Signals and Systems - Electrical Engineering

8.2 Discrete-Time Signals 453

Determine an appropriate sampling periodTsaccording to the Nyquist sampling rate condition,

and obtain the discrete-time signalx[n] corresponding to the largest allowed sampling period.

Solution

To samplex(t)so that no information is lost, the Nyquist sampling rate condition indicates that

the sampling period should be

Ts≤

π

max

=

π

2 π

=0.5

For the largest allowed sampling periodTs=0.5, we obtain

x[n]=3 cos( 2 πt+π/ 4 )|t=0.5n=3 cos(πn+π/ 4 ) −∞<n<∞

which is a function of the integern. n

nExample 8.2

To generate the celebrated Fibonacci sequence of numbers,{x[n]}, we use the recursive equation

x[n]=x[n−1]+x[n−2] n≥ 2

x[0]= 0

x[1]= 1

which is a difference equation with zero input and two initial conditions. The Fibonacci sequence

has been used to model different biological systems.^1 Find the Fibonacci sequence.

Solution

The given equation allows us to compute the Fibonacci sequence recursively. Forn≥2, we find

x[2]= 1 + 0 = 1

x[3]= 1 + 1 = 2

x[4]= 2 + 1 = 3

x[5]= 3 + 2 = 5

..

.

where we are simply adding the previous two numbers in the sequence. The sequence is purely

discrete as it is not related to a continuous-time signal. n

(^1) Leonardo of Pisa (also known as Fibonacci) in his bookLiber Abacidescribed how his sequence could be used to model the
reproduction of rabbits over a number of months assuming bunnies begin breeding when they are a few months old.