1.4 Wireless communications background  (Page 2/4)

$\mathrm{d(t)}={\sum }_k{I}_k\mathrm{p(t\; -\; kT);}{I}_kЄ\{\pm 1,\; \pm 3\};\mathrm{k\; =\; 1,\; 2,\; ...}$

Where:

$\mathrm{d(t)}$ = an analog waveform consisting of pulses at symbol time $\mathrm{kT}$ and the amplitude of the pulse is proportional to the associated symbol value.

Ideally, the pulse should be chosen so that the value of message at $k$ does not interfere with the message at any other time (no ISI) and makes efficient use of bandwidth.

ISI can manifest itself in two ways: when the pulse shape $\mathrm{p(t)}$ is wider in time than a single symbol time interval $T$ , and when the pulses experience channel distortions and multipath fading effects.

Consider a two-level system in which $s_0\mathrm{(t)}$ and $s_1\mathrm{(t)}$ are finite energy signals representing logical 0 and 1, respectively. These signals can be of any shape but must have finite energy in the signaling interval. Then you can construct a framework of representative basic modulation schemes. Some fundamental digital modulation schemes are below.

Amplitude shift keying

In amplitude shift keying (ASK), the information is conveyed by varying the amplitude of a carrier wave in accordance with the symbol stream. ASK can be expressed as Equation 4:

$s_k(t)=I_k\mathrm{p(t)cos(}{w}_{0}t+Ф);I_kЄ\{0,1\};\mathrm{0\le t\le T};$

In Equation 4, the phase term is an arbitrary constant. Binary ASK signaling, also called on-off keying, was one of the earliest forms of digital modulation used in radio telegraphy. ASK has a high peak-to-average ratio and is no longer widely used; however, TI’s low-power wireless radio-frequency integrated circuits support this modulation scheme for various data rates in sensor applications. Figures 1 - 4 shows waveforms for Amplitude Shift Keying.

Figure 1. Amplitude Shift Keying source symbols.

Figure 2. Amplitude Shift Keying modulated symbols.

Figure 3. Amplitude Shift Keying constellation plot.

Figure 4. Amplitude Shift Keying spectrum.

Frequency-shift keying

The general analytical expression for frequency shift keying (FSK) is given in Equation 5:

$s_k(t)=A\mathrm{cos(}{w}_0\mathrm{t+2\pi }{I}_k\mathrm{\Delta ft+Ф);}\mathrm{k\; =\; 1,...M;}\mathrm{0\le t\le T}$

Where:

The frequency Δf = the amount of shift in the carrier frequency corresponding to the alphabet I _k ε{±1, ±3. . .,±M}

Phase term is an arbitrary constant.

The FSK waveform sketch in the following figures show typical frequency changes at the symbol interval. The change from one frequency to another can be rather abrupt; this gives rise to spikes in the spectrum of FSK. The minimum required bandwidth for orthogonal FSK signals for coherent detection is 1/2T, whereas for noncoherent detection the bandwidth is 1/T. FSK does not have constellation plots because of constant rotation of the signal vector in the IQ plane. Figures 5 - 7 are typical waveforms for FSK.

Figure 5. Frequency Shift Keying symbol source.

Figure 6. Frequency Shift Keying modulated symbols.

Figure 7. Frequency Shift Keying spectrum.

Phase-shift keying

Developed the during early days of the space program, phase-shift keying (PSK) is widely used in commercial satellite links. The general expression for PSK is shown in Equation 6:

$s_k(t)=\mathrm{p(t)cos(}{w}_{0}t+{Ф}_k);\mathrm{k\; =\; 1,...,M}\mathrm{0\le t\le T}$

Where the phase term ${Ф}_i$ will have M discrete values typically given by Equation 7: