Two discrete-time systems are connected in series. Their input-output difference equations are:

\begin{align} w[k]&=0.25x[k-2]\\ y[k]&=0.5w[k-1]+0.5w[k-2]. \end{align}

  1. Determine the overall input-output difference equation relating $y[k]$ to $x[k]$
  2. Determine the impulse response of the overall system



You simply need to put $w[k]$ into $y[k]$:

$$w[k]=0.25x[k-2] \Rightarrow \begin{align}w[k-1]&=0.25x[k-3]\\ w[k-2]&=0.25x[k-4] \end{align}$$

\begin{align} y[k] &= 0.5w[k-1] + 0.5w[k-2]\\ &=0.5\cdot0.25x[k-3] + 0.5\cdot0.25x[k-4]\\ &= 0.125(x[k-3]+x[k-4]) \end{align}

You can directly read the impulse response from it.

  • $\begingroup$ okay, and then i know what the b3 and the b4 koefficients are (0.125), Okay this might be a very stupid quastion but, how do i read the impulse response from this? and Thank you $\endgroup$
    – julie
    Nov 10 '16 at 6:44
  • $\begingroup$ The impulse response is [0, 0, 0, 0.125, 0.125, 0, 0, 0, ....] $\endgroup$ Nov 10 '16 at 7:35

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