Signals and Systems - Electrical Engineering

Problems 111

1.10. Ramp in terms of unit-step signals
A ramp,r(t)=tu(t), can be expressed as

r(t)=

∫∞

0

u(τ)u(t−τ)dτ

(a) Show that the above expression forr(t)is equivalent to

r(t)=

∫t

0

dτ=tu(t)

(b)Compute the derivative of

r(t)=

∫∞

0

u(τ)u(t−τ)dτ

to show that

u(t)=

∫∞

0

u(τ)δ(t−τ)dτ

1.11. Sampling signal and impulse signal—MATLAB
Consider the sampling signal

δT(t)=

∑∞

k= 0

δ(t−kT)

which we will use in the sampling of analog signals later on.

(a) PlotδT(t). Find

ssT(t)=

∫t

−∞

δT(τ)dτ

and carefully plot it for allt. What does the resulting signalss(t)look like? In reference 17, Craig calls it

the “stairway to the stars.” Explain.

(b)Use MATLAB functionstairsto plotssT(t)forT=0.1. Determine what signal would be the limit as

T→ 0.

(c)A sampled signal is

xs(t)=x(t)δT(t)=

∑∞

k= 0

x(kTs)δ(t−kTs)

Letx(t)=cos( 2 πt)u(t)andTs=0.1. Find the integral

∫t

−∞

xs(t)dt

and use MATLAB to plot it for 0 ≤t≤ 10. In a simple way this problem illustrates the operation of a

discrete-to-analog converter, which converts a discrete-time into a continuous-time signal (its cousin

is the digital-to-analog converter or DAC).