7. Filter Implementation

• We will discuss in this section implementation of finite impulse response (FIR) and infinite impulse response (IIR) filters in HLS.

• Recall that both FIR and IIR filters can be defined by linear I/O difference equations in the time domain:

  • FIR filter of order \(M\):

    (7.1)\[\begin{align} y[n] = \sum_{k=0}^M b_k x[n-k] & \hspace{90pt} & (b_0 \neq 0) \end{align}\]

  • IIR filter of feedforward order \(M\) and feedback order \(N\) (or simply of order \(N\)):

    (7.2)\[\begin{align} \sum_{k=0}^N a_k y[n-k] = \sum_{k=0}^M b_k x[n-k] & \hspace{50pt} & (a_0 =1) \end{align}\]

    where \(x[n]\) and \(y[n]\) denote the input and output signals, respectively.

• Equivalently, in the \(z\)-domain, the above difference equations are transformed to the following algebraic equations:

  • FIR filter of order \(M\):

    (7.3)\[\begin{align} Y(z) = \sum_{k=0}^M b_k z^{-k} X(Z) & \hspace{80pt} & (b_0 \neq 0) \end{align}\]

  • IIR filter of feedforward order \(M\) and feedback order \(N\) (or simply of order \(N\)):

    (7.4)\[\begin{align} \sum_{k=0}^N a_k z^{-k} Y(z) = \sum_{k=0}^M b_k z^{-k} X(z) & \hspace{50pt} & (a_0 =1) \end{align}\]

    where \(x[n] \stackrel{z}{\longleftrightarrow} X(z)\) and \(y[n] \stackrel{z}{\longleftrightarrow} Y(z)\).