Wireless Ad Hoc and Sensor Networks

Background 53with x( 0)being the initial amount in the account. Using the standard seriessummation formula,k−1kj a∑ a = 1 − 1−aj=0one derives the standard formula for complex interest with constantannuities of d.kxk ( ) = ( 1+ i) x( 0) + d( 1+i)k= ( 1+ i) x( 0) + d( 1+i)∑k−1k−1k−1j=01j( 1+i)⎡ 1⎢1−⎢( 1+i)⎢ 1⎢1−⎣ ( 1+i)⎡k−1+ − ⎤k ( 1 i)1= ( 1+ i) x( 0)+ d⎢⎥⎣ i ⎦k⎤⎥⎥⎥⎥⎦SIMULATIONIt is very easy to simulate a discrete-time system. No numerical integrationdriver program is needed in contrast to the continuous-time case. Instead,a simple “do loop” can be used. A complete MATLAB program thatsimulates the compound interest dynamics is given by%Discrete-Time Simulation program for CompoundInterest Dynamicsd = 100; i = 0.08; % 8% interest ratex(1)= 1000;for k = 1:100x(k+1)=(1+i)*x(k)endk = [1:101];plot(k,x);2.2 Mathematical Background2.2.1 Vector and Matrix NormsWe assume the reader is familiar with norms, both vector and inducedmatrix norms (Lewis, Abdallah, and Dawson 1993). We denote any suitable