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Spread spectrum signals and systems
1.1 Basic definition
The term spread spectrum is today one of the most popular in the radio engineering and communication community. At the same time, it may appear difficult to formulate an unequivocal and precise definition distinctively separating the spread spectrum philosophy from a ‘non-spread spectrum’ one. Certainly, every expert in system design and every experienced researcher has an intuitive understanding of the core of the issue, but—unlike a newcomer—such a person does not need to think about definitions in order to respond successfully to his or her professional challenges. From the point of view of the target audience of the book it seems worthwhile to dedicate some space to elaborating an appropriate explanation of what is implied in the following text under the spread spectrum concept.
Let us start with a reminder of the basics of spectral analysis. Every signal s(t) of finite energy can be synthesized as a sum of an uncountable number of harmonics whose amplitudes and phases within the infinitesimal frequency range [f , f þ df ] are determined by a spectral density or spectrum s~(f ). It is the pair of inverse and direct Fourier transforms that expresses this fact mathematically:
sðtÞ ¼ |
Z1~sðf Þ expðj2 ftÞ df |
s~ðf Þ ¼ |
Z1sðtÞ expð j2 ftÞ dt |
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Due to the one-to-one correspondence between the signal representation in the time domain s(t) and in the frequency domain s~(f ), we are able to switch arbitrarily between these two tools, selecting the more convenient one for any specific task. To characterize the size of the zones occupied by signal energy in the time and frequency domains we use the notions of signal duration T and bandwidth W, respectively. A signal whose energy
Spread Spectrum and CDMA: Principles and Applications Valery P. Ipatov
2005 John Wiley & Sons, Ltd
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Spread Spectrum and CDMA |
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is concentrated within strictly limited space in the time domain cannot have finite (i.e. non-zero in only limited frequency interval) spectrum and vice versa. Because of this, to define at least one of the parameters T, W, or both, some agreement is necessary about what is meant by duration or bandwidth. In this way effective, root mean square, etc. duration and bandwidth came into existence, showing the size of a zone spanned by a substantial part of signal energy in the time and frequency domains, respectively [1].
It is absolutely obvious that one way or another, the word ‘spread’ is indicative of wide spectrum, i.e. broad bandwidth W of a signal. But against what is the spectrum wide? Where is the reference for comparison? To demonstrate how a definition of spread spectrum may provoke debate, let us consult with several excellent and world-renowned books.
A rather frequent way to explain the concept consists in the statement that a system or a signal is of spread spectrum type if its bandwidth significantly exceeds the minimum bandwidth necessary to send the information [1–6]. What may seem mentally problematic in this definition is the very idea of minimum bandwidth of information or message. According to the fundamental Shannon’s bound, spectral efficiency (the ratio between the data rate R and the signal bandwidth W ) of a communication system operating over the Gaussian channel obeys the inequality:
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where Eb is signal energy per bit of information and N0 is the one-side power spectral density of a Gaussian noise. Figure 1.1 represents bound (1.2) graphically, showing that any combinations of R/W and Eb/N0 falling below the curve are possible, at least in principle. But this means that the theoretical ‘minimum bandwidth necessary to send the information’ is zero and therefore any real system—which, of course, occupies some
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Figure 1.1 Shannon’s bound
Spread spectrum signals and systems |
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non-zero bandwidth—should be treated as a spread spectrum one! Undeniably, any attempt to use near-zero data transmission bandwidth would be rather demanding for signal energy. For one thing, to operate with R ¼ 100W one would need to provide bit signal-to-noise ratio Eb/N0 around 280 dB, which is quite unrealistic. However, data transmission within bandwidth, for instance, up to ten times smaller than data rate is quite typical and is practised in many digital communication links (radio-relay lines, modem communications etc.). This shows the vagueness of the very idea of the ‘minimal bandwidth’ and the arguable character of taking it as a starting point for explaining the notion of spread spectrum.
As an attempt to eliminate ambiguity we can try to use rate of data in bits per second as a substitute for the above-mentioned minimal necessary bandwidth [7,8]. It is not very logical, however, that one of many possible and, in principle, equal in rights units of measurement of data rate is rendered some conceptually prominent role. Besides, defining spread spectrum in terms of bandwidth significantly exceeding data rate in bits per second is risky of comprising systems which are in no way of spread-spectrum type. Take, for example, the uplink between a single user and a base station in a GSM mobile telephone. With the rate of primary digitized speech data of 9.6 kbits/s, the user’s signal has bandwidth around 200 kHz, which may mislead someone to classify GSM as a spread spectrum system. However, no genuine features of spread spectrum are involved in the band broadening in the GSM uplink: the only reason why bandwidth exceeds the data rate is time-division multiple access (TDMA) forcing operation with much shorter transmitted symbols in comparison with the actual average time interval per information bit.
There is still one more reason to look for alternative definitions. Even ignoring the troubles discussed earlier, linking a definition to data rate or ‘message bandwidth’ can serve only data transmission systems, whereas spread spectrum is widely employed in many others, like radar, sonar, navigation or remote control for time and distance measuring, signal resolution etc. Actually, these systems were among the first to adopt the advantages of the technology under discussion. In those applications such categories as ‘information rate’ or ‘data bandwidth’ are hardly meaningful or, at least, have nothing to do with the aims of spreading spectrum. In the wake of the endeavour to define ideas of spread spectrum in some universal way, matching not only communication aspects but the needs of other application areas as well, the following definition of spread spectrum seems more relevant.
Let us turn to the Gabor uncertainty principle, according to which the product of signal duration and bandwidth (time–frequency product) satisfies inequality WT a, where constant a depends on the exact way in which duration and bandwidth are specified; however, it is always of the order of 1. A signal for which WT 1, and therefore duration and bandwidth are tightly linked to each other can be called plain (non-spread spectrum). The only way to widen the bandwidth of a plain signal is to reduce its duration, i.e. to shorten it. On the other hand, a deterministic signal for which WT 1 and bandwidth can be governed independently of duration is a spread spectrum one. Putting it in other words, we may say that any spread spectrum signal occupies a rectangle in the time–frequency plane whose square is much greater than 1. This definition automatically defines a spread spectrum system, too: a system employing spread spectrum signals is a spread spectrum system.
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Note that in this definition the independence of duration and bandwidth is particularly emphasized, meaning that one can broaden the bandwidth (duration) without shortening the signal in time (frequency). This has a further implication for the critical role of angle (phase or frequency) modulation in all spread spectrum technology. Indeed, how can amplitude modulation help in widening the spectrum? The answer is: only by reducing the area over which signal energy is effectively spread in the time domain, i.e. by actually reducing the effective signal duration. It is only angle modulation that is capable of widening the signal spectrum with no influence on the time-distribution of signal energy.
As an illustration, Figure 1.2 gives the example of two rectangular pulses having the same duration T and carrier frequency f0: (a) a signal with no internal modulation and (b) a linearly frequency-modulated (LFM) signal with deviation Wd ¼ 20/T. The lower curves show the spectra of these signals. As is seen for signal (a), bandwidth W has the order W 1/T, meaning that the signal energy spans in the frequency domain an interval approximately equal to inverse pulse duration. Thereby, duration and bandwidth are strictly tied, the time–frequency product is fixed and widening the spectrum can be achieved only in exchange for pulse shortening. At the same time the bandwidth of pulse (b) is close to frequency deviation (W Wd ) and much greater than the inverse duration. As a result bandwidth can be easily controlled independently of signal duration by just varying the deviation. Accordingly, we classify the first signal as plain and the second as of spread spectrum type.
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Figure 1.2 Unmodulated (a) and frequency modulated (b) rectangular pulses and their spectra