Discrete spread spectrum signals |
145 |
|
|
where fFk, i, i ¼ 0, 1, . . ., N 1g is the frequency code sequence of the kth signal. It is obvious that computation of CCF is again just tallying up a number of coincident frequencies in the pair of signals time-shifted by m chip positions.
5.6 Processing gain of discrete signals
Let us revisit the general model (5.1) to discuss the issue of the processing gain of a discrete signal. Suppose that all Fi belong to the frequency alphabet of size M, in which consecutive frequencies are separated by a chip bandwidth providing orthogonality of chips with different frequencies. Then at each of M available frequencies we have a signal subspace whose dimension is N, since we are fully free in selection of the amplitudephase code sequence, i.e. N-dimensional vector a ¼ (a0, a1, . . . , aN 1). Orthogonality of these subspaces means that the dimension of the whole signal space covering all M frequencies is MN. In Section 2.5 it was shown that the dimension of bandpass signal space coincides with the total time–frequency resource allocated to signals. Being interested only in spread spectrum signals, each occupying the whole available resource, we may predict that the time–frequency product of a discrete signal, i.e. processing gain, equals MN.
Let us confirm this statement by a straightforward computation, setting Dc ¼ D. Estimating chip bandwidth as 1/D and taking into account that available bandwidth and time resources are then W ¼ M/D and T ¼ ND, we arrive at the result WT ¼ MN. Obviously, for APSK signals M ¼ 1, and processing gain WT ¼ N.
Problems
5.1. A discrete signal of length N ¼ 5 has complex amplitudes a0 ¼ 1 þ j, a1 ¼ 1 þ j, a2 ¼ 1 þ j, a3 ¼ 1 j, a4 ¼ 1 j and frequencies Fi ¼ 0, i ¼ 0, 1, 2, 3, 4. Evaluate the phases and amplitudes of its chips and classify the signal by its modulation mode.
5.2. A discrete signal has amplitude-phase and frequency codes ai ¼ exp [j i(i þ 1)/2], Fi ¼ 0, i ¼ . . . , 1, 0, 1, . . .. Calculate the amplitudes and phases of its chips. Classify the signal by its modulation mode. Is this signal periodic? If so, specify its period.
5.3.Prove the evenness of the periodic and aperiodic autocorrelation functions of code sequences of APSK signals.
5.4.An APSK signal is built of rectangular chips with Dc ¼ D and has the code sequence set by the vector a ¼ (1, 1, 0, 1, 0, 0, 1). Calculate and sketch its aperiodic and periodic ACF. Do the same for the case Dc ¼ D/2.
5.5.What happens to the periodic and aperiodic ACF of an APSK signal under the following transformations of a code sequence:
(a)Cyclic shift of elements?
(b)Changing the signs of all elements?
(c)Changing the signs of only the elements with even numbers?
(d)Multiplying all elements by the same constant?
(e)Mirror-like rearranging (i.e. reading from right to left)?
146 |
Spread Spectrum and CDMA |
|
|
5.6.Is the combination jRa(1)j ¼ 3, Rp(1) ¼ 1 possible for a PSK code? What about
the combinations Ra(1) ¼ 2:1, Rp(1) ¼ 0:8 0:6j; Ra(1) ¼ 0:6 þ 0:8j, Rp(1) ¼ |
|
|
|
1:1 þ j ? What is the possible value of Rp(1) Ra(1) for a PSK code?
5.7.The spaces between frequencies of an FSK signal are multiples of F ¼ 1/D. Prove that time-aligned rectangular chips with different frequencies are orthogonal.
5.8.An FSK signal of length N involves M < N frequencies. Chips of different frequencies are orthogonal. Is it possible that ACF is zero at all non-zero time shifts ¼ mD?
Matlab-based problems
5.9.Write a program displaying APSK, BPSK, QPSK and FSK discrete signals. Take
N ¼ 6 10, carrier frequency f0 ¼ (10 20)/D, chip duration Dc ¼ D and frequency step F ¼ 1/D. Run the program for various modulation modes and chip shapes (e.g. rectangular and half-wave sine), adjust the code sequences to provide satisfactory visualization and comment on the waveforms observed. Examples are given in Figure 5.4. Use the program also to demonstrate the finiteness or periodicity of signals.
t(s APSK),
s(t ), BPSK
s(t ), QPSK
s(t ), FSK
2
0
–2
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
1
0
–1
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
1
0
–1
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
1
0
–1
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
t/∆
Figure 5.4 Example waveforms of discrete signals
5.10.Write a program demonstrating that the ACF of an APSK signal is the APSK signal itself, its chip being the ACF of the original signal and its code sequence
being the ACF of the initial code sequence. Recommended steps:
Discrete spread spectrum signals |
147 |
|
|
(a)Form a plain chip (e.g. rectangular or half-wave sine).
(b)Form a real (e.g. binary, ternary etc.) code sequence containing 7–10 elements.
(c)Form and plot the signal with this chip and modulation law.
(d)Calculate and plot its ACF directly using an appropriate Matlab command.
(e)Calculate and plot the ACF of the initial signal chip.
(f)Calculate and plot the initial code ACF.
(g)Verify that the ACF of item (d) reproduces an APSK signal with the chip obtained in item (e) and the code obtained in item (f).
(h)Run the program, varying the chip shape and code of the initial signal, and give your comments. Example plots are shown in Figure 5.5.
S(t)
R(τ)
Rc(τ)
Ra(m)
1
0
–1
–
1
0
–1
–
1
0.5
0
–
1
0
–1
–8 |
–6 |
–4 |
–2 |
0 |
2 |
4 |
6 |
8 |
m
Figure 5.5 Aperiodic ACF of the ternary signal of length N ¼ 8
5.11.Write a program verifying the association between periodic and aperiodic ACF of discrete signals (see Figure 5.2). Recommended steps:
(a)Specify some real (more convenient to display versus a complex one) code sequence of length N ¼ 7 15.
(b)Calculate its periodic ACF directly from definition.
(c)Plot one period of it for the case when the chip is rectangular.
(d)Calculate the aperiodic ACF.
(e)Plot it and its N-shifted copy.
(f)Run the program for various codes and check the validity of (5.13).
148 |
Spread Spectrum and CDMA |
|
|
Frequency code
5
4
3
2
1
0
1
s(t)
R(τ)
0
–1
0
1
0.5
0
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
|
|
|
|
τ/∆ |
|
|
|
|
Figure 5.6 ACF of FSK signal of length N ¼ 8 (see Example 5.5.1)
5.12.Write and run a program calculating precisely a real envelope of the ACF of an FSK signal. Recommended steps:
(a)Form a rectangular chip envelope.
(b)Specify a frequency code of length N ¼ 7 10 and frequency alphabet size M ¼ 5 . . . N 1;
(c)Form the complex envelope of an FSK signal with a specified frequency code, setting the frequency step F ¼ 1/D.
(d)Calculate and plot the real envelope of the ACF of the signal obtained.
(e)Compare the obtained values of ACF at the points ¼ mD to the theoretically predicted ones.
(f)Run the program, varying the frequency code.
(g) Pay attention to the situations where the ACF level between the points¼ mD is higher than at these points (see Figure 5.6). How would you explain this phenomenon?