1.4 Wireless communications background  (Page 3/4)

${Ф}_k\mathrm{(t)\; =}\frac{\mathrm{2\pi k}}{M}\mathrm{k\; =\; 1,...M}$

In the basic case of binary PSK, the modulating data signal shifts the phase of the waveforms to one of two states: either zero or $\pi$ . The waveform sketch in the figure shows abrupt phase changes in the signaling interval. Figures 8 through 11 are typical waveforms for a PSK modulation system.

Figure 8. Phase Shift Keying symbol source.

Figure 9. Phase Shift Keying modulated symbols.

Figure 10. Phase Shift Keying constellation plot.

Figure 11. Phase Shift Keying spectrum.

Comparison metric for digital communication and BER curves

One of the most important metrics of performance in digital communication is the plot of bit error probability $P_b$ versus energy per bit over noise power spectral density ( $\frac{E_b}{N_0}$ ) . This dimensionless ratio is a standard quality measure for digital communication system performance. The smaller the $\frac{E_b}{N_0}$ required, the more efficient the detection process for a given probability of error. In digital communication systems, one discrete symbol is transmitted that may be one bit or more in a fixed signaling interval. In analog, where the information source is continuous, this discrete structure does not exist. In digital systems, a figure of merit should allow us to compare one system with another at a bit or symbol level; hence $\frac{E_b}{N_0}$ renders itself naturally for that purpose. A relationship between the SNR, data rate and bandwidth is expressed in Equation 8:

$\frac{E_b}{N_0}=\frac{S}{N}(\frac{B}{R})$

Figures 12 through 14 show typical $P_b$ vs. $\frac{E_b}{N_0}$ curves for orthogonal and multiphase signaling.

Figure 12. The curve to the left in the above chart represents M = 32 while the curve to the right represents M = 2.

Figure 13. The curve to the left in the above chart represents M = 2 and the curve to the right represents M = 32.

Figure 14. BER comparisons of ASK, 2FSK and 2PSK

Digital modulation methods can be classified in two ways, with opposite behavioral characteristics. The first class is orthogonal signaling; its error performance follows the curves in the first figure. The second class constitutes nonorthogonal signaling is shown in the second figure. Error performance improvement or degradation depends on signaling category.

Channel capacity

Claude Shannon's fundamental theorem states that it is possible (in principle, using some coding scheme) to transmit information with an arbitrarily small probability of error, provided that the data rate $R$ is less than or equal to the channel capacity $C$ . Shannon’s work showed that SNR and bandwidth set a limit on transmission rate but not on probability of error. The channel capacity of a white bandlimited Gaussian noise channel is expressed in Equation 9:

$C\; ={\mathrm{Blog}}_2(1+\frac{E_b}{N_0}\frac{B}{R})bits/sec/Hz$

Where:

$B$ = the channel bandwidth

$\frac{E_b}{N_0}$ = the energy/bit with an example SNR of 30 dB

Using Equations 8 and 9, the capacity of a circuit with 2.4-kHz bandwidth is approximately 24 kbps, whereas at 10-dB SNR the capacity drops to about 8.3 kbps. Thus, Shannon's theorem allows designers to apply trade-offs in bandwidth, signal power and various modulation methods to establish a communication link with a desired probability of error.