Correlator bank for preamble detection
A method for detecting a preamble waveform of a received signal is described. The method includes dividing a correlation into a plurality of sub-correlations, for a plurality of frequency offset indices, k, covering a range of frequency offsets. The correlation has a correlation length equal to N, and a plurality of sub-correlations has a sub-correlations quantity equal to I, each sub-correlation of the plurality of sub-correlations has a sub-correlation length equal to M. A complex oscillation for a template frequency offset associated with k is approximated, the approximation is to be constant over an M-sample interval and is a piece-wise approximation. The approximated complex oscillation has length I. A quantity I of sub-correlations at each k is assembled using the approximated complex oscillation of length I. The method further includes determining that the received signal comprises the preamble based on the assembled sub-correlations and a correlation threshold.
1. A method for detecting a preamble waveform of a received signal over a range of frequency offsets, the received signal comprising a preamble, the preamble having a preamble length corresponding to a plurality of complex samples in the preamble, the preamble comprising a plurality of waveforms in a sequence, each waveform having a plurality of samples, the method comprising:
dividing a correlation into a plurality of sub-correlations, the correlation being associated with the received signal and being for a plurality of frequency offset indices covering the range of frequency offsets, the correlation having a fixed correlation length and each sub-correlation of the plurality of sub-correlations having a fixed sub-correlation length;
approximating a complex oscillation for a template frequency offset associated with the plurality of frequency offset indices, the approximation to be constant over the fixed sub-correlation length, the approximation being a piece-wise approximation, the approximated complex oscillation having a length equal to the number of waveforms in the plurality of waveforms;
assembling a quantity of sub-correlations at each frequency offset index of the plurality of frequency offset indices using the approximated complex oscillation; and
determining that the received signal comprises the preamble based on the assembled sub-correlations and a correlation threshold.
2. The method of claim 1 , wherein, for a given sample rate, the fixed sub-correlation length is determined based on a predetermined level of signal energy loss due to the piece-wise approximation.
3. The method of claim 2 , wherein a signal energy loss of a waveform of the plurality of waveforms at a frequency offset, f, due to the piece-wise approximation is:
-
1
0
log
1
0
{
sin
(
π
M
f
1
/
f
s
)
sin
(
π
f
1
/
f
s
)
·
sin
[
π
Δ
min
(
f
1
)
/
f
s
]
sin
[
π
M
Δ
min
(
f
1
)
/
f
s
]
}
2
M being a quantity of the plurality of samples of the waveform;
f 1 being a maximum expected 1-sided received frequency offset;
f s being the given sample rate; and
Δ min (f) being a frequency distance from f to a nearest correlator bank frequency offset among a set of correlator bank frequency offsets.
4. The method of claim 3 , wherein Δ min (f) is expressed as:
Δ min (f)=min k∈ (|f−f k |)
K being a set of frequency offset indices;
k being a frequency offset index from K;
f k being correlator bank offset frequencies expressed as:
f k =Δf·k, for k∈[−K 1 , +K 1 ]
Δf being a uniform frequency spacing between the correlator bank offset frequencies; and
K 1 being the number of correlator bank offset frequencies on each side of 0 Hz.
5. The method of claim 4 , wherein:
Δf=f s /N
N being a quantity of the plurality of complex samples in the preamble.
6. The method of claim 1 , wherein assembling the quantity of sub-correlations is performed using a Fast Fourier Transform (FFT) in conjunction with a frequency bin de-mapping function.
7. The method of claim 5 , wherein the correlation is performed using a correlator, for detecting an orthogonal frequency division multiplex (OFDM) long preamble, that uses only two sub-correlators, each of fixed sub-correlation length M, with at least one delay, the correlator having the following parameters:
5 sub-correlations;
sub-correlation length M=32;
N=160;
sample rate f s =20 MHz; and
maximum 1-sided frequency offset f 1 =116.5 kHz.
8. The method of claim 7 , wherein the parameters further include;
Δf=78125 Hz;
and K 1 =1.
9. The method of claim 8 , for k=−1, 0, and 1, the correlation having N=160 is performed using a plurality of sums from outputs of one or both of the sub-correlators and the at least one delay, the plurality of sums being expressed as:
y
n
(
1
6
0
,
k
)
≈
∑
i
=
0
4
e
-
j
π
k
i
/
4
·
y
n
-
1
6
0
[
1
-
(
i
+
1
)
/
5
]
(
32
,
0
,
i
)
resulting in assembler phasors that are either ±1, ±j, (1±j)/√{square root over (2)}, or (−1±j)/√{square root over (2)}; and
y n (160,k) corresponding to the assembled sub-correlations for each k.
10. The method of claim 9 , wherein the correlation threshold is a preset correlation level, and determining that the received signal comprises the preamble includes:
determining a correlation peak associated with the received signal is above the preset correlation level corresponding to y n (160,k) .
11. A correlator apparatus configured for detecting a preamble waveform of a received signal over a range of frequency offsets, the received signal comprising a preamble, the preamble having a preamble length corresponding to a plurality of complex samples in the preamble, the preamble comprising a plurality of waveforms in a sequence, each waveform having a plurality of samples, the correlator apparatus comprising processing circuitry configured to:
divide a correlation into a plurality of sub-correlations, the correlation being associated with the received signal and being for a plurality of frequency offset indices covering the range of frequency offsets, the correlation having a fixed correlation length and each sub-correlation of the plurality of sub-correlations having a fixed sub-correlation length;
approximate a complex oscillation for a template frequency offset associated with the plurality of frequency offset indices, the approximation to be constant over the fixed sub-correlation length, the approximation being a piece-wise approximation, the approximated complex oscillation having a length equal to the number of waveforms in the plurality of waveforms;
assemble a quantity of sub-correlations at each frequency offset index of the plurality of frequency offset indices using the approximated complex oscillation; and
determine that the received signal comprises the preamble based on the assembled sub-correlations and a correlation threshold.
12. The correlator apparatus of claim 11 , wherein, for a given sample rate, the fixed sub-correlation length is determined based on a predetermined level of signal energy loss due to the piece-wise approximation.
13. The correlator apparatus of claim 12 , wherein a signal energy loss of a waveform of the plurality of waveforms at a frequency offset, f, due to the piece-wise approximation is:
-
1
0
log
1
0
{
sin
(
π
M
f
1
/
f
s
)
sin
(
π
f
1
/
f
s
)
·
sin
[
π
Δ
min
(
f
1
)
/
f
s
]
sin
[
π
M
Δ
min
(
f
1
)
/
f
s
]
}
2
M being a quantity of the plurality of samples of the waveform;
f 1 being a maximum expected 1-sided received frequency offset; and
f s being the given sample rate; and
Δ min (f) being a frequency distance from f to a nearest correlator bank frequency offset among a set of correlator bank frequency offsets.
14. The correlator apparatus of claim 13 , wherein Δ min (f) is expressed as:
Δ min (f)=min k∈ (|f−f k |)
K being a set of frequency offset indices;
k being a frequency offset index from K;
f k being correlator bank offset frequencies expressed as:
f k =Δf·k, for k∈[−K 1 , +K 1 ]
Δf being a uniform frequency spacing between the correlator bank offset frequencies; and
K 1 being the number of correlator bank offset frequencies on each side of 0 Hz.
15. The correlator apparatus of claim 4 , wherein:
Δf=f s /N
N being a quantity of the plurality of complex samples in the preamble.
16. The correlator apparatus of claim 11 , wherein assembling the quantity of sub-correlations is performed using a Fast Fourier Transform (FFT) in conjunction with a frequency bin de-mapping function.
17. The correlator apparatus of claim 15 , wherein the correlation is performed using a correlator, for detecting an orthogonal frequency division multiplex (OFDM) long preamble, that uses only two sub-correlators, each of fixed sub-correlation length M, with at least one delay, the correlator having the following parameters:
5 sub-correlations;
sub-correlation length M=32;
N=160;
sample rate f s =20 MHz; and
maximum 1-sided frequency offset f 1 =116.5 kHz.
18. The correlator apparatus of claim 17 , wherein the parameters further include:
Δf=78125 Hz; and
K 1 =1.
19. The correlator apparatus of claim 18 , for k=−1, 0, and 1, the correlation having N=160 is performed using a plurality of sums from outputs of one or both of the sub-correlators and the at least one delay, the plurality of sums being expressed as:
y
n
(
1
6
0
,
k
)
≈
∑
i
=
0
4
e
-
j
π
k
i
/
4
·
y
n
-
1
6
0
[
1
-
(
i
+
1
)
/
5
]
(
32
,
0
,
i
)
resulting in assembler phasors that are either ±1, ±j, (1±j)/√{square root over (2)}, or (−1±j)/√{square root over (2)}; and
y n (160,k) corresponding to the assembled sub-correlations for each k;
the correlation threshold being a preset correlation level; and
determining that the received signal comprises the preamble includes determining a correlation peak associated with the received signal is above the preset correlation level corresponding to y n (160,k) .
20. A system, the system comprising a correlator apparatus configured for detecting a preamble waveform of a received signal over a range of frequency offsets, the received signal comprising a preamble, the preamble having a preamble length corresponding to a plurality of complex samples in the preamble, the preamble comprising a plurality of waveforms in a sequence, each waveform having a plurality of samples, the correlator apparatus comprising processing circuitry configured to:
divide a correlation into a plurality of sub-correlations, the correlation being associated with the received signal and being for a plurality of frequency offset indices covering the range of frequency offsets, the correlation having a fixed correlation length and each sub-correlation of the plurality of sub-correlations having a fixed sub-correlation length;
approximate a complex oscillation for a template frequency offset associated with the plurality of frequency offset indices, the approximation to be constant over the fixed sub-correlation length, the approximation being a piece-wise approximation, the approximated complex oscillation having a length equal to the number of waveforms in the plurality of waveforms;
determine, for a given sample rate, f s , the fixed sub-correlation length based on a predetermined level of signal energy loss due to the piece-wise approximation;
assemble a quantity of sub-correlations at each frequency offset index of the plurality of frequency offset indices using the approximated complex oscillation; and
determine that the received signal comprises the preamble based on the assembled sub-correlations and a correlation threshold.