Artefactos

The method of estimating the pulse arrival time by noting the instant that the input signal crosses a given level is prone to frequent false alarms, since noise bursts or low energy extraneous signals could have a brief amplitude that exceeds the threshold. In order to achieve maximum probability of signal detection and minimum probability of false alarms for a given energy to noise density ratio, E /N₀, the detector should consist of a matched filter. In the case of a rectangular pulse, this filter is an integrator with a discharge switch that resets the integrator at the end of the pulse duration. Since the pulse arrival time is initially not known, the receiver searches for pulses in a pulse train by adjusting the instant of the start of integration until the output of the integrator reaches a peak value just before the closing of the switch. In this case, the minimum passband bandwidth is 2/T, where T is the pulse duration. Figure 2.5 compares the arrival time resolution of two pulses having the same energy. Both pulses give the same output of the matched filter if timing is precise. However, the short, high-amplitude pulse in Figure 2.5(b) gives better resolution because the slope of the integrated signal is greater than that of the pulse in Figure 2.5(a). The bandwidth needed to pass the signal in Figure 2.5(b) is four times greater than that needed for the signal in Figure 2.5(a), which is consistent with (2.8).

We see that the short pulse system has the higher range precision, even though both the short and long pulse systems have the same pulse energy. Average power per pulse is maintained by increasing peak power of the short duration pulse. The pulse repetition rate is the same in both cases.

It is possible to maintain a reasonable peak to average power ratio using a wide pulse, while increasing the bandwidth considerably in order to get good time-of-arrival precision. The method of doing this is called pulse compression. Two common pulse compression methods used for ranging are chirp modulation and

(a)

Time Time Input

Output

A₀ 4T₀

4A₀T₀

Input

Output 4A0

T₀

(b) 4A₀T₀

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 2.5 Received pulses at the input and output of an integrator matched filter. (a) Long pulse, T= T0; and (b) short pulse, T= T0/4.

direct sequence spread spectrum. In addition to giving increased range precision, they both discriminate against interference.

A chirp pulse is created by frequency modulating the pulse with a linearly changing (increasing or decreasing) sawtooth baseband signal, expressed as follows:

s(t)= sin冋^2␲⭈ 冉^{f0+k2 ⭈ t}冊^{⭈ t + ␾}册 ^{0≤ t ≤ T (2.10)}

where s(t) is the chirp pulse, f₀ is the start frequency, k is the rate of frequency change per unit time,␸is a random phase, and T is the pulse width. A chirp pulse with k = 20 is shown in Figure 2.6.

Figure 2.6 Chirp pulse. Rate of frequency change k= 20.

The chirp signal is detected using a matched filter. The impulse response h(t) of a matched filter is a delayed and reversed version of the input pulse, expressed as

h(t)= s(T − t) (2.11)

Figure 2.7 shows the single pulse spectrum and detector output of signals with the same pulse width T and start frequency f₀, but different values of chirp parame-ter k. Detector outputs are the squared outputs of matched filparame-ters. All signals have the same energy but the time resolution of the detector outputs is inversely proportional to the bandwidth, which is a function of the parameter k. The signal of Figure 2.7(a) has the widest spectrum and sharpest pulse arrival time resolution.

0 0.5 1 1.5 2

Time

0 10 20 30 40 50

Frequency

(a)

Figure 2.7 Chirp pulse spectrum and time resolution for different values of chirp parameter k.

(a) k= 20, (b) k = 5, and (c) k = 0.

0 0.5 1 1.5 2 Time

0 10 20 30 40 50

Frequency

(b)

0 10 20 30 40 50

Frequency

0 0.5 1 1.5 2

Time (c) Figure 2.7 (continued).

In Figure 2.7(b), the bandwidth is approximately one-fourth as large as that of Figure 2.7(a), and time resolution is around four times worse. A constant frequency pulse shown in Figure 2.7(c) has the same energy as the two chirp pulses but does not provide their advantages of time resolution and interference rejection. It is evident that in order to use the improved time resolution obtainable from the wide bandwidth chirp signals, the receiver clock rate must be high enough to detect the compressed matched filter output pulse.

The matched filter for chirp pulse generation and detection is commonly imple-mented using a SAW dispersive delay line that is fabricated specifically to match the known parameters of the signal. A dispersive delay line has a propagation time between input and output that is a function of signal frequency.

A second method for increasing bandwidth while maintaining a constant pulse duration at a given pulse energy is based on direct sequence spread spectrum. In the transmitter, the RF pulse carrier is modulated by a sequence of bits that have very good autocorrelation properties. These bits, which are used for pulse compression and not directly as data, are called chips. The received signal, r(t), which contains added noise and interference, is cross correlated with a locally generated sequence s(t) that corresponds to the expected chip sequence. The correla-tion process over a pulse duracorrela-tion T can be expressed as

Z(T)=冕^T

0

r(t) s(t) dt (2.12)

When r(t) and s(t) are similar over a period of T and are lined up in time phase, Z(t) will have a maximum value that is proportional to the energy of the received signal over the pulse width T. If s(t) has good autocorrelation properties, the output Z(T) will be relatively small compared to its maximum value when r(t) is shifted in time by one chip or more relative to the locally generated s(t). The output of the correlator, then, is a compressed pulse with average width of one chip that has the same equivalent energy as the input pulse of width T. Time-of-arrival resolution is± one chip.

An example of good pulse compression sequences are Barker codes, listed in Table 2.1 for N= 5, 7, 11, and 13, where N is the number of bits in a sequence.

The bits are bipolar and therefore are shown as sequences of plus and minus symbols. Note that the bits in a sequence may be inverted, or the sequence may be reversed, without affecting the cross correlation properties.

Table 2.1 Barker Codes for N= 5, 7, 11, 13

N Sequence

5 + + + − + 7 + + + − − + − 11 + + + − − − + − − + − 13 + + + + + − − + + − + − +

Equation (2.13) is a discrete expression for (2.12) where k is the shift in chips between the two sequences, where s_iis a chip of the local generated sequence and r_{i+ k}is a received chip at the sample time i.

Z_k= ∑^N

i= 1

r_{i+ k}s_i (2.13)

Table 2.2 shows the values of Z_kfor k= 0 . . . 6 using the 7-bit Barker code of Table 2.1, calculated with (2.13). The Z_k’s when k equals 1 to N− 1 are called side lobes. The first row is the prototype sequence {s_i} and the following rows are the shifted chip sequences {r_{i+ k}}, free of noise and interference The side lobes vary between−1 and 0, and the correlation when the sequences line up at k = 0 is 7. This shows that the input pulse with Barker code modulation whose energy is spread over seven chips has a seven times improvement in time of arrival resolution at the output of the correlation process. Similar improvement, in proportion to the value of N, is obtained with other Barker sequences and different codes with good autocorrelation properties.

In the simplified spread spectrum receiver block diagram Figure 2.8, a baseband matched filter implements the correlator. It is followed by a sliding window low-pass filter. When the expected spread spectrum pulse is received, the digitally filtered output of the matched filter exceeds the threshold of the detector, which

Table 2.2 Correlation Values for 7-Chip Barker Code

s1-1 s2-1 s3-1 s4-−1 s5-−1 s6-1 s7-−1 k Zk

−1 0 0 0 0 0 0 6 −1

1 −1 0 0 0 0 0 5 0

−1 1 −1 0 0 0 0 4 −1

−1 −1 1 −1 0 0 0 3 0

1 −1 −1 1 −1 0 0 2 −1

1 1 −1 −1 1 −1 0 1 0

1 1 1 −1 −1 1 −1 0 7

RF Demodulator Matched filter

Sequence coefficients

Lowpass filter

Threshold detector

Clock

Read clock

Figure 2.8 Spread spectrum pulse receiver with matched filter.

outputs the value of the real time clock, which is an estimate of the time of arrival of the pulse.

Figure 2.9 is a digital correlator based on the 7-chip Barker code. Note the direction of the input chips, and that the locally stored sequence is in reverse order to that direction. This is in conformance with the impulse response of the matched filter, given in (2.11). The matched filter consists of six 1-bit delay elements, which could be implemented by shift registers, multipliers and an accumulator.

A simulation output of the spread spectrum pulse arrival time estimator with a 7-chip Barker code is shown in Figure 2.10. Noise was added to the input signal

−Tc

−Tc −Tc −Tc −Tc −Tc

Demodulator

1 1 −1 −1 1 −1

1

Σ

Output 1 1 1 1 1 1 1− − −

Figure 2.9 A 7-chip digital matched filter.

Figure 2.10 Result of simulation of arrival time detection by matched filter of a seven chip spread spectrum sequence in noise.

for E /N₀= 18.5 dB. Note that the matched filter output peak is clearly distinguished from sidelobes and noise.

Frame synchronization in burst type data communication systems is often achieved using the spread spectrum correlation technique described in the preceding paragraphs. An 11-bit Barker code is the basis for despreading the data in 1 and 2-Mbps IEEE 802.11 DSSS physical layer used in WLAN. Other protocols include a frame synchronization sequence as part of the packet preamble, or of every data frame. Usually these sequences are more than 13 chips long, for which there is no Barker code, so other sequences with good correlation properties are employed.

The frame delineation epoch is a convenient place to make a time-of-arrival estima-tion for TOA and TDOA distance measuring and locaestima-tion methods.