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Filter Design for Multirate Systems

Module by: Douglas L. Jones

The filter design techniques learned earlier can be applied to the design of filters in multirate systems, with a few twists.

Example 1

Design a factor-of-LL interpolator for use in a CD player, we might wish that the out-of-band error be below the least significant bit, or 96dB down, and < 0.05 % < 0.05 % error in the passband, so these specifications could be used for optimal L ∞ L ∞ filter design.

In a CD player, the sampling rate is 44.1kHz, corresponding to a Nyquist frequency of 22.05kHz, but the sampled signal is bandlimited to 20kHz. This leaves a small transition band, from 20kHz to 24.1kHz. However, note that in any case where the signal spectrum is zero over some band, this introduces other zero bands in the scaled, replicated spectrum (Figure 1).

Figure 1
Figure 1 (imag001.png)
So we need only control the filter response in the stopbands over the frequency regions with nonzero energy. (Figure 2)
Figure 2
Figure 2 (imag002.png)
The extra "don't care" bands allow a given set of specifications to be satisfied with a shorter-length filter.

Direct polyphase filter design

Note that in an integer-factor interpolator, each set of output samples x 1 Ln+p x 1 L n p , p=01…L-1 p 0 1 … L 1 , is created by a different polyphase filter g p n g p n , which has no interaction with the other polyphase filters except in that they each interpolate the same signal. We can thus treat the design of each polyphase filter independently, as an NL N L -length filter design problem. (Figure 3)

Figure 3
Figure 3 (iamg003.png)
Each g p n g p n produces samples x 1 Ln+p= x 0 n+pL x 1 L n p x 0 n p L , where n+pL n p L is not an integer. That is, g p n g p n is to produce an output signal (at a T 0 T 0 rate) that is x 0 n x 0 n time-advanced by a non-integer advance pL p L .

The desired response of this polyphase filter is thus H D p ω=ⅇⅈωpL H D p ω ω p L for |ω|≤π ω , an all-pass filter with a linear, non-integer, phase. Each polyphase filter can be designed independently to approximate this response according to any of the design criteria developed so far.

Exercise 1

What should the polyphase filter for p=0 p 0 be?

Solution 1

A delta function: h 0 n=δ n ′ h 0 n δ n ′

Example 2: Least-squares Polyphase Filter Design

  • Deterministic x(n) - Minimize ∑n=-∞∞|xn- x d n|2 n x n x d n 2 Given xn=xn*hn x n x n h n and x d n=xn* h d n x d n x n h d n . Using Parseval's theorem, this becomes
    min{∑n=-∞∞|xn- x d n|2}=min{12π∫-ππ|XωHω-Xω H d ω|2dω}=min{12π∫-ππ|Hω- H d ω||Xω|2dω} n x n x d n 2 1 2 ω X ω H ω X ω H d ω 2 1 2 ω H ω H d ω X ω 2 (1)
    This is simply weighted least squares design, with |Xω|2 X ω 2 as the weighting function.
  • stochastic X(ω) -
    min{E|xn- x d n|2}=E|xn*hn- h d n|2=min{12π∫-ππ| H d ω-Hω|2 S x x ωdω} x n x d n 2 x n h n h d n 2 1 2 ω H d ω H ω 2 S x x ω (2)
    S x x ω S x x ω is the power spectral density of xx. S x x ω=DTFT r x x k S x x ω DTFT r x x k r x x k=Exk+lxl¯ r x x k x k l x l Again, a weighted least squares filter design problem.

Problem 1

Is it feasible to use IIR polyphase filters?

Solution 1

The recursive feedback of previous outputs means that portions of each IIR polyphase filter must be computed for every input sample; this usually makes IIR filters more expensive than FIR implementations.

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