Many signals in communication systems are real bandpass signals with a frequency response that occupies a narrow bandwidth 2B centered around a carrier frequency fc with 2B « fc, as shown in Figure A.1. Since bandpass signals are real, their frequency response has conjugate symmetry: a bandpass signal s(t) has |S(f)| = |S(−f)| and ∠S(f) = −∠S(−f). However, bandpass signals are not necessarily conjugate symmetric within the signal bandwidth about the carrier frequency fc; that is, we may have |S(fc + f)| ≠ |S(fc − f)| or ∠S(fc + f) ≠ −∠S(fc − f) for some f : 0 < f ≤ B. This asymmetry in |S(f)| about fc (i.e., |S(fc + f)| ≠ |S(fc − f)| for some f < B) is illustrated in the figure. Bandpass signals result from modulation of a baseband signal by a carrier, or from filtering a deterministic or random signal with a bandpass filter. The bandwidth 2B of a bandpass signal is roughly equal to the range of frequencies around fc where the signal has nonnegligible amplitude. Bandpass signals are commonly used to model transmitted and received signals in communication systems. These are real signals because the transmitter circuitry can only generate real sinusoids (not complex exponentials), and the channel simply introduces an amplitude and phase change at each frequency of the real transmitted signal.