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(Gallager) Assume that a communication channel first filters the trans- mitted passband signal before adding WGN. Suppose the channel is known and the channel filter has an impulse response h(t). Suppose that a QAM scheme with symbol duration T is developed without knowledge of the channel filtering. A baseband filter θ(t) is developed satisfying the Nyquist property that {θ(t - kT)}k is an orthonormal set. The matched filter θ(-t) is used at the receiver before sampling and detection.
If one is aware of the channel filter h(t), one may want to redesign either the baseband filter at the transmitter or the baseband filter at the receiver so that there is no intersymbol interference between receiver samples and so that the noise on the samples is i.i.d.
1. Which filter should one redesign?
2. Give an expression for the impulse response of the redesigned filter (assume a carrier frequency fc).
3. Draw a figure of the various filters at passband to show why your solution is correct. (We suggest you do this before answering the first two parts.)
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