Tomto Souboru.pdf

EE263 Autumn 2008-09 homework problems cover various topics in linear systems, including power control algorithms, linear mechanical systems, time-series models, linear functions, convolution systems, and matrix representations.

Problem 2.1 involves a simple power control algorithm for a wireless network, where the goal is to adjust the powers to ensure that the signal to interference plus noise ratio (SINR) exceeds a threshold value. The algorithm is given by pi(t + 1) = pi(t)(αγ/Si(t)), and the problem asks to show that this can be expressed as a linear dynamical system with constant input.

Problem 2.2 deals with state equations for a linear mechanical system, where the equations of motion can be expressed as M ¨q + D ˙q + Kq = f, and the goal is to write linear system equations with state x = [qT ˙qT ]T, input u = f, and output y = q.

Problem 2.3 involves expressing various time-series models, including moving average (MA), autoregressive (AR), and autoregressive moving average (ARMA) models, as linear dynamical systems with input u and output y.

Problem 2.4 asks to show that a linear function f: Rn → Rm can be represented as matrix multiplication, i.e., f(x) = Ax, and to describe how to get the coefficients Aij from f.

Problem 2.5 deals with a convolution system, where the input and output signals are related via convolution, and the goal is to find the input/output (Toeplitz) matrix and the Hankel matrix associated with the system.

Problem 2.6 involves finding the matrix representation of polynomial differentiation, where a polynomial p(x) is represented as a vector [a0 a1 ... an-1]T, and the goal is to find the matrix D that represents the linear transformation D that differentiates polynomials.

Problem 2.7 asks to find a matrix G that shows how the output at t = 0, ..., N depends on the initial state x(0) and the sequence of inputs u(0), ..., u(N) for a discrete-time linear dynamical system.

Problem 2.8 deals with sparsity patterns, including tridiagonal matrices and a specific linear mapping with a particular sparsity structure.

Problem 2.9 involves finding matrices A and B that represent linear mappings in signal flow graphs.