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The waveform S(t) can be a set of voltages or currents (over a wire), a set of optical signals (over an optical medium), a set of electric ﬁeld intensities (over the air), and so on. 2 Simply, the modulator emits signals at a rate of RS = 1/TS symbols per second. 3 If the symbol rate of this channel model is RS , then the bit rate is mRS or m/TS bps. 1, the modulated signal is distorted by attenuation, fading, noise, and other interference. A(t) expresses attenuation and fading while n(t) expresses added noise.

Notice also that this noise component has a zero mean and its variance is expressed as N0 /2. Similar to the binary antipodal signal, if this noise variable is more than half of the distance between S0 and S1 , a decision error occurs. 11 showing the probability of bit error is, p(ε) = ∞ d/2 1 2π N0 /2 e−n 2 /N 1/2 0 dn = Q d 2N0 1/2 . 12 emphasize the fact that the only factor to minimize the probability of error detection is the squared signal-space distance. So when we design modulation techniques, we simply need to maximize the squared signal-space distance given the boundaries of the signal energy that we can use.

20) which yields a 3 dB loss comparing to antipodal signaling (this is expected since envelope fading has no impact on phase). 21) which gives poorer performance than that of coherent orthogonal signals (4–6 dB factor). 5 Power Spectrum The power spectral density (power spectrum) of the modulated signal is important to understand and is related to the signal energy efﬁciency studied so far. Note that the power spectrum is measured as an averaged property of the signal and is in the order of seconds (which can hide certain properties of the signal).