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Detection of Signals Transmitted over Complicated Channels

Module by: Don Johnson. E-mail the author

In considering the additive, white Gaussian noise channel, we found that the performance of the optimum receiver depended only on the signal-to-noise ratio. Consequently, details of the signal waveforms do not matter to a great degree: Energy is what counts. We continue with the study of channels which corrupt the transmitted signal in different and more complicated (from a theoretical viewpoint) ways. We shall see that signal waveforms do matter. Consequently, the design of signal sets become more involved.

Dispersive Channels

We first consider the corruption of the transmitted signal by a linear filter.

Figure 1
Figure 1 (dispersive.png)
Such channels are said to be dispersive channels. h CH ⁢tτ h CH t τ denotes the impulse response of the deterministic, possibly time-varying, linear filter. s i * ⁢t=∫0T h CH ⁢tτ⁢ s i ⁢τd τ s i * t τ 0 T h CH t τ s i τ As far as the receiver is concerned, the transmitter is using the signal set s i * ⁢t s i * t . Consequently, the solution of the optimum receiver is straightforward using the theory as it stands. However, a complication arises in the detailed consideration of these signals' durations. The channel tends to increase the duration of s i ⁢t s i t , hence the origin of the term dispersive. We first consider the problem where a sufficient amount of time is allowed for the transmission of a "bit" so that successive transmissions do not overlap each other.

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