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Pulse Amplitude Modulation Through Bandlimited Channel

Module by: Behnaam Aazhang

Summary: A description of the functionality of a pulse amplitude modulated transmission system in a bandlimited channel. The bandlimited nature of the channel causes it to have memory which causes intersymbol interference, which can be dealt with in a number of ways.

Consider a PAM system b -10 b -10 ,…, b -1 b -1 , b 0 b 0 b 1 b 1 ,…

This implies

∀ a n , a n ∈M levels of amplitude: x t =∑n=-∞∞ a n st-nT a n a n M levels of amplitude x t n a n s t n T (1)

The received signal is

r t =∫-∞∞∑n=-∞∞ a n st-τ-nTgτdτ+ N t =∑n=-∞∞ a n ∫-∞∞st-τ-nTgτdτ+ N t =∑n=-∞∞ a n s ˜ t-nT+ N t r t τ n a n s t τ n T g τ N t n a n τ s t τ n T g τ N t n a n s ˜ t n T N t (2)

Since the signals span a one-dimensional space, one filter matched to s ˜ t=s¯gt s ˜ t s g t is sufficient.

The matched filter's impulse response is

∀t: h opt t=s¯gT-t t h opt t s g T t (3)
The matched filter output is
yt=∫-∞∞∑n=-∞∞ a n s ˜ t-τ-nT h opt τdτ+νt=∑n=-∞∞ a n ∫-∞∞ s ˜ t-τ-nT h opt τdτ+νt=∑n=-∞∞ a n ut-nT+νt y t τ n a n s ˜ t τ n T h opt τ ν t n a n τ s ˜ t τ n T h opt τ ν t n a n u t n T ν t (4)

The decision on the k th k th symbol is obtained by sampling the MF output at kT k T :

ykT=∑n=-∞∞ a n ukT-nT+νkT y k T n a n u k T n T ν k T (5)
The k th k th symbol is of interest:
ykT= a k u0+∑n=-∞∞ a n ukT-nT+νkT y k T a k u 0 n a n u k T n T ν k T (6)
where n≠k n k .

Since the channel is bandlimited, it provides memory for the transmission system. The effect of old symbols (possibly even future signals) lingers and affects the performance of the receiver. The effect of ISI can be eliminated or controlled by proper design of modulation signals or precoding filters at the transmitter, or by equalizers or sequence detectors at the receiver.

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