# Understanding digital geometry

Designing a digital radio system can be challenging, especially with transmitter power limits and bandwidth restrictions imposed on modern wireless systems. These restrictions can influence the radio’s modulation design — particularly when a large quantity of data is transmitted over a narrowband channel.

The most critical modulation design parameter is *encoded symbol separation*. When several encoded symbols (*M*) are needed in a narrow bandwidth application, they should be carefully separated to minimize the cross-correlation — or symbol overlap — between them, defined as the *cross-correlation metric* (r). A well-designed symbol separation ensures that a system will have a consistent performance and operating range without demanding excessive transmitter power.

*Orthogonal symbol sets* are necessary to achieve optimal symbol separation. To clarify, imagine that the modulation symbols are vectors in a geometric signal space. Each symbol has its own *vector magnitude,* or bit energy (*Eb*), and direction with respect to other symbols being transmitted. When the angle (f) between any two of these vectors is 90°, the symbols are *orthogonal* (perpendicular). The cross-correlation metric is equivalent to the cosine of the angle (f) between any two symbol vectors, r 5 cos(f), with values of -1¶r¶11.

This geometry can be easily visualized with a four-symbol (*M*54) *quadrature phase modulation* design. When the symbols are orthogonal, each occupies a point at the center of the quadrant of a phase circle, separated by 90°. With *frequency-shift keyed* modulation, however, this analogy does *not* apply. Each FSK symbol has a unique frequency, with some spectral overlap between adjacent symbols in the set. These overlapping FSK symbols will be orthogonal only when the maximum signal level of one symbol waveform occurs at the zero-crossing of another, as shown in Figure 1 on page 44. Two FSK symbols are orthogonal when separated by a specific frequency (*Rs*) determined by the multiple inverse of the symbol period (*Ts*). For coherent FSK systems, orthogonal separation occurs at *Rs* = 1/(2*Ts*). For noncoherent FSK systems, this separation is *Rs* = 1/*T*.

### Accounting for RF environment

If symbols are transmitted one at a time, non-simultaneously, why does this overlap matter? To detect symbols, a digital receiver will correlate the energy of the phase, frequency or amplitude, depending on the modulation scheme. This is a straightforward process when the symbol energy is well above the noise — there will be few errors in choosing one symbol from another. However, in a low-signal, high-interference environment, this gets difficult, unless enough separation exists between symbols to make the proper choice.

The description of a simple FSK receiver’s operation is useful for illustrating how symbol cross-correlation can degrade link performance. A symbol is received when its specific frequency detector has more energy output than noise coming from the other detectors. Within each symbol period, the detectors build to a maximum, then gradually decline, due to *filter hysteresis*, holding some energy into the next symbol period. When signals are weak compared to noise, this creates an uncertainty in the receiver’s ability to choose which detector has the peak symbol energy during each sample period — especially if these symbols are close together with a large spectral overlap.

Symbol separation is a key parameter for predicting the probability of bit error (*P _{b}*) of a digital system. This

*P*is calculated using the statistical Q function to relate the cross-correlation and bit energy-to-noise ratio (

_{b}*Eb/No*) (neglecting interference) to the bit error performance (BER):

The Q function is maximum, *P _{b}* 5 ½ (50% errors), when its argument is zero (no signal), and approaches its minimum value (almost zero errors) when the signal is much greater than the channel noise. The r metric in the equation scales the

*Eb/No*ratio up or down, as determined by the symbol separation, and adjusts the Q function’s argument, improving or degrading

*P*(

_{b}. Binary antipodal symbols*M*=2), like BPSK, have a maximum symbol separation angle of 180° (r =-1) that scales the

*Eb/No*ratio by a factor of 2 and improves the

*P*.

_{b}BPSK antipodal modulation has one of the best *P _{b}* performance curves in digital design because of this large symbol separation. For multiple symbol schemes, however (

*M*>2), optimal separation is defined when all symbol vectors are

*mutually orthogonal*(r50). For FSK systems, mutual orthogonality requires critical spacing that increases the bandwidth by the number of symbols times the symbol separation frequency (

*M*3

*Rs*). For this reason, mutually orthogonal FSK designs are not feasible in narrow bandwidths.

If all the symbols cannot be orthogonal, then at least they should be consistently spaced to minimize differences between the r metric values. Large spreads of r values degrade system performance. This can be illustrated by modeling a set of BER curves using test patterns consisting of chosen symbol pairs, or pattern duets (as opposed to pseudo-random noise [PN] symbols). Each pattern duet generates a BER performance curve that, compared to an ideal orthogonal curve, can measure the effect these cross-correlation values have on the system.

When orthogonal symbols are tested in this way, each of the pattern duets has identical BER curves — each laying on top of the ideal orthogonal curve as shown in Figure 2 above, labeled r50. When nonorthogonal symbols with large spreads of r values are tested in this way, the pattern duets will have different BER curves — each stretching farther from the ideal orthogonal curve as their r values approach 11.

Consider a specific case of a noncoherent, *M*54 FSK digital modulation scheme designed for a narrowband channel. Each per-symbol frequency is modulated by *k*3600Hz (*k* 5 61 and 63) resulting in the symbol set (6600Hz, 61,800Hz) transmitted at a rate of 4800 baud. If this symbol set were orthogonal, the difference between any two symbols (*k _{i}*2

*k*) would be equal to an integer multiple of the symbol rate: (

_{j}*k*2

_{i}*k*) 5

_{j}*n*3 4800. Because no two symbols satisfy this condition, it is clear that this 4-FSK example is nonorthogonal.

There are three values for the frequency spacing between these symbols: 1,200Hz, 2,400Hz and 3,600Hz, with three r metric values: 0.9, 0.67 and 0.3, respectively. It is interesting to note that the lowest r value corresponds to the largest symbol spacing (3,600Hz) between the 61,800Hz deviation points. This design case was used to generate the set of nonorthogonal pattern duet test curves with the large spread of r values labeled in Figure 2. When this nonorthogonal system is tested using the standard PN test pattern, the red PN curve in Figure 2 results, showing a system performance several decibels poorer than ideal. This specific-case 4-FSK-modulation design suffers a degraded BER system performance due to its varied nonorthogonal r values.

In a Rayleigh-faded channel, the BER performance predictions use the equation:

where *E(*a* ^{2})* is the Rayleigh-fading term. This Rayleigh term predicts performance degradations similar to the (1-r) cross-correlation term. Therefore, nonorthogonal systems like the 4-FSK example above may have static BER performance curves similar to the faded-channel performance of an ideal orthogonal system.

### Applications to system purchase

Buyers should know how the performance of any prospective digital system is characterized. It’s unrealistic to expect the performance of a nonorthogonal modulation design to be predicted from the standard orthogonal symbol performance curves found in textbooks and industry references. This is especially important when considering narrowband FSK modulation systems having *M*>2 symbols.

When analyzing such systems, be wary of those with unequal symbol separations having several r values greater than zero. Such modulation schemes can suffer from reduced link performances when passing typical random data patterns.

Bartlett is a wireless systems engineer and consultant for terrestrial and space applications. His email address is [email protected].