Multistage Multirate Filters (Digital Filter Design Toolkit)

The filters you design in the Designing Floating-Point Multirate Filters and the Designing Fixed-Point Multirate Filters books are single-stage multirate filters. In single-stage multirate filters, the normalized transition bandwidth of the lowpass FIR filter H(z) is inversely related to the filter order. The narrower the normalized transition bandwidth, the higher the filter order. A lowpass FIR filter with a narrow normalized transition bandwidth therefore requires more resources to implement.

In decimation and interpolation multirate filters, the normalized transition bandwidth inversely relates to the decimation factor M and the interpolation factor L. The order of a decimation or interpolation filter increases as M or L increases, and the resulting multirate filter uses more resources to implement. You can use multistage multirate filters to simplify multirate filters that have large sampling frequency conversion factors.

A multistage filter gradually increases or decreases the sampling frequency by passing the signal through two or more resampling stages. Each stage has a lower decimation or interpolation factor than the corresponding single-stage multirate filter and contains fewer operations. Except when the sampling frequency conversion factor is a prime number, multistage filtering is more efficient than single-stage filtering because you can change the sampling frequency in multiple stages rather than in a single stage. Using multiple stages reduces the computation operations and memory usage. Refer to the book Multirate Systems and Filter Banks for more information about multistage multirate filter design.

In a multistage decimation system, the overall decimation factor M is equal to M₁M₂...M_N, where M_i is the decimation factor of stage i. The following figure illustrates this N-stage decimation process.

In a multistage interpolation system, the overall interpolation factor L is equal to L₁L₂...L_N, where L_i is the interpolation factor of stage i. The following figure illustrates the N-stage interpolation process.

You can use the DFD NStage MRate Filter Design VI to design multistage multirate filters with either of the following approaches:

• Specify the overall sampling frequency change factor and the factors for every stage.
• Specify only the overall sampling frequency change factor and use the DFD NStage MRate Filter Design VI to determine the factors for every stage.

Use the following guidelines when you manually specify factorizations.

• Use two or three stages for optimal or near optimal results.
• Use the largest factor at the highest sampling frequency. Decimate in order from the largest to the smallest factor and interpolate in order from the smallest to the largest factor.

When you implement a fixed-point multistage multirate filter, the output signal word length of the previous filter stage must be the same as the input signal word length of the next filter stage. Refer to the Multistage Multirate Filter Design VI in the labview\examples\Digital Filter Design\Floating-Point Filters\Multirate directory for an example that demonstrates how to use the DFD NStage MRate Filter Design VI to design a multistage multirate filter.

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