Method for demodulation and demodulator

Abstract

A method is provided for demodulation of an analog receive signal carrying information, wherein a number of more than two analog signals is formed from the receive signal in separate channels such that the receive signal is multiplied in each case by a period function, the phase thereof respectively differing in the channels, and wherein the multiple signals are each low-pass filtered.

Claims

1. A method for demodulating an analog receive signal carrying information, the method comprising: generating more than two separate channels; forming more than two analog signals from the analog receive signal by multiplying the analog receive signal by a periodic function in each of the more than two separate channels, a phase of the periodic function respectively differing in each of the more than two separate channels; and low pass filtering and separately digitizing each of the more than two analog signals to form filtered analog signals.

2. The method according to claim 1, wherein the periodic function differs from the analog receive signal and is selected as a sine function.

3. The method according to claim 1, wherein the filtered analog signals are each digitized to form digital signals.

4. The method according to claim 3, wherein the filtered analog signals are each digitized using a word width of less than 4 bits or a word with of 1 bit.

5. The method according to claim 3, wherein a bandwidth of the low pass filtering of the more than two analog signals and a sampling rate of the digitization are selected such that the sampling rate corresponds to at least twice the bandwidth.

6. The method according to claim 1, wherein the analog receive signal is of a given carrier frequency, which carries the information in a modulated manner.

7. The method according to claim 6, wherein the filtered analog signals are each low-pass-filtered by separating higher frequency products with respect to the carrier frequency.

8. The method according to claim 6, wherein the periodic function of the carrier frequency is selected as a repetition frequency.

9. The method according to claim 1, wherein phases of the periodic function are each selected equidistantly apart in the separate channels.

10. The method according to claim 1, wherein at least one parameter of a transmission path is estimated based on the filtered analog signals.

11. The method according to claim 10, wherein a phase shift, a runtime, a time shift, a Doppler shift or a signal strength is estimated as the parameter.

12. The method according to claim 1, wherein the information carried by the analog receive signal is estimated or decoded based on the filtered analog signals.

13. The method according to claim 10, wherein a maximum likelihood method is used for the estimation.

14. A demodulator for demodulating an analog receive signal carrying information, the demodulator comprising: an input channel for the analog receive signal; a demodulation unit configured to carry out the method according to claim 1; at least two output channels; a multiplication device; and a low pass filter device, wherein the multiple signals are each made available at the input channels.

15. The demodulator according to claim 14, wherein the demodulation unit includes an AD converter.

16. The demodulator according to claim 15, wherein the AD converter includes AD converters assigned to the channels, each of which configured for digitization with a word width of less than 4 bits or a word width of 1 bit.

17. The demodulator according to claim 14, wherein an estimation or decoding device, which is connected by information technology to the output channels of the demodulation unit, is also included, which is configured to estimate or decode at least one parameter of a transmission path and/or the information carried by the receive signal from the multiple digitized signals.

18. The demodulator according to claim 17, wherein the estimation or decoding unit is a maximum likelihood estimator.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The present invention will become more fully understood from the detailed description given hereinbelow and the accompanying drawings which are given by way of illustration only, and thus, are not limitive of the present invention, and wherein:

(2) FIG. 1 shows a schematic view of the structure of a demodulator for carrying out an overdemodulation;

(3) FIG. 2 shows an AD converter implemented as a circuit;

(4) FIG. 3 shows the performance of the overdemodulation method as a comparison of the average squared error width and an ideal QAM method as a function of the signal-to-noise ratio and the number of separate channels with respect to the ascertainment of the phase difference of the transmission path;

(5) FIG. 4 shows the ratio of the mean squared error width with respect to the ascertainment of the runtime on the transmission path in a representation according to FIG. 3; and

(6) FIG. 5 shows the maximum transmission rate achievable in each case by the overdemodulation according to Shannon as a function of the signal-to-noise ratio and for the number of separate channels.

DETAILED DESCRIPTION

(7) The starting point for the following observations is the known QAM method, wherein two independent baseband signals are modulated on a carrier signal by means of multiplicative mixing. To demodulate the carrier signal to the baseband, a quadrature demodulator having two output channels is used on the receiver side. The carrier or receive signal is multiplied by a sinus function oscillating with the carrier frequency in each of the two channels, the phase of the sine function differing by /2 in both channels. After the higher frequency multiplication products have been filtered out, the particular baseband signal remains directly in the orthogonal channels. The baseband signal obtained in this manner on the receiver side may be digitized for further processing. Complex AD converters, which sample using the sufficient word width, are needed for a high resolution and a high sampling rate. The achievable sampling rate decreases along with the resolution accuracy of the digitization. Complex AD converters also demonstrate a high energy consumption.

(8) The present invention solves the problem associated with the use of complex AD converters of a reduced sampling rate and a high energy consumption on the receiver side by generating more than two separate channels, the receive signal being multiplied by a periodic function in each channel, in particular a sinus function, the phase of the periodic function respectively differing in the channels. In contrast to the WAM modulation method, more than two linearly dependent signals are generated from the receive signal.

(9) To explain the specified method of overmodulation, a transmit signal of the form
x(t)=x.sub.1(t){square root over (2)} cos(.sub.ct)x.sub.2(t){square root over (2)} sin(.sub.ct) (1)
is assumed as an example, where .sub.c is the carrier frequency and x.sub.1/2(t) is two independent input or information signals. The receive signal
y(t)=x.sub.1(t){square root over (2)} cos(.sub.ct)x.sub.2(t){square root over (2)} sin(.sub.ct)+(t) (2)
results on the receiver side, where is an attenuation coefficient and is a time shift due to the signal propagation. designates a phase shift in the receive channel. (t) would be noise caused by the receiver.

(10) For demodulation to the baseband, the receiver generates, by particular multiplication, m=1, . . . , M channels from the receive signal of the form

(11) $\begin{array}{cc}\begin{array}{c}{}_m^{}\left(t\right)=y^{}\left(t\right).Math.\sqrt{2}\cos \left({}_{c}t+l_m\right)\\ =x_1^{}\left(t-\right)\left(\cos \left(2{}_{c}t-+l_m\right)+\cos \left(+l_m\right)\right)-\\ -x_2^{}\left(t-\right)\left(\sin \left(2{}_{c}t-+l_m\right)+\sin \left(+l_m\right)\right)+\\ +{}_2^{}\left(t\right)\sqrt{2}\cos \left({}_{c}t+l_m\right),\end{array}& \left(3\right)\end{array}$
with the particular phases .sub.m of the sine or cosine function used for multiplication. After a low pass filter h(t; B) with a bandwidth B, the particular signal may be written in the mth output channel as

(12) $\begin{array}{cc}\begin{array}{c}{y}_m\left(t\right)=x_1\left(t-\right)\left(\cos \left(\right)\cos \left(l_m\right)-\sin \left(\right)\sin \left(l_m\right)\right)+\\ +x_2\left(t-\right)\left(\sin \left(\right)\cos \left(l_m\right)+\cos \left(\right)\sin \left(l_m\right)\right)+\\ +\cos \left(l_m\right){}_1\left(t\right)+\sin \left(l_m\right){}_2\left(t\right),\end{array}& \left(4\right)\end{array}$
where
.sub.1(t)=h(t;B)*({square root over (2)} cos(.sub.ct)(t))
.sub.2(t)=h(t;B)*({square root over (2)} sin(.sub.ct)(t)) (5)
then describes two independent random processes of a spectral power density (). The notation * used here designates the convolution operator. If the different phases in the channels are each described as a vector of the form
=[.sub.1.sub.2 . . . .sub.M].sup.T, (6)
the signals in the M separate channels may be indicated as
y(t)=A()(B()x(t)+(t)), (7)
with the analog signals
y(t)=[y.sub.1(t)y.sub.2(t) . . . y.sub.M(t)].sup.T
x(t)=[x.sub.1(t)x.sub.2(t)].sup.T
(t)=[n.sub.1(t).sub.2(t)].sup.T (8)
and the matrices

(13) $\begin{array}{cc}A\left(\right)=\left[\begin{array}{cc}\cos \left({}_1\right)& \sin \left({}_1\right)\\ \cos \left({}_2\right)& \sin \left({}_2\right)\\ \mathrm{.Math.}& \mathrm{.Math.}\\ \cos \left({}_M\right)& \sin \left({}_M\right)\end{array}\right].& \left(9\right)\end{array}$

(14) After digitizing each of the M channels at a sampling rate of f.sub.s=2B for a duration of T=N/f.sub.s and by defining the parameter vector ==[ ].sup.T, the digital receive signal N includes temporarily ascertained sampled values y.sub.n.sup.m of the form

(15) $\begin{array}{cc}\begin{array}{c}{y}_n=A\left(\right)B\left(\right)x_n\left(\right)+A\left(\right){}_n\\ =8_n\left(\right)+{}_n\end{array}& \left(10\right)\end{array}$
with the digital values

(16) $\begin{array}{cc}{y}_n={\left[\begin{array}{cccc}{y}_1\left(\frac{(n-1}{f_s}\right)& y_2\left(\frac{(n-1}{f_s}\right)& \mathrm{.Math.}& y_M\left(\frac{(n-1}{f_s}\right)\end{array}\right]}^T{x}_n\left(\right)={\left[\begin{array}{cc}{x}_1\left(\frac{(n-1}{f_s}-\right)& x_2\left(\frac{(n-1}{f_s}-\right)\end{array}\right]}^T& \left(11\right)\\ {}_n={\left[\begin{array}{cc}{}_1\left(\frac{(n-1}{f_s}\right)& {}_2\left(\frac{(n-1}{f_s}\right)\end{array}\right]}^T& \end{array}$
and stochastic Gaussian noise .sub.n, .sub.n. It follows from Equation 5 that

(17) $\begin{array}{cc}E\left[{}_n{}_n^T\right]=\frac{1}{2}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]& \left(12\right)\end{array}$
and the noise covariance matrix for the sampled value is constituted by the equation

(18) $\begin{array}{cc}C=E\left[{}_n{}_n^T\right]=\frac{1}{2}A\left(\right)A^T\left(\right).& \left(13\right)\end{array}$

(19) Parameter vector , which in the present case indicates the phase and time shift caused by the signal propagation and signal reception, is unknown on the receiver side.

(20) The model described above is apparent from FIG. 1. In this figure, a demodulator 1 is schematically illustrated by the particular corresponding signals. Demodulator 1 receives a receive signal y(t) on an input channel 2. M receive signals y.sub.m(t) are generated from receive signal y(t) in M separate channels with the aid of a multiplication device 3 by multiplying receive signal y(t) by a cosine function of a different phase in each case. Each of the generated signals y.sub.m(t) is low-pass filtered by a low pass filter device 5 of a corresponding bandwidth B. M low-pass-filtered output signals y.sub.m(t) are present in the M output signals of demodulator 1. The design variant illustrated in FIG. 1 corresponds to a demodulator 1 having the smallest possible structural unit, which is designated demodulation unit 4.

(21) The method of overdemodulation illustrated by FIG. 1 results in the known QAM method when two is selected for the number M of channels and [0 /2].sup.T is selected for the phase vector corresponding to Equation 6. This case results in the matrix

(22) $\begin{array}{cc}A\left({}_c\right)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]& \left(14\right)\end{array}$
so that, without a phase shift due to propagation, the individual signals may be described as
y.sub.1/2(t)=x.sub.1/2(t)+.sub.1/2(t), (15)
the noise in both channels being non-correlated according to

(23) $\begin{array}{cc}C=\frac{1}{2}A\left({}_c\right)A^T\left({}_c\right)=\frac{1}{2}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right].& \left(16\right)\end{array}$

(24) For the present method of overdemodulation, it is to be further assumed by way of example that the AD converter used in each individual M channel for digitizing the signals is a symmetrical 1-bit converter, so that the resulting digitized receive data r.sub.n{1,1}.sup.M may be described as
r=sign(y.sub.n), (17)
the sign function being defined by

(25) 0$\begin{array}{cc}\mathrm{sign}\left(z\right)=\{\begin{array}{cc}+1& \mathrm{if}z0\\ -1& \mathrm{if}z<0\end{array}.& \left(18\right)\end{array}$

(26) FIG. 2 shows a general AD converter device 10, a corresponding AD converter being arranged for each of channels 8 shown in FIG. 1. In the present case, quantization function Q is given as an example by the above sign function, whereby the 1-bit converter results. A correspondingly modified demodulator 1 thus outputs M digitized signals in the M output channels, the digital data only representing the sign of the sampled signals.

(27) The advantages of the overdemodulation method using a number of M>2 output channels is illustrated below by the measures of information according to Fisher and Shannon. The given channel problem is examined for this purpose, wherein vector , i.e. the phase and time shift resulting from the signal propagation and signal reception, is deterministically constituted but unknown as such to the receiver. Parameter may be indicated or ascertained, for example, using the maximum likelihood estimator (MLE) according to the maximum likelihood method

(28) $\begin{array}{cc}\begin{array}{c}\hat{}\left(r\right)=\arg \underset{}{\max }\ln p\left(r;\right)\\ =\arg \underset{}{\max }\underset{n=1}{\overset{N}{.Math.}}\ln p\left(r_n;\right),\end{array}& \left(19\right)\end{array}$
the digitized receive signal with N sampled values having the following form
r=[r.sub.1.sup.Tr.sub.2.sup.T . . . r.sub.N.sup.T].sup.T. (20)

(29) For a sufficiently large number N of sampled values, matrix R.sub. of the mean squared error deviation may be analytically indicated according to the Cramer-Rao inequality as the inverse Fischer Information Matrix (FIM)

(30) $\begin{array}{cc}\begin{array}{c}{\stackrel{\_}{R}}_{\hat{}}=\underset{N.fwdarw.}{\lim }E\left[\left(\hat{}\left(r\right)-\right){\left(\hat{}\left(r\right)-\right)}^T\right]\\ =F^{-1}\left(\right)\end{array}& \left(21\right)\end{array}$

(31) The FIM is defined by

(32) $\begin{array}{cc}F\left(\right)={}_{R}p\left(r;\right){\left(\frac{\ln p\left(r;\right)}{}\right)}^{2}dr,& \left(22\right)\end{array}$

(33) R being the mathematical carrier of digitized receive vector r, For temporary sampled values r.sub.n, the FIM may be additively written as

(34) $\begin{array}{cc}F\left(\right)=\underset{n=1}{\overset{N}{.Math.}}{F}_n\left(\right),& \left(23\right)\end{array}$
the sampled value-specific FIM being constituted as

(35) $\begin{array}{cc}{F}_n\left(\right)={}_{R_n}p\left(r_n;\right){\left(\frac{\ln p\left(r_n;\right)}{}\right)}^{2}d{r}_n.& \left(24\right)\end{array}$

(36) For a pessimistic measure of information according to Fisher, an approximation of F.sub.n() FIM may be assumed in the form
F.sub.n()F.sub.n(). (25)
With the moments
.sub.n()=.sub.R.sub.nr.sub.np(r.sub.n;)dr.sub.n
R.sub.n()=.sub.R.sub.n(r.sub.n.sub.n())(r.sub.n.sub.n()).sup.Tp(r.sub.n;)dr.sub.n, (26)
the pessimistic FIM results as

(37) $\begin{array}{cc}{\stackrel{\_}{F}}_n\left(\right)={\left(\frac{{}_n\left(\right)}{}\right)}^T{R}_n^{-1}\left(\right)\left(\frac{{}_n\left(\right)}{}\right).& \left(27\right)\end{array}$

(38) The first moment may be calculated element by element via

(39) $\begin{array}{cc}\begin{array}{c}{\left[{}_n\left(\right)\right]}_m=p\left({\left[r_n\right]}_m=1;\right)-p\left({\left[r_n\right]}_m=-1;\right)\\ =\mathrm{erf}\left(\frac{{\left[s_n\left(\right)\right]}_m}{{\sqrt{2}\left[C\right]}_{\mathrm{mm}}}\right),\end{array}& \left(28\right)\end{array}$

(40) where erf (z) is the error function. The following furthermore results for the second moment
[R.sub.n()].sub.mm=1[.sub.n()].sub.m.sup.2, (29)
with the off-diagonal entries
[R.sub.n()].sub.mk=4.sub.mk()(1[.sub.n()].sub.m)(1[.sub.n()].sub.k), (30)
where .sub.mk() is the cumulative density function (CDF) of the bivariate Gaussian distribution

(41) $\begin{array}{cc}p\left({\left[{}_n\right]}_m,{\left[{}_n\right]}_k\right)=N\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}{\left[C\right]}_{\mathrm{mm}}& {\left[C\right]}_{\mathrm{mk}}\\ {\left[C\right]}_{\mathrm{km}}& {\left[C\right]}_{\mathrm{kk}}\end{array}\right]\right)& \left(31\right)\end{array}$
with the upper bounds of integration
[[s.sub.n()].sub.m[s.sub.n()].sub.k].sup.T.

(42) The derivation of the first moment may be indicated element by element via

(43) $\begin{array}{cc}{\left[\frac{{}_n\left(\right)}{}\right]}_m={\frac{2}{\sqrt{2{\left[C\right]}_{\mathrm{mm}}}}\left[\frac{s_n\left(\right)}{}\right]}_m{e}^{-{\left(\frac{{\left[s_n\left(\right)\right]}_m}{\sqrt{{2\left[C\right]}_{\mathrm{mm}}}}\right)}^2},& \left(32\right)\end{array}$
where

(44) 0$\begin{array}{cc}\begin{array}{c}\frac{s_n\left(\right)}{}=\left[\frac{s_n\left(\right)}{}\frac{s_n\left(\right)}{}\right]\\ =\left[A\left(\right)\frac{B\left(\right)}{}{x}_n\left(\right)A\left(\right)B\left(\right)\frac{x_n\left(\right)}{}\right],\end{array}& \left(33\right)\end{array}$
and where

(45) $\begin{array}{cc}\frac{B\left(\right)}{}=\left[\begin{array}{cc}-\sin \left(\right)& \cos \left(\right)\\ -\cos \left(\right)& -\sin \left(\right)\end{array}\right]\frac{x_n\left(\right)}{}=-{\left[\frac{d{x}_1\left(t\right)}{dt}\frac{d{x}_2\left(t\right)}{dt}\right]}^T{|}_{t=\left(\frac{\left(n-1\right)}{f_s}-\right)},& \left(34\right)\end{array}$

(46) For performance statements of the overdemodulation method according to the pessimistic measure of information according to Fisher, a transmit signal of the following form is furthermore examined by way of example

(47) $\begin{array}{cc}{x}_{1/2}\left(t\right)=\underset{k=-}{\overset{}{.Math.}}{\left[b_{1/2}\right]}_{\mathrm{mod}\left(k,K\right)}g\left(t-{\mathrm{kT}}_b\right),& \left(35\right)\end{array}$
where b.sub.1/2{1,1}.sup.K binary vectors with K=1023 are symbols of a particular duration T.sub.b=977.52 ns and g(t) is a rectangular pulse having a bandwidth of B=1023 MHz. The sampling rate would be f.sub.s=2B to obtain a corresponding temporary sampling rate. If the signal is sampled for a period T=1 ms, N=2046 sampled values are obtained on the receiver side. The unknown vector is assumed as

(48) $={\left[\frac{}{8}0\right]}^{T^{}},$
the phase differences in the channels are situated equidistantly apart with [].sub.m=(m1)*/M and the performance of the method would be assumed to be M=2 and an unlimited AD resolution with regard to an ideal reference system. With regard to an ideal M=2 (QAM) system, the ratio of the mean squared error deviation of the overdemodulation method in decibels is then constituted by

(49) $\begin{array}{cc}{X}_{/}\left(\right)=10\log \left(\frac{{\left[{\stackrel{\_}{F}}^{-1}\left(\right)\right]}_{11/22}}{{\left[F_{\mathrm{ref}}^{-1}\left(\right)\right]}_{11/22}}\right),& \left(36\right)\end{array}$
where the FIM of the reference system is constituted by

(50) $\begin{array}{cc}{F}_{\mathrm{ref}}\left(\right)={}^2\underset{n=1}{\overset{N}{.Math.}}{\left(\frac{s_n\left(\right)}{}\right)}^T{C}^{-1}\left(\frac{s_n\left(\right)}{}\right).& \left(37\right)\end{array}$

(51) In the case of M=2, the noise is independent in both demodulation channels. Under this condition, Equation 36 results in the exact information or performance loss due to a 1-bit conversion.

(52) In FIGS. 3 and 4, the particular result is illustrated against an assumed carrier-to-noise ratio (C/N.sub.0), where C indicates the signal power of modulated signal x(t) (see Equation 1) in watts prior to receipt, and N.sub.0 indicates a noise power density in watts per hertz.

(53) The result with regard to the ascertainment of phase shift is represented in FIG. 3 and with regard to time shift in FIG. 4 by the signal propagation and signal receipt. In both figures, the result is also plotted for a different number M of channels.

(54) For both parameters, it is apparent that the quantization loss may be reduced by the employed 1-bit converter from 1.96 dB to 1.07 dB for M=16, due to the specified overdemodulation method within a range of a low signal-to-noise ratio. In the case of 75 dB Hz, the loss may even be reduced from 9.69 dB to 0.57 dB for the parameter of phase shift . For the parameter of time shift , the loss here may be reduced from 7.12 dB to 3.44 dB. With a high signal-to-noise ratio, in particular, the overdemodulation method therefore delivers a much better performance than the known QAM method with regard to an estimate of transmission parameters.

(55) In information theory, the indicated overdemodulation method may be interpreted as a so-called MIMO method (multiple input, multiple output), two inputs and M outputs being present with regard to the example examined. As a result, the output may be described as equation
y=Hx+, (38)
an AD converter of the form r=sign (y) being connected downstream.

(56) For a system of this type, the measure of information according to Shannon I (x; r), which permits a statement to be made about the maximum possible transmission rate, may be estimated by

(57) $\begin{array}{cc}I\left(x;r\right)\frac{1}{2}{\log }_2\det \left(1_M+R_{{}^{}{}^{}}^{-1}{H}^{}{R}_{\mathrm{xx}}{H}^T\right),& \left(39\right)\end{array}$
where R.sub.xx is the second moment of input signal x, and

(58) $\begin{array}{cc}\begin{array}{c}{R}_{{}^{}{}^{}}=\frac{2}{}\left(\arcsin \left({\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}{R}_{\mathrm{yy}}{\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}\right)\right)-\\ \frac{2}{}({\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}{R}_{\mathrm{yy}}{\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}+\\ \frac{2}{}({\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}{R}_{}{\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}\\ H^{}=\sqrt{\frac{2}{}}{\mathrm{diag}\left(R_{\mathrm{yy}}\right)}^{-\frac{1}{2}}{R}_{\mathrm{yy}}.\end{array}& \left(40\right)\end{array}$
applies.

(59) Assuming random input signals, which are independent from each other, and the covariance matrix

(60) $\begin{array}{cc}{R}_{\mathrm{xx}}=\frac{1}{2}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right].& \left(41\right)\end{array}$

(61) FIG. 5 shows the transmission rate achievable with the aid of the overdemodulation method, plotted against the signal-to-noise ration (SNR) for a different number M of demodulation channels, 1-B AD converters being used in each case. It is apparent that, with the aid of the overdemodulation method, the transmission rate for M with respect to co may be increased approximately 23% over the QAM method according to the prior art in the low SNR regime. For high signal-to-noise ratios, approximately 51% more data may be transmitted with the aid of the specified method than is possible with the aid of a known QAM method.

(62) The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications as would be obvious to one skilled in the art are to be included within the scope of the following claims.