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Pulse Amplitude Modulation through Bandlimited Channel

Module by: Tuan Do-Hong. E-mail the author

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 ⁢s⁢t−n⁢T 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 ⁢s⁢t−(τ−n⁢T)⁢g⁢τdτ+ N t =∑n=−∞∞ a n ⁢∫−∞∞s⁢t−(τ−n⁢T)⁢g⁢τdτ+ N t =∑n=−∞∞ a n ⁢ s ˜ ⁢t−n⁢T+ 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¯⁢g⁢t s ˜ t s g t is sufficient.

The matched filter's impulse response is

∀t: h opt ⁢t=s¯⁢g⁢T−t t h opt t s g T t (3)
The matched filter output is
y⁢t=∫−∞∞∑n=−∞∞ a n ⁢ s ˜ ⁢t−(τ−n⁢T)⁢ h opt ⁢τdτ+ν⁢t=∑n=−∞∞ a n ⁢∫−∞∞ s ˜ ⁢t−(τ−n⁢T)⁢ h opt ⁢τdτ+ν⁢t=∑n=−∞∞ a n ⁢u⁢t−n⁢T+ν⁢t 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 k⁢T k T :

y⁢k⁢T=∑n=−∞∞ a n ⁢u⁢k⁢T−n⁢T+ν⁢k⁢T y k T n a n u k T n T ν k T (5)
The k th k th symbol is of interest:
y⁢k⁢T= a k ⁢u⁢0+∑n=−∞∞ a n ⁢u⁢k⁢T−n⁢T+ν⁢k⁢T 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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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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