6.2 Efficient multirate filter structures

Rate-changing appears expensive computationally, since for both decimation and interpolation the lowpass filter is implementedat the higher rate. However, this is not necessary.

Interpolation

For the interpolator, most of the samples in the upsampled signal are zero, and thus require no computation. ( [link] )

Computational cost

• No simplification: $\frac{N}{T_1}=\frac{LN}{T_0}\frac{\mathrm{computations}}{\sec }$
• Polyphase structure: $L\frac{L}{N}\frac{1}{T_0^o}\frac{\mathrm{computations}}{\sec }=\frac{N}{T_0}$ where $L$ is the number of filters, $\frac{N}{L}$ is the taps/filter, and $\frac{1}{T_0}$ is the rate.

Efficient decimation structures

We only want every $M$ th output, so we compute only the outputs of interest. ( [link] ) $$x_1(m)=\sum_{k=N_1}^{N_2} x_0(Lm-k)h(k)$$

Polyphase decimation structure

Efficient l/m rate changers

Interpolate by $L$ and decimate by $M$ ( [link] ).

Computational cost