Electricity: measuring and testing – Measuring – testing – or sensing electricity – per se – Analysis of complex waves
Reexamination Certificate
2000-11-17
2002-09-17
Le, N. (Department: 2858)
Electricity: measuring and testing
Measuring, testing, or sensing electricity, per se
Analysis of complex waves
Reexamination Certificate
active
06452376
ABSTRACT:
The invention relates to a method of estimating the carrier frequency of a phase-modulated signal.
BACKGROUND OF THE INVENTION
Digital radio transmissions usually employ phase modulation. Phase-modulated transmission consists of transmitting a sinusoidal carrier, for example of constant amplitude, with a particular angular frequency &ohgr;
0
and applying to that carrier in each symbol period a phase shift that depends on the value of the symbol to be transmitted. If the symbol to be transmitted is a binary digit 0 or 1, the phase shift can take only two values, for example 0 or &pgr; radians. If the symbol is a two-digit binary number, the phase shift can take four values, for example 0, &pgr;/2, &pgr; and 3&pgr;/2 radians. The zero phase shift corresponds to the symbol 00, for example, the phase shift of &pgr;/2 radians corresponds to the symbol 01, the phase shift of &pgr; radians corresponds to the symbol 10, and the phase shift of 3&pgr;/2 radians corresponds to the symbol 11. Phase modulation of this kind is often referred to as M-PSK modulation, where PSK denotes “phase shift keying” and M is the total number of phases.
When the phase-modulated signal is received, the receiver determines the values of the phase shifts in order to determine the values of the symbols. To determine the phase shifts, i.e. the information part of the received signal, the receiver (modem) must know other parameters of the signal, namely its amplitude, frequency, and original phase.
The received signal is generally expressed by the complex number Z having the following value:
Z
(
t
)
=
A
ⅇ
j
(
ω
0
t
+
ϕ
0
+
m
π
2
)
(
1
)
In the above equation, t is time, A is the amplitude of the carrier, &ohgr;
0
=2&pgr;f
0
is the angular frequency of the carrier, &psgr;
0
is an original phase shift and m is an integer which represents the symbol. In the above example, m can take the values 0, 1, 2 and 3 and those values respectively correspond to the symbols 00, 01, 10 and 11.
i is the imaginary number such that:
j
2
=−1 (2)
If the values A (amplitude), &psgr;
0
(initial phase) and &ohgr;
0
are known, the number m can easily be deduced. All that is required is to multiply Z(t) by:
1
A
ⅇ
-
j
(
ϕ
0
+
ω
0
t
)
(
3
)
This yields
Z
(
t
)
·
1
A
ⅇ
-
j
(
ϕ
0
+
ω
0
t
)
=
ⅇ
j
m
π
2
(
4
)
The receiver generally has available to it information about the expected carrier frequency. However, the carrier frequency often does not have exactly the expected value; this is why it must be possible to estimate it. There are various causes of carrier frequency error: they include the Doppler effect, for example, when the receiver or the transmitter is moving, and inaccuracy of the local clock of the transmitter, which determines the carrier frequency.
A Fourier transform is applied to the received signal to determine the carrier frequency. The Fourier transform provides the spectrum of the received signal, i.e. the curve of signal amplitude variation as a function of frequency. The spectrum theoretically has a center line at the frequency f
0
and of high amplitude compared to the remainder of the spectrum. However, because the Fourier transform is computed over a finite period, the line has a finite width. The frequency position of the center line is also inaccurate because of additive noise on the transmission channel and because the measurement is based on sampling.
It is known in the art to use a Nyquist bandpass filter with a relatively wide passband to reduce noise before computing the Fourier transform. In this way the noise which is uniformly distributed across the spectrum is eliminated outside the passband of the Nyquist filter.
Then, before computing the Fourier transform, and in order to obtain a pure carrier, the phase of the received signal Z(t) is multiplied by the number N of symbols, i.e. by 4 in the above example. For this purpose, the signal Z(t) can be raised to the power N, for example. Accordingly, when N=4:
&AutoLeftMatch;
[
Z
(
t
)
A
]
4
=
ⅇ
j
(
4
ω
0
t
+
m
4
π
2
+
4
ϕ
0
)
=
ⅇ
j
(
4
ω
0
t
+
4
ϕ
0
)
(
5
)
It can therefore be seen that the signal raised to the power N does not depend on m, i.e. it does not depend on the modulating value.
The Fourier transform is then computed from the pure signal, which supplies the value 4&ohgr;
0
. All that then remains is to divide the value obtained by 4.
Despite the presence of the Nyquist filter, this method of estimating the carrier frequency of the receive signal is very sensitive to noise, which constitutes a particularly serious problem for a radiocommunications system, for which there are many sources of interference.
OBJECTS AND SUMMARY OF THE INVENTION
In a first aspect, the invention relates to a method of estimating the carrier frequency which achieves better elimination of noise than the prior art method.
In a second aspect, the invention relates to a method of estimating the carrier frequency which, for a given noise level, supplies a more accurate estimate than the prior art method.
Accordingly, in the first aspect of the invention, the method of estimating the carrier frequency of a phase-modulated digital signal entails Nyquist filtering the received signal, then multiplying the phase of the received signal by the number N of symbols, or by a multiple of that number N, and then computing the Fourier transform of the signal whose phase has been multiplied by N, and after multiplying the phase of the signal by the number N, or by a multiple of N, applying filtering using a bandpass filter whose characteristic has steep flanks and a flat top in the passband and whose passband is substantially equal to the range of uncertainty as to the carrier frequency and centered in that range.
The bandpass filter whose characteristic has steep flanks and a flat top in the passband is preferably a Hermite filter as described in U.S. Pat. No. 5,886,913.
The invention is based on the observation that multiplying the phase by the number N increases the noise significantly, which noise is significantly reduced by the filter whose characteristic has steep flanks. Using a filter whose characteristic has steep flanks produces a passband that is significantly narrower than that obtained with a Nyquist filter and therefore improves noise elimination.
Note that the filter is used after multiplying the phase of the signal by N because, prior to multiplication, since the received signal is not a signal at a pure frequency, filtering narrower than the main half-lobe would distort the signal, whereas after multiplying the phase by N the signal is at a pure frequency and the passband of the filter can be as narrow as is permitted by the range of uncertainty as to the frequency, intersymbol distortion no longer applying.
A Hermite filter has the advantage of requiring a relatively modest quantity of computation.
In a second aspect of the invention, which can be used independently of or in combination with its first aspect, the invention also provides a method of estimating the frequency of a carrier of a phase-modulated digital signal in which the received signal is Nyquist filtered, the phase of the received signal is multiplied by the number N of symbols, or by a multiple thereof, the Fourier transform of the signal whose phase has been multiplied by N, or by a multiple of N, is computed, and the frequency is determined for which the spectrum obtained in this way has a maximum, wherein, to determine the frequency of the maximum of the spectrum, the derivative of that spectrum and the frequency for which that derivative is zero are determined.
The derivative can be obtained by calculating the following quantity:
D=Im[FFT
(
Z
t
)·
FFT*
(
tZ
t
)] (6)
In the above equation, Im signifies the imaginary part of the quantity between square
Bertrand Pierre
Marguinaud Andre
Alcatel
Le N.
LeRoux Etienne P
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