EP1044543A1

EP1044543A1 - Modulating a digital signal with narrow spectrum and substantially constant envelope

Info

Publication number: EP1044543A1
Application number: EP98962536A
Authority: EP
Inventors: Jean-Louis Dornstetter; Eric Georgeaux
Original assignee: Nortel Matra Cellular SCA
Current assignee: Nortel Networks France SAS
Priority date: 1997-12-22
Filing date: 1998-12-21
Publication date: 2000-10-18
Also published as: JP2001527330A; WO1999033238A1; CN1284228A; US6868126B1; FR2773032B1; CA2315902A1; FR2773032A1

Abstract

The invention concerns a transmission signal (S) resulting from the modulation of a data digital signal by a modulation function dependent on time (t), said data signal being formed by a series of bits each (bk) identified by its rank k and having a duration T, said transmission signal (S) consisting in a summation indexed on the rank k of the product of the constant complex j to the power of k, of the modulation function h(t-kT) and of an input signal. The input signal F(k) being a function of the data signal (bk), the modulation function h(t-kT) is a Gaussian function of the time t.

Description

Modulation of a digital signal with a narrow spectrum and a substantially constant envelope

The present invention relates to a technique for modulating a digital signal. It therefore applies in the field of transmissions, in particular in that of radio frequencies.

A typical application is aimed at radiocommunication systems, in particular so-called "broadband" systems. Such systems are naturally designed to offer a high capacity and it is therefore necessary to adopt a modulation with high spectral efficiency, which amounts to saying that the spectrum of a given channel must be as narrow as possible given the technical specifications.

In addition, constant envelope modulation is generally used, which makes it possible to minimize the complexity of the transmitters. Indeed, if the signal has relatively large amplitude variations, the amplification stages, in particular the power amplifier, must be perfectly linear. However, it appears that all of the constant envelope modulations known up to now have a spectrum which has a lateral lobe.

This lateral lobe, if it is located in the spectrum of a neighboring channel, will increase the level of interference in this neighboring channel.

For example, the GMSK modulation (for "Gaussian Minimum Shift Keying") used in the GSM system has a side lobe located at -40 dBc and 200 kHz from the main lobe, while the spacing between two adjacent channels is also 200 kHz. It is clear that the spectral efficiency is reduced accordingly.

We also know the QAM modulations (for "Quadrature Amplitude Modulation") whose spectrum does not have a side lobe as long as an appropriate shaping filter is used. However, these modulations cause large variations in the amplitude of the modulated signal. As mentioned above, it is appropriate in this case to use more complex amplifiers and therefore more expensive. The present invention thus relates to a modulation technique which presents a spectrum without lateral lobe while retaining a practically constant envelope.

The invention therefore applies to a transmission signal resulting from the modulation of a digital data signal by a modulation function dependent on time t, this data signal being formed of a series of bits each bjç identified by its rank k and having a duration T, this transmission signal consisting of a summation indexed on rank k of the product of the complex constant j at the power k, of the modulation function h (t-kT) and of a signal d 'Entrance. According to the invention, the input signal being a function of the data signal, the modulation function is a Gaussian function of time t. Advantageously, this modulation function is thus defined: t ²

2σ ² T ² h (t) = - e, where

G 27T the parameter σ is a form factor which determines the spread of a bit. The simplest form of implementation of

The invention consists in defining the input signal as being equal to the data signal.

However, by adopting this solution, the transmission signal still presents amplitude variations which, although minimal, always bring some constraints on the amplifiers of the transmitter. Thus, preferably, the input signal is worth:

NF (k) = ∑a "B ⁿ , where n = 0

- N, level of correction, is a strictly positive natural integer, - a is a constant of positive correction,

- the polynomial _ is defined as follows:

L 2 M

B _k ⁿ = Σ (π " _{k + p} ><^" *

1 = 1 i = 0 ¹ > a family of relative integers pτ_ j_ is constructed so that there exists a natural integer M which makes it possible to verify the following relationships:

M M

Y p - ∑ p = 2n

^ 1, 2 i ^ 1, 2 i -1 i = 0 i = 1 p <p whatever i

1, i 1, i +1

- L represents the total number of these families.

For example, the correction constant has for 1_ value e σ2 • The invention also relates to a modulator for producing the transmission signal.

According to a preferred embodiment, the modulator comprises a digital processor which receives the input signal to produce the real part and the imaginary part of the transmission signal, a first mixer to multiply the real part by a carrier, a phase shifter receiving this carrier for the phase shift of π / 2, a second mixer to multiply the imaginary part by the output signal of the phase shifter, and an adder to sum the output signals of the two mixers.

Advantageously, when the level of correction is greater than zero, the digital processor comprises a first module for producing the polynomials B ^ ¹¹ .

In addition, the digital processor comprises a second module for producing digital samples of the transmission signal, four samples E ^ + i being associated with the bit bjς for i varying from 0 to 3 and being equal to: ^{k ~ 5} q 1 i 1 ₇ E = V j. (B + -. B + .B). h

4k + i ^ q ₈ q ₅₄ q 4 (k ~ q) + iq = k

The invention also relates to a demodulator for restoring the data signal from the transmission signal.

Preferably, this demodulator comprises a base band transposition member which receives a signal which has been the subject of a modulation by the modulation function, a complex multiplier for multiplying the output signal

_ ^• JEÈ. of this transposition organ by the expression e ^2T , a convolution operator which performs the convolution of the output signal of the complex multiplier and of the modulation function, and a decision organ which restores the data signal as a function of the sign of the real part of the result of this convolution.

This transposition organ is commonly a Hubert filter.

The present invention will now appear in more detail in the context of the following description of exemplary embodiments given by way of illustration with reference to the appended figures which represent: FIG. 1, a diagram of a modulator according to the invention,

FIG. 2, a diagram of a first module of an embodiment of the modulator, FIG. 3, a diagram of a second module of this same embodiment of the modulator,

- Figure 4, a diagram of a demodulator 1 according to the invention. It is therefore a question of modulating a digital data signal which consists of a series of bits bjς taking the value + or -1.

The following expression of a modulated transmission signal S is known which covers several types of modulation:

S = ∑j ^k .h (t-kT) .b _k , where kk is the index of the current bit bjς, j, the complex constant such that j = -l, T, the duration of a bit, t represents time, and h the modulation function. When h is a rectangle function, we speak of "QPSK offset" modulation.

According to the invention, h is a Gaussian function which for example takes the following form: t ²

1 2σ T h (t) = e

The parameter σ is a form factor which determines the spread of a bit.

By noting B the half bandwidth at 3dB of the spectrum and by adopting the term In to represent the natural logarithm, we have the following relation:

Wine 2

B. T =

2 πσ

Adopting a Gaussian as a modulation function makes it possible to suppress the lateral lobe which is present in the constant envelope modulation spectra.

Note however that the modulated signal is still subject to amplitude variations although these are significantly reduced compared to a QAM type modulation.

Thus, according to another aspect of the invention, a corrective term C is added to the modulated signal as defined above:

• N is an integer representing a level of correction, 1_

• a is a constant which is equal to e 2 •

The polynomials B _k ⁿ are constructed as follows. We search, for a given value of n, all families of relative integers pτ_ j_ so that there is a natural integer M which allows us to verify the following relationships: MM

1 = 0 1 = 1

M M

= 2 n (2)

1, 2 i - Σ 1, 2 i —1

1 = 0 1 = 1 p <p whatever i (3)

1, i 1, i +1

The limit values of p ^ ^ should be sought, ie pi Q and pτ_ 2M *

From equation (1) it follows that pj _Q ^is ^ negative or zero, in fact: M

According to equation (3), this expression is negative or zero.

Similarly, the term Pi, 2M ^is positive or zero.

Indeed, equation (1) can be written:

M

P ⁼ Σ (P. ^~~ P)

1, 2 M l, 2 i -1 1, 2 i -2 i = 1 Now according to equation (3), this expression is positive or zero. Furthermore, the relative integers pi - ^ constituting a strictly increasing sequence, there exists a single natural integer z for which the product Pi, z "Pl, z- _l ^is negative or zero.

If z is even, equation (2) can be written: z 1 _{million. 5} „ _{. .} , _. 2 „ _. ,.

-r.2 4- r.2 + fn ² - pr.?2 _Λ . ,) = 2n - ^ ^P l, 2i ^P l, 2i -1 ^P l, z ^ ^ ^P l, 2i ^r l, 2i + li = - + l ^{i = 0}

2

If z is odd, equation (2) can be written: z-3

! _{+ 1} < ^P Ï, 2i- ^P l, ₂ iJ ^{+ P} ï, _Z -l ⁺ i ₌

2 In the two expressions above, the left side of the equation is in the form of a sum of positive terms, which implies that each of these terms is at most equal to 2n. So : IF Z <2M, 2

1.2M 1.2M <2n

(P 1.2M '1.2M) (P 1, 2M + P) ≤ 2n,

1.2M - 1 by setting (p _{1 2M} - P _{1 / 2M} ) = a, with 1 <a <2n

We can easily verify that (n / a + a / 2) (n + l / 2) when a is between 1 and 2n.

It follows that pi 2M ^is ^ less than or equal to n.

We prove in the same way that p ^ Q ^is t greater than or equal to - n.

It follows from the above that the set of families of relative integers p ^ j_ is a finite set.

For a given value of n, we now consider the first family pi i which is obtained for 1 = 1. To build this family, we start from p ^ o ^{= -n} then we empirically search for the sequence of relative integers p ^ ι, ••• 'Pi 2M ¹ ^ satisfies equations (1), (2) and (3).

For example, when n = l, there is one family F = {p ₁ _1/0, p _lfl, Pι _f2} = {- 1, 0, 1}.

If n is greater than l, we find all families in the same way by successively incrementing all pτ_ j. In this case, 1 varies from 1 to L.

For the first values of n, these families are:

- n = 2, F ₁ = {p _1/0 = {- 2, -1, +1}

^F 2 = ^{ P2,0 = {- l, +1, +2}

"I, 0, +1, +2}

^F 3 = ^{ P3,0 = {- l, +2, +3}

- _n = 4, F! = {p _1/0 = {- 4, -3, +1}

^F 2 = ^{ P2,0 P2,3 'P2,4} ⁼ {" ³ '" 2, 0, +1, +2} ^F 3 = {P3,0 = {- 2, 0, +2} ^F 4 = ⁽ P4.0 P4.3 'P, 4} = {- 2. "I, 0, +2, +3} 5 = ⁽ P5.0 = {- !, +3, +4} Finally, the polynomial B _k ⁿ is obtained by the following expression:

Using the previous examples:

Bk ¹ = ^b kl- ^b k- ^b k + l

^B k ² = ^b k-2- ^b kl- ^b k + l + ^b kl- ^b k + l- ^b k + 2

^B k ^{3 ≈ b} k-3- ^b k-2- ^b k + l ^{+ b} k-2- ^b kl- ^b k- ^b k + l- ^b k + 2 ⁺ ^b kl- ^b k + 2- ^b k +3 ^B k ⁴ = ^b k-4- ^b k-3- ^b k + l + ^b k-3- ^b k-2- ^b k- ^b k + l- ^b k + 2 + ^b k-2- ^b k- ^b k + 2 + ^b k-2- ^b kl- ^b k- ^b k + 2- ^b k + 3 ⁺ ^b kl- ^b k + 3- ^b k + 4 We can now resume the equation of the modulated signal S:

should have written:

We can thus define an input signal F (k):

F (k) = ∑ a ⁿ B _k ^π n = 0 When N = 0, we find the simplest embodiment of the invention while, the larger N, the more the amplitude variations of the modulated signal S are limited.

Note that the spectrum of this signal is independent of N. It is worth:

The invention naturally relates to a modulator for producing the modulated signal S and injecting it on a carrier. Although the realization of a modulator thus specified is within the reach of one skilled in the art, we now give one example among many of such an embodiment.

With reference to FIG. 1, the modulator comprises a digital processor PR which receives the bits bjς to produce the real part I and the imaginary part Q of the modulated signal S: S = I + jQ.

It also includes a first mixer Ml to multiply the real part I by the carrier C and a second mixer M2 to multiply the imaginary part Q by the carrier phase-shifted by π / 2. To this end, a DEP phase shifter receives this carrier to inject it on the second mixer M2.

It also includes an adder AD to sum the output signals of the two mixers Ml, M2.

Finally, the modulator comprises a time base BDT which supplies on the one hand the clock signals Ck to the digital processor PR and, on the other hand, the carrier to the first mixer M1 and to the phase shifter DEP. It operates for the most varied values of the various constants and in particular with a level of correction N equal to 0. However, in order to obtain good performance and to facilitate the task of the PR processor, the following values are retained as 'example:

1 - form factor: Q ^" - - =; therefore, the 3 In 2 1_ constant a = e 2 is 1/8, which allows a multiplication by a by shifting three bits towards the right,

- correction level N = 2, - value of the bits bjς: + or - 1,

- modulated signal S expressed on 12 bits,

- oversampling factor: 4. The modulated signal S therefore results from a series of digital samples produced at the rate of four per bit time T.

The modulation function h (t) is also represented by a series of positive numbers hq of 11 bits. An appropriate scale factor is chosen so that the modulated signal S can indeed be coded on 12 bits:

( ⁿ q) _θ <q _ll = {0,1,5,17,47,116,253,485,816,1205,1563,1780} The function h (t) is even, so that for all q included in 0 and 11, h2 ₃ - q = hq. Given the scale factor adopted, q is zero for q <0 or q> 23: the function is stored for -3T <t <3T.

Furthermore, due to oversampling, we can assume that q = 4.k + i, i varying from 0 to 3; in other words, k is the integer part of q / 4.

With reference to FIG. 2 and by way of example, the processor PR comprises a first module for calculating the expressions Bj _ζ ¹ and Bj. The corresponding calculations are here carried out by means of a shift register which, at a reference instant, comprises the bits bjç + 2 to bjς-2 • B ^ ¹ is obtained by a first multiplier PI which produces the product of the bits, bjζ_ι, bjς, and bjς + i • To obtain Bj _ζ , ϋ is provided a second multiplier P2 which makes the product of bits bj _ζ -2, ^b kl 'and bjς + i a third multiplier P3 which makes the product of bits bjζ -i, bjç + i, and bjς + 2 and a summator R to sum the outputs of the second P2 and third P3 multipliers.

The processor PR also includes a second module shown in FIG. 3. This second module is responsible for calculating the digital samples of the modulated signal S, this by filtering the oversampled input signals by means of an impulse response filter h (t ). The four samples E ^ + i associated with the bit bjς for i varying from 0 to 3 therefore being worth: ^k ~ ⁵ Q 1 i 1 ₂

E = ∑ j. (B + -. B +. B). h

4K + i ^ Q ₈ q ₆₄ q 4 (k ^~ q) + iq = k

This expression can still be written:

^F 4k + i = E _ki ° + E _ki ^l + E _ki 2, with k -5

^E , ki- ⁼ Σ ^j ' ^b n' ^h 4 _Λ ( _t , kq x) _ + ι_ι- ^{= X} 0 ^{υ +} ^ 0 ^υ < ⁴ > q = kk -5

E = ∑ J ^q .-- B _q ¹ . _{4 (k} _ _{q) +} . = x ₁ + jy ₁ (5) ki q = k ^{k "~ 5} q 1

= ∑ j • -B -h = x ₂ + jy (6) ki ^ ₆₄ q 4 (kq) + ι ^z q = k

The numbers XQ, YO, ^x i Yi / ^x 2 'and y2 are real numbers.

By way of example, the second module comprises a first sampler EQ which receives the bit b _k to supply it to a first switcher A _Q synchronized with this sampler. The first switcher produces as output signal I _Q successively the first sample of the bit b _k then the third sample of this same bit b _k changed sign. It also produces as output signal Q _Q successively the second sample of bit b _k then the fourth sample of this same bit b _k changed sign. The second module then correlates (marked by the operator * in the figure) the output signal I _Q with the modulation function h according to equation (4) to produce the first real component x ₀ . Note that only the terms corresponding to an even index q are non-zero.

The discrete correlation operation is not more detailed since it is a technique well known to one skilled in the art. The second module also correlates the output signal Q _Q with the modulation function h according to

1 equation (4) to produce the first imaginary component Y _Q. Note that only the terms corresponding to an odd index q are non-zero.

Similarly, the second module includes a second sampler E ^ which receives the signal Bjç ¹ to supply it to a second switcher A ^ synchronized with this sampler. The second switcher produces as output signal 1 ^ successively the first sample of the term Bjς ¹ then the third sample of this same term changed sign. It also produces as output signal Q ^ successively the second sample of the term Bjç ¹ then the fourth sample of this same term changed sign. The second module then correlates the output signal Iχ with the modulation function h multiplied by the constant a (1/8 in this case) according to equation (5) to produce the second real component x ^.

The second module also correlates the output signal Q ^ with the modulation function h multiplied by 1/8 according to equation (5) to produce the second imaginary component y ^.

Similarly, the second module produces the third real 2 and imaginary component y2 from the expression B _k ² according to equation (6).

The real part I of the modulated signal results from the sum of the three real components x ₀ , xχ _f X ₂ and its imaginary part Q results from the sum of the three imaginary components yo, y and γ ₂ . The invention also quite naturally relates to a demodulator for recovering the data signal from the modulated signal S. Although the production of a demodulator thus specified is within the reach of those skilled in the art, an example is now given among others of such an achievement. With reference to FIG. 4, the demodulator comprises a transposition member in base band FIL which receives a signal r (t) which has been the subject of a modulation as described above. This transposition device is commonly produced using a Hubert filter.

The demodulator also includes a complex multiplier MUL to multiply the output signal of the organ

. πt_ of transposition FIL by the expression e ^2T and thus produce a signal of frequency equal to the quarter of bit time. It also includes a convolution operator CONV which performs the convolution of the output signal of the complex multiplier MUL and of the modulation function h (t) defined above.

The result of this convolution is interpreted by a decision-making body DEC which restores the bit b _k as a function of the sign of the real part of this result.

The invention therefore relates to a digital modulation technique which applies regardless of how the modulation function is represented, including using a compression law. It is not limited to the examples of embodiments described above. In particular, it is possible to replace any means with equivalent means.

Claims

1) Transmission signal (S) resulting from the modulation of a digital data signal by a time-dependent modulation function t, said data signal being formed of a series of bits each (b _k ) identified by its rank k and having a duration T, this transmission signal (S) consisting of a summation indexed on said rank k of the product of the complex constant j at the power k, of said modulation function h (t-kT) and of a input signal, characterized in that, said input signal F (k) being a function of said data signal (b _k ), said modulation function h (t-kT) is a Gaussian function of time t. 2) Transmission signal according to claim 1, characterized in that said modulation function is thus defined: t ²

2σ ² T ² h (t) = e, where the parameter σ is a form factor which determines the spread of a bit.

3) Transmission signal according to claim 1 or 2, characterized in that said input signal is said data signal (F (k) = b).

4) Transmission signal according to claim 1 or 2, characterized in that said input signal E (k) is:

NOT

F (k) = ∑ a B, where k n = 0

- N is a strictly positive natural integer,

- a is a positive correction constant,

- the polynomial B _k ⁿ is thus defined:

L 2 M

B _k ⁿ = ∑ (n _{k + p} ), ^WHERE

1 = 1 i = 0! 'i - a family (F ^) of relative integers pi j_ is constructed so that there exists a natural integer M which makes it possible to verify the following relationships:

M M

p <p whatever i

1, i 1, i +1 - L represents the total number of these families.

5) transmission signal according to claim 4, characterized in that said correction constant has for 1_ value e 2 • σ

6) Modulator for producing a transmission signal (S) according to any one of claims 1 to 3.

7) Modulator according to claim 6, characterized in that it comprises a digital processor (PR) which receives said input signal F (k) to produce the real part (I) and the imaginary part (Q) of said signal transmission (S), a first mixer (Ml) to multiply said real part (I) by a carrier (C), a phase shifter (DEP) receiving this carrier to phase it by π / 2, a second mixer (M2) to multiply said imaginary part (Q) by the output signal of the phase shifter, and an adder (AD) for summing the output signals of the two mixers (Ml, M2).

8) Modulator for producing a transmission signal (S) according to claim 4 or 5.

9) Modulator according to claim 8, characterized in that it comprises a digital processor (PR) which receives said input signal F (k) to produce the real part (I) and the imaginary part (Q) of said signal transmission (S), a first mixer (Ml) for multiplying said part real (I) by a carrier (C), a phase shifter (DEP) receiving this carrier to phase it by π / 2, a second mixer (M2) to multiply said imaginary part (Q) by the output signal of the phase shifter, and an adder (AD) for summing the output signals of the two mixers (Ml, M2).

10) Modulator according to claim 9, characterized in that said digital processor (PR) comprises a first module for producing said polynomials B _k ⁿ . 11) Modulator according to claim 10, characterized in that said digital processor (PR) comprises a second module for producing digital samples of said transmission signal (S), four samples E _{4k +} j_ being associated with the bit b _k for i varying from 0 to 3 and worth: ^{k ~ 5} q 1 i 1 ₂ E = Y j. (B + -. B + .B). h

4k + i ^ q ₈ q ₆₄ q 4 (k ~ q) + iq = k

12) Demodulator receiving a transmission signal (S) according to any one of claims 1 to 5 to restore said data signal (b _k ).

13) Demodulator according to claim 12, characterized in that it comprises a base band transposition member (FIL) which receives a signal r (t) having been the subject of a modulation by said modulation function h ( t-kτ), a complex multiplier (MUL) to multiply the output signal of this transposition device (FIL) by the expression _ ^• -ϋE e ^2T , a convolution operator (CONV) which performs the convolution of the signal of output of the complex multiplier (MUL) and of said modulation function, and a decision member (DEC) which restores said data signal (b _k ) as a function of the sign of the real part of the result of said convolution.

14) Demodulator according to claim 13, characterized in that said filter (FIL) is a Hubert filter.