Tutorials - Digital Signal Processing: An Introduction with MATLAB and Applications - page 232

5. If |x(t)| B c Vt (bounded), then we have |x(t - T 0 )| B c Vt.
Hence, |y(t)| B c Vt and the system is BIBO-stable.

Tutorial 4

Q: Find the output of the following linear time-invariant system for a unit-step
input (i.e., find the step response). Verify that the impulse response is the time
derivative of the step response.

h ( t ) = e − at u ( t )

x ( t ) = u ( t ) y ( t )

( a > 0 )

We have: y ð t Þ¼ u ð t Þ h ð t Þ¼ R 1 1 u ð k Þ h ð t k Þ dk :

Solution:

Step 1: Formulas of functions using t:

:

h ð t Þ¼ e at u ð t Þ¼ e at ; u ð t Þ¼ 1 ;

t 0 t 0

and

0 ; t\0 0 ; t\0

Step 2: Formulas of functions using k:

u ð k Þ¼ 1 ;

k 0

and

0 ; k\0

( )

h ð t k Þ¼ e a ð t k Þ ;

t k 0 ! k t

:

0 ; k [ t

Step 3: Integrate to find the convolution. The range of integration can be found

graphically as the zone of non-zero overlap between the two functions.
This can be done by moving h(t - k) from left to right (i.e., by varying
t from -? to ?) while keeping u(k) fixed (see the plot below).
Case 1: t [ 0

y ð t Þ¼ Z

t

exp ½ a ð t k Þ dk

0

¼ð 1 = a Þ½ exp f a ð t k Þg t 0 ¼ð 1 = a Þ½ 1 exp ð at Þ:

Case 2: t \ 0

No overlap ) u(k)h(t - k) : 0 Vk ) y(t) : 0.