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Digital Transmission over Baseband Channels

Module by: Behnaam Aazhang

Summary: I have no idea whatsoever.

Until this point, we have considered data transmissions over simple additive Gaussian channels that are not time or band limited. In this module we will consider channels that do have bandwidth constraints, and are limited to frequency range around zero (DC). The channel is best modified as gt g t is the impulse response of the baseband channel.
Consider modulated signals x t = s m t x t s m t for 0≤t≤T 0 t T for some m∈12…M m 1 2 … M . The channel output is then
r t =∫-∞∞ x τ gt-τdτ+ N t =∫-∞∞ S m τgt-τdτ+ N t r t τ x τ g t τ N t τ S m τ g t τ N t (1)
The signal contribution in the frequency domain is
∀f: S m ˜ f= S m fGf f S m ˜ f S m f G f (2)
The optimum matched filter should match to the filtered signal:
∀f: H m opt f= S m f¯Gf¯ⅇ-ⅈ2πft f H m opt f S m f G f 2 f t (3)
This filter is indeed optimum (i.e., it maximizes signal-to-noise ratio); however, it requires knowledge of the channel impulse response. The signal energy is changed to
E s ˜ =∫-∞∞| S m ˜ f|2df E s ˜ f S m ˜ f 2 (4)
The band limited nature of the channel and the stream of time limited modulated signal create aliasing which is referred to as intersymbol interference. We will investigate ISI for a general PAM signaling.

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