Signals and Systems - Electrical Engineering

476 C H A P T E R 8: Discrete-Time Signals and Systems

obtained by sampling a continuous-time signal formed by ramp and unit-step signals with a

sampling periodTs=0.15. Write MATLAB functions to generate the ramp and the unit-step signals

and obtainy[n]. Then write a MATLAB function that provides the even and the odd decomposition

ofy[n].

Solution

The real-valued signal is obtained by sequentially adding the different signals as we go from−∞

to∞:

y(t)=















0 t<− 3

3 r(t+ 3 )= 3 t+ 9 − 3 ≤t<− 1

3 t+ 9 − 6 r(t+ 1 )=− 3 t+ 3 − 1 ≤t< 0

− 3 t+ 3 + 3 r(t)= 3 0 ≤t< 3

3 − 3 = 0 t≥ 3

The three functionsramp, ustep, andevenoddfor this example are shown below. The following

script shows how they can be used to generate the ramp signals, with the appropriate slopes and

time shifts, as well as the unit-step signals with the desired delay, and then how to compute the

even and the odd decomposition ofy[n].

%%%%%%%%%%%%%%%%%%%%

% Example 8.18

%%%%%%%%%%%%%%%%%%%%

Ts = 0.15; % sampling period

t =−5:Ts:5; % time support

y1 = ramp(t, 3, 3); y2 = ramp(t,−6, 1); y3 = ramp(t, 3, 0); % ramp signals

y4 =− 3 ∗ustep(t,−3); % unit-step signal

y = y1 + y2 + y3 + y4;

[ze, zo] = evenodd(t, y);

We choose as support− 5 ≤t≤5 for the continuous-time signaly(t), which translates into a sup-

port− 5 ≤0.15n≤5 or− 5 /0.15≤n≤ 5 /0.15 for the discrete-time signal. Since the limits are not

integers, to make them integers (as required becausenis an integer) we use the MATLAB function

ceilto find integers larger than− 5 /0.15 and 5/0.15 giving a range [−33, 33]. This is used when

plottingy[n].

The following function generates a ramp signal for a range of time values, for different slopes and

time shifts.

function y = ramp(t, m, ad)

% ramp generation

% t: time support

% m: slope of ramp

% ad : advance (positive), delay (negative) factor