from 1 to 6/64, as predicted by equation (2.48). In other words, to maintain a rate of 9.6 kbps bandwidth wider than 100 kHz will be involved. This is not a prohibitive figure for many applications and, for instance, the cdmaOne cellular telephone standard exploits exactly this principle in the uplink (for more details see Section 11.3).

Let us now imagine a designer who is quite impressed by the figure above and plans to go further along the same way, targeting a 10-times (10 dB) reduction of transmitted power. To realize that, blocks of m ¼ 20 bits should be converted into M ¼ 220 > 106 orthogonal signals. This will lead to spectral efficiency less than 2 10 5 or bandwidth occupied wider than 480 MHz, which looks wholly impractical versus the data rate 9.6 kbps.

The discussion undertaken illustrates the very tough character of the trade-off between energy efficiency and spectral efficiency inherent in orthogonal signalling. At the same time it is pertinent to note that, although big energy gains are unattainable in practice with orthogonal signals due to the enormous demand for bandwidth, asymptotic orthogonal coding gain may serve as a good reference point, being the upper border of theoretical efficiency of any m-bit-block coding.

Return now to equation M ¼ WtTt and consider the question: when the number of orthogonal signals and thus the product WtTt is measured in the tens or more, does it point to a spread spectrum? In other words, is a system exploiting numerous orthogonal signals always a spread spectrum one? As the discussion in the next section shows, the answer to this question is in general negative.

2.7 Examples of orthogonal signal sets

Throughout this section we will again ignore the opportunity of doubling the number of orthogonal signals by quadrature carrier shifts, which is always present in a coherent bandpass system, and concentrate only on the dependence between M and equivalent baseband system time–frequency resource WtTt. We will demonstrate first how to build the simplest orthogonal sets based on fragmentation of an available resource.

2.7.1 Time-shift coding

It is obvious that the inner product of any two non-overlapping time-shifted signals is zero. Consider M signals shown in Figure 2.10a, which occupy jointly time period Tt. With signal duration no greater than T ¼ Tt/M and the time shift between successive signals no smaller than the signal duration, this time-shift coding produces orthogonal signals. The estimated bandwidth W of each of these signals is inverse to its duration and all the signals are permitted to occupy the same bandwidth with no violation of orthogonality: W ¼ Wt. Hence, the maximal number of orthogonal signals of this sort which can be accommodated within a given total time–frequency resource Tt, Wt is M ¼ Tt/T ¼ WtTt, i.e. as is easily foreseen, it is equal to the signal space dimension ns ¼ WtTt. A high necessary number of signals M 1 implies a large product WtTt ¼ M, which may seem to point to spread spectrum. However, for any individual signal the time–frequency product is WT ¼ WtT ¼ WtTt/M ¼ 1, so that the signals are not of spread spectrum type. In the

value of the peak-factor , which is the ratio between peak and average powers. Because an individual signal occupies only an Mth part of the available time resource, average power is M times smaller than peak power and ¼ M 1. At the same time, in designing a transmitter power amplifier, a small value of is crucial: the closer it is to 1, the softer are the demands on the linearity of an amplifier and the better is its power performance.

2.7.2 Frequency-shift coding

The other straightforward way to provide orthogonality is frequency-shift coding. Due to time–frequency duality or Parseval theorem, the inner products of signals u(t), v(t) and of their spectra u~(f ), v~(f ) coincide:

which allows transfer of the idea discussed above into the frequency domain (see Figure 2.10b). With an entire overlap of the signals in time (T ¼ Tt) each of them has bandwidth W ¼ 1/Tt at the least. Thus the maximum number of orthogonal signals formed by shifting the spectra is again M ¼ Wt/W ¼ WtTt ¼ ns. As in the previous case, the total resource is again ‘sliced’, but differently: the strips are horizontal, meaning that the total time resource Tt but only an Mth part of the entire frequency resource Wt are utilized by every signal (Figure 2.11b). Clearly, each individual signal is again non-spread-spectrum since its time–frequency product WT ¼ (Wt/M)Tt ¼ 1, and any system with however large a number of orthogonal signals of this sort is certainly not a spread spectrum one.

The peak-factor of this mode of orthogonal signalling, unlike time-shift coding, is ¼ 1 and synchronization errors are not that dramatic because orthogonality is provided by signal non-overlap in the frequency domain. Instead, spectra drifts (e.g. because of Doppler shifts) may sometimes be destructive. Still, this transmission mode is extremely popular and the conventional M-ary FSK modulation is its direct embodiment.

The examples considered explain why employing even a great number of orthogonal signals and, hence, the necessity for a total resource WtTt 1 does not automatically mean the involvement of spread spectrum technology.

2.7.3 Spread spectrum orthogonal coding

Fragmentation of the total time–frequency resource inherent to the two discussed modes of orthogonal signalling may in some cases be a preferable solution in connection with hardware implementation aspects. However, with M increasing reasons of this sort are getting more doubtful since, as mentioned above, time-shift coding demands a high peak-factor while frequency-shift coding implies optimal processing with a bank of numerous parallel frequency-detuned filters.

Under such circumstances spread spectrum orthogonal signalling can prove very competitive, allowing all signals to share a total time–frequency resource with no distribution or slicing of the latter. Consider a simple example of realization of the idea

where the second equality expresses the rule in terms of matrix Kronecker product . The orthogonality of rows of H2M is obvious: if two rows have numbers differing by any integer but M, they have zero inner product, since their two M-element halves are orthogonal. Otherwise, the first M components of the rows coincide, while the rest of the components are opposite, which again gives zero inner product.

To make use of the Sylvester algorithm one can start with matrix

1 1

H2 ¼ 1 1

which is evidently a Hadamard one, and construct H4 (using the symbols ‘þ’ and ‘ ’ in place of þ1 and 1 for brevity), then from H4 produce H8, and so forth:

Thereby a Hadamard matrix of any order M ¼ 2m (2, 4, 8, 16, 32, . . . ) can be built up. Rows of Hadamard matrices of this kind are also known as Walsh functions.

Figure 2.12 shows baseband orthogonal BPSK signals (2.50)—Walsh functions— generated with the aid of Hadamard matrix H8.

Figure 2.13 illustrates that within this signalling mode there is no resource distribution: all signals share the common resource, fully overlapping in both the time and frequency domains. Indeed, the bandwidth of each signal is estimated as W ¼ 1/D while duration T ¼ MD, thus producing WT ¼ M ¼ WtTt. Orthogonality is now achieved at the cost of an appropriate signal modulation, rather than either time interval or bandwidth fragmentation.

Analysing the benefits of spread spectrum orthogonality, one can note that methods of generation and processing of signals (2.50) are quite well matched to modern digital microchip circuitry (ASIC, VLSI, microprocessors). Another factor is the automatic

s1(t)

t

s2(t)

t

s3(t)

t

s4(t)

t

s5(t)

t

s6(t)

t

s7(t)

t

s8(t)

t

Figure 2.12 Baseband Walsh functions

M

k

Wt = W

2

1

Tt = T

Figure 2.13 Resource allocation in orthogonal spread-spectrum signalling

acquiring of those merits of the spread spectrum which cannot be seen directly within the classical reception framework but are numerous and very valuable in practice (for details see Chapter 3). This gives an explanation of the great popularity of orthogonal signalling of this sort in advanced telecommunication systems (e.g. cdmaOne, UMTS, cdma2000; see Chapter 11).

Now the moment has come to draw an overall conclusion on the results of Sections 2.5–2.7. As one may see, theoretically the classical M-ary transmission problem does not lean implicitly towards the spread spectrum, and in principle optimal signals can be realized as plain ones. On the other hand, there are implementation reasons, along with the desire to gain the numerous advantages pertaining to spread spectrum beyond the classical reception model. Because the latter opportunity is potentially promised by a large total necessary time–frequency resource WtTt 1, this can incline a system designer to prefer spread spectrum signals to plain ones.