Transmitters
3.4 PULSE COMPRESSION
3.4.3 Other types of modulation and their spectra
Much effort has been expended to find codes that are as simple as the Barker codes, give a narrow spike with low range sidelobes on recompression, have little degradation for echo signals with Doppler frequency shift, and can be implemented using digital circuitry. Digital circuitry is reproducible using standard hardware and no adjustment is
-1 -0.5
0.5 1
0 0.5
2 4
6 8
10 12
-1 -0.5
0.5 1
20 40
60 80
100
(a) Barker 13 element code (b) 100 element random binary code
Clock Reset to one
Shift register
Exclusive OR
Output
Figure 3.18. Two examples of the modulation for binary codes
Figure 3.19 An example of the generation of maximum length, pseudo-random, or Galois codes.
necessary and it is more flexible as changing the clock frequency changes the pulse compression ratio and switching the delays may be used to change the length.
In the search for codes that may be recombined to give narrow compressed pulses, a number of polyphase codes have been developed. These codes have low time sidelobes without resorting to tapering giving the resulting widening and losses.
In the sections that follow a number of codes are described that are used for pulse compression with examples with lengths of 16 and 100. For completeness the diagrams start with binary codes in Figure 3.18.
3.4.3.1 Binary code examples
Figure 3.18 shows a 13 element Barker code in comparison with a 100 element random binary code.
Binary codes may be stored in read only memory. A number of long codes with acceptable peak sidelobes may be found in [3, p. 108] or be generated using shift registers and exclusive OR blocks as shown in the example in Figure 3.19 for a maximum length sequence, pseudo-random sequence or Galois code [4, p. 693] of length 15 (four steps, length 2 – 1). The code in Figure 3.19 is4
1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0
and phase shifts are B, B, B, B, 0, 0, 0, B, 0, 0, B, B, 0, B, 0 radians.
Other binary codes can to be found in [6, p. 457].
A simple tapped transformer may be used to split the coherent oscillator signal into an in phase and antiphase signal, and a semiconductor switch is used to choose the phase as in Figure 3.20.
Alternating voltage from the coherent oscillator (COHO)
Switch to choose the in phase or antiphase component
Binary phase modulated out put
Digital to analogue converter
+
X
90 degree phase shift
X
In phase component (I)
Quadrature phase component (Q)
Modulated output (Coherent)
oscillator
Digital to analogue converter
NFrank ' 2B
N (i&1)(j&1) Figure 3.20 One way of modulating a binary coded signal.
Figure 3.21 A two phase vector modulator.
(3.11) Where more than one phase is used the modulating signals are generated either by the systems shown below or are held in memory as two phase signals (I and Q). In both cases a vector modulator is used to generate the waveform for the transmitter as shown in Figure 3.21.
Binary codes are sensitive to Doppler frequency shifts and their ambiguity functions are discussed in Chapter 8.
Polyphase codes described in the next sections bridge the gap between binary codes and linear frequency modulation and have been constructed so that no tapering is required in the compression stages (Chapter 8).
3.4.3.2 Frank code
The Frank code was the first code to give an accelerating phase using digital components. The phase of each element is given by
where the indices i and j are valued from 1 to N. The compression ratio D is N².
For N = 4 the phases are
0,0,0,0, 0, B/2, B, 3B/2, 2B, 3B, 4B, 5B, 6B, 8B, 10B, 12B
and it can be seen that these phases are added to the initial frequency f to give blocks of accelerating phase shown in0 Figure 3.22.
0 2B 4B 6B 8B 10B 12B
2 4 6 8 10 12 15
Element number
1 3 5 7 9 11 13 14 16
Frequency 0 Frequency 1 Frequency 2 Frequency 3 0 0 0 0 0 B/2 B 3/B2 2B 3B 4B 5B 6B 8B 10B 12B
Frequency
Added phase at frequency 0
(a) 16 element Frank code N = 4 (b) 100 element Frank code N = 10 (c) Phases in a 100 element Frank code
Figure 3.22 The accelerating phase and frequency blocks in a 16 element Frank code (N = 4).
Figure 3.23 Examples of the modulation used for Frank codes.
The modulation for the 16 element (N =4) code in Figure 3.22 is shown in Figure 3.23(a). With greater pulse compression ratios, D = 100 and N = 10, in Figure 3.23(b) and (c). N phases are used.
The Cartesian values (I and Q) may be stored in a read only memory (ROM) but historically digital circuit components used for the fast Fourier transform may be used to expand and compress pulses; an example is shown in Figure 3.24. The components are delay elements for one chip width (top) and four chip widths (bottom), multiplier and adder blocks denoted by x +, and adders, +. Where complex multiplication is used, denoted by j at the inputs, both the I and Q phases must be implemented: the conjugates of the echo signals are always complex. Synchronous logic is used and the clock frequency may be changed as necessary for each pulse length.
To expand the pulse a single pulse is clocked along the top row of delays implemented as a shift register. Before entering the first delay the pulse is present at the input of the first and third multipliers from the left where they are both multiplied by the weighting factors +1 and are passed to the adder so that there are two pulses present simultaneously at the outputs of the first row of multipliers and adders. The second row of multipliers includes weights of j and -j and the adders combine the weighted inputs to give an output for each position of the pulse in the top row shift register.
The bottom row consists of three shift registers that have a delay of four clock pulses so that the first four clock pulses the outputs are 1 from the leftmost multiplier and adder block. The next group of four pulses come from the third multiplier-adder, as would be the case with a four point fast Fourier transform circuit and have the values 1, j, -1, -j or phases angles 0, B/2, B, 3B/2 radians. Note that the add symbol means OR. As there is only one input to each of the multiplier-adder blocks.
+
X X
+
X
+
X
+
X X
+
X
+
X
+
X X
+
X
+
X
+
X X
+
X
+
X
+
X X
+
X
+
X
+
X X
+
X
+
X
+
X X
+
X
+
X
+
X X
+
X
+
X
+ + + +
Delay Jc
Delay 4Jc
Delay Jc Delay Jc
Delay 4Jc Delay 4Jc
1
1 1
1
1 -1 1 -1 1
1 1 1 -1 j 1 -j
Short pulse expander input for the transmitter
Conjugates of the returning echo signals
Compressed output (to signal processor) Expanded
output to the transmitter
A B C D
E F G H
NP1 ' &B
N (N & (2j & 1)) (N(j & 1) % (i & 1))
Figure 3.24 A circuit to expand and compress signals for a 16 element (N = 4) Frank code.
(3.12) The complete run of 16 phases is thus
0, 0, 0, 0 0, B/2, B, 3B/2 0, B, 2B, 3B 0, 2B, 3B, 4B
The Frank code does not tolerate bandwidth clipping in the transmitter and receiver circuits [1, p. 15], and attempts to overcome this led to the development of the U.S. Naval Research Laboratory P-codes [1]. The first was the P1 code, see [1, p. 68].
3.4.3.3 P1 code
With the P1 code the frequency groups are rearranged in time to place the smallest phase jumps in the center of the pulse to give the phase run, giving
where the indices i and j are valued from 1 to N. The compression ratio D is N².
P1 code sequences for 16 and 100 elements are shown in Figure 3.25.
(a) 16 element P1 code (b) 100 element P1 code N = 10 (c) Phases in a 100 element P1 code
NP2 ' B 2
N&1 N &B
N (i&1) (N%1&2j)
(a) 16 element P2 code (b) 100 element P2 code N = 10 (c) Phases in a 100 element P2 code
NP3 ' B (i&1)2 D
Figure 3.25 P1 code phase sequences for lengths of 16 and 100 elements.
(3.13)
Figure 3.26 The modulation phases for 16 and 100 element P2 codes.
(3.14) 3.4.3.4 P2 code
The P2 code is a modification of the P1 code using the weighting in a Butler matrix for antennas to give a symmetrical code and has the bandwidth tolerance of the P1 code. The phases of the individual elements are given by
where the indices i and j are valued from 1 to N. The compression ratio D is N².
The phases of the modulation for 16 and 100 element P2 codes are shown in Figure 3.26.
3.4.3.5 P3 code
Linear frequency modulation has much better Doppler frequency characteristics than stepped frequency modulation. The phases of the elements are then given by
where the index i is valued from 1 to the compression ratio D.
The accelerating phases are shown for 16 and 100 element codes in Figure 3.27.
(a) 16 element P3 code (b) 100 element P3 code D = 100 (c) Phases in a 100 element P3 code
-1 -0.5
0 0.5
1
-1 -0.5
0.5 1
20 40
60 80
100
NP4 ' B (i & 1)2
D & B(i & 1) Figure 3.27 The phases for the modulation for 16 and 100 element P3 codes
Figure 3.28 The run of phases in a 100 element P3 code.
(3.15) The run of phases can be better illustrated by a line joining the phase vectors as in Figure 3.28.
3.4.3.6 P4 code
The P4 code has the same relationship to the P3 code as the P1 code has to the Frank code and is given by
where the index i is valued from 1 to the compression ratio D.
The phase changes are greatest at the start and end of the expanded pulse, and phase vectors for 16 and 100 element P4 codes are given in Figure 3.29.
(a) 16 element P4 code (b) 100 element P4 code D = 100 (c) Phases in a 100 element P4 code
0 1
-1 -0.5
1
20 40 60 80 100
+ +
+ + + + + + + + + +
+ Delay 4Jc + Delay 4Jc + Delay 4Jc + Compressedoutput (to signal processor) Expanded
output to the transmitter
B B
E F G H
Figure 3.29 The phase vectors for 16 and 100 element P4 codes.
Figure 3.30 The run of phases in a 100 element P4 code.
Joining the tips of the phase vectors shows its relation to the P3 code and is shown in Figure 3.30.
The phase shifts necessary in Figure 2.24 are shown in Table 3.3.
Table 3.3
Phasing for the generation of the P codes
Code A B C D E F G H
Frank 0 0 0 0 0 0 0 0
P1 0 5B/4 B/2 7B/4 B B 0 0
P2 0 5B/4 B/2 7B/4 9B/8 -3B/8 3B/8 -9B/8
P3 0 B/16 4B/16 9B/16 0 0 B B
P4 The strapping from the lower row of adders is different and is shown below.
0 B/16 4B/16 9B/16 0 0 B B
Energy '
I
over the pulse widthJt
Pt dt (3.16)
3.4.3.7 Other polyphase codes
Barker codes may be nested but they can have awkward peak sidelobes, whereas examples of polyphase Barker codes may be found in [3, p. 110]. Frank, P1, and P2 codes have a length of N ² and these have been extended to include any length by Zadoff and Chu [3, p. 122]. The reference also describes other types of codes.
3.4.3.8 Costas codes
Costas codes are the result of rearranging the order of the frequency steps in stepped frequency linear frequency modulation [3] so that the parts may be recombined to give a thumbtack ambiguity function. The spectrum of the signal resembles linear frequency modulation with additional sidebands for the frequency switching function.
Other forms of phase modulation using less abrupt phase changes give narrower spectra. Examples are:
• Widest spectrum, sinx/x spectrum width between the first nulls 2/Jc; Barker codes, pseudo-random and random binary phase codes;
P1 to P4 codes;
Frank codes, stepped frequency modulation [4];
• Narrower spectra with relatively steep sides;
Nonlinear frequency modulation, Costas codes;
Linear frequency modulation (depends on time-bandwidth product).
The transmitter must amplify the full signal bandwidth with minimal distortion in amplitude and phase so that when the expanded pulses are passed through a compression filter, the original narrow pulse is restored with the resolution in range. The distorted components, such as amplitude and phase distortion, limited bandwidth, and analogue-to-digital converter errors, go to increasing the time sidelobes that decrease resolution and probability of detection (see Chapter 8, Matched and matching filters). This process takes time and gives rise to time lobes before and after the delayed restored pulse. These time lobes are reduced by windowing or weighting.