Figure 7.4 Despreading of BPSK data-modulated signal

Figure 7.5 Spreading (a) and dispreading (b) a BPSK data-modulated signal

~

Sb( f )

~

Sbs( f )

Figure 7.6 Power spectra of original and spread datastreams

7.1.2 DS spreading: general case

The idea of direct spreading considered above for the BPSK data transmission and a binary signature may be readily generalized to comprise a much wider range of data modulation modes and signatures. Let B_ k(t) denote a complex waveform corresponding to a datastream bk of the kth user transmitted in some digital data modulation format (ASK, M-ary PSK, QAM etc.). With M-ary digital data transmission B_ k(t) consists of contiguous rectangles having duration T ¼ ( log M)Tb and manipulated by complex

symbols belonging to the specific M-ary modulation alphabet. For example, in the case of 8-PSK, rectangles of duration T ¼ 3Tb are manipulated by complex amplitudes from the alphabet fexp (jl /4): l ¼ 0, 1, . . . , 7g shown in Figure 2.6c; if 16-QAM is preferred, then the rectangles have duration T ¼ 4Tb, their complex amplitudes taking the values defined by Figure 2.6b and so on. In the case of an ordinary (non-spread) M-ary modulation the transmitted signal carrying a datastream bk (it is convenient to assume now that components bk, i of bk are M-ary complex symbols: B_ k(t) ¼ bk, i, (i 1)T < t iT ):

skðt; bkÞ ¼ Re B_ kðtÞ expðj2 f0tÞ

The received signal:

skrðt; bkÞ ¼ Re B_ kðt kÞ exp½jð2 f0t þ kÞ&

has a complex envelope:

The transmitted ith data symbol bk, i is just the complex amplitude of a CW carrier, which is constant within the interval ((i 1)T, iT ]. Consequently, to recover this symbol the receiver should decide between M replicas of the rectangular pulse existing at ((i 1)T þ k, iT þ k] and having different competitive values of complex amplitude. To fulfil this, the observed complex envelope Y_ (t) retrieved from the noisy observation y(t) should be correlated with exp (j k), resulting in:

used (after normalization to the reference symbol energy) to get the necessary estimate

^

bk, i of bk, i.

The demodulator of Figure 7.7, where the double circle symbolizes complex multiplication, implements the procedure. This scheme just generalizes the correlator

Figure 7.7 M-ary demodulator

complex envelope into the one characteristic of an ordinary (non-spread) M-ary data modulation, i.e. performs the despreading. Thanks to this the receiver may again be thought of as a two-stage one: fulfilling first despreading and then a conventional M-ary demodulation, using, e.g., the scheme of Figure 7.7.

Let us dwell now on implementing complex multiplication and extracting the complex envelope Y_ (t) from a truly observed real waveform y(t). Recalling the basic rules of complex arithmetic:

ReðxyÞ ¼ ReðxÞReðyÞ ImðxÞImðyÞ; ImðxyÞ ¼ ReðxÞImðyÞ þ ImðxÞReðyÞ

the multiplier of two complex entities x and y contains four conventional multipliers and two adders (Figure 7.8). Input complex operands x, y specified by their real and imaginary parts produce two outcomes, being the real and imaginary parts of the product xy.

Obtaining the observed complex envelope is based on the definition of Y_ (t): y(t) ¼ Re[Y_ (t) exp (j2 f0t)]. Applying the complex multiplication rule above and the Euler formula we have y(t) ¼ [ReY_ (t)] cos (2 f0t) [ImY_ (t)] sin (2 f0t). Multiplying both parts of this equation by 2 cos (2 f0t) and 2 sin (2 f0t), along with using trigonometric identities, leads to:

2yðtÞ cosð2 f0tÞ ¼ ReY_ ðtÞ þ ½ReY_ ðtÞ& cosð4 f0tÞ ½ImY_ ðtÞ& sinð4 f0tÞ

2yðtÞ sinð2 f0tÞ ¼ ImY_ ðtÞ ½ImY_ ðtÞ& cosð4 f0tÞ ½ReY_ ðtÞ& sinð4 f0tÞ ð7:9Þ

The first terms of the right-hand sides of equations (7.9) are baseband waveforms (since the complex envelope is a modulation law, i.e. baseband), while the rest are bandpass waveforms, having a central frequency 2f0. The bandwidth of the modulation law is smaller than f0 (see Figure 7.9a), therefore the lowpass filters may easily filter out highfrequency components in (7.9), preserving only the real and imaginary parts of a desired complex envelope Y_ (t). This principle of restoring the complex envelope from the real observation y(t) is realized in the scheme shown in Figure 7.9b.

To conclude this discussion, Figure 7.10 illustrates the operations run by the transmitting and receiving sides in a generic DS spread spectrum system. The modulator (Figure 7.10a) implements (7.6) using only the real part of the complex product. In the demodulator (Figure 7.10b) the observed complex envelope retrieved according to the scheme of Figure 7.9b is first despread by multiplication with the reference S_k(t k)

Figure 7.8 Complex multiplication

Figure 7.9 Retrieving the complex envelope

Figure 7.10 Modulation (a) and demodulation (b) in a generic DS spreading

and is then input to a conventional M-ary demodulator (see Figure 7.7) to produce decisions on the received symbols.

Note in passing that the method of spreading–despreading just described is not unique and a variety of concrete circuitry solutions exists to implement those operations. For instance, multiplying complex envelopes may be done indirectly in the course of heterodyning. Specifically, if ui(t) ¼ Ui(t) cos [2 fit þ i(t)], i ¼ 1, 2 are two bandpass signals with carrier frequencies fi and complex envelopes U_ i(t) ¼ Ui(t) exp [j i(t)], their product:

1

u1ðtÞu2ðtÞ ¼ 2 U1ðtÞU2ðtÞ cos½2 ðf1 f2Þt þ 1ðtÞ 2ðtÞ&

1

þ 2 U1ðtÞU2ðtÞ cos½2 ðf1 þ f2Þt þ 1ðtÞ þ 2ðtÞ&

The two terms here are bandpass signals of carrier frequencies f1 f2 and f1 þ f2. If the lower carrier frequency f1 f2 exceeds the bandwidth of the product U_ 1(t)U_ 2 (t), then after filtering out the higher-frequency term the remaining lower-frequency bandpass signal will have complex envelope U_ 1(t)U_ 2 (t), i.e. exactly the product necessary in