Discrete-Time Linear Systems : Theory and Design with by Guoxiang Gu

By Guoxiang Gu

Discrete-Time Linear platforms: thought and layout with functions combines approach concept and layout so that it will exhibit the significance of procedure thought and its position in method layout. The booklet specializes in procedure idea (including optimum kingdom suggestions and optimum country estimation) and method layout (with functions to suggestions keep an eye on platforms and instant transceivers, plus process identity and channel estimation).

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52) k=0 where H(k) is a matrix of size p × m and is the impulse response at time t = k. In obtaining the impulse response of the system, the m impulse inputs need be applied one by one. The corresponding m output signals of size p can then be packed together column-wise to form {H(t)}t=0 . Since the impulse response dies out in finitely many samples, it acquires the name FIR (finite impulse response). 52). Let {u(t)} and {y(t)} be the associated input and output signals, respectively. Then y(t) = t ∑ H(k)u(t − k) = ∑ k=0 H(t − k)u(k).

44 2 Signals and Systems With variable substitution γ = α − β , the above results in E{y(t)y(t − k)∗ } = = = ∞ ∞ ∑ ∑ β =−∞ γ =−∞ ∞ ∑ β =−∞ ∞ ∑ τ =−∞ H(t − β − γ )Ru (γ )H(t − β − k)∗ R˜ y (t − β )H(t − β − k)∗ R˜ y (τ )H(τ − k)∗ = Ry (k), which is independent of time t, where R˜ y (τ ) = H(τ ) Ru (τ ) with τ = t − β . Hence, it is concluded that the output of an LTI system is a WSS random process, provided that the input is. Let Ψu (ω ) be the PSD associated with input. 9 in Exercises) Ψy (ω ) = H e jω Ψu (ω )H e jω ∗ .

An important application content of this text focuses on design of wireless transceivers which are the building block and the physical layer of the wireless networks. A digital communication system is linear and operates in discrete time. Its purpose lies in communication of digital data or symbols which have finite alphabet table. These data cannot be transmitted and received directly. They have to be modulated with continuous-time waveforms and converted to radio frequency (RF) signals prior to transmission and reception, leading to linear discrete-time systems for wireless data communications.

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