ECE 280/Roberts13

Chapter 2

• Unit step $$u(t)$$
  • $$u(at)=u(t)$$
• Unit ramp ramp($$t$$)=$$r(t)$$=$$t\,u(t)$$
  • $$r(at)=at\,u(at)=at\,u(t)=a\,r(t)$$
• Unit rectangle rect($$t$$)=$$u(t+1/2)-u(t-1/2)$$ is a rectangular pulse centered on 0 of height 1 and width 1 (nonzero from $$t=$$ $$-1/2$$ to $$1/2$$)
  • rect$$\left(\frac{t}{a}\right)$$ is a rectangular pulse centered on 0 of height 1 and width $$|a|$$ (nonzero from $$t=$$ -$$|a|/2$$ to $$|a|/2$$)
  • rect$$\left(t-t_0\right)$$ is a rectangular pulse centered on $$t_0$$ of height 1 and width 1 (nonzero from $$t=$$ $$-1/2+t_0$$ to $$1/2+t_0$$)
  • rect$$\left(\frac{t-t_0}{a}\right)$$ is a rectangular pulse centered on $$t_0$$ of height 1 and width $$|a|$$ (nonzero from $$t=$$ $$-|a|/2+t_0$$ to $$|a|/2+t_0$$)
• Unit triangle tri($$t$$)=$$r(t-1)-2r(t)+2(t+1)$$ is a triangular pulse centered on 0 of height 1 and width 2 (nonzero from $$t=$$ $$-1$$ to $$1$$)
  • tri$$\left(\frac{t}{a}\right)$$ is a triangular pulse centered on 0 of height 1 and width $$2|a|$$ (nonzero from $$t=$$ -$$|a|$$ to $$|a|$$)
  • tri$$\left(t-t_0\right)$$ is a triangular pulse centered on $$t_0$$ of height 1 and width 2 (nonzero from $$t=$$ $$-1+t_0$$ to $$1+t_0$$)
  • tri$$\left(\frac{t-t_0}{a}\right)$$ is a triangular pulse centered on $$t_0$$ of height 1 and width $$2|a|$$ (nonzero from $$t=$$ $$-|a|+t_0$$ to $$|a|+t_0$$)
• Unit impulse $$\delta(t)$$ has an area of 1 at $$t=0$$ and is 0 otherwise
  • $$\delta(at)=\frac{1}{|a|}\delta(t)$$
• Unit periodic impulse train $$\delta_T(t)=\sum_{k=-\infty}^{\infty}\delta(t-kT)$$ (pulse train with period $$T$$)
  • $$\delta_T(t-t_0)=\sum_{k=-\infty}^{\infty}\delta(t-kT-t_0)$$ (shifted pulse train with period $$T$$)

Chapter 3

• [3.1] Unit impulse $$\delta[n]$$ has a value of 1at $$n=0$$ and is 0 otherwise
  • $$\delta[an]=\delta[n]$$
• [3.2] Unit step $$u[n]$$ is 0 for $$n<0$$ and 1 for $$n\geq 0$$
• [3.5] Unit ramp ramp[$$n$$]=$$r[n]=n\,u[n]$$
• [3.6] Unit periodic impulse train $$\delta_N[n]=\sum_{m=-\infty}^{\infty}\delta[n-mN]$$ (pulse train with period $$N$$)