### Discrete and continuous FIR filter design

The discrete time filter design problem is treated as a weighted Chebyshev approximation problem and is solved using the Remez Multiple Exchange algorithm to compute the filter coefficients. The algorithm builds a discrete time representation of the filter.

In Embed, the Discrete parameter in the Transfer Function Setup dialog box controls whether the generated FIR filters are discrete or continuous. When you design a discrete FIR filter, you must also specify a time step in the dT box.

To implement a continuous FIR filter, de-activate the Discrete parameter. In this case, the filter is initially designed as a discrete time filter. Bilinear transformation is subsequently used to produce a continuous time equivalent. For more information on the Remez algorithm, see Theory and Application of Digital Signal Processing (Prentice Hall).

Tapped delay is a method of transfer function implementation that has linear computational and storage requirements with respect to model order. Because most FIR filters tend to be high order, it makes sense to design FIR filters with tapped delay implementation. To do so, activate Tapped Delay in the Transfer Function Properties dialog box.

To add a band

Enter the band specification and click Add.

The band information is added to the list box. Each row in the list box corresponds to a single band. For FIR filters, if the number of bands increases, the filter order must be increased correspondingly, to maintain the same approximation error. For differentiator and Hilbert transformers, the number of bands is limited to one. The gain on the differentiator implies the gain achieved at the end frequency. The weight in either case is optimally adjusted to give the best error characteristics.

To delete a band

Select the band to be deleted from the list box and click Delete.

To change a band’s specifications

1.    Select the band to be changed from the list box.

2.    The band’s data appears in the edit boxes.

3.    Make the desired changes.

4.    Click Change. The band data is modified in the list box to reflect the changes.

Calc Filter command button. This generates the appropriate filter coefficients. Before computing the filter coefficients, the algorithm computes the maximum approximation error. This error is usually referred to as delta (δ) and is defined as the weighted difference between the actual and the desired magnitude response. A band with a weight of one will have delta as its absolute approximation error, while a band with a weight of 10 will have its absolute error 0.1 times δ. The value of δ is displayed in the message box.

1.    Click Calc Filter. The coefficients are displayed in the Num (numerator) and Den (denominator) boxes. If the delta displayed is too large, increase the order of the filter and re-calculate the filter.

2.    Click Done to close the FIR Filter Properties dialog box and transfer the filter numerator and denominator coefficients to the Transfer Function Properties dialog box.