

In accordance with the present invention, the QMSK signal basis set common to these signaling methods is not conventionally orthogonal. Because of their joint reliance on a multi-dimensional, non-orthogonal signal basis set, QMSK, QMSK(CE), and M-QMSK signals can not be detected successfully by a conventional correlating demodulator. Therefore, in accordance with an additional aspect of the invention, a power-efficient method, that is, an augmented correlating demodulator for detecting multi-dimensional, non-orthogonal signals such as QMSK, QMSK(CE), and M-QMSK signals, is provided.

It will be shown, in the detailed discussion which follows, that the QMSK signal basis set is orthogonal with respect to a particular class of weight functions. Thus, one arrives at the concept of a modified correlating demodulator in which the set of waveforms which multiply the incoming signal in parallel is not simply the signal basis set, but is instead the product set of the QMSK signal basis set and a selected weight function. A preferred embodiment of the weight function is provided in the detailed discussion.

The QMSK signal basis set and the weighted QMSK signal basis set are orthogonal over any required half-symbol interval, and therefore the weighted correlating demodulator can successfully detect and resolve QMSK-based signals. It will be shown in the detailed discussion that, with both a QMSK signal and broadband AWGN present at its input, the weighted correlating demodulator produces an output signal-to-noise power ratio (SNR) which is inferior to that of conventional antipodal signaling by 3 dB. However, it will also be shown that specified band-limiting of both the signal and AWGN, prior to their being input to the weighted correlating

demodulator, allows essentially antipodal power efficiency to be achieved. Thus, the augmented correlating demodulator, which consists of an input filter of specified bandwidth followed by the weighted correlating demodulator, is provided as the preferred detector for QMSK-based signals.

In contrast with the optimum signal detector known in the prior art (i.e., the conventional correlating demodulator), the weighted correlating demodulator requires an input filter of specified bandwidth in order to achieve power-efficient detection. Therefore, it has been necessary to investigate other effects of such specified filtering on the detection of QMSK-based signals. In particular, an analysis of the effect of such filtering on the inter-symbol interference (ISI) properties of QMSK-based signaling has been performed. The analysis is given in the detailed discussion, but its results are summarized here. Appropriately band-limited QMSK and QMSK(CE) signaling suffer barely-discernible power-efficiency degradation due to ISI. Band-limited M-QMSK, because it incorporates discrete amplitude modulation of the QMSK signal basis set components, predictably suffers decreasing power efficiency due to ISI as the number of amplitude bits carried per signal basis set component increases. The analysis, however, also suggested that ISI could theoretically be eliminated by suitably modifying the QMSK symbol waveforms. Therefore, in accordance with a still additional aspect of the invention, a separate embodiment of the QMSK signal basis set is provided. The QMSK_O, QMSK_O(CE), and M-QMSK_O signal basis set, in conjunction with the augmented correlating demodulator, inhibits ISI in the detection of appropriately band-limited, QMSK_O-based signals. The QMSK_O signal basis set can be realized by appropriately combining specified phase-modulated carrier components.

In accordance with a yet additional aspect of the invention, maximum-likelihood reconstruction of a correctly-phased, weighted QMSK signal basis set from modulated

QMSK-based signals is provided. Because MSK-like signals inherently contain timing information, the reconstruction method also provides clock re-generation and integrate-sample-dump control signals. As has been shown in the prior art, maximum-likelihood reconstruction methods provide superior phase noise performance in comparison to methods based on n -power loops or Costas loops. At first glance, implementation of the maximum-likelihood phase detector appears to be somewhat hardware-intensive; the required hardware, however, operates at baseband, and can be realized by application-specific integrated circuits (ASICs).

In accordance with another aspect of the invention, maximum-likelihood construction of a phase-coherent, automatic gain control (CAGC) signal used for leveling signal power input to the augmented correlating demodulator is provided.

Maximum-likelihood phase detection, in conjunction with maximum-likelihood CAGC, may provide some, if not all, of the benefits of adaptive channel equalization.

In accordance with still another aspect of the invention, application of direct-sequence, spread-spectrum concepts to QMSK and QMSK(CE) signaling is provided. Direct-sequence QMSK signaling, for example, achieves twice the underlying data rate of direct-sequence QPSK or MSK.

DESCRIPTION OF THE DRAWINGS FIG. 1 is a timing diagram which illustrates symbol waveforms of certain prior art signaling methods.

FIG. 2 is a graph which illustrates equivalent baseband power density functions of prior art QPSK, MSK, and Q²PSKH signaling methods.

FIG. 3 is a graph which illustrates transmission efficiencies of certain prior art signaling methods.

FIG. 4 is a timing diagram which illustrates QMSK and M-QMSK symbol waveforms.

FIG. 5 is a timing diagram which illustrates QMSK and M-QMSK symbol information coefficient timing.

FIG. 6 is a functional block diagram which illustrates the generation of QMSK and M-QMSK symbol information coefficients from an input serial data stream. FIG. 7 is a functional block diagram which illustrates a canonical QMSK and M-QMSK modulator.

FIG. 8 is a functional block diagram which illustrates an alternate QMSK modulator.

FIG. 9 is a functional block diagram which illustrates phase-coherent generation of QMSK and M-QMSK carrier and clock waveforms.

FIG. 10 is a functional block diagram which illustrates the generation of QMSK and M-QMSK signal basis set waveforms.

FIG. 11 is a graph which illustrates the equivalent baseband power density function of the QMSK signal.

FIG. 12 is a timing diagram which illustrates the weight function waveform requirement for successful detection of QMSK and M-QMSK signals.

FIG. 13 is a functional block diagram which illustrates a weighted correlating demodulator channel.

FIG. 14 is a functional block diagram which illustrates an augmented correlating demodulator channel.

FIG. 15 is a graph which illustrates noise power density functions at strategic points in an augmented correlating demodulator channel.

FIG. 16 is a graph which illustrates the noise power output of an augmented correlating demodulator channel as a function of its input bandwidth.

FIG. 17 is a functional block diagram which illustrates a practical augmented correlating demodulator for QMSK and M-QMSK signals.

FIG. 18 is a functional block diagram which illustrates the generation of clock waveforms and phase-modulated carrier waveforms used to construct the zero-ISI embodiments of QMSK and M-QMSK signals.

FIG. 19 is a graph which illustrates the equivalent baseband power density function of the zero-ISI embodiment of the QMSK signal.

FIG. 20 is a graph which illustrates probability of bit error as a function of the ratio (E_b/N_o) for QMSK and M-QMSK signals and their zero-ISI embodiments. FIG. 21 is a graph which illustrates the transmission efficiencies of QMSK and M-QMSK signals and their zero-ISI embodiments; for comparison, the transmission efficiencies of certain prior art signaling methods are also illustrated.

FIG. 22 is a functional block diagram which illustrates both a QMSK and M-QMSK correlating demodulator and generation of signals jointly required as inputs to a maximum-likelihood phase detector and a maximum-likelihood, phase-coherent AGC detector.

FIG. 23 is a functional block diagram which illustrates a maximum-likelihood phase detector for QMSK and M-QMSK signals.

FIG. 24 is a functional block diagram which illustrates phase-coherent recovery

(re-generation) of waveforms required for successful detection of QMSK and M-QMSK signals.

FIG. 25 is a timing diagram which illustrates sample and dump control waveforms recovered for application to the QMSK and M-QMSK correlating demodulator.

FIG. 26 is a functional block diagram which illustrates a maximum-likelihood, phase-coherent AGC detector for QMSK and M-QMSK signals.

FIG. 27 is a functional block diagram which illustrates a receiver for QMSK and M-QMSK signals.

FIG. 28 is a functional block diagram which illustrates a direct sequence, spread-spectrum QMSK modulator.

FIG. 29 is a functional block diagram which illustrates the despreading of a direct sequence, spread-spectrum QMSK signal.

DETAILED DESCRIPTION OF THE INVENTION

Detailed Description Outline

A. Signal Definition

B. Signal Envelope: M-QMSK & QMSK(CE)

C. Intermodulation Products

D. Average Signal Power

E. Power Density Spectrum F. Signal Geometry

G. Signal Detection

H. Band-Limited Signal

I. Zero-ISI Signal

J. Error Probability

K. Transmission Efficiency

L. Signal Recovery

M. Spread-Spectrum QMSK

N. Baseband Signaling

O. Summary of Some of the Benefits of the Present Invention

A. SIGNAL DEFINITION

As discussed above, the present invention defines methods of digital data transmission wherein a set of appropriately-timed digital data streams modulates a multi-dimensional, non-orthogonal signal basis set. The modulated signal basis set components are combined to provide a composite modulated signal for transmission.

The composite modulated signals of the present invention are described in their most general form by the canonical equations

during the m^th symbol interval. In a particular embodiment of the invention, the basic symbol waveforms p_n(t) are defined by the equations

ρ₁(t) = ρ₃(t) = cos(ω_ot) -T ≤ t ≤ +T (6a) p₂(t) = p₄(t) = sin(ω_ot) 0 ≤ t ≤ 2T (6b) ω_o = π/T_s = π/2T = 2πf_o. (6c)

This particular set of basic symbol waveforms, which exemplifies the staggered nature of all such sets of basic symbol waveforms relevant to the invention, is illustrated in

Figure 4. T_s = 2T is the symbol interval common to the basic symbol waveform set. R_s = 1/T_s = 2f_o, in units of symbols per second, is the common symbol rate. The quadrature carrier waveform set [c_n(t)] is illustratively defined by the equations C₁(t) = C₄(t) = cos(ω_ct) (7a)

C₂(t) = C₃(t) = sin(ω_ct) (7b) ω_c = 2πf_c, (7c) where f_c is the fundamental carrier frequency (signal center frequency) common to all carrier waveforms. The signal basis set [v_n(t,m,T_s)] is defined by the product equations v_n(t,m,T_s) = p_n(t-mT_s)c_n(t) 1≤n≤N = 4. (8)

The coefficients C_mn of Equation (5) may equivalently be defined as symbol information coefficients or as signal basis set information coefficients. When annotated as (C_mn), they are defined as symbol or basis set information sequences.

The information sequences (C_mn) are time-coincident with their associated symbol waveforms. Figure 5 illustrates their timing relationships near the origin. In most general form, C_mn = B_mnA_mn, (9) where each B_mn is a one-bit antipodal (± 1) coefficient, and where each A_mn is a discrete-valued amplitude coefficient whose levels are determined by an additional k data bits. Thus, during the m^th symbol interval, each information coefficient C_mn carries (k+1) data bits. Four such C_mn are effective during each (staggered) symbol interval, so the total signal data bit rate, in units of bits per second, is defined by R_b = N(k+1)R_s = 4(k+1)R_s = log₂(M)R_s. (10) M is the number of signal states possible during a given (staggered) symbol interval.

Figure 6 illustrates a method for generating the sequences (C_mn) from an input serial data stream of bit rate R_b. The Gray encoder of Figure 6 ensures that a one-bit data state change results in an output signal state which is geometrically adjacent to the original signal state. The symbol waveform set defined by Equation (6) can be modified to allow the polarity of each symbol waveform to alternate in adjacent symbol intervals. The modified symbol waveform set no longer has an explicit functional dependence on the parameter ml_s, and is defined by the equations ρ₁(t) = ρ₃(t) = cos(ω_ot) (2m-1)T≤ t≤ ( 2m+ 1)T (11 a) ρ₂(t) = ρ₄(t) = sin(ω_ot) 2mT≤t≤ (2m +2)1 (11b) ω_o = π/T_s = π/2T = 2πf_o. (11c)

The corresponding modified signal basis set waveforms [v_n(t)] also lose their functional dependence on mT_s, and are defined by the equations v₁(t) = cos(ω_ot)cos(ω_ct) = (1/2) [cos(ω_Lt)+cos(ω_Ht)] (12a) v₂(t) = sin(ω_ot)sin(ω_ct) = (1/2)[cos(ω_Lt)-cos[ω_Ht)] (12b) v₃(t) = cos(ω_ot)sin(ω_ct) = (1/2)[sin(ω_Ht)+sin(ω_Lt)] (12c) v₄(t) = sin(ω_ot)cos(ω_ct) = (1/2)[sin(ω_Ht)-sin(ω_Lt)] (12d) ω_H = (ω_c+ ω_o) ; ω_L = (ω_c-ω_o). (12e) The composite signal which results from summing the independently-modulated components or members of this particular signal basis set can be defined in canonical form by the equation

The information coefficients C_mn can be restricted to the form C_mn = B_m A, (14) where each B_mn is again a one-bit antipodal coefficient and where A is a constant which is functionally related to signal power. Thus restricted, the C_mn contain no information in the form of intentional amplitude variation. The signal s_m(t) of Equation (13), when it incorporates both the modified signal basis set of Equation (12) and the restricted information coefficients C_mn of Equation (14), can be expanded by appropriate substitution and expressed in the form s_m(t) = A[B_m1cos(ω_ot)cos(ω_ct)+B_m2sin(ω_ot)sin(ω_ct)] (15) + A[B_m3cos(ω_ot)sin(ω_ct)+B_m4sin(ω_ot)cos(ω_ct)].

The first bracketed term of Equation (15) can individually be identified as s_ml(t) = A[B_m1cos(ω_ot)cos(ω_ct)+B_m2sin(ω_ot)sin(ω_ct)] (16a)

= AB_m1cos(ω_ct-B_m1B_m2ω_ot). (16b)

The second bracketed term of Equation (15) can individually be identified as s_mQ(t) = A[B_m3cos(ω_ot)sin(ω_ct)+B_m4sin(ω_ot)cos(ω_ct)] (17a)

= AB_m3sin(ω_ct+B_m3B_m4ω_ot). (17b)

Equations 16(b) and (17b) are obtained from Equations 16(a) and 17(a), respectively, by using trigonometric identities and the fact that B² _mn = +7 for any (m,n).

In the form of Equation (16b), the signal s_ml(t) is understood to represent an MSK modulation of an in-phase (cosinusoidal) carrier waveform of center frequency f_c. The signal s_mQ(t), in the form of Equation (17b), is understood to represent an independent, though time-synchronous, MSK modulation of a quadrature (sinusoidal) carrier waveform of center frequency f_c. Thus the signal defined by s_m(t) = AB_m1cos(ω_ct-B_m1B_m2ω_ot) + AB_m3sin(ω_ct+B_m3B_m4ω_ot) (18a) = s_ml(t) + s_mQ(t) (18b) is understood to be indistinguishable from the sum of two synchronized, but otherwise independent, MSK signals, each of whose carrier waveforms have identical center frequency f_c and a quadrature phase relationship with respect to that of the other. In this context, the signal s_m(t) of Equation (13), when it incorporates both the modified signal basis set of Equation (12) and the restricted information coefficients C_mn of Equation (14), is henceforth designated to be a Quadrature Minimum-Shift Keyed (QMSK) signal of the present invention.

The canonical structure of the QMSK signal s_m(t) represented by Equation (13) suggests a canonical signal generation method such as that illustrated in Figure 7. Equivalently, the dual synchronized MSK structure of the QMSK signal s_m(t) represented by Equation (18) suggests a signal generation method such as that exemplified by the alternate QMSK modulator illustrated in Figure 8.

The QMSK signal, like its individual MSK components, is a continuous-phase signal. As such, phase coherence of its waveform components is important. Figure 9 illustrates a method for generating the phase-coherent clock waveforms and QMSK carrier waveforms required to implement the alternate QMSK modulator illustrated in Figure 8. Note that in Figure 9, the signal center frequency is implicitly defined as f_c = (2r+1)f_o , r≥ 1 , (19) where r is a positive integer. With this assignment, the higher QMSK shift frequency f_H is given by f_H = (f_c+f_o) = (2r+2)f_o , (20) and the lower QMSK shift frequency f_L is given by f_L = (f_c-f_o) = 2rf_o. (21)

For any assignment of center frequency, the frequency difference Δf = (f_H-f_L) = 2f_o = 1/T_s (22) is maintained in order to provide phase continuity. B. SIGNAL ENVELOPE: M-QMSK & QMSK(CE)

As shown by the identities of Equation (12), the output QMSK carrier waveforms of Figure 9 can be combined by a method such as that illustrated in Figure 10 to provide the QMSK signal basis set waveforms. The signal s_m(t) of Equation (13), when it incorporates both the QMSK signal basis set of Equation (12) and the generalized information coefficients C_mn of Equation (9), is henceforth designated to be an M-ary Quadrature Minimum-Shift Keyed (M-QMSK) signal. The canonical structure of the M-QMSK signal s_m(t) represented by Equation (13) suggests that it can be generated by any method which is functionally equivalent to the canonical M-QMSK modulator illustrated in Figure 7.

As briefly discussed above, each member of the QMSK signal basis set can be jointly modulated by an appropriately-timed, independent, antipodal data stream and a sequence of discrete amplitude levels determined by other independent data streams which are in time-coincidence with the antipodal data stream. Summing a QMSK signal basis set modulated as so described provides a composite M-ary QMSK

(M-QMSK) signal which has improved bandwidth efficiency. Appropriately bandlimited M-QMSK signaling achieves bandwidth efficiency equal to 2(k+1), where k is the number of amplitude bits carried by each member of the QMSK signal basis set. Its power efficiency, when demodulated in accordance with other aspects of the invention, is consistent with that of M-MSK when identical numbers of amplitude bits are carried by their respective signal basis components. For identical bandwidth efficiency, the M-QMSK signal carries fewer amplitude bits on each of its four signal basis components than does the M-MSK signal on each of its two signal basis components. Thus for identical bandwidth efficiency, M-QMSK power efficiency is dramatically improved with respect to that of M-MSK.

The M-QMSK signal, whose symbol information coefficients C_mn contain

information in the form of intentional discrete amplitude variation, is easily understood to have a time-varying signal envelope. The QMSK signal, even though its C_mn are antipodal, also has a non-constant signal envelope. The QMSK signal s_m(t) expressed in the form of Equation (15) can also be expressed in the equivalent form s_m(t) = A[B_m1cos(ω_ot)+B_m4sin(ω_ot)]cos(ω_ct) (23a) +A[B_rn3cos(ω_ot)+B_m2sin(ω_ot)]sin(ω_ct)

= l(t)cos(ω_ct) + Q(t)sin(ω_ct) , (23b) where l(t) represents the multiplier of cos(ω_ct) and where Q(t) represents the multiplier of sin(ω_ct). The squared envelope function of the QMSK signal is then defined by

E²(t) = l²(t) + Q²(t) (24a)

= A²[2+(B_m1B_m4+B_m2B_m3)sin(2ω_ot)]. (24b)

For completely independent sequences (B_mn), Equation (24b) shows that the QMSK signal envelope is time-dependent, and is therefore non-constant. However, one-half of the sixteen possible QMSK signal states result in a constant envelope value equal to

A√2

Equation (24b), which illustrates the non-constant envelope of the QMSK signal, also suggests a method by which it can be modified to provide a signal having constant envelope. As also briefly discussed above, when the QMSK signal basis set is modulated by appropriately-timed antipodal data streams, three of which are independent and the fourth of which is dependent on the first three, a constant-envelope QMSK(CE) signaling method is provided. The method consists of

guaranteeing that the antipodal B_mn always obey the constraining equation B_m1B_m4 + B_m2B_m3 = 0 . (25) Multiplying both sides of Equation (25) by the quantity B_m2B_m4, one obtains the equivalent constraining equation B_m1B_m2 + B_m3B_m4 = 0 , (26) which can also be expressed as -B_m1B_m2 = B_m3B_m4. (27)

Substituting the results of Equation (27) into the QMSK signal expression of

Equation (18a) shows that the constant-envelope QMSK signal is simply the sum of two synchronized, quadrature MSK signals, each of which always has instantaneous frequency f_i equal to that of the other. The joint instantaneous frequency is given by f_i = (f_c-B_m1B_m2f_o) = (f_c+B_m1B_m4f_o) . (28)

The QMSK signal whose B_mn are constrained to obey Equation (25) is henceforth designated to be a Quadrature Minimum-Shift Keyed (Constant Envelope) signal, and is assigned the acronym QMSK(CE). Equation (25) can also be expressed in the form B_m4 = -B_m1B_m2B_m3 , (29) which shows that, for the QMSK(CE) signal, only three of the original four antipodal data bits remain independent. Thus the bit rate of QMSK(CE) is reduced with respect to that of QMSK, and is given by

R_b[QMSK(CE)] = (3/4)R_b[QMSK] = 3R_s. (30) This is the penalty paid for modifying a QMSK signal to have constant envelope.

Subject to the constraint of Equation (25), the QMSK(CE) signal can be generated by any method which is functionally equivalent either to the canonical QMSK modulator illustrated in Figure 7 or the alternate QMSK modulator illustrated in Figure 8.

C. INTERMODULATION PRODUCTS

The QMSK signal, because of its composition as the sum of two fully-independent, though synchronous, MSK signals, can generate intermodulation products when processed in a non-linear channel. It is desirable to avoid low-order intermodulation products. The QMSK(CE) signal, whose dual MSK components always have identical instantaneous frequency, cannot generate such products when non-lineariy processed. Assuming a non-linear channel, QMSK intermodulation products occur only when

B_m1B_m4 + B_m2B_m3≠ 0. (31 )

Thus on an average basis, for IID (independent, identically-distributed) sequences (B_mn), these products occur 50% of the time. It will be shown later in this discussion that

W = 2R_s = 4f₀ (32) is a reasonable channel bandwidth in which to process the QMSK signal.

Having defined W = 4f₀ as the signal processing bandwidth of interest, a simple analysis allows the prediction of the order of instantaneous, in-band intermodulation products. In this analysis, in-band products are considered to occur when they fall within the signal processing frequency band whose lower and upper band-edge frequencies (f ,f₊) are defined as f- = (f_c-W/2 ) = (f_c-2f₀) = (2r-1)f₀, (33) f₊ = (f_c+W/2) = (f_c+2f₀) = (2r+3)f₀. (34)

In Equations (33) and (34), the signal center frequency has again been defined to be f_c = (2r+1)f₀. Recall that the lower and higher shift frequencies (f_L,f_H) of the QMSK signal have previously been defined by Equations (20) and (21).

Let (i,j) be arbitrary integers. QMSK intermodulation products thus occur when if_L + jf_H≤f₊ , (35) and when if_L + jf_H≥ f-. (36) Making appropriate substitutions, the inequalities

2irf₀ + 2j(r+1)f₀ ≤ (2r+3)f₀, (37) 2irf₀+ 2j(r+1)f₀ ≥ (2r-1)f₀ (38) can be derived from the inequalities (35) and (36), respectively. Cancelling common factors and combining the remaining terms, inequalities (37) and (38) become i + [(r+1)/r]j ≤ [1 +(3/2r)], (39) i + [(r+1)/r]j ≥ [1-(1/2r)], (40) respectively. Inequalities (39) and (40) can be solved simultaneously, giving two sets of integer values (i₁,j₁), (i₂,j₂) for each specific integer value of the parameter r. Low-order solutions for given values of r are tabulated in Table 1.

TABLE 1

Table 1 supports the conclusion that the QMSK signal center frequency f_c = [r + (1/2)]R_s (41) should be large compared to the symbol rate R_s in order to avoid low-order intermodulation products.

D. AVERAGE SIGNAL POWER

M-QMSK average signal power will be shown to depend on the quantity

C² = E(A² _mn) , (42) where E ( · ) is the expectation operator. In order to obtain computationally-useful results, it is necessary to specify a particular method by which the amplitude factors A_mn of the generalized information coefficients C_mn of Equation (9) can be assigned their discrete amplitude levels. The particular method specified herein consists of requiring that each A_mn, during each appropriate symbol interval, be selected from the discrete amplitude level set defined by

A_mn = A,3A,5A,7A...,JA; J = [2^(k+1) - 1]. (43)

With this assignment, the parameter k defines the number of data bits carried by each amplitude coefficient A_mn. Thus, during each appropriate symbol period, each A_mn can occupy one of 2^k possible amplitude states. This particular assignment is valid for both M-QMSK and QMSK (for which k = 0) signals, and is known in the prior art to allow efficient signal detection.

The following derivations are made for the generalized M-QMSK signal; all results also apply to the QMSK signal when k = 0. For odd, positive values of the integer index j [see Equation (43)],

For odd, positive integers j, it can be shown that

Substituting J from Equation (43) into Equation (45), one obtains

Substituting Equation (46b) into Equation (44), one obtains

C² = E(A² _mn) = (A²/3) [2^(2k+2)-1] = (A²/3)[4^(k+1)-1). (47)

Thus C² = E(A² _mn) is independent of the indices (m,n), and in fact only depends on the parameters A and k.

Average M-QMSK signal power P is defined by the equivalent expressions

For antipodal (±1) B_mn, one can show that E(B_miB_mj) = d_ij, (49) where d_ij is defined by d_ij = 1 i = j (50) d_ij = 0 i ≠ j.

Thus Equation (48d) reduces to

For the M-QMSK signal basis set of Equation (12), it is easy to show that

Thus substituting the results of Equation (52) into Equation (51), one finally obtains

These results will be used to advantage in succeeding sections of the detailed description. E. POWER DENSITY SPECTRUM

During the m^th symbol interval, using Equations (5), (6), and (7), the M-QMSK signal can be expressed as s_m(t) = [C_m1p₁(t-mT_s)+C_m4p₄(t-mT_s)]cos(ω_ct) (54)

+ [C_m3p₃(t-mT_s)+C_m2p₂(t-mT_s))sin(ω_ct) . From Equation (6), p₁(t) = p₃(t) = cos(ω_ot), and p₂(t) = p₄(t) = sin(ω_ot). Since p₁(t-T) = cos[ω_o(t-T)] = cos(ω_ot-π/2) = sin(ω_ot), it is evident that Equation (54) can also be expressed as s_m(t) = [C_m1p₁(t-2mT)+C_m4p₁(t-[2m+1]T)]cos(ω_ct) (55a)

+[C_m3p₁(t-2mT)+C_m2p₁(t-[2m+1] T)}sin(ω_ct) = I_m(t)cos(ω_ct) + Q_m(t)sin(ω_ct) . (55b)

Define the truncated signal f_T' (t) by the equations

Let (7'/2) correspond to Ll_s, such that

T' = 2LT_s . (57)

The Fourier transform of x₁(t) is given by

The Fourier transform of y₁(t) is given by

The Fourier transform of f_τ, (t) is given by

F_T'(ω) = (1l2)[X₁(ω₊)+X₁(ω_{_} )] + (i/2)[Y₁(ω₊)-(Y₁ω_{_} )] , (60) where ω₊ = (ω+ ω_c) , ω_{_} = (ω-ω_c) . (61 )

The complex conjugate of F_τ' (ω) is given by

F_τ'(ω)F*_T (ω)

The product is given by

In the derivation of Equation (63) it has been assumed that individual spectral components centered at ω₊ and ω_, respectively, do not overlap.

From Equations (49) and (50), one can derive

E (C_mnC_ij) = C²d_mid_nj..

(64)

Thus from Equations (58) and (59) one can show that

and that

Using Equations (63), (65), and (66), one obtains

A formal definition of the M-QMSK power spectral density function is given by

Noting from Equation (57) that L approaches infinity as T' approaches infinity, and substituting Equation (67) into Equation (68), one obtains

Thus the M-QMSK equivalent low-pass power density function is

The Fourier transform of the basic QMSK symbol waveform

is given by

(72a) /

For the QMSK signal,

(73)

T_s = log₂(M)T_b = 4T_b . Substituting ω = 2πf and Equation (73) into Equation (72b), one obtains

Since P₁(f) is real, one obtains

For the QMSK signal (k=0),

Thus substituting Equations (73), (75), and (76) into Equation (70), one obtains

Converting to the normalized frequency variable

(78) x = f/R_b = fT_b , and noting that, in general,

(79)

S(x) = R_bS(f) , one finally obtains

The QMSK equivalent low-pass power density function of Equation (80), normalized to A² = 1 and plotted as a function of the normalized frequency variable of Equation (78), is illustrated in Figure 11. MSK and QMSK signals each utilize the basic symbol waveform defined in Equation (71). When each signal is band-limited to an RF bandwidth defined by

W = 2R_s , (81) both the MSK and QMSK signals retain approximately 96.9% of their total power. From Equation (81), the RF (radio frequency) half-bandwidth (equivalent low-pass bandwidth) is given by ∆f = WI2 = R_s . (82)

The normalized equivalent low-pass bandwidth is

Δx =∆f/R_b = 1/[log₂(M)] . (83)

Thus for MSK (see Figure 3), M = 4 and one computes (∆x)_MSK = 1/2 = 0.5 , (84) and for QMSK (see Figure 11), M = 16 and one computes (∆x)_MSK = 1/4 = 0.25 . (85)

F. SIGNAL GEOMETRY

In analogy to the scalar product of two arbitrary vectors

and , define the

scalar product of two arbitrary real signals x_i(t) and xdt) to be

The vectors

and are defined to be orthogonal when their scalar product is

identically equal to zero. Analogously, two signals x_i(t) and x_j(t) are defined to be orthogonal when (X_i, X_j ) = 0 . (87)

For the QMSK signal basis set of Equation (8), with its basic symbol waveforms defined by Equation (6) and with its quadrature carrier waveforms defined by

Equation (7), it can be shown that

(88a) (v₁,v₁) = (v₂,v₂) = (v₃,v₃) = (v₄,v₄) = T_s/4 , (88b)

(v₁,v₂) = (v₁,v₃) = (v₂,v₄) = (v₃, v₄) = 0 , (v₁,v₄) = (v₂,v₃) = T_s/2 π . (88C)

The individual equations of Equation (88) are each satisfied for any carrier frequency f_c which satisfies Equation (19).

The non-zero result of Equation (88c) shows that the QMSK signal basis set is not orthogonal. Equation (88b) shows that the vector signal basis set components v₁ and v₄ are each orthogonal to both components v₂ and v₃. Since v₁ and v₄ are not identical (i.e., parallel in the vector signal space), they can each be orthogonal to the plane defined by v₂ and v₃ only if the signal space dimension is greater than three. Equation (88) is fully compatible with a signal space of dimension N = 4. Though the signal basis set pairs [ιv₁(t),v₄(t)] and [v₂(t),v₃(t)] are not individually orthogonal, each such pair defines and occupies a distinct hyperplane in a four-dimensional vector signal hyperspace, and these distinct hyperplanes are themselves orthogonal. The non-orthogonality of the QMSK signal basis set implies that QMSK signals cannot be detected successfully by a conventional correlating demodulator, in which successful detection occurs only when a particular signal basis set satisfies (V_i,V_j) = Kd_ij ; i,j = 1,2,...,N . (89)

G. SIGNAL DETECTION

The previous description of the geometry of the QMSK signal basis set in accordance with the present invention showed that this set is not conventionally orthogonal. Thus, the present invention provides a non-conventional, augmented correlating demodulator which enables power-efficient detection of the

multi-dimensional, non-orthogonal signals. Also, this invention provides band-limited, transmission-efficient digital signaling methods wherein inter-symbol interference (ISI) is inhibited through use of preferred, non-orthogonal symbol waveforms in conjunction with the augmented correlating demodulator. Moreover, this invention provides maximum-likelihood reconstruction of waveforms required by the augmented correlating demodulator for power-efficient signal detection. In addition, this invention provides maximum-likelihood construction of a phase-coherent, automatic gain control signal used for leveling the signal power input to the augmented correlating demodulator. Furthermore, the present invention provides multi-dimensional direct-sequence, spread-spectrum signaling methods wherein the underlying data rate can be significantly increased. Below, the details of these features of the present invention are explained in detail.

Equations (88c), (6), and (7) illustrate the cause of the non-orthogonality. Recall that for conventional signal basis sets which have staggered symbol waveforms, orthogonality is required over each joint half-symbol interval. Signal basis set components v₁(t) and v₄(t) have identical carrier waveforms and non-orthogonal symbol waveforms. The same statement is also true of the signal basis set components v₂(t) and v₃(t). Equation (88b) shows that such QMSK signal basis set orthogonality as does exist is due entirely to that provided by the quadrature carrier waveforms. The non-orthogonality of the identical symbol waveform pairs [p_r(t),p₄(t)] and [p₃(t),p₂(t)], over a particular joint half-symbol interval, is illustrated in Figure 12A. From this figure it is easily understood that the basic QMSK symbol waveform product P₁(t)p₂(t) is not orthogonal (i.e., does not integrate to zero) over any joint half-symbol interval.

Figure 12A, which illustrates the non-orthogonality of the basic, staggered QMSK symbol waveforms there defined as p₁(t) = cos(ω_ot) and p₂(t) = sin(ω_ot), also suggests a method by which QMSK symbol orthogonality can be accomplished. The method consists of multiplying the symbol waveform product p₁(t)p₂(t) by any third waveform x(t) which is an odd function with respect to the mid-point of each joint half-symbol interval of length T = T_s/2. Among the class of waveforms [x(t)] which satisfy this definition, the waveform w(t) = cos(2ω_ot) (90) is harmonically the most simple, and is illustrated in Figure 12B. Figure 12C illustrates that the product p₁(t)p₂(t) w(t) = (1/4)sin(4ω_ot) (91) does in fact integrate to zero over any joint half-symbol interval of length T. The set

[x(t)], of which w(t) is a member, is thus identified to be a set of weight functions with respect to the basic QMSK non-orthogonal symbol waveform pair. The general concept of orthogonality with respect to weight functions is well-known in

mathematics. Because it is harmonically the most simple, the particular weight function defined by Equation (90), with the possible exception of its polarity, is here identified as the preferred weight function for any QMSK symbol waveform set.

Let the preferred QMSK weight function set be defined by

W_n(t) = (-1)^(n-1)W(t) = (-1)^(n-1)cos(2ω_ot) , (92a) n = 7,2,...,N=4 , (92b)

(92c) ω_o = π/T_s = π/2T . Using standard trigonometric identities it can be shown that, for appropriate choice of carrier frequency f_c, and for either the specific QMSK signal basis set of Equation (8) or the specific QMSK signal basis set of Equation (12),

The individual integrations of Equation (93) are understood to be performed over appropriate (staggered) symbol intervals. Thus from Equation (93), the weighted QMSK signal basis sets v_iW(t) = vi(t)Wi(t) , i = 1,2,...,N=4 , (94) are understood to provide a method by which QMSK-based signals can successfully be detected. The method consists of detecting QMSK-based signals with a weighted correlating demodulator exemplified by the parallel operations

It is important to note that, because the product waveform v_iw(t) decomposes by trigonometric identities into various algebraic sums of constituent waveforms, and because the order of terms in a product waveform can be changed without changing results, eaghted correlating demodulator operations indicated by Equation (95) can be performed in various equivalent configurations and in various equivalent orders.

A specific functional block diagram of the f channel of such a weighted correlating demodulator is illustrated in Figure 13. The peak signal output power of such a channel is given by

(96) P_s = V ² _mi = [ (1/8)C_miT_s ]² , i = 1,2,...,N=4. The weighted correlating demodulator defined by Equation (95) does successfully detect QMSK signals, but it does not provide antipodal output signal-to-noise ratio in the presence of AWGN (additive white Gaussian noise). Let n(t) of Figure 13 be an AWGN signal with two-sided power spectral density S_n(f) = N_o/2 . (97)

Its autocorrelation function is given by

R_n (t-x) = E [n (t)n (x)] = (N_o/2) δ(t-x) . ⁽⁹⁸⁾

Let N_i, given by

be the integrated, sampled output noise voltage of the i^th channel of the weighted correlating demodulator. The corresponding average output noise power is defined by

\ j

For either the specific QMSK signal basis set of Equation (8) or that of Equation (12), it can be shown that

so Equation (100d) reduces to

P_no = N_oT_s /16 , (102) which is the average output noise power of any weighted correlating demodulator channel due to input AWGN.

The weighted correlating demodulator channel output signal-to-noise ratio (SNR) for QMSK signals can now be given as

For QMSK signals,

is given by

Substituting Equation (104) into Equation (103c), one obtains the weighted correlating demodulator output signal-to-noise ratio (Ps /P_no )_QMSK = E_b/N_o . ⁽¹⁰⁵⁾

It is known in the art that conventional correlating detection of antipodal signals achieves an ideal output signal-to-noise ratio

(P_s/P_no ) = 2E_b/ N_o , ^{(1 06)} so the weighted correlating detection of QMSK signals provides an output

signal-to-noise ratio (SNR) which is inferior to that of conventional correlating detection of antipodal signals by 3 dB. A better signal-to-noise ratio is preferable. It will now be shown that QMSK signals can be detected with essentially antipodal output signal-to-noise ratio if the weighted correlating demodulator is preceded by a band-pass filter of pre-determined bandwidth. The cascade of an input band-pass filter of specified bandwidth followed by a weighted correlating demodulator is henceforth designated to be either an augmented correlating demodulator or, more simply, an augmented demodulator. A functional block diagram of an i^th channel of such an augmented demodulator is illustrated in Figure 14.

Let the input band-pass filter of Figure 14 have an ideal transfer function defined by

where u(.) is the unit step function. As before, let the input signal n(t) be AWGN with power spectral density given by Equation (97). The band-pass filtered noise signal n₁(t) has power spectral density given by

which is illustrated in Figure 15A. In Figure 14, v_i(t)w_i(t) represents any appropriate weighted QMSK signal basis set member. For random signals x(t) having power spectral density S_x(f), the identical power spectral densities of x(t)cos(ω_at) and x(t)sin(ω_at) are given by ~

By repeated application of Equation (109), one can show that the low-pass component of the power spectral density of the signal (110) n₂(t) =n₁(t)v_i(t)W_i(t) i = 1,2,.., N=A is given by

Substituting the specific value

(112)

Δf = 2f_o into Equation (111), one obtains

This specific power spectral density function is illustrated in Figure 15B.

The combined integrator and sample/hold (S/H) function block of Figure 14 has a transfer function H_ID(f) whose squared absolute value is given by

The output noise power of the augmented demodulator channel of Figure 14 is given by

Due to the rapid frequency roll-off of

, for large f_c one can ignore the band-pass components of

centered near

and make the equivalent computation

For the specific value of Δf given by Equation (112), one computes

Using the identity

and understanding from Figure 15B that the maximum value of

is given by

Equation (117) allows us to establish the upper bound

For

, the input RF bandwidth is

Band-pass filtering the input signal

yields a filtered output signal

From previous results we know that, filtered to an RF bandwidth

, the

QMSK signal retains approximately 96.9% of its original signal power. Let the augmented demodulator channel output signal power be given by

where the constant C₁ must be determined for augmented detection of M-QMSK signals. It will soon be shown that, for the RF bandwidth given by Equation (121 ),

C₁ ≅ 1/8 .

(123) 0n this basis, using Equations (104), (120), and (122), augmented detection of QMSK signals provides the output signal-to-noise ratio

Thus it is understood that, for properly-chosen RF bandwidth, augmented detection of QMSK signals provides essentially antipodal output signal-to-noise ratio.

For arbitrary values of Δf, the augmented demodulator channel output noise power

has been computed by substituting Equation (111 ) into Equation (116b) and performing numerical integration. The result, computed and plotted as a function of the ratio

, is illustrated in Figure 16. For

, Figure 16 indicates that the actual output noise power is

Thus the use of the upper bound of Equation (120) gives a slightly pessimistic result in Equation (124). Note that for large

asymptotically approaches

the value

previously obtained [see Equation (102)] for the weighted demodulator. One concludes from this discussion that the augmented correlating demodulator, when its input band-pass filter is properly specified, can provide essentially antipodal detection of QMSK signals. A functional block diagram of a practical augmented demodulator for QMSK-based signals is illustrated in Figure 17. Means for recovering the weighted QMSK signal basis set from an M-QMSK signal in the presence of AWGN are provided in a later section of the detailed description.

H. BAND-LIMITED SIGNAL

The fundamental QMSK symbol waveform

of Equation (6a) can be expanded in a trigonometric Fourier series and expressed as

The Fourier coefficients A_r of this expansion are given by

Thus p₁(t) can be expressed as

If p,(f) is filtered by an ideal low-pass filter having a cut-off frequency just slightly greater than 2f_o, the band-limited fundamental symbol waveform p_1LP(t) is given by

(129b) a = 2/π ; b = 2/3 . The filter response to the staggered fundamental symbol waveform

is given by

The band-limited QMSK symbol waveform set can now be expressed as

The band-pass filtered QMSK signal basis set, band-limited to the RF bandwidth W of Equation (121), can then be expressed as

where the carrier waveforms c_n(t) are given by Equation (7). Thus band-pass filtering an M-QMSK signal s_m(t) to the RF band-width W of Equation (121) yields the band-limited M-QMSK signal

Referring to the augmented demodulator block diagram of Figure 17, let the sum of the band-pass filtered M-QMSK signal and the band-pass filtered AWGN signal be given by

The band-limited M-QMSK signal yjt), in the presence of the band-limited noise signal n₁(t), is detected by the parallel operations 36)

The low-pass filters (LPF) of Figure 17 allow us to ignore first multiplier products centered near 2f_c in the subsequent evaluation of Equation (136). The low-pass filter cut-off frequency f_m is not critical provided that it is greater than 2f_o; it could, for example, be specified as

Making appropriate substitutions from Equations (7), (12), (92), (132), and (133), it can be shown that

( )

Thus the approximation of Equation (123) gives a pessimistic (smaller) value for C₁ It can further be shown that

Because the low-pass filters eliminate the double-frequency components centered near 2f_c, it can yet further be shown that

Using Equations (134), (138), (139), and (140), it can now be shown that

From Equations (134) and (138), the terms v_mJ of Equation (141) can be identified as

Thus the terms v_mj represent the detected signal information components. From Equations (134) and (139), and from an understanding of the staggered nature of the symbol information coefficients C_mn (see Figure 5), the terms x_mj of Equation (141) can be identified as

( )

X_m1 = C₂T_S [C_{(m-1 )4} + C_m4] , X_m2 = C₂T_S [C_m3 + C_{(m+1 )3} ] ,

(143b) (143c)

X_m3 = C₂T_s [C_{(m- 1)2} + C_m2 ] , (143d)

X_m4 = C₂T_s [C_m1 + C_(m+1)1 ] . The terms X_mj are thus understood to represent inter-symbol interference (ISI) components introduced by band-limiting the input M-QMSK signal s_m(t). Let N_j, given by

be the integrated, sampled output noise voltage of the j^th channel of the augmented demodulator. Using the upper bound of Equation (120), the output noise power of each such channel due to input AWGN is given by

P_no = E (N_lN_j)≤ N_oT_s/32 . (145)

The individual z_mj of Equation (136) can now be expressed as

(146) z_mj = v_mj + x_mj + N_j ; j = 1,2, ...,N =4 . The integrated, sampled outputs z_mj of each channel thus consist of the desired signal components v_mj distorted both by ISI components x_mj and Gaussian noise components

For random symbol information sequences (C_mn), the ISI components x_mn of Equation (143) are each random, bounded variables. Their effect on signal detection statistics is best modeled by computing their variance (effective average power). From

Equation (143a), X_i1 = C₂T_S[C_(i-1)4 + C_i4] ^; (147a) X_{j 1} = C₂T_S [C_(j-1)4 + C_j4] ^. ( 147b)

The product x_i1x_j1 is given by (148)

X_i1X_j1 = (C₂T_s)² [C_(i-1)4C(_{j-1 )4} + C_{(i-1 )4}C_j4]

+ (C_cT_s) ²[ C_i4C_(j-1)4 + C_i4C_j4] . Using Equations (47), (49), and (148), it can be shown that the variance of x_m1 is given by

Although Equation (149) was derived for a particular channel of the augmented demodulator, it is easily shown that the result is valid for each such channel. Because x_mj and N_j are independent random variables, their combined effect on signal detection statistics can be determined by adding their individual average powers. Thus the total combined power P_rT of these random variables is given by

P_rT = P_no + P_ISI = P_no (1 + P_ISI /P_no) .

Substituting Equation (149) and the upper bound of Equation (120) into Equation (150), one obtains

In analogy to Equation (51), the average power of the band-limited M-QMSK

signal y_m(t) of Equation (134) is given by

Substituting v_nBP(t) of Equation (133) for v_n(t) in Equation (52), it can be shown that

(153b) = K₁ = 22/9h² ≅0.24767.

Substituting Equation (153b) into Equation (152), one obtains

(154b)

= 4(k+1)E_bR_s . Thus one derives

Substituting Equations (155), (153b), and (139c) into Equation (151), one obtains

(156b)

≅ (N_OT_S/32)[1 + (1.31×10^-3)(k+1)(E_b/N_o)] .

Thus P_Isl and P_rT increase directly with k, the number of amplitude bits carried by each signal basis set component of the M-QMSK signal. The total random signal power P_rT of Equation (156) will later be used in the computation of bit error probabilities P_b(e) of M-QMSK signals. I. ZERO-ISI SIGNAL

From the previous discussion it is understood that band-limiting the QMSK signal, while necessary in order to obtain essentially antipodal detection, also causes undesirable ISI components to be present in the output of each augmented

demodulator channel. The ISI components are the result of signal geometry distortion caused by band-limiting. Equation (143) shows that the ISI components are each proportional to the constant C₂ derived in Equation (139). Equation (139c), however, also suggests a method by which the ISI components can be eliminated. Referring to Equations (126), (128), and (129), postulate a fundamental QMSK symbol waveform q₁(t) having a band-limited Fourier expansion given by

(157a) q_1LP (t) = a₀ [1 ₊ b_ocos (2ω₀ t)] ; -T≤ t ≤ +T ;

(157b) b_o = 5/7 .

Substituting Equation (157b) into Equation (139c), one sees that for this specific value of b_o, the constant C₂ is identically zero. Thus one also understands from Equation (143) that, for this specific value of b_o, all ISI components vanish. Fortunately, the task of finding a suitable fundamental QMSK symbol waveform q₁(t) is relatively easy.

The waveform

(158a) q(t) = cos[θ(t)] , -T≤ t≤+T ,

(158b) θ(t) = (π/4) [1 - gcos (2ω_ot)] , has the Fourier expansion

In Equation (159),

(160b)

X = g π/4 .

In Equations (158b) and (160b), g is an arbitrary variable; the J_n( ·) of Equation (159) are Bessel functions of the first kind. Low-pass filtering the waveform q(t) to a cut-off frequency just slightly greater than 2f₀ eliminates all terms in the first infinite sum of Equation (159) and retains only the first term in the second infinite sum. The band-limited waveform q_LP(t) is then given by

q_LP (t) = RJ₀ (X) + 2RJ ₁ (X)cos (2ω₀t) (161a)

The waveform q_LP(t) of Equation (161 b) has the same general expression as the waveform q_1LP(t) of Equation (157); the two waveforms can be made identical if one can determine a specific value X₀ of the variable x such that

Equation (162b) can be solved numerically to provide the approximate values

(163a) g₀≅0.85696 , χ_o =g_o π/4≅ 0.67305 . (163b)

The results of Equation (163) can be substituted into Equation (162a) to obtain

(164) a₀≅0.62927 .

The desired fundamental QMSK symbol waveform q₁(t) can now be identified as (165a) q₁(t) =cos[θ_o(t)] , -T≤ t≤ +T , θ₀(t) = (π/4) [1 -g_ocos(2ω₀t)] , (165b) where g₀ is the exact solution of Equation (162b), having approximate value given by Equation (163a).

Define the staggered waveform (166a) q₂(t) = q₁(t-T) = sin[θ_o(t)] , 0≤ t≤ 2T , θ₀(t) = (π/4)[1 -g_ocos(2ω_ot)] . ^(166b)

A further embodiment of the QMSK symbol waveform set, henceforth designated as the QMSK₀ symbol waveform set, is now defined by

(167a) q₁(t) = q₃(t) =cos[θ_o(t)] , -T≤ t ≤ +T ;

(167b) q₂(t) = q₄(t) = sin[θ₀(t)] , 0 ≤ t≤ 2T ;

(167c) θ₀(t) = (π/4)[1 - g_ocos(2ω_ot)] = π/4 -m(t) ; (167d) m(t) =m₀cos (2ω_ot) ; (167e) m₀ = g₀π/4 . The QMSK_o signal basis set is now defined by v_no(t) = q_n(t)c_n(t) , n = 1 , 2, ...,N=4 . (168)

The carrier waveforms c_n(t) of Equation (168) are defined by Equation (7). The generalized M-QMSK_o signal is thus defined by

Based on Equations (139c), (157b), and (143), one can assert that the M-QMSK_o signal, when detected by the augmented demodulator of input bandwidth W = 2R_s = 4f₀, is free of ISI components at each channel output. All previous formulas derived for band-limited M-QMSK signals which depend on the parameters a and b are also valid for M-QMSK_o signals, provided that a₀ and b₀ are substituted for a and b, respectively.

Waveforms required in the generation of M-QMSK_o signals can be provided by the method illustrated in the block diagram of Figure 18 in conjunction with the signal basis set generator of Figure 10. Referring to Figure 18, the phase-modulated (PM) waveforms cos [θ₊(t)] = cos [ω_ct + θ_o(t)] (170a) sin [θ₊(t)] - sin [ω_ct + θ₀ (t)] (170b)

(170c) cos [θ-(t)] = cos [ω_ct - θ_o (t)] sin [θ-(t)] = sin [ω_ct - θ_o (t)] (170d) can be combined in the dual 180° -hybrid structure of Figure 10 to generate the QMSK_o signal basis set. In the method of Figure 10 it is necessary to make the input substitutions cos [θ-(t)]→ cos [ω_Lt) (171a) cos [θ₊(t)]→ cos [ω_Ht) (171 b) sin [θ₊(t)]→ sin [ω_Ht) (171c) sin [θ-(t)]→ sin [ω_Lt) (171d) in order to obtain the QMSK_o signal basis set as outputs. Note that this combined method of generating the QMSK_o signal basis set is not unique. Having obtained the QMSK_o signal basis set, the M-QMSK_o signal can be generated as illustrated in the canonical modulator block diagram of Figure 7.

The Fourier transform of the fundamental QMSK_o symbol waveform q₁(t) of Equation (165) can be obtained by numerical methods. Substituting the results of such a method into Equation (70), and making the transformation to the normalized frequency variable of Equation (78), the equivalent baseband power density function of the QMSK_o signal is obtained; this power density function is illustrated in Figure 19. MSK and QMSK signals are each characterized by the modulation index m = 1/2; QMSK_o signals are characterized by the modulation index m₀ ^≅ 0.67305. This difference in modulation indices accounts for the difference in the sidelobe structures of the QMSK and QMSK_o power density functions observed when comparing figures 11 and 19. When band-limited to the RF bandwidth W = 2R_s = 4f₀, the QMSK_o signal retains approximately 96.4% of its original signal power. Note that a constant-envelope QMSK_o(CE) signal can be provided if the antipodal information sequences

(β_mn) of the QMSK_o signal are constrained to obey Equation (25).

J. ERROR PROBABILITY

From Equation (142), recall that the M-QMSK output signal component of the n^th channel of the augmented demodulator is

v_mn = C₁T_SC_mn = B_mn(C₁T_SA_mn) , n = 1 ,2, ..., N=4 . (172) Also recall that

A_mn = jA ; (173a) j = 1,3,5,7,...,J ; (173b)

J = [2^(k+1)-1] . (173c)

Define the quantity x = C₁T_SA , (174) such that

V_mn = B_mn(jx) = B_mnY_mn , (175a)

Y_mn = jx . (175b) For signal detection, there are only two statistically-significant possibilities, given by

(176a)

Y_mn =jx , j ≠ J ;

Y_mn = Jx . (176b)

Let

be the conditional probability of a correct decision event C_n in

the n^th augmented demodulator channel given that

Let

be the conditional probability of a correct decision event C in the n^th channel given that Y_mn = Jx. The decision region for the event is bounded by

(j-1)x≤ r≤ (j+1)X .

(177)

The decision region for the event (Y_mn = Jx) is bounded by

(J-1)X≤ r≤∞.

(178)

Assume a Gaussian probability density function having variance given by v² = P_rT . (179)

One can now state that

(180c) = 1 -2Q (X/V) ;

One can also state that v 7

(181c)

= 1 - Q (x/v)

Let C_s be the event that a correct signal state decision is made for the m^th symbol period. Let E_S be the event that an erroneous signal state decision is made. Then

P(E_S) = 1 - P(C_S) . (182)

A correct signal state decision requires that all channel decisions be simultaneously correct. Decisions in any channel are made independently of decisions in any other channel. Further assume that the individual data bits carried by the information coefficients C_mn have been encoded such that two signal states which differ by only one data bit are geometrically adjacent. For independent channel decisions,

C_n is a compound event in that a correct decision can occur both for Y_mn≠ J_x and for Y_mn = Jx. The events (Y_mn≠ Jx) and (Y_mn = Jx) are mutually exclusive. Define the probabilities

(184a)

P_jx = P (Y_mn ≠ Jx )P (C_n | Y_mn≠Jx ) = [(2^k-1)/2^k][1 -2Q(x/v)] ,

(184b) and

P_Jx =P(Y_mn =Jx)P(C_n|Y_mn=Jx) (185a)

(185b) = (1/2^k) [1 -Q(x/v)] . The probability P(C_n) is given by

(186)

P(C_n) =P_jχ +P_Jx .

Note from Equations (184) and (185) that P_jx and P_jx are independent of the channel index n. Thus Equation (183) can be expressed as

(187a)

P(C_s) = [P(C_n)]^N = (P_jx +P_Jx)^N

where B(N,r) is the binomial coefficient

B(N,r) = N!/[r!(N-r)!] . (188)

Substituting Equations (184b), (185b), and (188) into Equation (187b), one obtains

z = X/V . (189b)

For properly encoded data bits, the probability of bit error is P_b(e) = [1/log₂(M)]P(E_s) = [1/N(k+1)] [1 - P(C_S)] . (190)

Combining Equations (47), (155), and (174), one obtains

(191b)

F (k) = (k+1)/[4^(k+1)-1] . Combining Equations (156a) and (191a), one obtains

Using Equations (138b), (139c), (153), (157b), (189), (191b), and (192), one can now compute P_b(e) as a function of (E_b/N₀) for M-QMSK and M-QMSK₀ signals. Table 2 summarizes the derived constants and formulas used for computing P_b(e). Note that, in the formulas for P_b(e), only the first-power terms Q(z) have been retained; this is justified for large (E_b/N₀).

The formulas summarized in Table 2 have been used to generate the P_b(e) vs. (E_b/N₀) curves illustrated in Figure 20. The broadening of the M-QMSK curves due to ISI is clearly indicated. Also note from Figure 20 that, due to ISI, arbitrarily low PJe) can not be achieved for 65,536-QMSK.

K. TRANSMISSION EFFICIENCY

From Equations (10) and (81), one derives

Bandwidth Efficiency = R_b/W = 2(k+1) (193)

for appropriately band-limited M-QMSK and M-QMSK₀ signals. Using the bit error probability formulas of Table 2 and the bandwidth efficiencies of Equation (193), one can derive the M-QMSK and M-QMSK₀ transmission efficiencies indicated in Table 3. Recall that the constant-envelope signal bit rates are only three-fourths those of the original signals.

Figure 21 illustrates all but the M = 65,536 transmission efficiencies of Table 3. For comparison, the transmission efficiencies of certain prior art signaling methods are also illustrated. Figure 21 illustrates the decisive transmission efficiency advantages of QMSK-based signaling methods in comparison to those of previously-known methods. Note that the E_b/N₀ values illustrated in Figure 21 for M-PSK and M-MSK signals were computed for the case of ideal detection (i.e., by ignoring ISI degradation). In contrast, the data points plotted for M-QMSK signals include the effects of ISI.

L SIGNAL RECOVERY

Define the N×1 (4×1) signal waveform column vector

where the vector component waveforms v_n(t) are the members of the particular QMSK signal basis set defined by Equation (12). Then the 4x1 signal waveform column vector [x_n(t)]W(t) = [x_n(t)]cos(2ω_ot) (195) is identical to the weighted waveform vector [v_nw(t)] of Equation (94). At the end of the m^th symbol period the analog output vector of the ideal augmented demodulator is given by the vector equation

In practice, [x_n(t)] and w(t) cannot be recovered from s_m(t) with perfect knowledge of symbol, carrier, and weight function phase. Let the recovered symbol/carrier and weight function waveforms be given by

where (Ø_S ,Ø_C ,Ø_W) represent the phase uncertainties of the recovered symbol, carrier, and weight function waveforms, respectively. Due to these phase uncertainties, the augmented demodulator analog output vector becomes

Define the 4x1 column vector

where D(·) is the channel decision operator exemplified by the D/A (digital-to-analog) converters of Figure 22. Thus

is the receiver estimate of the original symbol

information vector [C_mn]; this estimate is formed by D/A converter operations on each set of symbol information data bits recovered by the augmented demodulator.

The scalar quantity

where ( ·)^T is the matrix transposition operator, is the cross-correlation between the receiver estimate

and the recovered analog information vector

X For reasonable signal-to-noise ratio, low ISI and perfect phase recovery [i.e., low P_b(e)],

Under these conditions, the quantity

is understood to be proportional to symbol energy during the m^th symbol period. Using Equations (154a) and (202), one computes

where E_S is the average symbol energy. Thus it is understood that E[R_m(Ø_S ,Ø_C ,Ø_W)] is a measure of the average symbol energy recovered by the decision-assisted augmented demodulator of Figure 22 in the presence of the phase uncertainties (Ø_S ,Ø_C ,Ø_W). Optimization of E[R_m(Ø_s ,Ø_C ,Ø_W)] thus corresponds to maximizing the average detected symbol energy and, as a consequence, also corresponds to maximum-likelihood (least probability of error) signal state decisions.

One can now write

J

Using Equation (197) and some, standard trigonometric identities, it can be shown that

= [F (Ø_S,Ø_C)] [x_n(t)] (205b)

where the matrix elements of [F(Ø_S ,Ø_C)] are given by a cos(Ø_s)cos(Ø_C) , (206a) b sin(Ø_S)sin(Ø_C) , (206b) c cos(Ø_S)sin(Ø_C) , (206c) d sin(Ø_S)cos(Ø_C) . (206d)

Substituting Equation (205b) into Equation (204b), one obtains

For small Ø_W, make the approximation

Substituting Equations (208), (196), and (205) into Equation (207), one obtains

For low P_b(e),

Using equations (210), (203), and (206a) in Equation (209b), one obtains

Let d( ·) be the partial differential operator. Maximum-likelihood, decision-assisted phase detection for the symbol phase variable∅_s and the carrier phase variable∅_C can now be defined by the joint operations

Phase detector realization for recovery of ∅_s is defined by

Phase detector realization for recovery of θ_C is defined by

Equations (213d) and (214d) represent direct (un-filtered) phase detector outputs, and also define phase detector implementation methods. When each output is filtered

(averaged) by an appropriate loop filter, the average phase detector outputs and

of Equations (212a) and (212b), respectively, are obtained.

Having established the maximum-likelihood symbol and carrier phase-recovery methods, these methods can be used in various ways to reconstruct the weighted QMSK signal basis set [v_n(·)w_n(·)] required for efficient augmented demodulator operation. One particular method is suggested by the equations

and the equations

Thus e_H and e_L are understood to be direct phase detector outputs which can be used in a compound phase-locked loop to recover the signals sin(ω_H t+θ_H) and sin(ω_L t+θ_L), respectively, where f_H = (f_c+f_o) and f_L = (f_c-f_o) are the upper and lower QMSK shift frequencies. Equations (214d), (213d), (215), and (216) provide the basis for understanding the compound phase detector block diagram of Figure 23. Note that all of the inputs required to implement the maximum-likelihood phase detector of Figure 23 are available as outputs from the decision-assisted M-QMSK demodulator of Figure 22.

In Figure 22, the

are delayed in order to compensate for the A/D and D/A converter delay in the processing paths.

The compound phase-detector outputs e_H and e_L of Figure 23 are used as inputs to the dual phase-locked loop structure of Figure 24 in order to construct the

phase-coherent signals sin(ω_H t+θ_H) and sin(ω_L t+θ_L). The indicated loop filters provide the averaged oscillator phase control signals and . The 90°-hybrid

and 180°-hybrid structure of Figure 24 then combines the signals

sin(ω_H t+∅_H) and sin(ω_L t+∅_L) to provide the output QMSK signal basis set components [v₃(·),v₄( ·), -v₁( ·),-v₂( ·)] as indicated. The weight function

w (t,∅_w ) = cos (2ω_ot+∅_w ) = cos (2ω_ot+2∅_s ) (217) is constructed from the signals sin(ω_H t+∅_H) and sin(ω_L t+∅_L) by the combined operation of the multiplier and band-pass filter (BPF) of Figure 24. Thus optimal recovery of ∅_s provides near-optimal recovery of ∅_w. The logic circuit of Figure 24 provides the staggered sample and dump control signals required by the augmented demodulator of Figure 22. The sample and dump timing diagram of Figure 25 enhances understanding of the sample and dump logic circuit.

Equation (211b) suggests a method of constructing a maximum-likelihood, phase coherent automatic gain control (CAGC) signal. Using Equation (211 ), define

Thus is understood to be the filtered (averaged) output of a phase-coherent

AGC detector; it can be used as the control signal in an AGC signal-leveling loop. The

CAGC detector is operationally defined by Equation (204a), which can be expanded as

Equation (219b) can bjs implemented by the method indicated in the coherent AGC detector block diagram of Figure 26.

Simultaneous maximum-likelihood phase detection and CAGC detection provide a sound basis for successful construction of an efficient M-QMSK receiver. The augmented M-QMSK demodulator of Figure 22, the maximum-likelihood phase detector of Figure 23, the M-QMSK signal recovery mechanism of Figure 24, and the

maximum-likelihood AGC detector of Figure 26 are combined with other necessary elements to provide the M-QMSK receiver block diagram of Figure 27. Although QMSK signal basis set recovery, weight function recovery, and AGC detection can be accomplished by other methods, the maximum-likelihood methods discussed herein are considered to be preferred methods. M. SPREAD-SPECTRUM QMSK

Direct-sequence, spread-spectrum modulation techniques are easily applied to QMSK and QMSK(CE) signals. Spread-spectrum techniques are used, for example, in

CDMA (code-division multiple access) and anti-jam applications. Using QMSK as the base modulation in a spread-spectrum application allows one to double the underlying information bit rate in comparison to spread-spectrum systems using MSK or QPSK as base modulations.

There are several possible methods for applying direct-sequence spreading codes to

QMSK signals. For example, let [c_n(t)] be a set of four antipodal direct-sequence spreading codes chosen to have low cross-correlation. The spread-spectrum QMSK signal

is easily generated by the canonical QMSK modulator of Figure 7 provided that the C_mn of Figure 7 are each replaced by the product c_n(t)B_mn. An alternative method of applying spreading codes to the QMSK signal is suggested by the equations

(221 a)

SSjt ) = A c ₁ (t )B_{m 1} COS (ω_c t-B_{m 1}B_{m 2} ω_o t )

+ A c ₂ (t )B_{m 3} sin (ω_c t+B_{m 3}B_{m 4} ω₀ t ) =ss_{m l} (t ) + ss_{m Q} (t ) , (221b) where c₁(t) and c₂(t) are antipodal direct-sequence codes. The direct-sequence technique of Equation (221 ) takes advantage of the dual synchronized MSK composition of QMSK signals. The direct-sequence spread-spectrum QMSK signal of Equation (221) is easily generated by a method such as that illustrated in the functional block diagram of Figure 28; this signal can be de-spread and demodulated by a method such as that illustrated in the functional block diagram of Figure 29. Note that a constant-envelope spread-spectrum QMSK(CE) signal can be generated by the method of Figure 28 provided that the B_mn of Equation (221a) have been constrained to obey Equation (26). This brief overview of the application of direct-sequence spread-spectrum techniques to

QMSK signals is not exhaustive, but is intended simply to illustrate the general concept.

N. BASEBAND SIGNALING

Baseband embodiments of QMSK-based signaling methods are obtained by deleting the carrier waveform components from any particular QMSK signal basis set previously described herein. As depicted in Figure 4, mathematically, this is equivalent to setting the carrier frequency f_c equal to zero. Thus a four-dimensional QMSK signal basis set with carrier frequency equal to zero is equivalent to a two-dimensional baseband signal basis set; for example in Equation 12 if ω_c = 0, then cos (ω_ct) = 1 and sin (ω_ct) = 0, so we are left with two terms which are the original QMSK signal basis set symbol waveforms. Thus the baseband signal basis set components are simply the two staggered, non-orthogonal basic symbol waveforms of the original QMSK signal basis set. For example, either the two basic symbol waveforms of Equation 6 or those of Equation 11 can constitute a two-dimensional baseband signal basis set, such capable of transmission by line such as coaxial cable, wire optical means, or other modes of transmission.

Signaling methods which utilize staggered, non-orthogonal QMSK basis symbol waveforms as a two-dimensional, baseband signal basis set are henceforth designated to be QMSK (Baseband) signaling methods, and are assigned the generic acronym QMSK (BB). For appropriately-paired, non-orthogonal basic QMSK symbol waveforms p₁(t) and p₂(t) , as illustrated in Figure 4, the QMSK (BB) signal is defined by the equation

during the m^th symbol period. The information coefficients C_mn can be chosen either in accordance with Equation 9 for multi-level signaling or in accordance with Equation 14 for antipodal signaling. QMSK (BB) signaling provides the same quantitative improvements for baseband transmission efficiency that QMSK signaling provides for carrier-based transmission efficiency.

QMSK (BB) signals can be generated by the canonical method illustrated in Figure 7 provided that the four-dimensional, carrier-based QMSK signal basis set [v_n(t)] is replaced by the two-dimensional QMSK (BB) signal basis set [p_n(t)]. The augmented demodulator of Figure 17 must be modified in obvious ways to provide efficient detection of QMSK (BB) signals. Referring to Figure 17, the modifications consist of replacing the input band-pass filter (BPF) by a low-pass filter of bandwidth W_BB = 2f_o, replacing the QMSK signal basis set [v_n(t)] by the QMSK (BB) signal basis set [p_n(t)], and deleting the indicated low-pass filters (LPF).

The maximum-likelihood methods provided earlier in this discussion for signal basis set reconstruction and coherent automatic gain control of QMSK signals are easily extended to provide similar methods for QMSK (BB) signals. QMSK (BB) modulation thus provides robust, transmission-efficient baseband signaling.

0. SUMMARY OF SOME OF THE BENEFITS OF THE PRESENT INVENTION

The present invention provides band-limited digital signaling methods wherein, in a practical sense, the Nyquist symbol efficiency of two symbols per second per Hertz of effective bandwidth is achieved. Moreover, band-limited digital signaling methods are provided wherein transmission efficiency is improved by modulation and subsequent detection of multi-dimensional, non-orthogonal signal basis sets. Also, band-limited digital signaling methods are provided wherein bandwidth efficiency is improved by a factor of two in comparison to most previously-known antipodal signaling methods while retaining essentially antipodal power efficiency. Furthermore, the present invention provides constant-envelope digital signaling methods wherein bandwidth efficiency is increased by 50% in comparison to most previously-known antipodal signaling methods while retaining essentially antipodal power efficiency.

This invention also provides band-limited digital signaling methods wherein power efficiency, for a pre-determined bandwidth efficiency, is dramatically improved in comparison to previously-known signaling methods.

Carrier-based QMSK and M-QMSK signals can be transmitted either by various forms of transmission line or by radiative methods. QMSK (BB) signals, having frequency spectra centered at the frequency origin, can be transmitted by conductive forms of transmission line.

Although the present invention has been described in terms of specific embodiments and applications, persons skilled in the art, in light of this teaching, can generate additional embodiments without exceeding the scope or departing from the spirit of the claimed invention. Accordingly, it is to be understood that the drawings and descriptions of this disclosure are provided to facilitate comprehension of the invention and should not be construed to limit the scope thereof.

Other benefits of this invention will derive from consideration of the drawings and detailed description contained herein.

Claims

IN THE CLAIMS:

1. A method for generating a composite modulated signal for transmission, said method comprising the steps of:

generating distinct components of a multi-dimensional, non-orthogonal signal basis set;

modulating said distinct components of said multi-dimensional, non-orthogonal signal basis set with appropriately-staggered or timed data streams to form a set of modulated signals; and

combining said set of modulated signals to form said composite modulated signal for transmission.

2. A method as recited in Claim 1 , wherein said multi-dimensional,

non-orthogonal signal basis set is formed by generating distinct products of a waveform selected from a non-orthogonal, staggered symbol wave form pair and a waveform selected from a quadrature carrier waveform pair.

3. A method as recited in Claim 1 , wherein said multi-dimensional, non- orthogonal signal basis set includes a pair of non-orthogonal staggered symbol waveforms.

4. A method as recited in Claim 2, wherein members of said non-orthogonal, staggered symbol waveforms have alternating polarity in adjacent symbol intervals.

5. A method as recited in Claim 3, wherein members of said pair of non-orthogonal, staggered symbol waveforms have alternating polarity in adjacent symbol intervals.

6. A method as recited in Claim 2, wherein said non-orthogonal, staggered symbol waveform pair comprises a cosinusoidal and sinusoidal symbol waveform, respectively.

7. A method as recited in Claim 3, wherein said pair of non-orthogonal, staggered symbol waveforms includes a cosinusoidal and a sinusoidal symbol waveform, respectively.

8. A method as recited in Claim 1 , wherein said respective sets of

appropriately-staggered or timed data streams individually include a single

appropriately-staggered or timed data stream.

9. A method as recited in Claim 1 , wherein said respective sets of

appropriately-staggered or timed data streams individually include multiple

appropriately-staggered or timed data streams.

10. A method as recited in Claim 1 , wherein a data stream is demultiplexed to form multiple data streams, and wherein selected ones of said multiple data streams are delayed in time with respect to remaining ones of said multiple data streams to form said respective sets of appropriately-staggered or timed data streams.

11. A method for demodulating a composite modulated signal, said method comprising the steps of:

multiplying said composite modulated signal by members of a set of complex waveforms to form a corresponding set of information signals; and

multiplying said corresponding set of information signals by a weight function to form a further corresponding set of demodulated information streams.

12. A method as recited in Claim 11 , wherein said set of complex waveforms is identical to the signal basis set of said composite modulated signal.

13. A method as recited in Claim 12, wherein said signal basis set is non-orthogonal.

14. A method as recited in Claim 12, wherein said signal basis set is formed by generating all distinct products of a waveform selected from a non-orthogonal, staggered symbol waveform pair and a waveform selected from a quadrature carrier waveform pair.

15. A method as recited in Claim 12, wherein said signal basis set comprises a pair of non-orthogonal, staggered symbol waveforms.

16. A method as recited in Claim 14, wherein members of said non-orthogonal, staggered symbol waveform pair have alternating polarity in adjacent symbol intervals.

17. A method as recited in Claim 15, wherein members of said pair of non-orthogonal, staggered symbol waveforms have alternating polarity in adjacent symbol intervals.

18. A method as recited in Claim 14, wherein said non-orthogonal, staggered symbol waveform pair includes a cosinusoidal and a sinusoidal symbol waveform, respectively.

19. A method as recited in Claim 15, wherein said pair of non-orthogonal, non- orthogonal symbol waveforms includes a cosinusoidal and a sinusoidal symbol waveform, respectively.

20. A method as recited in Claim 14, wherein said weight function is an odd function with respect to the midpoint of the half-symbol interval common to said pair of non-orthogonal, staggered symbol waveform pair.

21. A method as recited in Claim 15, wherein said weight function is an odd function with respect to the midpoint of the half-symbol interval common to said pair of non-orthogonal, staggered symbol waveforms.

22. A method as recited in Claim 11 , comprising the further step of:

passing said composite modulated signal through a band-pass filter of a pre-determined bandwidth.

23. A method as recited in Claim 11 , comprising the further step of:

passing said composite modulated signal through a low-pass filter of a pre-determined bandwidth.

24. A method as recited in Claim 11 , comprising the further steps of:

extracting from said demodulated information streams a set of maximum-likelihood, decision-assisted phase error signals; and

generating said set of complex waveforms and said weight function in response to said set of maximum-likelihood, decision-assisted phase error signals.

25. A method for generating a composite information signal for transmission, said method comprising of steps of:

generating a first Minimum-Shift Keyed information signal in response to a first pair of staggered data streams;

generating a second Minimum-Shift Keyed information signal in response to a second pair of staggered data streams; and

combining said first and said second Minimum-Shift Keyed information signals to form a four-dimensional, composite information signal for transmission.

26. A method as recited in Claim 25, wherein said first and said second Minimum-Shift Keyed information signals are generated by appropriately modulating cosinusoidal and sinusoidal carrier waveforms, respectively, in response to said first pair and said second pair of staggered data streams.

27. A method as recited in Claim 25, wherein said four-dimensional, composite information signal is equivalent to the sum of a set of four antipodally-modulated, non-orthogonal signal basis set components.

28. A method as recited in Claim 25, wherein a data stream is demultiplexed to form four data streams, and wherein two of said four data are delayed in time with respect to the remaining two of said four data streams to form said first pair and said second pair of staggered data streams.

29. A method for recovering demodulated information streams from a composite modulated signal, said method comprising the steps of:

receiving said composite modulate signal; and

resolving said composite modulated signal into said demodulated information streams in response to an augmented correlating demodulator.

30. A method as recited in Claim 29, wherein modulated members of a non-orthogonal signal basis set are summed to form said composite modulated signal.

31. A method for generating a composite information signal for carrier-based transmission, said method comprising the steps of:

generating first and second carrier waveforms having a common frequency and a quadrature phase relationship with one another;

combining said first and said second carrier waveforms with first and second symbol waveforms to form first, second, third and fourth complex waveforms, wherein specific paired combinations of said complex waveforms are orthogonal to one another and wherein specific other paired combinations of said complex are

non-orthogonal to one another;

modulating said first, second, third and fourth complex waveforms in response to respective first, second, third and fourth sets of appropriately-staggered or timed data streams to form first, second, third and fourth information signals; and

combining said first, second, third and fourth information signals to form said composite information signal for carrier-based transmission.

32. A method as recited in Claim 31 , wherein said first and said second symbol waveforms have alternating polarity in adjacent symbol intervals.

33. A method for generating a composite information, signal for baseband transmission, said method comprising the steps of:

generating first and second symbol waveforms which are staggered in time with respect to one another;

modulating said first and said second symbol waveforms in response to respective first and second sets of appropriately-staggered or timed data streams to form first and second information signals; and

combining said first and said second information signals to form said composite information signal for baseband transmission.

34. A method as recited in Claim 33, wherein said first and said second symbol waveforms have alternating polarity in adjacent symbol intervals.

35. A method as recited in Claim 33, wherein said first and said second symbol waveforms comprise a two-dimensional, non-orthogonal signal basis set.

36. A method for generating signal basis set components said method comprising the steps of:

generating first and second carrier waveforms having a common frequency and a quadrature phase relationship with one another;

combining said first and said second carrier waveforms with a first symbol waveform to form first and third signal basis set components; and

combining said first and said second carrier waveforms with a second symbol waveform to form and second and fourth signal basis set components, wherein said first, second, third and fourth signal basis set components form a four-dimensional, non-orthogonal signal basis set.

37. A method as recited in Claim 36, wherein said first and second symbol waveforms are cosinusiodal and sinusoidal, respectively.

38. A method as recited in Claim 36, wherein said four-dimensional non-orthogonal signal basis set components are modulated to form a composite modulated signal for transmitting information, said method comprising the steps of:

modulating said first, second, third and fourth signal basis set components with individual sets of appropriately-staggered or timed digital data streams to form individually-modulated signal basis set components; and

combining said individually-modulated signal basis set components to form said composite modulated signal for transmitting information.

39. A method as recited in Claim 38, wherein said individual sets of appropriately-staggered or timed digital data streams include a single appropriately-staggered or timed digital data streams.

40. A method as recited in Claim 38, wherein said individual sets of appropriately staggered or timed digital data streams include multiple appropriately-staggered or timed digital data streams.

41. A further step in a method as recited in Claim 38 for demodulating said composite modulated signals as recited in Claim 38, comprising:

multiplying said composite modulated signal by a weight function.

42. A method as recited in Claim 41 , wherein said first and second symbol waveforms have the same interval length and wherein said weight function is an odd function with respect to the mid-point of one half of that interval length.

43. A method for generating signal basis set components, said method comprising the steps of:

additively combining first and said second carrier waveforms having respective first and second carrier frequencies to form a first signal basis set component;

subtractively combining said first and said second carrier waveforms to form a second signal basis set component;

additively combing third and fourth carrier waveforms having said second and said first carrier frequencies, respectively, to form a third signal basis set component; and

subtractively combining said third and said fourth carrier waveforms to form a fourth signal basis set component, wherein particular ones of said first, second, third, and fourth signal basis set components are non-orthogonal to one another.

44. A method as recited in Claim 43 wherein said signal basis set components are modulated to form a composite modulated signal for transmitting information, said method comprising the steps of:

modulating said four signal basis set components with individual sets of appropriately staggered or timed digital data streams to form individually-modulated signal basis set components; and

combining said individually-modulated signal basis set components to form said composite modulated signal for transmitting information.

45. A method as recited in Claim 44, comprising the further steps of:

receiving said transmission of information; and

multiplying said composite modulated signal by a weight function.

46. A method as recited in Claim 45, wherein said symbol waveforms have the same interval length and wherein said weight function is an odd function with respect to the mid-point of one half of that interval length.

47. A method as recited in Claim 45, further comprising the step of passing said composite modulated signal through a band-pass filter of a predetermined bandwidth.

48. A method for generating signal basis set components, said method comprising the steps of:

generating a first symbol waveform to form a first signal basis. set component; and

generating a second symbol waveform which is staggered in time with respect to said first symbol waveform to form a second signal basis set component, wherein said first and said second signal basis set components form a two-dimensional, non-orthogonal signal basis set.

49. A method as recited in Claim 48, wherein said two-dimensional, non-orthogonal signal basis set is modulated to form data streams.

50. A method as recited in Claim 49, wherein said data streams are transmitted by line.

51. A method as recited in Claim 29, comprising the further steps of:

receiving said data stream; and

multiplying said data stream by a weight function.

52. A system as recited in Claim 51 , wherein said symbol waveforms have the same interval length and wherein said weight function is an odd function with respect to the mid-point of one half of that interval length.

53. A system as recited in Claim 51 , wherein prior to said multiplying step, said method includes the step of passing said single data stream through a band-pass filter of a predetermined bandwidth.

54. A system for transmitting information, said system comprising:

means for generating a set of complex waveforms, wherein said set of complex waveforms comprises a multi-dimensional, non-orthogonal signal basis set;

means for modulating said set of complex waveforms with sets of appropriately-staggered or timed data streams to form a corresponding set of information signals; and

means for combining said corresponding set of information signals to form a composite information signal for transmission.

55. A system as recited in Claim 54 further comprising means for demodulating said composite information signal.

56. A system as recited in Claim 55, wherein said means for demodulating said composite information signal is an augmented correlating demodulator.

57. A system for demodulating a composite information signal, said system comprising:

means for receiving said composite information signal; and

means for resolving said composite information signal into demodulated information streams in response to an augmented correlating demodulator.

58. A system as recited in Claim 57, wherein said augmented correlating demodulator includes means for multiplying said composite information signal by a weight function.

59. A system as recited in Claim 57, wherein said composite information signal includes a set of modulated complex waveforms.