reasoning for a doubling of the number of orthogonal bandpass signals against baseband ones follows directly from equations (2.24) and (2.25): building up ns orthogonal signals of the form (2.24), we can add to them ns more, obtained by just shifting the carrier frequency phase by angle /2. This possibility is practical only when all signals are deterministic or coherent, which means that their carrier phases are controllable and can actually be used for message identification. In reality, however, this may often not be the case because either a transmitter itself or a channel may destroy the coherence of signals in such a manner that their phases become random and as a consequence cannot be used for distinguishing messages. This case is addressed in Section 2.5.

2.4 Complex envelope of a bandpass signal

Before extending our discussion to the more complicated models of the M-ary transmission, it is reasonable to diverge from the main line in order to recollect some more facts about handling bandpass signals.

Let us begin with the observation that the real envelope S(t) in equation (2.24) is fictitious, i.e. is just a suitable artificial instrument, whereas only the signal s(t) itself is an observable physical reality. More than this, equation (2.24) does not give any unique definition of the envelope of s(t). In fact, it follows from (2.24) that one may take an arbitrary ‘phase modulation’ law (t), and then ‘envelope’ S(t) ¼ s(t)/ cos [2 f0t þ (t)] will produce a given signal s(t). Therefore, some special agreement is needed on how to interpret the notion of an envelope or amplitude modulation S(t).

A universally adopted basis for determining the envelope is the Hilbert transform. By its physical content, the Hilbert transform is just filtering which rotates the phases of all harmonic components independently of frequency through the same angle, /2, and does not change the amplitudes of the harmonics. In the frequency domain such a transform simply means multiplication of the signal spectrum by j /2 for positive frequencies and by j /2 for negative ones, and, therefore, the transfer function of a

Straightforward calculation of the inverse Fourier transform of this leads to a filter pulse response hg(t) ¼ 1/ t. Hence, in the time domain the Hilbert transform s?(t) of signal s(t) may be presented via the convolution integral:

Using the definition of the Hilbert transform and the Parseval theorem, the reader may easily prove the following relations:

Z1

sðtÞ ¼ 1 s?ð Þ d

t

0

which is nothing but the inverse Hilbert transform, and:

where Re stands for taking the real part of the complex entity, and the second cofactor in the square brackets is a complex notation of the CW of a carrier frequency f0 through the Euler formula. Turning again to Figure 2.8, we can see that with s(t) treated according to (2.37) as a real part of the complex signal S_(t) exp (j2 f0t), the imaginary part of the latter is the Hilbert transform of s(t):

s?ðtÞ ¼ Im½S_ðtÞ expðj2 f0tÞ&

This leads to one more complex substitute of the real signal, called the analytic signal:

Formally, the analytic signal uses complex notation to advance factorization model (2.24) of a bandpass signal so that the first factor covers all modulation (not only amplitude modulation) and the second is responsible for only the unmodulated CW of a carrier frequency f0.

Using the basic rule of spectral analysis, it may be easily proved that the spectrum of the complex envelope of a bandpass signal (2.37) is located around zero frequency. Therefore—since, given the carrier frequency, the signal is entirely presented by its complex envelope—the latter is a baseband equivalent of a bandpass signal, simplifying the analytical and computational work by getting rid of the carrier frequency dependence.

In what follows we will need a generalized version of the inner product (2.5), which is applicable not only to real signals u(t), v(t) but also to their complex substitutes— analytic signals u(t), v(t) or complex envelopes U_ (t), V_ (t). This modified inner product is defined as:

ZT ZT

00

where a complex conjugation is used to preserve the equality between the inner product of the vector by itself and the vector squared length (always real and non-negative!), while coincidence of the inner products of analytic signals and complex envelopes follows from definition (2.38). Specifically, for a signal s(t) of energy E, according to equations (2.36) and (2.35):

since the Hilbert transform does not affect the amplitude–frequency spectrum and therefore the energies of s(t) and s?(t) are always the same.

Let us now take two signals sk(t), sl(t) and calculate the squared distance between their complex envelopes S_k(t), S_l(t):

To calculate the distance between the signals sk(t; k), sl(t; l) we can make use of equation (2.43) and evaluate the distance between the complex envelopes S_k(t; k), S_l(t; l) instead of the signals themselves. Doing this, consistent with the generalization of the cosine theorem (2.41) and under the assumption that all signals have identical energies E, we get:

d2ðS_ k ; S_ l Þ ¼ 4E½1 Re klð Þ&

where the additional subscript underlines the correspondence of the vector S_ k to the complete complex envelope S_k(t; k), independence of signal energy E of initial phase k is taken into account, and:

is the correlation coefficient of the complete complex envelopes S_k(t; k), S_l(t; l). Phases k, l being independent of time, the latter quantity can be rewritten as kl( )

¼ kl exp [j( k l)], where the correlation coefficient kl covering only the deterministic

(random-phase-free) complex envelopes S_k(t), S_l(t) of the signals is introduced:

where kl ¼ arg ( kl):

Now, the squared distance above takes the form:

The trouble about this distance is its dependence on unknown signal phases k, l. Due to this, there are a lot of distances for the fixed deterministic modulation laws S_k(t), S_l(t) governed by a random phase difference k l. With a perceptible level of the correlation modulus j klj the hazard is always present that due to unfavourable combination of phases (that is, when kl þ k l is small enough) the distance (2.45) may prove very small. To fully ensure against it, the correlation modulus should be as low as possible and the best signal set should obey the condition:

As is seen, we again, as in Section 2.3, come to orthogonal signals. This time, however, the orthogonality condition is much more binding, forcing complex envelopes, or in other words modulation laws, of signals to be orthogonal, not just the signals themselves. Because of this, bandpass signals retain orthogonality under any combinations of their phases, since kl ¼ 0 entails kl( ) ¼ 0. On the other hand, condition (2.46)

excludes the opportunity to provide orthogonality by means of a quadrature phase shift