than of radar systems, where the terminology stems from. For instance, a great number of regular algorithms for constructing Costas arrays [54], i.e. square (M ¼ N) sonar arrays or FSK sequences having equal length and number of frequencies, are known.

Problems

6.1. The frequency of a rectangular pulse drops linearly throughout its duration T ¼ 10 ms from 110 to 90 MHz. Calculate the processing gain of the signal. What is the approximate duration of the signal at the matched filter output? Sketch the ambiguity function and ambiguity diagram.

6.2.The frequency of a rectangular pulse drops linearly over the first half of its

duration T ¼ 10 ms from 110 to 90 MHz and then grows linearly from 90 to 110 MHz over the second half. Calculate the processing gain of the signal. Sketch the ambiguity function and its low-level and high-level horizontal sections.

6.3.Calculate aperiodic and periodic ACF for a binary Barker code of length N ¼ 11. Try to do it the most economical way.

6.4.Take a periodic sequence faig of period N and form a new sequence fbig picking each dth element of faig: bi ¼ adi, where multiplication in the index is modulo N. Such a transform is called the decimation of faig with index d. Prove that if faig has perfect periodic ACF and d is co-prime to N, fbig also has perfect periodic ACF.

6.5.The binary f 1g code of length N ¼ 5 has periodic ACF Rp(m) ¼ 1, m 6¼0 mod5. Values of its aperiodic ACF are Ra(1) ¼ 0, Ra(2) ¼ 1. Find Ra(3) and Ra(4).

6.6. The binary f 1g code of length N ¼ 5 has constant component a~0 ¼ þ3 and Ra(4) ¼ 1. Find Rp(m) and the rest of the values of Ra(m).

6.7.Can a binary f 1g code of an odd length N > 5 have Ra(5) ¼ 1? Can a binary code of an even length N > 6 have Ra(6) ¼ 1? For an arbitrary binary code formulate and prove the relation between parities of the three values: length N, shift m and level of Ra(m).

6.8.Is it possible for a binary f 1g sequence that Ra(m) ¼ 1, Ra(m þ 1) ¼ 3 for some m? What about parities of Ra(m) and Ra(m þ 1)?

6.9.Is it possible for a binary f 1g sequence that Ra(2) ¼ 2, Rp(2) ¼ 1?

6.10.Suppose someone has found that each of the PSK sequences of length N ¼ 100 at his disposal has non-normalized periodic ACF taking on values 12 at some shifts

m 2 f1, 2, . . . , N 1g. Can a code with a, max < 0:05 be present among them?

fþ þ þ g and show waveforms at its characteristic points when aperiodic and periodic versions of the signal are applied to its input.

6.13. A student has calculated periodic ACF of the binary f 1g sequence of length

6.14.Prove the non-existence of minimax binary sequences (Rp(m) ¼ 1, m ¼ 1, 2, . . . , N 1) for lengths N ¼ 17, 21, 29, 33, 37, 45. (Hint: use the same technique as in deriving the necessary conditions for perfect ACF of binary sequences.)

6.15.Prove that decimation of a minimax binary sequence again produces a minimax binary sequence whenever the decimation index is co-prime to the period N.

6.16.Prove that decimation of any periodic sequence does not change the maximal periodic sidelobe, whenever the decimation index is co-prime to the period N.

6.17.Calculate (3 þ 7) 5(6 7 þ 4) þ 1 in GF(11).

6.18.Solve the equation 6x þ 7(5 þ 4 2) 1 ¼ 1 in GF(11).

6.19.Prove that GF(4) cannot be built on the basis of modulo 4 operations.

6.20.Is the sequence of length L ¼ 7 f0100110g a binary m-sequence? What is the

answer if all zeros are replaced by ones and vice versa?

6.21. Construct a binary m-sequence of length L ¼ 15 with an initial loading d0 ¼ 1, d1 ¼ d2 ¼ d3 ¼ 0, draw the structure of its generator and compile a table exhibiting its state changes.

6.22.Prove that in one period of a binary m-sequence of memory n the number of series of successive symbols (01), (10), (11) is 2n 2 while the number of series (00) is smaller by one.

6.23.Someone observes an m-sequence knowing its alphabet and memory but not its coefficients of recurrence (6.13). What is the minimal necessary and sufficient number of observed symbols to recover the coefficients?

6.24.Prove the independence of binary character of a specific choice of primitive element.

6.25.Find a primitive element of GF(13). Build a table of logarithms and binary characters of all non-zero elements of GF(13).

6.26.Build the Legendre sequence of length N ¼ 11, calculate its periodic ACF and compare it with the theoretically predicted one.

6.27.Build the Legendre sequence of length N ¼ 13, calculate its periodic ACF and compare it with the theoretically predicted one.

6.28.Find the cyclic shift of the sequence of Problem 6.26 with minimal aperiodic ACF sidelobe.

6.29.Prove the perfection of periodic ACF for Chu codes of odd lengths.

6.30.Prove the perfection of periodic ACF for Frank codes. (Hint: use representation

i ¼ i1h þ i2, m ¼ m1h þ m2; 0 i1, i2, m1, m2 h 1 and summation over i1, i2 in (5.9).)

6.31.Does an 8-PSK sequence of length N ¼ 64 with perfect periodic ACF exist? If so, construct it.

6.32.Prove the non-existence of QPSK sequences with perfect periodic ACF for odd lengths N.

6.33.Prove the non-existence of QPSK sequence of length N ¼ 30 with perfect periodic ACF.

6.34.Prove that for a ternary f0, 1g sequence with perfect periodic ACF, the number of non-zero elements per period is always a square of an integer.

6.35.Prove the non-existence of ternary f0, 1g sequences having perfect periodic PACF and a single zero per period for any odd length.

6.36.Construct an SLSF, show the effect of sidelobe suppression and find SNR loss for a binary sequence fþ þ þ þ þ g.

(6.1). Run the program for rectangular and bell-shaped pulses and explain the results.

6.42.Write a program calculating and plotting the ambiguity function and its horizontal sections (ambiguity diagrams) at different levels for the LFM (see Figure 6.6) and V-LFM rectangular pulses, whose complex envelopes are:

>

>

>

:

Take three values of deviation (e.g. Wd T ¼ 10, 25, 40), comparing for each of them ambiguity functions and diagrams on high and low levels for both signals.

6.43. Write a program to calculate and plot aperiodic and periodic autocorrelation functions of an arbitrary APSK sequence. Use it to verify the optimality of binary Barker codes. Calculate the ACF of the ternary sequence fþ þ þ þ þ þ0 þ 0 þ þ 0 0 þ 0 g and find the maximal level of its sidelobe relative to the mainlobe. Verify that for binary codes the following properties of periodic ACF are true:

N 1

X

RpðmÞ RpðlÞ ¼ 0 mod4; 8m; l; RpðmÞ ¼ ðNþ N Þ2

m¼0

6.44.Write a program for an exhaustive search for the optimal binary code of given length, minimizing the maximum level of the aperiodic ACF sidelobe. Provide for measuring execution time and try to optimize the processing speed of the program. How does operating time grow when length is incremented by one? What is the maximal length you have managed to find an optimal code for?

6.45.Using Matlab, illustrate time-resolution of three copies of the bandpass periodic APSK signal manipulated with the ternary code ( þ þ þ þ þ þ 0 þ 0 þþ0 0 þ 0 ) at the matched filter output (Figure 6.26). Take time shifts between consecutive copies of 2–3 and 5.5–6.5 chip durations D correspondingly. Recommended steps are:

(a)Form several (3–4) periods of ternary code and oversample them 100 times to simulate rectangular baseband chips manipulated by the given code: the signal complex envelope.

6.53.Write a program finding SLSF and calculating its SNR loss for a given binary sequence. Plot the signal manipulated by this sequence, its periodic ACF and

Figure 6.28 Rectangular-chip signal manipulated by a binary sequence (a), its periodic ACF (b) and SLSF response (c)

6.54.Write a program to calculate aperiodic ACF of a radar array. Run the program to verify the properties of the FSK code in Example 6.13.1.