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 $)