This blog discusses a problematic situation that can arise when we try to implement certain digital filters. Occasionally in the literature of DSP we encounter impractical digital IIR filter block diagrams, and by impractical I mean block diagrams that cannot be implemented. This blog gives examples of impractical digital IIR filters and what can be done to make them practical.

Implementing an Impractical Filter: Example 1

Reference [1] presented the digital IIR bandpass filter shown in Figure 1(a) where H NO (z) is a FIR notch filter and variable K is a constant.

Figure 1: Reference [1]'s proposed IIR bandpass filter: (a) high-level depiction; (b) detailed signal path depiction.

The idea behind this bandpass filter is to subtract the output of the notch filter from the X(z) input to produce a narrow bandpass filter output. The Figure 1(a) filter's z-domain transfer function is:

$$H_{\mathsf{BP}}(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 + KH_{\mathsf{NO}}(z)}\tag{1}$$

NO

where the bandpass filter's) term is a 2nd-order FIR notch filter defined by

$$H_{\mathsf{NO}}(z) = 1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}.\tag{2}$$

In Eq. (2) variable $\omega_c$, where $0<{\omega_c}<\pi$, represents the center-frequency of the FIR filter's notch measured in radians/sample.

Based on Eq. (2) the proposed bandpass filter can be depicted as shown in Figure 1(b). The Figure 1 filter seems like a plausible design but is in fact impossible to implement (i.e., it's impractical). Here's why.

The Problem

The trouble with the above bandpass filter is that it contains a feedback signal path containing no delay element (a delay-less signal path) as shown by the bold arrows in Figure 2.

Figure 2: Time-domain depiction of the impractical bandpass filter with the delay-less signal path highlighted in bold.

Upon the arrival of an x(n) input sample, in order to compute a y(n) output sample we first need to compute the w(n) feedback sample. But to compute w(n) we need y(n)! That delay-less signal path means the Figure 2 filter cannot be implemented.

The Solution: Algebra to the Rescue

As it turns out, algebraic expansion of the H BP (z) equation produces a new filter z-domain transfer function that can be implemented. As shown in Appendix A, we can plug Eq. (2) into Eq. (1) to obtain a practical transfer function of:

$$H_{\mathsf{BP,Practical}}(z) = \frac{\alpha}{1+2K{\alpha}\mathsf{cos}(\omega_c)z^{-1}+K{\alpha}z^{-2}}\tag{3}$$

where ${\alpha}=1/(1+K)$, and $K≠-1$. Unlike the Figure 2 filter, the Eq. (3) bandpass filter can be implemented with no problem as shown in Figure 3.

Figure 3: Practical (implementable) version of the impractical Figure 1 bandpass filter.





Implementing an Impractical Filter: Example 2

A second example of an impractical filter, proposed in Reference [2], is the narrowband digital notch filter shown in Figure 4.

Figure 4: High-level depiction of Reference [2]'s proposed IIR notch filter.

where C(z) is a notch filter and variable a is a constant. The z-domain transfer function for the proposed Figure 4 IIR filter, derived in Appendix B, is:

$$H_\mathsf{N}(z) = \frac{Y(z)}{X(z)} = \frac{(1+\alpha)C(z)}{1+{\alpha}C(z)}.\tag{4}$$

The) function is a traditional 2nd-order narrowband IIR notch filter whose transfer function is:

$$C(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\rho\mathsf{cos}(\omega_c)z^{-1}+\rho^2z^{-2}}\tag{5}$$

where variable $\omega_c$, $0<{\omega_c}<\pi$, is the center-frequency of the filter's notch measured in radians/sample, and $\rho$ is the radii of the conjugate poles' z-plane locations. Based on Eq. (5) the proposed notch filter can be depicted in more detail as shown in Figure 5.

Figure 5: Detailed signal path depiction of the H N (z) notch filter.

The Figure 5 bandpass filter contains a delay-less feedback signal path as shown by the bold arrows in Figure 6.

Figure 6: Time-domain depiction of the impractical H N (z) notch filter with the delay-less signal path highlighted in bold.

That delay-less signal path makes the Figure 6 filter impractical because to compute a y(n) output sample we first need to compute the u(n) feedback sample. But to compute u(n) we need y(n). Similar to the Figure 2 filter, the Figure 6 filter cannot be implemented.

Again, as we did in Example 1, algebraic expansion of the H N (z) equation produces a new filter z-domain transfer function that can be implemented. Substituting Eq. (5)'s C(z) expression into Eq. (4), as shown in Appendix C, we produce a practical transfer function of:

$$H_{\mathsf{N,Practical}}(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\frac{\rho+\alpha}{1+\alpha}\mathsf{cos}(\omega_c)z^{-1}+\frac{\rho^2+\alpha}{1+\alpha}z^{-2}}\tag{6}$$

where ${\alpha} ≠ -1$. This new H N,Practical (z) function is a traditional 2nd-order IIR filter that can be implemented using the block diagram shown in Figure 7.

Figure 7: Practical (implementable) version of the impractical Figure 4 notch filter.

Conclusion

The bottom line of this blog is: When you encounter a digital network's block diagram that is impractical (i.e., impossible to implement because it contains a delay-less feedback signal path), algebraic analysis of the network's z-domain transfer function may well lead to a new and equivalent block diagram that is possible to implement.

Postscript: December, 2017

In my original July 2016 version of this blog I presented the following Figure 8 network and claimed it to be impractical.



Figure 8: Reference [3]'s spectrum analysis network.



But, as was pointed out to me by Prof. Zsolt Kollár (Budapest University of Technology and Economics, Hungary) I was mistaken. If we draw the details of the resonators in Figure 8 we have Figure 9, and that network can indeed be implemented due to the delay elements in each path.



Figure 9: Expanded version of the Figure 8 network.



Upon arrival of an x[n] input sample we compute all the outputs of the z k multipliers, shift those products through their delay elements, and wait for the next x[n+1] input sample. No problem. As the cowboys in Texas would say, "The Figure 8 system works real fine."



References

[1] S.Engelberg, “Precise Variable-Q Filter Design,” IEEE Signal Processing Magazine, Sept. 2008, pp. 113–119.

[2] S. Pei, B, Guo, and W. Lu, "Narrowband Notch Filter Using Feedback Structure", IEEE Signal Processing Magazine, May 2016, pp. 115-118.

[3] G. Peceli, G. Simon, "Generalization of the Frequency Sampling Method", IEEE Instr. and Meas. Conference, Brussels Belgium, June 1996.

https://www.researchgate.net/publication/3632671_Generalization_of_the_frequency_sampling_method

Appendix A: Derivation of Eq. (3)

We derive Eq. (3) by starting with the above Eq. (1), repeated here as:

$$H_{\mathsf{BP}}(z) = \frac{1}{1 + KH_{\mathsf{NO}}(z)}.\tag{A-1}$$

Substituting the notch filter's H NO (z) from Eq. (2) into Eq. (A-1) yields:

$$H_{\mathsf{BP}}(z) = \frac{1}{1 + K[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}$$

$$= \frac{1}{(1 + K)-2K\mathsf{cos}(\omega_c)z^{-1}+Kz^{-2}}.\tag{A-2}$$

Next, dividing Eq. (A-2)'s numerator and denominator by (1+K) we produce the desired expression for the Figure 1 proposed bandpass filter's equivalent and practical transfer function of:

$$H_{\mathsf{BP,Practical}}(z) = \frac{\alpha}{1+2K{\alpha}\mathsf{cos}(\omega_c)z^{-1}+K{\alpha}z^{-2}}\tag{A-3}$$

where ${\alpha}$ = 1/(1+K), and K ≠ -1.

Appendix B: Derivation of Eq. (4)

The following derivation was graciously supplied to me by Reference [2]'s co-author Bo-Yi Guo. Master's student Guo derived Eq. (4) by redrawing Figure 4 as shown in Figure B1.

Figure B1: Reference [2]'s proposed IIR notch filter.

From Figure B1, we start with

$$W(z) = X(z)-Y(z)\tag{B-1}$$

and

$$V(z) = X(z)+{\alpha}W(z).\tag{B-2}$$

Replacing W(z) in Eq. (B-2) with Eq. (B-1), we have:

$$V(z) = X(z) +{\alpha}(X(z)–Y(z)).\tag{B-3}$$

The Y(z) output can be expressed as:

$$Y(z) = V(z)C(z).\tag{B-4}$$

Replacing V(z) in Eq. (B-4) with Eq. (B-3) gives:

$$Y(z) = [X(z) + {\alpha}(X(z)–Y(z))]C(z)$$

$$= (1+{\alpha})X(z)C(z)-{\alpha}Y(z)C(z).\tag{B-5}$$

Rearranging Eq. (B-5) yields our desired H N (z) transfer function as:

$$H_\mathsf{N}(z) = \frac{Y(z)}{X(z)} = \frac{(1+\alpha)C(z)}{1+{\alpha}C(z)}.\tag{B-6}$$

Appendix C: Derivation of Eq. (6)

We derive Eq. (6) by starting with the above Eq. (4) repeated here as:

$$H_\mathsf{N}(z) = \frac{(1+\alpha)C(z)}{1+{\alpha}C(z)}\tag{C-1}$$

or after dividing through by C(z),

$$H_\mathsf{N}(z) = \frac{1+\alpha}{\frac{1}{C(z)}+\alpha},\tag{C-2}$$

recalling from Eq. (5) that

$$H_{\mathsf{N,Practical}}(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\rho\mathsf{cos}(\omega_c)z^{-1}+\rho^2z^{-2}}.\tag{C-3}$$

Next we rewrite Eq. (C-2) by separating C(z) into its numerator and denominator parts as:

$$H_\mathsf{N}(z) = \frac{1+\alpha}{\frac{1}{C_\mathsf{numer}(z)/C_\mathsf{denom}(z)}+\alpha}= \frac{1+\alpha}{\frac{C_\mathsf{denom}(z)}{C_\mathsf{numer}(z)}+\alpha}$$

$$= \frac{(1+\alpha)C_\mathsf{numer}(z)}{C_\mathsf{denom}(z)+{\alpha}C_\mathsf{numer}(z)}.\tag{C-4}$$

Substituting Eq. (C-3)'s numerator and denominator terms into Eq. (C-4) we have:

$$H_{\mathsf{N}}(z) = \frac{(1+\alpha)[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}{1-2\rho\mathsf{cos}(\omega_c)z^{-1}+\rho^2z^{-2}+\alpha[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}$$

$$= \frac{(1+\alpha)[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}{(1+\alpha)-2(\rho+\alpha)\mathsf{cos}(\omega_c)z^{-1}+(\rho^2+\alpha)z^{-2}}\tag{C-5}$$

which is Reference [2]'s Eq. (15). Finally, dividing Eq. (C-5)'s numerator and denominator by (1+$\alpha$) we produce the desired equivalent and practical expression for the proposed notch filter's transfer function of:

$$H_{\mathsf{N,Practical}}(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\frac{\rho+\alpha}{1+\alpha}\mathsf{cos}(\omega_c)z^{-1}+\frac{\rho^2+\alpha}{1+\alpha}z^{-2}}\tag{C-6}$$

where ${\alpha} ≠ -1$.

